The Universe of Conics: From the Ancient Greeks to 21st Century Developments 366270305X, 9783662703052

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Georg Glaeser Hellmuth Stachel Boris Odehnal

The Universe of Conics From the Ancient Greeks to 21st Century Developments Second Edition

The Universe of Conics

Georg Glaeser • Hellmuth Stachel Boris Odehnal

The Universe of Conics From the Ancient Greeks to 21st Century Developments Second Edition

Georg Glaeser Department of Geometry University of Applied Arts Vienna Department of Geometry Vienna, Austria

Hellmuth Stachel Inst. of Disc. Mathematics and Geometry Vienna University of Technology Inst. of Disc. Mathematics and Geometry Vienna, Austria

Boris Odehnal Department of Geometry University of Applied Arts Vienna Department of Geometry Vienna, Austria

ISBN 978-3-662-70305-2 ISBN 978-3-662-70306-9 (eBook) https://doi.org/10.1007/978-3-662-70306-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2016, 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer-Verlag GmbH, DE, part of Springer Nature. The registered company address is: Heidelberger Platz 3, 14197 Berlin, Germany If disposing of this product, please recycle the paper.

Preface

You are now holding the second edition of “The Universe of Conics”, which ended up in roughly 15% more pages. This update was deemed necessary for several reasons: Firstly, to correct certain typos and minor errors found in the previous edition; secondly, to include additional content not covered so far; and thirdly, to incorporate recent findings. Still, we are convinced that it is nearly impossible to present a complete collection of results on conics. Nevertheless, we have endeavored to explore the geometry of conics as comprehensively as possible. The book has been written for people who love geometry, and it is mainly based on figures and synthetic conclusions rather than on pure analytic calculations. In many proofs, illustrations help to explain ideas and to support the argumentation, and in a few cases, the picture can display a theorem at a glance together with its proof. Large portions of the book can be understood by undergraduate students. Some sections, however, require a little more mathematical background, and we hope that they will be a treasure trove even for professional mathematicians and geometers. 4

5

3

7

1

2

6

8

10

9

Many of the results presented in this book naturally reflect the authors’ interests. The order in which the topics are presented has not been changed: After a short historical summary (1), we explain the classical definitions of conics (2). Then, differential geometric properties are discussed (3), including the non-trivial fact that the orbits of planets are ellipses. Chapter 4 shows that conics can be planar intersections of quadratic cones. The framework of Projective Geometry is the proper setting for the v

vi

Preface

study of conics (5 and 6). Still fitting into this context are polar systems and pencils of conics (7). After that we switch to the affine properties of conics (8). Chapter 9 deals with special problems. The last chapter is devoted to conics in various two-dimensional geometries: on the sphere, in Pseudo-Euclidean planes, and in hyperbolic planes. When Isaac Newton proved Johannes Kepler’s conjecture that the paths of the planets are ellipses, he utilized properties of ellipses which have in the mean-time almost been forgotten. The 23-years-old Carl Friedrich Gauß gained his reputation by rediscovering the dwarf planet Ceres which had only been observed on a very few occasions. He used the fact that an ellipse is uniquely determined by a single focal point (the Sun) and three additional points. Gauß himself considered that this type of problem “commended itself to mathematicians by its difficulty and elegance”. 215 years later, with Ceres still on its elliptic path, NASA’s Dawn spacecraft has orbited the dwarf planet, sending stunning images of this icy remnant of our early solar system. F C1

C3

C2

A geometrical homage to a famous problem: Ceres in its icy beauty (©NASA_Dawn). The path of a planet is fully determined by three locations, due to fact that an ellipse is uniquely defined when we know its focal point F and three points C1 , C2 , and C3 .

We wish all readers a pleasant journey into the universe of conics! Vienna, April 2024 Acknowledgements We say thanks to Bob Bix, David Brannan, Fritz Manhart, Gerhard Pillwein, Andreas Rüdinger, and Norman Wildberger for assistance, suggestions, and proof reading.

Table of Contents

1 Introduction

1

2 Euclidean plane 2.1 Classical definitions . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Tangent lines of conics . . . . . . . . . . . . . . . . . . . . . . 2.3 Mechanisms tracing conics . . . . . . . . . . . . . . . . . . . .

11 12 30 56

3 Differential Geometry 69 3.1 Conics as orbits of planets . . . . . . . . . . . . . . . . . . . . 70 3.2 Conics in planar differential geometry . . . . . . . . . . . . . . 85 3.3 Conics in differential geometry of surfaces . . . . . . . . . . . 128 4 Euclidean 3-space 4.1 Planar intersections of cones of revolution . 4.2 Pairs of focal conics, Dupin cyclides . . . . . 4.3 Perspective images of conics . . . . . . . . . 4.4 Spatial interpretation of conic constructions 5 Projective Geometry 5.1 Projective planes . . . . . . . 5.2 Perspectivities, projectivities 5.3 Coordinatization . . . . . . . 5.4 Harmonic quadruples . . . .

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137 138 147 163 172

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187 188 199 205 217

6 Projective conics 6.1 Steiner’s definition of a conic . . . . . . . . 6.2 Pascal, Pappus, Brianchon . . . . . . . . . . 6.3 Equation and parametrization of a conic . . 6.4 There is only one conic in a projective plane

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229 230 232 240 252

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7 Polarities and pencils 273 7.1 Polar system of a conic . . . . . . . . . . . . . . . . . . . . . . 274 7.2 Definition of a conic according to von Staudt . . . . . . . . . 299

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Table of Contents

7.3 7.4 7.5

Pencils of conics . . . . . . . . . . . . . . . . . . . . . . . . . . 306 Desargues’s involution theorem . . . . . . . . . . . . . . . . . 336 Quadratic Cremona transformations . . . . . . . . . . . . . . . 356

8 Affine Geometry 377 8.1 Conjugate diameters of ellipses and hyperbolas . . . . . . . . 379 8.2 Conics are rational quadratic Bézier curves . . . . . . . . . . . 400 8.3 Conics and number theory . . . . . . . . . . . . . . . . . . . . 405 9 Special problems 9.1 Conics in triangle geometry . . . . . . . . . . . 9.2 Isoptic curves of conics . . . . . . . . . . . . . 9.3 Pedal points on conics, Apollonian hyperbola 9.4 Pedal curves of conics . . . . . . . . . . . . . . 9.5 Poncelet porisms . . . . . . . . . . . . . . . . . 9.6 Billiards in ellipses . . . . . . . . . . . . . . . .

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411 412 434 442 446 453 479

10 Other geometries 515 10.1 Spherical conics . . . . . . . . . . . . . . . . . . . . . . . . . . 516 10.2 Conics in non-Euclidean planes . . . . . . . . . . . . . . . . . . 549

References

565

Index

573

1 Introduction

Our Universe is full of conics, even if we cannot always see them – like the orbits of the planets. It needed very accurate observations to detect (Johannes Kepler), and a great physicist (Isaac Newton) to prove that.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2024 G. Glaeser et al., The Universe of Conics, https://doi.org/10.1007/978-3-662-70306-9_1

1

2

Chapter 1: Introduction

Conics play a substantial role in the whole universe. On a large scale, the orbits of planets look like conics. Astronomers have recognized this fact and physicists have proved it. Many older and newer visualizations clearly show the ellipses as orbits of planets of our solar system with the Sun as the common focus. At this point, we should mention the efforts of the ancient Greeks. Among them, especially Apollonius of Perga (262 BC–190 BC) collected many results which were well known at his time. In his famous eight books, entitled Konika, he showed that all “four types” of conics can be obtained from the same cone. He even gave the names ellipse, parabola, and hyperbola to these curves. The first four books collect many elementary results which were known at that time.

FIGURE 1.1. A long history and a back and forth between languages: The 5th book of Apollonius, here in a translation from Arabic into German.

The books five to seven present material which is probably due to Apollonius. Therein, he discussed the problem of normals to a conic and prepared even the construction of the evolute of a conic. However, Apollonius’s books only survived in form of translations: first from Greek into Latin, then into Arabian, and back into Latin or other languages once again (cf. Figure 1.1, https://ia601408.us.archive.org/1/items/4737397/ 4737397.pdf). Unfortunately, the eighth book is lost.

3

Chapter 1: Introduction y

p2

2a

√ x0= 3 2 y0 = √ 3

a

22 x a

2a

p1

FIGURE 1.2. The Delian problem: The volumes of the two cubes on the left have ratio 2 ∶ 1. How can one construct the scaling factor of the edge length?

The Greeks even used conics as auxiliary curves for the solution of mathematical problems: Menaichmos (380 BC–320 BC), e.g., suggested to solve the classic “Delian problem” or “cube duplication” by means of parabolas (cf. Figure 1.2, for more details see p. 306). Conics have a perfect shape, and with a bit of practice it is easy to judge whether a curve is a conic or just a “similar curve”. This is why designers and architects use “true conics” instead of “just some freeform curves” when they deal with certain shapes.

Curved, but still entirely in a plane There is only one type of algebraic curve of degree one: the straight line with an equation of the form Ax + By + C = 0, and zero curvature. A conic is an algebraic curve of degree two, described by the equation Ax2 + Bxy + Cy 2 + Dx + Ey + F = 0.

4

Chapter 1: Introduction S2

S1

s

c

S1=

s

S2

S2

c

S1

=S

2

=S

1

s

c

FIGURE 1.3. In the algebraic sense, a conic c always has two points S1 and S2 of intersection with a given straight line s. Left: two real common points. Middle: s is the tangent at the point S1 = S2 . Right: The points S1 and S2 of intersection are complex conjugate. We cannot locate them in the real plane. However, we sometimes indicate a pair of complex conjugate points by a pair of crosses on a line or curve, just in order to point out that there are two further points though we cannot see them.

Conics are always planar curves (there is no “true space curve” of degree two). One says that they have zero torsion. They are of degree 1 two, i.e., they have two intersection points with a straight line – counted in the algebraic sense, where a point of contact counts twice (has multiplicity two) and complex conjugate solutions are also considered (Figure 1.3).

One possible classification of three types Depending on the number of real intersection points with the line at infinity, we can distinguish three types of conics: Ellipses (no real intersection points), parabolas (the line of infinity touches the curve) and hyperbolas (two real intersection points).

Duality – points become straight lines and vice versa Conics are also curves of class two which means that, from each point in the conic’s plane, we have two tangents to the conic (in the algebraic sense). When the point lies on the conic, the tangents coincide (Figure 1.4). The set of all points, where there are no real tangents to the conic, can be called the interior of a conic. The points where the tangents are complex conjugate and perpendicular are called focal points. 1

Sometimes, and especially, in the older literature, the word order is used instead of degree. However, we prefer the term degree.

5

Chapter 1: Introduction T t1

t2

T c

t1=

t1 =t2

t2

c

c T t2 =t1

FIGURE 1.4. Left: In the algebraic sense, a conic c always sends two tangents t1 and t2 through a given point T . For an exterior point T , we find two real tangents. Middle: At a point T on c, the two tangents coincide (one tangent with multiplicity two). Right: Through an interior point, there are two complex conjugate tangents. In order to make clear that there are such tangents, we sometimes indicate such tangents by dashed lines.

FIGURE 1.5. Dual curves of concis: the set of tangents.

Each theorem about conics that only uses the terms point, straight line, to touch, and to intersect remains, therefore, true if we interchange the terms point and straight line (and vice versa) but keep the relation of being incident. A conic considered as its set of points and tangents is a self-dual term. The dual of a conic is still a conic, viewed as a set of lines, cf. Figure 1.5. We will explain this in more detail later in Sec. 6.2.

Ellipses and hyperbolas in nature As we know, the Universe is “full of ellipses” (hyperbolas also occur sometimes, whereas parabolas are very unlikely), since all planets and comets have such trajectories, at least in good approximation (Figure 1.6). As we mentioned earlier, conics usually appear as orbits, i.e., they are not directly visible. If we push a stick with endpoint P into the horizontal

6

Chapter 1: Introduction

Mercury Mars Sun

Venus Earth

FIGURE 1.6. Ellipses as path curves of the inner planets. This image allows us to explain why one can see Mercury and Venus only close to the Sun (and thus around sunset and sunrise): We have two tangents to each of the inner orbits with comparatively small angles. To the orbit of Mars and all the other planets, there are no real tangents.

ground and mark the position of all shadows P s of P on the planar ground throughout a day, the set of all shadow points will almost exactly trace a hyperbola. Ellipses can only occur close enough to the poles, parabolas only on certain days in this region. Only on two days of the year (the equinoctials), the conic “degenerates” into a straight line. The reason for this is the following: During one day, the Earth rotates only approximately 1○ around the Sun. It fully spins, however, around the Earth axis during that time. Relatively speaking, the sun rays through a point P , therefore, form – in a good approximation – a cone of revolution, whose axis is parallel to the Earth’s axis. The intersection of the cone with the base plane results in a conic. Since the inclination of the spin axis is the geographic latitude, the cone’s axis is only steep enough to generate an ellipse when our location is close enough to the poles (above the Arctic Circle or below the Antarctic Circle). Else we get a (branch of a) hyperbola. The angle between the oriented sun rays and the oriented rotation axis varies between 90○ ±ε, where

7

Chapter 1: Introduction North equinoctials

Star

P Ps South

North

winter half-year

summer half-year

FIGURE 1.7. On each day of the year, the shadow of a point moves along a branch of a hyperbola. On the equinoctials, the curve degenerates into a straight line (Ottoman sundial, Istanbul).

ε ≈ 23.44○ is the Earth’s axial tilt. Thus, the cone can also degenerate into a plane – which leads to a straight shadow orbit. Figure 1.7 shows how this was used for sun dials and calenders: The photo was taken at the Topkapı-Palace in Istanbul2 .

FIGURE 1.8. The “gardener’s construction” of an ellipse.

The ancient Greeks drew their geometric figures mainly in sand. Whether they also drew ellipses, we do not know. Maybe they already knew about the “gardener’s construction” of the ellipse (see Figure 1.8). 2

http://www.muslimheritage.com/topics/default.cfm?ArticleID=942

8

Chapter 1: Introduction

FIGURE 1.9. High precision perspectives of circles produced 310 BC: Paintings in the tomb of Philipp III of Macedon. If this perspective wall painting was a few hundred years old, it would already be impressive. However, it is 2300 years old – and thus a little sensation.

For sure, they knew many facts about conic sections. We have good reasons to suppose that the painter of the scenes shown in Figure 1.9 (about 310 BC) knew something about ellipses, because otherwise he would not have been able to draw perspective views of circles of such a high precision.

Chapter 1: Introduction

9

It is uncertain if Menaichmos already knew earlier than Apollonius that planar intersections of cones (and cylinders) of revolution were probably identical with the conics defined in the plane. Albrecht Dürer (1471–1528) was probably aware of that, although his famous drawings in his book Underweysung der Messung, mit dem Zirckel und Richtscheyt, in Linien, Ebenen unnd gantzen corporen (1525) lack a little bit of symmetry (cf. Figure 1.10). In fact, Dürer believed that the curvature at the upper vertex is larger than that in the lower one.

FIGURE 1.10. Dürer’s constructions of conic sections.

It was the Belgian mathematician Germinal Pierre Dandelin 1794– 1847 who gave a very elegant proof (see Figure 1.11).

FIGURE 1.11. Dandelin spheres and focal points of conic sections.

10

Chapter 1: Introduction

Many elementary properties of conics have been discovered in the meantime and ever since. A fundamental theorem was discovered by Blaise Pascal (1623–1663) in 1639. His theorem, as illustrated in Figure 1.12, is a generalization of Pappus’s theorem and relates conics and Projective Geometry, although this kind of geometry was introduced approximately 170 years later. The advent of Projective Geometry created a real “boom” in the theory of conics. Meanwhile, the chains of thought were extended to three-dimensional space which led to the natural generalization of the conics, namely the quadrics. A2 A1 A1 A2 A3

A3 B1 B2

B3

p B1

B3

p

B2

FIGURE 1.12. Pascal’s theorem: The three pairs of opposite sides of a hexagon inscribed into a conic section meet in three collinear points.

It was only natural that this generalization was extended to higher dimensions (Felix Klein, 1849–1925). Nowadays, we quite often use such hyperquadrics as point models for complicated geometries. Besides the geometric importance of conics, there is an additional “physical component”: According to Johannes Kepler (1571–1630), the orbits of the planets are ellipses (Kepler’s First Law, 1609). The final proof took another half century and was given by Isaac Newton (1642–1727). Primarily, the proof uses techniques from calculus. Newton, however, did it in an elementary way with the help of certain properties of conics (Sec. 3.1). Knowledge about conics and quadrics probably reached its peak point at the beginning of the twentieth century. Since then, there seems to be an increasing loss of knowledge. This book will help to sum up and preserve more or less known properties of these fascinating curves.

2 Euclidean plane

A pencil of planes meets a cone of revolution in a family of conics which maps to a pencil of conics in the top view. These conics in the top view share a focal point and the associated directrix.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2024 G. Glaeser et al., The Universe of Conics, https://doi.org/10.1007/978-3-662-70306-9_2

11

12

Chapter 2: Euclidean plane

2.1 Classical definitions There are various definitions of conics. Depending on whether the underlying geometry is Euclidean, affine or projective, conics can be defined by their focal properties, as planar sections of cones of revolution, as perspective images of circles, by their quadratic equation, or from the viewpoint of Projective Geometry as the set of intersection points of projective pencils, or as the sets of self-conjugate points of a polarity in the plane. Also the ranges of the different definitions differ. Sometimes also the degenerate cases are included, sometimes only the regular ones, sometimes the circles are excluded. In the sequel, the term conics stands for ellipses (including circles), hyperbolas, and parabolas. Whenever we want to include degenerate cases like lines or pairs of lines, we will mention this. y

F1

b

y

y

b x

M a

F1

F2

x

a M

R

F2

F p/2 p/2

P Q

l

x

p

FIGURE 2.1. The standard definitions of ellipses, hyperbolas and parabolas.

We start with conics in the Euclidean plane E2 , the plane of intuitive geometry and idealization of planes in our physical world. The standard definition of conics in E2 is based on the focal properties. There is a strict difference between ellipses, hyperbolas, and parabolas: ellipses are defined according to the “gardener’s construction” (Figure 1.8) as the sets of points P in a plane with a constant sum of distances1 P F1 + P F2 = 2a with 2a > F1 F2 ≥ 0 1

We use the overline to denote distances between two points or between a point and a line. [P, Q] denotes the line connecting the points P and Q, while P Q is the segment terminated by P and Q.

13

2.1 Classical definitions

from their focal points F1 and F2 . Points Q of a hyperbola with focal points F1 and F2 are defined by a constant difference of distances ∣QF1 − QF2 ∣ = 2a > 0. The points R of a parabola are characterized by equal distances to its focal point F and its directrix l provided p = Rl ≠ 0.

Not only the definitions of ellipses, hyperbolas, and parabolas are different but also their shapes, as Figure 2.1 reveals. Hence, from this point of view it is surprising that there is a common metric definition for all three types, the definition due to Apollonius of Perga (around 200 BC). Later, when dealing with projective properties of conics, we will even notice that all conics are the projective transforms of one single conic (Sec. 6.4).

Apollonian definition of conics Our approach to conics in E2 is based on the Apollonian definition given below. In the sequel, we will learn step by step that this definition is equivalent to all other definitions of conics other than circles. We prefer this definition as it works for all three types of conics simultaneously. Also concerning proofs and graphical constructions, we prefer those which need no separation into different cases. Definition 2.1.1 In the Euclidean plane E2 , let F be a fixed point and l be a line not passing through F . For any positive constant ε > 0, the set of points c = {P ∣ P F = ε ⋅ P l } is called a conic with focal point (or focus) F and associated directrix l. The constant ε is the numerical eccentricity of c ; the distance p ∶= ε ⋅ F l is the parameter of the conic c (Figure 2.2). According to the eccentricity ε, we distinguish between ellipses with ε < 1, parabolas with ε = 1, and hyperbolas with ε > 1. Remark 2.1.1 The notion ‘parameter’ is standard in the geometry of conics and should not be mixed with a ‘parameter’ used for the parametrization of curves.

The normal line s drawn from F to l is an axis of symmetry of the conic c. Figure 2.2 shows how on the axis s the point A ∈ c between F and l

14

Chapter 2: Euclidean plane

p/ε

c

A

F

s p

L

l

P p/ε

FIGURE 2.2. The Apollonian definition of the conic c with the focus F and the associated directrix l, with numerical eccentricity ε and parameter p.

can be obtained. The point A divides the segment F L on s between F and l in the ratio p AF ∶ AL = ε ∶ 1 = p ∶ . ε We call A ∈ s a vertex of c. Figure 2.3 shows for fixed F and l how the numerical eccentricity ε influences the shape of the conic.2 The parameter p defines the size of c. Lemma 2.1.1 Ellipses and hyperbolas have a second axis of symmetry s2 , and therefore, a second focus F2 with the associated directrix l2 . The corresponding numerical eccentricity ε and the parameter p remain unchanged. Proof: We are going to demonstrate that for ε ≠ 1. On all lines parallel to the axis s there are two points of intersection with the conic c . To this end, we introduce a coordinate frame with the x-axis along s and the y-axis along the directrix l (Figure 2.4). Then, the focal point F has the coordinates (p/ε, 0). For any point P = (x, y), the condition P F = ε ⋅ P l is equivalent to 2p p2 x2 − x + 2 + y 2 = ε 2 x2 , ε ε hence 2p p2 (1 − ε2 )x2 − x + ( 2 + y2 ) = 0 (2.1) ε ε which, in the case ε ≠ 1, can be solved for x by 1 p √ x= [ ± p2 + (ε2 − 1)y 2 ] . 2 1−ε ε 2

Using the terminology of Section 7.3, we can state: The conics displayed in Figure 2.3 belong to a pencil of the third kind with two complex conjugate base points and tangents (compare with Figure 7.23, right).

15

2.1 Classical definitions ε =1

l ε =1.80

ε =0.75 ε =0.56

F

s ε =0.40

ε =2.50 FIGURE 2.3. Conics sharing the focal point F and the associated directrix l. In the hyperbolic case, we get for each y ∈ R a pair of real points P, P2 ∈ c. In the elliptic case, we have to take care of the limits − √ p 2 ≤ y ≤ √ p 2 . For y = 0, we obtain the two points 1−ε

p A=( , 0) , ε(1 + ε)

1−ε

A2 = (

p , 0) . ε(1 − ε)

The midpoints of all pairs (P, P2 ) have the same x-coordinate p xM = . ε(1 − ε2 )

(2.2)

(2.3)

Hence, there is a second axis s2 of symmetry. The point M = (xM , 0) is the crossing point of the two axes of symmetry, and consequently, a center of symmetry. By reflection in s2 , we obtain a second focal point F2 and a second directrix l2 (Figure 2.4). Let P, P2 ∈ c be symmetric with respect to s2 . Then, P2 F2 = P F = ε P l = ε P2 l2 .

This reveals that the same ellipse or hyperbola c can also be defined by the condition P F2 = ε ⋅ P l2 . Hence, the numerical eccentricity remains the same. The same holds for the parameter p, since F l = p/ε = F2 l2 .

◾

The second axis s2 of symmetry is called minor or secondary axis of the conic c, in comparison with the major or principal axis s through F . The point M = s ∩ s2 is called the center of the ellipse or hyperbola c. With regard to ellipses and hyperbolas we speak of central conics, in contrast to parabolas.

16

Chapter 2: Euclidean plane y y

l P

P2

P2

P

c

l L

s A

F

M

F2

x A2

M F2

A2

x

A L

F

s2 p/ε

c

p/ε

FIGURE 2.4. Conics with ε ≠ 1 have a second axis s2 of symmetry, and therefore, also a second focal point F2 with associated directrix l2 (not shown here).

The distances from l and F to the center M are as follows: M l = ∣xM ∣,

p pε e ∶= M F = ∣xM − ∣ = ∣ ∣. ε 1 − ε2

The distance e = M F = M F2 is a new type of eccentricity, called linear eccentricity of the conic c. The vertex A as well as the second vertex A2 given in (2.2) yield the same ratio AF ∶ AL = A2 F ∶ A2 L = ε ∶ 1

with L = s ∩ l. The vertex A lies between L and F , and A2 outside (see Figure 2.4). We say, A2 is the harmonic conjugate of A with respect to (‘w.r.t.’ by short) F and L.3 In the elliptic case ε < 1 the two vertices A and A2 lie on the same side of the directrix l, in the hyperbolic case ε > 1 on opposite sides. For a parabola, the harmonic conjugate A2 of A w.r.t. F and L lies at infinity.

Theorem 2.1.1 The Apollonian definition of conics is equivalent to the union of the standard definitions of ellipses without circles, of hyperbolas, and of parabolas. 3

See Section 5.4 on page 217. As a consequence of this harmonic relation, the directrix l is the polar of the focal point F (see Exercise 7.1.5 in Section 7.1).

17

2.1 Classical definitions Proof: Here, we need to separate the different types of conics.

(1) Obviously, in the case ε = 1 the Apollonian definition is identical with the standard definition of a parabola. (2) In the elliptic case (Figure 2.4, left), we conclude from the existence of the second axis of symmetry, by virtue of (2.3), P F + P F2 = P F + P2 F = ε(P l + P2 l) = 2ε ⋅ xM =

2p = const. 1 − ε2

Hence, for ε < 1, the standard definition of ellipses is a consequence of the Apollonian definition. The semimajor axis a, the semiminor axis b, the parameter p, and the numerical eccentricity ε of an ellipse satisfy a=

p , 1 − ε2 2 b p= , a

√ p pε b= √ , e = MF = = a2 − b2 , 2 2 1−ε 1−ε e b2 p a2 ε = , Fl = , Ml = = . a e ε(1 − ε2 ) e

(2.4)

The triangle inequality, applied to any triangle F F2 P with P ∈ c, implies that 2e = F F2 < P F + P F2 = 2a. The circles are not among the ellipses satisfying the Apollonian definition since e > 0, hence F2 ≠ F . Conversely, if any ellipse c is given according to the standard definition by its two different focal points F1 , F2 and the constant 2a, we can compute consecutively the linear √ eccentricity e = F1 F2 /2 ≠ 0, the numerical eccentricity ε = e/a, the semiminor axis b = a2 − e2 , the parameter p = b2 /a, and the distance a2 /e of the directrices l1 , l2 from the midpoint M of F1 F2 . The focal point Fi lies between M and the associated directrix li , i = 1, 2 . Then, the conic c′ , defined according to Apollonius by F1 , the associated directrix l1 , and the numerical eccentricity ε, shares with c the focal points and the constant 2a which implies c′ = c. (3) In the hyperbolic case ε > 1 (see Figure 2.4, right), the x-coordinates of the two points P and P2 have different signs. Therefore, we obtain ∣P F − P F2 ∣ = ∣P F − P2 F ∣ = ε ∣P l − P2 l∣ = 2ε ⋅ ∣xM ∣ =

2p = const. ε2 − 1

Also for hyperbolas, the standard definition follows from the Apollonian definition. The mutual dependencies between the two semiaxes a and b, the linear and the numerical eccentricity e and ε, and the parameter p are √ p p pε , b= √ , e = MF = 2 = a2 + b2 , 2 ε2 − 1 ε − 1 ε −1 (2.5) b2 e b2 p a2 p= , ε = , Fl = , Ml = = . 2 a a e ε(ε − 1) e √ The lines through the center M with the slope ± ab = ± ε2 − 1 are the asymptotes of the hyperbola (see Figures 2.1 and 2.4, right). In the following sections, we will learn that the asymptotes are the limits of tangent lines when the point of contact tends to infinity. The formulas above reveal that the directrix l passes through the pedal points of the asymptotes w.r.t. F . a=

With arguments analogous to the elliptic case, we can confirm that, conversely, each hyperbola c given according to the standard definition is identical with a hyperbola c′ defined by the Apollonian definition.

◾

18

Chapter 2: Euclidean plane

Equations of conics A first equation of conics was already presented in (2.1) on page 14. The corresponding coordinate frame (depicted in Figure 2.4) had its origin at the intersection point L between the directrix l and the axis s of symmetry which served as x-axis. Now, we translate the origin to the vertex A (see Figure 2.5, left) in order to eliminate the constant term in the quadratic polynomial on the left-hand side of (2.1). This means by (2.2) that we p have to replace x by x + ε(1+ε) . Thus, we obtain 2

p 2p p p2 (1 − ε ) (x + ) − (x + ) + ( 2 + y2) = 0 ε(1 + ε) ε ε(1 + ε) ε 2

which reduces to

y 2 = 2px − (1 − ε2 )x2 .

(2.6)

This quadratic equation is usually called a vertex equation, since one vertex is taken as the origin. The vertex equation generalizes the standard equation y 2 = 2px (2.7)

of parabolas to the other conics. In the limiting case ε = 0, we get the vertex equation of a circle with radius p. y P (x, y)

r

c

y

c

x

l

p/ε

F

ϕ

A

l

x

p

F p

A

P (r, ϕ)

p/ε

FIGURE 2.5. The coordinate frames for the vertex equation (left) and for the polar equation (right) of the conic c.

On the other hand, in the case ε ≠ 1, we can eliminate the linear term in (2.1) by choosing the center M as the origin of our coordinate frame

19

2.1 Classical definitions

(Figure 2.1). This means by (2.3) that we replace x in (2.1) by x + xM = p x + ε(1−ε 2 ) . This gives (1 − ε2 )2 x2 + (1 − ε2 )y 2 − p2 = 0 .

(2.8)

We call this the center equation of the conic. Now, we apply formulas from (2.4) or (2.5), respectively, in order to express the coefficients of x2 and y 2 in terms of the semiaxes a and b. Thus, we obtain the standard equations Ellipse:

x2 y 2 + − 1 = 0, a2 b2

x2 y 2 − − 1 = 0. a 2 b2

Hyperbola:

(2.9)

y

Pa

τ

b

P x

M

cb

c

a

ca

Pb

FIGURE 2.6. The construction of points P of an ellipse, named after Proclus as well as after De la Hire.

The standard equation (2.9), left, of the ellipse c reveals that an ellipse with the semimajor axis a and the semiminor axis b can be transformed into its principal circle or circumcircle ca , i.e., the circle with center M and radius a (Figure 2.6) by the scaling (affine transformation) α ∶ E2 → E2 ,

(x, y) ↦ (x,

a y) . b

(2.10)

20

Chapter 2: Euclidean plane

Conversely, one obtains the ellipse by starting with the circumcircle ca and multiplying all y-coordinates by the factor ab < 1 . In an analogous way, we can start with the incircle cb with radius b and stretch the x-coordinates by the factor ab > 1 . This results in a construction attributed to Proclus (Greek philosopher, 412–485), as well as to De la Hire (French mathematician, 1640–1718), which is depicted in Figure 2.6. Here, the axial dilations are performed by parallel projection of the ratio ab formed by the aligned points Pa ∈ ca , Pb ∈ cb , and the center M .4 A curve of degree two in the Euclidean plane E2 is the set of points whose coordinates in a Cartesian coordinate system satisfy a quadratic equation a11 x2 + 2a12 xy + a22 y 2 + 2a1 x + 2a2 y + a = 0.

(2.11)

The degree is invariant under changes (x, y) ↦ (x′ , y ′ ) of the Cartesian coordinate system, since the new coordinates x′ and y ′ are linear functions of the initial coordinates, and vice versa. When in (2.11) the polynomial on the left-hand side is reducible, i.e., equals the product of two linear factors, the curve is called reducible or degenerate, otherwise irreducible. A reducible curve either consists of two lines, or it is a single line. The curve x2 + y 2 + 1 = 0 is irreducible, but contains no real points. The curve x2 + y 2 = 0 contains only the origin (0, 0); it is irreducible over R, but not over the field C of complex numbers, since x2 + y 2 = (x + iy)(x − iy). Obviously, the conics are non-degenerate curves of degree two. The theorem below states the converse of this statement. Theorem 2.1.2 Each non-empty curve of degree two which is irreducible over C is a conic. If (2.11) is the equation of a conic c, then the sign of the discriminant D = a11 a22 − a212 determines the type of the conic c. In the case D > 0 the conic is an ellipse, for D = 0 a parabola, and D < 0 characterizes hyperbolas. 4

Later we will learn that the axial dilations are perspective affine transformations which, e.g., can be used to transform tangents of the circles ca or cb into tangents of the ellipse c (see Figure 2.6).

21

2.1 Classical definitions

Proof: The first part of this statement is a particular case of a standard result from linear algebra, the principal axes transformation, which holds for all quadrics. This will be shown in volume 2. Here, we confine ourselves to a short description of how to obtain a suitable coordinate system in the two-dimensional case which reduces the general equation (2.11) to one of the three standard equations given above. (i) The substitution

x = x′ cos ϕ − y ′ sin ϕ, y = x′ sin ϕ + y ′ cos ϕ represents a rotation of the coordinate frame about the origin through the angle ϕ. There is always an appropriate ϕ such that after the substitution in (2.11) the coefficient of the mixed term x′ y ′ vanishes. (ii) By an appropriate substitution x′ = x′′ − x0 and y ′ = y ′′ − y0 we can eliminate linear terms: if the monomials with x′ 2 and y ′ 2 have non-vanishing coefficients we can eliminate both linear terms, otherwise only one of them, which yields the standard equations (2.9) or (2.7), respectively. The substitutions (ii) do not influence the coefficients a11 , a12 , and a22 of the quadratic terms in (2.11). In order to figure out the effect of the substitutions (i), we rewrite the quadratic form a11 x2 + 2a12 xy + a22 y 2 in matrix form as (x y) A (

x ) y

with

A=(

a11 a12

a12 ). a22

Hence, the substitutions (i) transform the quadratic form into (x′ y ′ ) A′ (

x′ ) y′

with

A′ = RT AR, where

R=(

cos ϕ sin ϕ

− sin ϕ ). cos ϕ

The discriminant D = det A remains unchanged since the matrix R is orthogonal, and therefore, det A′ = (det R)2 det A = D. The standard equations of the conics indicate which sign of D corresponds to which conic type.

The substitution x = r cos ϕ and y = r sin ϕ in (2.11) shows, after the division by r 2 in the limit r → ∞, that hyperbolas (D < 0) have two real points at infinity, parabolas (D = 0) one, while ellipses (D > 0) have no real ideal point. Of course, this follows also immediately by inspection of related figures.

◾

We continue our discussion on equations of conics by providing some parameter representations. Let us first return to the affine transformation α in (2.10) between an ellipse ce and its circumcircle ca (see Figure 2.6). The inverse transformation α−1 maps the parametrization ca = {(a cos τ, a sin τ ) ∣ 0 ≤ τ < 2π}

of ca by its polar angle τ onto a parameter representation of the ellipse ce with semiaxes a, b: ce = {(a cos τ, b sin τ ) ∣ 0 ≤ τ < 2π}.

(2.12)

Herein, it does not matter whether a > b or not. The ‘half-angle substitution’ (see Figure 2.7) cos τ =

1 − t2 , 1 + t2

sin τ =

2t τ with t = tan 2 1+t 2

22

Chapter 2: Euclidean plane y P

(0, t) τ /2

τ M

1

x

FIGURE 2.7. The half-angle substitution t = tan τ2 .

gives rise to a parametrization of the ellipse, ce = {(a

1 − t2 2t , b ) ∣ t ∈ R} , 2 1+t 1 + t2

(2.13)

which is called rational since it contains only rational functions. However, the point (−a, 0) is only obtained as the limit t → ∞.

An analogous parametrization, where the trigonometric functions are replaced by hyperbolic functions hr = {(a cosh τ, b sinh τ ) ∣ − ∞ < τ < ∞}

(2.14)

describes one branch hr of a hyperbola. This is true, since on the one hand each point of hr satisfies the hyperbolic standard equation because of (a cosh τ )2 (b sinh τ )2 − = cosh2 τ − sinh2 τ = 1. a2 b2 On the other hand, for each y ∈ R there is a unique τ with y = sinh τ , while cosh τ > 0. By the same token, the branch hr of the hyperbola can also be parametrized by hr = {(

a π π , b tan σ) ∣ − < σ < } . cos σ 2 2

(2.15)

As an analogue to (2.13), there is also a rational parametrization of hyperbolas: 1 + t2 2t ch = {(a , b ) ∣ t ∈ R {1, −1}} . (2.16) 1 − t2 1 − t2

23

2.1 Classical definitions

It represents the right branch for −1 < t < 1. The lower part of the left branch is obtained for t > 1 and the upper part for t < −1. The point (−a, 0) is again missing. For the sake of completeness, we add the parametrization cp = {(2pτ 2 , 2pτ ) ∣ − ∞ < τ < ∞}

(2.17)

for the parabola satisfying the vertex equation y 2 = 2px. There is another important equation for conics: Theorem 2.1.3 Polar equation: The set of points with coordinates (x, y) = (r cos ϕ, r sin ϕ) satisfying the polar equation r=

p 1 + ε cos ϕ

(2.18)

with constant p, ε ∈ R {0} is a conic with the focus F = (0, 0) and the associated directrix x = p/ε. The conic has the parameter ∣p∣ and the numerical eccentricity ∣ε∣. In the case ε = 0, we obtain a circle with center F and radius p. Proof: For p, ε > 0, the given polar equation is equivalent to r + εr cos ϕ = p, and further, to r = ε(

p − r cos ϕ) . ε

This implies that ∣r∣ = ε ∣ pε − r cos ϕ∣, which means, in view of the coordinate frame depicted in Figure 2.5, right, that p P F = ε ⋅ P l with F = (0, 0) and l ∶ x = . ε Also, the converse of the implication above is valid for the following reason: When the sign of r differs from that on the right-hand side in the equation above, i.e., r = −ε (

p −p − r cos ϕ) , hence r = , ε 1 − ε cos ϕ

we replace simultaneously r by −r and ϕ by ϕ + π . This does not change the defined set of points, but converts the above given polar equation into that presented in (2.18). Hence, for p, ε > 0, the given polar equation is equivalent to the Apollonian definition. Changing the sign of ε in the polar equation (2.18) is equivalent to the replacement of ϕ by ϕ + π. Therefore, it causes a rotation of the conic about F through the angle π. The same is true for changing the sign of p, since in this case r is replaced by −r. Thus, we have proved that for all p, ε ∈ R {0} the polar equation (2.18) represents a conic.

◾

We conclude with a comment on the domain of ϕ in the function on the right-hand side of the polar equation (2.18):

24

Chapter 2: Euclidean plane

• We obtain the entire ellipse by −π < ϕ ≤ π for ε < 1.

• For parabolas ε = 1, we need to exclude ϕ = π, hence −π < ϕ < π. • In the hyperbolic case ε > 1, we have to take into account that when ϕ tends to ϕ0 ∶= arccos(−1/ε), the radius r tends to infinity. Therefore, we need to restrict the polar angle to −ϕ0 < ϕ < ϕ0 for obtaining the first branch of the hyperbola. For ϕ0 < ϕ < 2π − ϕ0 , the radius r in (2.18) becomes negative, and we get the second branch. The two asymptotes have the slopes ± tan ϕ0 .

Velocity and acceleration vectors In order to comprehend why the trajectories of planets are conics, one needs to be familiar with velocity and acceleration vectors of curves. Therefore, we provide a brief introduction into the differential geometry of parametrized curves in the Euclidean 3-space E3 . With respect to any Cartesian coordinate frame (x, y, z), let c(τ ) = (x(τ ), y(τ ), z(τ )),

τ ∈ I,

be the trajectory of a point or a particle moving in E3 . This curve is parametrized by time τ , which passes the interval I. Then, the first derivative v ∶= c˙ =∶ ve1 with ∥e1 ∥ = 1

˙ equals the instantaneous velocity, is the velocity vector. Its norm v = ∥c∥ i.e., the derivative ds/dτ of the curve’s arc length s by the time τ . For v ≠ 0, the vector e1 is the unit tangent vector to the curve, and the spanned line {c(τ ) + λe1 ∣ λ ∈ R} is the tangent line to the curve c(I) at the point c(τ ). The second derivative

a ∶= ¨c = v˙ e1 + v e˙ 1

is the acceleration vector. According to results from elementary differential geometry, we have e˙ 1 =

de1 ds v ⋅ = vκ e2 = e2 , ds dτ ̺

(2.19)

where κ ≥ 0 is the curvature at the point c(τ ), and, for κ ≠ 0, ̺ = 1/κ is the radius of curvature. For κ ≠ 0, the unit vector e2 is unique and

25

2.1 Classical definitions

̺ c∗ (τ0 )

v

a

c(τ0 ) an t a

FIGURE 2.8. The geometric equivalent to the mutual dependence between the velocity vector v, the acceleration vector a = an + at , and the center of curvature c∗ is the shaded right-angled triangle.

orthogonal to e1 : A positive quarter turn in the plane spanned by c˙ and c¨ carries e1 to e2 . The orthogonal unit vectors e1 and e2 span the Frenet frame of the curve c at c(τ ). We call e2 the principal normal vector of the curve. Thus, we obtain a = ¨c = v˙ e1 + v 2 κ e2 . The first term in this vector sum defines the tangential acceleration vector at , the second one the normal acceleration vector an . We summarize a = an + at with ∥at ∥ = ∣v∣, ˙

∥an ∥ =

v2 . ̺

(2.20)

For constant speed (v˙ = 0) the tangential acceleration ∥at ∥ vanishes. Then, the second derivative c¨ is orthogonal to c˙ everywhere in I along the curve. In Figure 2.8, a planar curve is depicted. Beside the velocity vector v and the acceleration vector a = an + at , also the center of curvature c∗ (τ0 ) = c+̺ e2 ∣τ =τ0 , i.e., the center of the osculating circle at the point c(τ0 ), is shown. As a consequence of the last equation in (2.20), there is a right triangle with the altitude ∥v∥ subdividing the hypotenuse in segments of lengths ̺ and ∥an ∥. As demonstrated in Figure 2.8, it is quite useful to visualize velocity vectors as well as acceleration vectors by arrows. However, this needs an

26

Chapter 2: Euclidean plane

agreement about how to represent velocities and accelerations by lengths of arrows. The simplest way is to define an appropriate scaling factor σ and to depict the scaled vector σv. For example, σ = 0.005 s converts a velocity of 10 m/s into an arrow of length 50 mm. The last equation in (2.20) reveals that, if σ is used for velocities, then σ 2 should be the scaling factor for accelerations. This guarantees that, because of (σ 2 ∥an ∥)̺= (σv)2 , the shaded right triangle in Figure 2.8 remains right after scaling. Often we are mainly interested in the geometry of curves, i.e., in properties which are independent of the way how we traverse the curve. Such properties do not alter when we replace the time τ by any other parameter τ which is related to τ by a differentiable and monotonic function τ = f (τ ),

df τ ∈ I, with f˙ = ≠ 0. dτ

The new parameter τ needs not be the time anymore. Nevertheless, we want to retain the notations ‘velocity’ and ‘acceleration’ since they provide the derivatives with an intuitive meaning. The derivatives of the new parametrization c(τ ) ∶= c (f (τ )), τ ∈ I, are dc dc dτ c˙ = = ⋅ = f˙ c˙ ≠ 0, dτ dτ dτ

2 ¨c = d c = f¨c˙ + f˙2 ¨c . dτ 2

We note that the velocity vectors v = c˙ and v = c˙ are linearly dependent. The spanned tangent line remains the same. The vector a ∶= c¨ is a linear combination of c˙ and c¨. The decomposition a = at + an

into components parallel and orthogonal to the velocity vector v = c˙ shows that an = f˙2 an . Therefore, by virtue of (2.20), also the radius of curvature ̺ = ∥v∥2 /∥an ∥ is invariant under to parameter transformations. This statement is also a consequence of the fact that among all circles passing through the point c(τ0 ), the osculating circle with center c∗ (τ0 ) = c + ̺ e2 , with radius ̺ = 1/κ, and located in the osculating plane spanned ˙ 0 ) and c¨(τ0 ), is the best approximating circle of the given curve by c(τ c(I). We prove this by showing that for any fixed parameter value τ0 ∈ I the distance d(τ0 ) between c(τ0 ) and a fixed center m in the osculating

27

2.1 Classical definitions

plane at the point c(τ0 ) is stationary of order ≥ 2 if, and only if, m(τ0 ) is the center of curvature c∗ (τ0 ). Proof: We differentiate the equation d2 (τ ) = (c(τ ) − m)2 twice and obtain ˙ )⟩, dd˙ = ⟨c(τ ) − m, c(τ

˙ 2 + ⟨c(τ ) − m, c ¨(τ )⟩. d˙ 2 + dd¨ = ∥c∥

We denote the Frenet frame at c(τ0 ) by (e1 , e2 ), extend it by e3 to an orthonormal frame in E3 , and set ̺0 = ̺(τ0 ) and v0 = v(τ0 ). Then, by virtue of (2.20), we get at τ = τ0 dd˙ ∣τ =τ0 = v0 ⟨c(τ0 ) − m, e1 ⟩,

v2 d˙ 2 + dd¨∣τ =τ0 = v02 + ⟨c(τ0 ) − m, v(τ ˙ 0 ) e1 + 0 e2 ⟩ . ̺0

˙ 0 ) = d(τ ¨ 0 ) = 0 is equivalent to c(τ0 ) − m = λ e2 + µ e3 and λ + ̺0 = 0, i.e., m = Hence, d(τ c(τ0 ) + ̺0 e2 − µ e3 = c∗ (τ0 ) − µ e3 for all µ ∈ R. As m lies, according to the assumption, in the osculating plane at c(τ0 ), µ must vanish, and m is the center of curvature c⋆ (τ0 ).

◾

Summing up, each reparametrization τ = f (τ ) of a given curve acts on the velocity vectors v like a scaling with factor σ = f˙, and on the normal acceleration vectors an like a scaling with factor σ 2 = f˙2 . The arc length s is often called a natural parameter of the curve. This particular parametrization is characterized by the fact that all tangent vectors dc/ds are unit vectors. The second derivatives d2 c/ds2 are orthogonal to the curve and have length κ; they are called curvature vectors. After this brief introduction to the elementary differential geometry of curves, we return to conics. We recall the polar equation (2.18) and the related coordinate frame depicted in Figures 2.5 (right) and 2.9. By choosing the polar angle ϕ as parameter τ , we obtain the parametrization c(τ ) = r(τ ) (cos τ, sin τ ) with r(τ ) =

p , if 1 + ε cos τ ≠ 0. 1 + ε cos τ

This yields the velocity vector v(τ ) = vr + vr⊥ with vr = r˙ (cos τ, sin τ ) , vr⊥ = r (− sin τ, cos τ ) , (2.21) and

r˙ =

p ε sin τ εr2 sin τ = . (1 + ε cos τ )2 p

The first component vr of the velocity vector v(τ ) has the direction of the position vector c(τ ), while the second component vr⊥ is orthogonal to the first. This gives rise to a universal graphical construction of tangent lines for conics (Figure 2.9).

28

Chapter 2: Euclidean plane y

σvr =−c(τ ) F

l P =c(τ )

τ

A

σv

σvr⊥

p

x T tP

P1

p/ε

FIGURE 2.9. Construction of the tangent line tP at the point P = c(τ ) to the conic c with focal point F and directrix l. We use the scaling factor σ = −r/r˙ for velocity vectors.

Theorem 2.1.4 Let c be a conic with focus F and associated directrix l. The tangent line tP to c at any point P different from the vertices A and A2 intersects l at a point T such that [F, P ] is orthogonal to [F, T ]. Proof: Under the assumption sin τ ≠ 0 we have r˙ ≠ 0. Therefore, we can choose the scaling factor r −rp −p σ=− = 2 = . r˙ εr sin τ ε r sin τ Then, p p σv = σvr + σvr⊥ = −r(cos τ, sin τ ) − (− sin τ, cos τ ) = −c(τ ) + (1, − cot τ ). ε sin τ ε In the final vector sum, the second component is orthogonal to the position vector, and its first coordinate equals that of the directrix l.

◾

Remark 2.1.2 We have already noted that F is the pole of the directrix l. According to the properties of polarities (see Section 7.1), the point T is the pole of the connecting line [F, P ]. This is obvious when we pay attention to the second point of intersection P1 between the conic and the line [F, P ] (Figure 2.9).

Differentiating the conic’s parametrization a second time w.r.t. the polar angle τ yields the acceleration vector

with

a(τ ) = (¨ r − r) (cos τ, sin τ ) + 2r˙ (− sin τ, cos τ ) r¨ =

pε cos τ 2pε2 sin2 τ + . (1 + ε cos τ )2 (1 + ε cos τ )3

29

2.1 Classical definitions

In order to compute the velocity and acceleration at the vertex A, we set τ = 0, to get p −p v(0) = (0, 1), a(0) = (1, 0) = an (0). 1+ε (1 + ε)2 We obtain v = ∥v∥ = p/(1 + ε) and, by virtue of (2.20), the curvature radius ̺(0) = v 2 /∥an ∥ = p, which gives a new geometric meaning to the parameter p:

Theorem 2.1.5 The parameter p of any conic c equals the radius of curvature of c at the vertices on the principal axis s. The vector an (0) of the normal acceleration points from the vertex A towards the center of curvature A∗ (Figure 2.8). Therefore, both A∗ and the focal point F lie on the same side of the corresponding vertex A (Figure 2.10).

F1 A1 A∗1

F2 A∗2

c

A2 p F1

c

A∗1

p

A1

A2 p

p

p

c

F2 A∗2

p

l A p

F

A∗

FIGURE 2.10. For all conics c, the parameter p equals the curvature radius at the vertices Ai on the principal axis.

●

Exercise 2.1.1 Osculating circle at the vertex.

Confirm Theorem 2.1.5 in the following way: Based on the vertex equations (2.6) of a conic c and a circle c∗ (ε = 0), try to specify c∗ in such a way that all (real and complex) intersection points between c and c∗ coincide with the origin.

30

Chapter 2: Euclidean plane

2.2 Tangent lines of conics The optical property We now are going to apply the general construction of tangents for conics to ellipses and hyperbolas, and pay attention to the fact that Theorem 2.1.4 holds for both focal points F1 and F2 .

P ϕ

T1 A1

F1

ϕ

α

tP v

P vr2 F2 c

l1

vr1

T2

tP

A2 l2

F1

F2 c

FIGURE 2.11. The tangent line tP at any point P to the ellipse c bisects the exterior angle of the triangle F1 P F2 .

Figure 2.11, left, shows for an ellipse c how the tangent tP to c at any point P can be constructed via F1 as well as via F2 . Of course, P must be different from the vertices A1 and A2 . The slope angle of tP is denoted by α. The intersection point of tP with the directrix li associated with Fi (for i = 1, 2) is denoted by Ti .

As a consequence, the angle ϕ between tP and the segment P F1 is congruent to the one between tP and P F2 , since according to the Apollonian definition (page 13), we have cos ϕ =

P F1 ε P l1 ε P l2 P F2 = = ε cos α = = . P T1 P l1 / cos α P l2 / cos α P T2

There is another way to prove this symmetry. Let any parametrization of the ellipse be given. According to (2.21), the velocity vector v at P can be decomposed into a radial component vr of signed length r˙ and an orthogonal component denoted by vr⊥ (Figure 2.9). If ri (τ ) for i = 1, 2 denotes the distance of points P ∈ c to the focal point Fi , then the standard definition r1 + r2 = const. implies that r˙1 = −r˙2 . Consequently, the

31

2.2 Tangent lines of conics

radial components vr1 and vr2 of v in direction of F1 P and F2 P , respectively, have the same length but different signs. Therefore, independently of the scaling of velocity vectors, the triangles formed by v and the two decompositions are symmetric with respect to the tangent tP (see Figure 2.11, right). We can rephrase this property by claiming that the tangent tP bisects the exterior angle of the triangle F1 P F2 at P . Similar arguments hold for tangents tP of hyperbolas. Again, we detect mutually similar right-angled triangles P F1 T1 and P F2 T2 due to the Apollonian definition (Figure 2.12, left). On the other hand, the standard definition ∣r1 − r2 ∣ = const. implies that r˙1 = r˙2 . This time, the lengths of the radial components vr1 and vr2 of v have equal signs (Figure 2.12, right). Consequently, the tangent tP at P to the hyperbola c bisects the interior angle of the triangle F1 P F2 at P . t2 P

l1

c

l2

P

c

vr2

vr1

v

T1 F1

A1

A2 tP

F2

F1

F2 tP

T2 FIGURE 2.12. The tangent line tP at any point P to the hyperbola c bisects the interior angle of the triangle F1 P F2 .

If the point P ∈ c tends to infinity, the tangent tP tends to one of the two asymptotes, since they intersect the directrices at the pedal points with respect to F1 or F2 (note the asymptote t2 and F2 in Figure 2.12, left). In the case of a parabola (Figure 2.13), we have equal distances P F = P l. Let G denote the pedal point of the directrix l w.r.t. P . Then, the triangles P F T and P GT are congruent, as both are right, share the hypotenuse P T , and have equal side-lengths P F = P l. Hence, the tangent tP is the axis of symmetry of the kite P F T G, and therefore, it bisects the interior angle between P F and P G.

32

Chapter 2: Euclidean plane

G

P tP N

c

T s A l

F

k

tA

FIGURE 2.13. The tangent line tP at any point P to the parabola c bisects an angle between [P, F ] and the parallel to the axis s through P . When reflecting the focal point F in the tangent line tP , we obtain a point G of the directrix. The pedal point N of tP w.r.t. F is located on the tangent line tA to c at the vertex A.

Theorem 2.2.1 At each point P of a conic c, the tangent tP bisects an angle between the connecting lines [P, F1 ] and [P, F2 ] of P with the focal points, provided that, in the case of a parabola, the point F2 is defined as the ideal point of the axis s.

F F1

F2

F

FIGURE 2.14. The optical property of ellipses and parabolas.

This is called the optical property of conics. In the case of an ellipse it means that rays of light going out from one focal point F1 and being reflected in the ellipse meet all at the second focal point F2 (Figure 2.14,

33

2.2 Tangent lines of conics

left). Analogously, after reflection in a parabola all rays parallel to the axis concentrate at the focus F (Figure 2.14, middle). This is no longer true for rays being not parallel to the axis (Figure 2.14, right).

a

P tP

m

a a

P

a

F1 m

M

F2

F1

M

F2

tP c c FIGURE 2.15. The line m through the center M and parallel to tP intersects the focal lines [P, F1 ] and [P, F2 ] in points at distance a to the point P .

●

Exercise 2.2.1 Construction of the semimajor axis.

Prove the following statement which holds for ellipses and hyperbolas c with focal points F1 and F2 and with the semimajor axis a : Let the line m through the center M of c be parallel to the tangent tP at P . Then, M intersects the lines [P, F1 ] and [P, F2 ] at points P1 and P2 at distance a to P (Figure 2.15).

Hint: Project the equidistant points F1 , M , and F2 parallel to tP onto the line [P, Fi ].

Pedal curves and orthotomics There are other consequences of the bisecting property of tangent lines, as stated in Theorem 2.2.1: In the case of a parabola, we learn immediately from Figure 2.13, that the reflection of the focal point in the tangent line tP is a point G on the directrix l. A dilation with center F and scaling factor 1/2 maps G onto the pedal point N of tP with respect to F ; this pedal point is located on the tangent tA at the vertex A. The analogues for ellipses and hyperbolas are depicted in Figure 2.16. The reflection in the tangent tP maps the focus F1 onto a point G1 on the line [P, F2 ] such that P F1 = P G1 . The standard definitions of ellipses and hyperbolas imply G1 F2 = 2a. Hence, the reflection G1 lies on a circle with center F2 , the orthotomic circle g1 , for which we write (F2 ; 2a) in brief.

34

Chapter 2: Euclidean plane

A dilation with center F1 and scaling factor 1/2 maps G1 onto the pedal point N1 of tP w.r.t. F1 , and the second focus F2 onto the center M of c. The orthotomic circle g1 with radius 2a is transformed into the principal circle ca = (M ; a). k

k

N2

G1

tP

c ca

P

P

a

G1 N1

ca g1

a 2a

A2 F2

M

F1 A1 N1

M

c

A2

F2

2a

A1 F1

N2 tP

g1

FIGURE 2.16. The reflection of the focal point F1 in the tangent tP of the ellipse or hyperbola c gives a point G1 on the orthotomic circle g1 = (F2 ; 2a). The pedal point N1 of tP with respect to F1 is located on the principal circle ca = (M ; a), which contacts c at the vertices A1 and A2 . Remark 2.2.1 The set of pedal points of the tangent lines of a curve d with respect to any point P is called the pedal curve of d with respect to P . The locus of reflections of P in the tangent lines of d is called the orthotomic of d with respect to P . A dilation with center P and scaling factor 1/2 maps the orthotomic of d w.r.t. P onto the pedal curve.

Theorem 2.2.2 The pedal curve of an ellipse or hyperbola c with respect to a focus is the principal circle ca = (M ; a). The reflections G1 of the focus F1 in the tangent lines tP to c are located on the orthotomic circle g1 = (F2 ; 2a). The line [G1 , F2 ] passes through the point P of contact between c and tP (Figure 2.16).

The pedal curve of a parabola c with respect to the focus F is the tangent tA at the vertex; the locus of reflections of F in the tangent lines of c is the directrix l (Figure 2.13).

35

2.2 Tangent lines of conics

This theorem as well as the following statements are also valid for circles, when they are seen as limiting cases of ellipses with F1 = F2 = M .

Conversely to Theorem 2.2.2, each tangent line tP of a conic c is the perpendicular bisector of the segment F1 G1 when G1 traces the circle g1 = (F2 ; 2a) or the line l. The latter holds for parabolas, of course. Corollary 2.2.1 The envelope of the axes of symmetry between a fixed point F1 and a point G1 tracing a line l or a circle g1 is a conic c.

Note the particular case: When F1 happens to be the center of g1 , then c is a circle. The equation P F1 = P G1 together with the collinearity of the points F2 , P , and G1 implies that P is the center of a circle k which contacts g1 at G1 and passes through F1 (see Figure 2.16, left and right). In the case of a parabola (see Figure 2.13), P is the center of a circle k which passes through F and is tangent to l.

c

g1 F1 F2

FIGURE 2.17. The displayed circles contact g1 and pass through F1 . Their centers are located on the hyperbola c.

36

Chapter 2: Euclidean plane

Corollary 2.2.2 A conic c is the locus of centers P of circles k which are tangent to a given circle g1 = (F2 ; 2a) or to a given line l and pass through a given point F1 ∈/ g1 , l (Figure 2.17). In other words, the conic c is the bisector of the point F1 and the circle g1 or the line l, i.e., the set of points at equal distance to F1 and g1 or l. Proof: The conic c cannot be a proper subset of the bisector, since in the case of a circle g1 both the bisector and the conic share exactly two points on each diameter of g1 . Similarly, in the case of a line l, both curves share exactly one point on each perpendicular to l.

◾

g2 g1

FIGURE 2.18. The displayed circles which contact g1 and g 2 , have their centers on an ellipse.

There are two kinds of contact between two different circles (M1 ; r1 ) and (M2 ; r2 ). Either one circle includes the other, or the point of contact lies between the two centers M1 and M2 . In the first case we speak of interior contact, in the second of exterior contact. It makes sense to endow the radii of circles with a sign such that at interior contact the signs of the involved radii are equal, and otherwise different. Then, the contact of the two circles is equivalent to M1 M2 = ∣r1 − r2 ∣.

(2.22)

37

2.2 Tangent lines of conics

This equation reveals that the contact is preserved when we add an arbitrary constant r0 ∈ R to both signed radii while the centers remain fixed. Of course, when one radius changes its sign, the type of contact changes. In the case r1 = 0 the center M1 lies on the second circle. When, according to Corollary 2.2.2, the circles k pass through a fixed point F1 , then it depends on whether F1 is inside or outside g1 = (F2 ; 2a). In the first (= elliptic) case, all k’s are in the interior of g1 and their signed radii are positive. Otherwise (= hyperbolic case), we have an exterior contact, and the radii of the circles k must be negative (Figure 2.17) in order to satisfy (2.22). Now, we can generalize the definition of conics c given in Corollary 2.2.2 in the following way: We add a constant r0 ∈ R to the signed radii of all k’s, replace at the same time the point F1 by the circle g 2 = (F1 ; r0 ) and the circle g1 = (F2 ; 2a) by the circle g 1 = (F2 ; 2a + r0 ). This yields a set of circles k being tangent to two fixed circles g 2 , g 1 with respective centers F1 and F2 . When traversing the k’s, the types of contact with g 1 and g 2 either remain the same or change simultaneously. The transition from one type of contact to the other takes place when the signed radius of k either becomes zero (note Figure 2.18) or tends to ±∞, while the center changes from one branch of the hyperbola to the other.

g2

k

g1

FIGURE 2.19. The lower half of a Dupin horn cyclide.

When the circles k serve as equators of spheres, then the surface enveloped by these spheres is a Dupin cyclide, a horn cyclide (Figure 2.19) or a ring cyclide. A needle cyclide arises when g 1 contacts g 2 . The parabolic case

38

Chapter 2: Euclidean plane

FIGURE 2.20. Two parabolic Dupin cyclides as plaster models, selected from the collection Martin Schilling, Leipzig 1880 (note [143]).

with a line g 1 leads to parabolic Dupin cyclides (Figure 2.20). We will meet the Dupin cyclides again in Section 4.2 on page 157 (note also [109, Fig. 10.18]). ●

Exercise 2.2.2 Pedal curve of an imaginary focus.

In Chapter 7, we will learn that for ellipses and hyperbolas it makes sense to define, besides the (real) foci with standard coordinates (±e, 0), also the pair of complex conjugate points (0, ±ie) as focal points. Prove analytically the following counterpart to Theorem 2.2.2: The pedal curve of an ellipse c with respect to its complex conjugate focal points is the circle cb = (M ; b). What is the analogue for hyperbolas?

Hint: Use the standard equations (2.9). At the point P = (xP , yP ) ∈ c the tangent line tP to the ellipse satisfies the equation x2P y2 xP yP x + 2 y = 1, where + P2 = 1. 2 2 a b a b The pedal point of tP with respect to an imaginary focus is (0, ±ie) + λ(xP /a2 , yP /b2 ), with an appropriate parameter λ.

Confocal conics Another consequence of the bisecting property of tangents to conics, as stated in Theorem 2.2.1, deals with families of confocal (or homofocal ) conics: Given two focal points F1 and F2 , all ellipses and hyperbolas sharing these focal points form an orthogonal net (Figure 2.21). Note that through each point P off the symmetry axes there passes one ellipse and one hyperbola, and the corresponding tangent lines at P are the two bisectors of the lines [P, F1 ] and [P, F2 ], and therefore, perpendicular. As limiting curves also the two axes of symmetry can be added: the segment F1 F2 is the limit of an ellipse with vanishing semiminor axis b; the secondary axis is the limit of a hyperbola with vanishing semimajor axis a; the

39

2.2 Tangent lines of conics

M F1

F2

FIGURE 2.21. Confocal central conics constitute an orthogonal net.

half-lines starting from F1 or F2 and pointing away from the center M are the limit of a hyperbola with vanishing b. Similarily, there is also an orthogonal net formed by confocal parabolas, i.e., by parabolas sharing the focal point F and the axis (Figure 2.22). On the axis, the two half-lines starting from F complete this orthogonal net. The nets of confocal conics have an additional property which is named after J. Ivory5 . This theorem turns out to be valid also in other geometries (see, e.g., Theorem 10.1.10). Theorem 2.2.3 Ivory’s theorem in E2 . In each curvilinear quadrangle formed by two pairs of conics of the same type within a confocal net, the two diagonals have the same lengths (Figure 2.23).6

5

Sir James Ivory (1765–1842), Scottish mathematician and astronomer. His famous theorem is only a by-product in his publication [75]. 6 To be precise, Ivory’s Theorem (Euclidean version) is valid only for curvilinear quadrangles X1 X1′ X2′ X2 where opposite sides (‘arcs semblables’ according to [29]) are corresponding

40

Chapter 2: Euclidean plane

F

FIGURE 2.22. The net of confocal parabolas is orthogonal, too.

We leave the proof of this planar Euclidean version as an exercise. ●

Exercise 2.2.3 Ivory’s theorem in E2 .

Verify Ivory’s theorem (Figure 2.23) for confocal ellipses and hyperbolas as well as for confocal parabolas in the following way: The parametrization x = e cos u cosh v, y = e sin u sinh v,

0 ≤ u < 2π, v ∈ R

defines as lines v = const., v ≠ 0, and u = const., u ≠ 0, ellipses and branches of hyperbolas, respectively, which share the focal points (±e, 0) (compare with Figure 2.21). On the other hand, the parameter lines u = const. and v = const. of the parametrization x = u2 − v2 , y = 2uv,

(u, v) ∈ R2

belong to the family of confocal parabolas (Figure 2.22). Show that for given parameter values u1 , u2 , v1 , v2 the diagonals in the curvilinear quadrangle bounded by the curves u = u1 , v = v1 , u = u2 , and v = v2 have the same length. Figure 2.23 shows the mapping c → c′ with Xi ↦ Xi′ between conics of the same type. This mapping is induced by v1 ↦ v2 , while u remains invariant. It acts like simultaneous scalings x ↦ x cosh v2 / cosh v1 and y ↦ y sinh v2 / sinh v1 of both coordinates.

The statement of Ivory’s theorem, X1 X2′ = X1′ X2 , is still valid when the ellipse c′ degenerates into the segment F1 F2 . When the principal vertices A or B serve as points X2 with X2′ = F1 or F2 , then Ivory’s theorem shows directly the standard definition of the ellipse c (Figure 2.24): XF1 + XF2 = XA′ + XB ′ = X ′ A + X ′ B = AB. under an affine transformation which fixes the axes of the confocal conics (note Lemma 8.1.3). In this case we speak of an Ivory quadrangle. For further details see page 388.

41

2.2 Tangent lines of conics

X2 c

X1

X1′ F1

X2′

c′

F2

FIGURE 2.23. Ivory’s theorem in the Euclidean plane E2 : X1 X2′ = X1′ X2 . Remark 2.2.2 Figure 8.12 reveals that the extended diagonals in each Ivory quadrangle are tangent to the same confocal conic. This result is well-known in the literature, and the content of Theorem 8.1.4. Another proof of Ivory’s theorem is presented on page 392 and can immediately be generalized to higher dimensions (note [109, Theorem 7.2.2]).

●

Exercise 2.2.4 Confocal conics obtained by a conformal map.

Show that in the complex plane the analytic function z ↦ cos z transforms the rectangular coordinate grid into a net of confocal ellipses and hyperbolas. The function z ↦ z 2 produces confocal parabolas.

Constructions with ruler and compass, Problem 1 In this section, we present graphic solutions for two basic geometric problems related to a conic c: Which tangents of c pass through a given point Q, and in which points intersects a given line g the conic c? These constructions will provide a deeper insight into the geometry of conics.7 7

Permanent users of geometry software might question the importance of these problems, as conics can be drawn with high precision, and points of intersection and tangents can be “snapped” accurately. When one can draw the conic with sufficient precision, one can immediately ‘see’ the points of intersection or ‘draw’ the tangents. Nevertheless, there is an interest in a theoretically exact solution, i.e., in an algorithm using only ruler and compass for finding the solutions, even without depicting the conic.

42

Chapter 2: Euclidean plane X c

A

A =F1 ′

c′

X′

B ′ =F2

B

FIGURE 2.24. Ivory’s theorem remains valid when the conic c′ degenerates into the segment F1 F2 . Then, we obtain XA′ = X ′ A, XB ′ = X ′ B and, hence, XA′ + XB ′ = AB.

Problem 1: Let a conic c be given — in the case of an ellipse or hyperbola by its focal points F1 , F2 and the length 2a, in the case of a parabola by its focal point F and the directrix l. How can we draw tangents from any given point Q to the conic c ? Where are the points of contact? Let c be an ellipse. By virtue of Theorem 2.2.2, the orthotomic of c with respect to the focal point F1 is the circle g1 = (F2 ; 2a). Each tangent t to c is the bisector of F1 and any point G1 ∈ g1 . When t passes through Q, then Q must be equidistant to F1 and G1 . Therefore, we can solve our problem in the following steps (Figure 2.25): (i) Draw the auxiliary circle h = (Q; QF1 ) and intersect it with the orthotomic circle g1 = (F2 ; 2a). (ii) For each point G1 of intersection, the bisector t of G1 and F1 is tangent to c and passes through Q. (iii) The point T of contact between t and c is aligned with G1 and F2 . In the case displayed in Figure 2.25, the circle h intersects g1 at two different points G1 and G1 . We obtain two different tangents t and t through Q. The point Q lies in the exterior of c. When Q happens to be a point of c, then the two intersection points G1 and G1 coincide. The line t = t equals the tangent tQ to c at Q. The circle h is an element of

43

2.2 Tangent lines of conics h

Q α

G1

α

T t c

β β F2

F1

G1 T

2a t

g1 FIGURE 2.25. Construction of the tangents t and t through the given point Q to the conic c. The lines t and t share the angle bisectors with the lines connecting Q with the two focal points. On the other hand, [F2 , Q] bisects the angle 1), the origin O lies outside k ′ . When the velocity vector c˙ is seen as a position vector tracing the circle k ′ with angular velocity ϕ, ˙ then ¨c is a tangent vector to k ′ . Hence, there is also a graphical way to determine ¨c, as shown in Figure 3.5 using the scaling factor σ 2 . Consequently, we can also find the normal acceleration an , and by Figure 2.8 the center of curvature P2∗ of k at the point P2 .

8

The velocities of the Earth around the Sun range between 29.3 km/s in summertime and 30.3 km/s in wintertime (w.r.t. the Northern hemisphere). The data for the orbit of the Earth (Figure 3.2) are as follows (in metrical units): a = 149 600 000 km = 1 astronomic unit, ε = 0.01672, b = 149 580 000 km, p = 149 560 000 km, T = 365.25964 days.

78

Chapter 3: Differential Geometry

α A′

σ 2 an σ 2 ¨c σ c˙ α P 2

hodograph

ϕ P1

P1′ ϕ

A r ϕ˙

p

B P2∗

k′

r

ϕ˙

orbit

l

ε

σ c˙

rϕ˙

P2′

k

p/

p

rϕ˙

O

p/ε B′

σ 2 ¨c

FIGURE 3.5. The acceleration vector c¨ is the velocity vector of the hodograph. This gives rise to a graphical construction of the orbit’s center of curvature P2∗ . We use the scaling factor σ = p2 /Cε for velocities and σ 2 for accelerations.

This is equivalent to the following analytic approach: Let α denote the angle between c and c˙ (Figure 3.5, left). The two components of c˙ in (3.3) ˙ as can be expressed in terms of v ∶= ∥c∥ r ′ ϕ˙ = v cos α and r ϕ˙ = v sin α .

The second equation implies C = r 2 ϕ˙ = rv sin α, hence v =

C . r sin α

On the other hand, by (2.20) we obtain the radius of curvature ̺ as ̺ = v 2 /an with an = ∥an ∥ as the normal acceleration. This yields, together with (3.1), v2 Gm1 = ∥an ∥ = ∥¨ c∥ sin α = 2 sin α, ̺ r and therefore, by (3.10) ̺=

r2 v2 r2 C2 1 C2 1 p ⋅ = ⋅ 2 2 ⋅ = ⋅ = . 3 Gm1 sin α Gm1 r sin α sin α Gm1 sin α sin3 α

79

3.1 Conics as orbits of planets

Theorem 3.1.4 Let k be any conic with focal point F and parameter p . For any given point X ∈ k, let α denote the angle between the line [F, X] and the tangent line at X (see Figure 3.1 and Figure 3.5, left). Then, the radius of curvature of k at X is p ̺= . sin3 α Note that this formula holds for all types of conics. It is even independent of the numerical eccentricity ε. The formula includes in particular the statement that the parameter p of the conic equals the radius of curvature at the principal vertex with α = π2 (Theorem 2.1.5).

Kepler’s equation For each Kepler orbit there is a simple geometric relation between the ˙ position c(t) and the corresponding velocity vector c(t), as Figure 3.5 demonstrates. However, the computation of c(t) as a function of time t can only be carried out numerically since the function ϕ(t) cannot be expressed in closed form.

We present a method valid for ellipses k, i.e., in the case 0 < ε < 1. Let our moving point P2 pass the vertex A at minimal distance to P1 at time t0 . Due to the constant areal velocity, the ratio between (t − t0 ) and the orbital period T equals the ratio between the areas of the elliptical sector from c(t0 ) to c(t) (shaded in Figure 3.6) and the area abπ of the ellipse.

We compute the area of the elliptical sector by inspecting its image under the affine transformation from the ellipse k to its principal circle k˜ with radius a. Under this transformation, areas are scaled by the factor a/b. Let β denote the center angle of the circular arc from c(t0 ) to the affine image c˜(t). For 0 < β < π we can compute the area of the affine image by subtracting the area of a triangle from the area of the circular sector. Thus, we obtain the expression a2 β ea sin β − , 2 2 which turns out to be valid also for π < β < 2π. This leads to the ratio b (a2 β − ea sin β) 2a

abπ

=

t − t0 , T

80

Chapter 3: Differential Geometry ˜ c(t)

k˜

c(t)

k

(0, 0)

β

r

ϕ

a

A (aε, 0) c(t0 )

FIGURE 3.6. The Kepler equation allows the computation of the polar angle ϕ for a given time t.

and hence, to the Kepler equation

t − t0 . (3.12) T The procedure for computing the polar angle ϕ for a given t runs as follows. First, we determine β from (3.12), where the function of β on the left-hand side is monotonic. Now, the comparison of the Cartesian coordinates of c(t) with those of the affine preimage of c˜(t), i.e., β − ε sin β = 2π

(a cos β, b sin β) = (aε + r cos ϕ, r sin ϕ) ,

yields r cos ϕ = a(cos β −ε). We substitute r and a from the polar equation (2.18) of conics and relations (2.4), respectively, and obtain finally cos ϕ =

cos β − ε , while sin β sin ϕ ≥ 0. 1 − ε cos β

Historical geometrical proofs revisited It is worthwhile to check how Newton in his “Philosophiae Naturalis Principia Mathematica” in 1686 could prove Kepler’s First Law without any support from calculus. The Nobel Prize winner Richard P. Feynman demonstrated this in his famous 1964 lecture at the California Institute of Technology [46].

81

3.1 Conics as orbits of planets

FIGURE 3.7. From Newton’s “De motu corporum in gyrum”, early draft — reproduced with the kind permission of the Syndics of Cambridge University Library.

At the time of Newton, and still decades later, ‘differential geometry’ was rather ‘difference geometry’. One imagined an orbit as a polygon where the segments represented the velocity vectors of the moving point. We demonstrate this way of reasoning in a proof for the constancy of the areal velocity. c(t4 )

a4 v4 v3 a3

P1

c(t3 )

v3 v2 c(t2 ) a2 v2 v1 a1

c(t1 ) v1

c(t0 )

FIGURE 3.8. Newton’s proof of the constancy of the areal velocity

We assume the fixed particle P1 as origin of our coordinate frame and start with a uniform subdivision t0 , t1 , . . . , tn of the given time interval, i.e., with ti − ti−1 = h for i = 1, . . . , n. The polygon c(t0 )c(t1 ) . . . c(tn ) approximates the orbit (Figures 3.7 and 3.8). The difference vectors vi ∶= c(ti ) − c(ti−1 ) approximate scaled velocity vectors (with scaling factor h)

82

Chapter 3: Differential Geometry

since ˙ i ) = lim c(t

1 1 1 1 (c(ti ) − c(ti − h)) = lim (c(ti ) − c(ti−1 )) = lim vi ≈ vi . h→0 h h→0 h h→0 h h

Furthermore, we set ai ∶= vi+1 − vi for i = 1, . . . , n − 1, and note that they approximate scaled acceleration vectors (with scaling factor h2 ) since 1 1 1 1 ¨c(ti ) = lim (c(t ˙ i+1 ) − c(t ˙ i )) = lim 2 (vi+1 − vi ) = lim 2 ai ≈ 2 ai . h→0 h h→0 h h→0 h h

Since the attractive force on the point c(ti ) acts along the connecting line with P1 , we can assume that ai is parallel to the position vector c(ti ). Hence, the vector c(ti ) spans with vi as well as with vi+1 parallelograms of equal area (note Figure 3.8)9 , as the have the same base length ∥c(ti )∥ and the same height. Therefore, adjacent halves of these parallelograms, the triangles P1 c(ti−1 )c(ti ), and P1 c(ti )c(ti+1 ) have the same area, too. Hence, all sectors P1 c(ti−1 )c(ti ) of the orbit are equi-areal. The limit h → 0 shows that the areal velocity must be constant. Now, we recall Hamilton’s elegant geometric explanation why the hodograph is circular and the orbit is a conic [65]. However, we will rephrase his way of reasoning in the language of differential geometry and modify some parts.

Theorem 3.1.5 (Hamilton, 1847) Let P1 be a fixed particle in space Ð → and let a particle P2 move under the influence of a single force F . Ð → 1. The force F acts along the connecting line [P1 , P2 ] if, and only if, (i) the orbit of P2 is located in a plane through P1 , and (ii) P2 moves with constant areal velocity about P1 . Ð → 2. Let F always act along [P1 , P2 ]. Then, in the case of non-vanishing areal velocity, the following three statements are equivalent: Ð → a) The force ∥ F ∥ is proportional to the inverse of the squared distance r between P1 and P2 . b) The hodograph of the orbit of P2 is a circular arc. c) The orbit of P2 is a conic with P1 as a focal point.

9

By the “parallelogram spanned by vectors a and b” we understand the parallelogram with the vertices 0, a, a + b, and b.

83

3.1 Conics as orbits of planets

¨} implies that Proof: According to our proof of Theorem 3.1.1, the linear dependence of {c, c c × c˙ = n is constant. In the case n ≠ 0, the orbit is planar and the areal velocity is constant. Otherwise, the point P2 moves along a line through P1 .

Conversely, any planar curve can be represented in polar coordinates. The condition r 2 ϕ˙ = C = const. shows by (3.5) the linear dependence of c and ¨ c. a) ⇔ b): In the case of a central attractive force there is a kind of symmetry between the orbit and the hodograph (see Figure 3.5): The position vector of one curve is a tangent vector to the other. For any planar curve, the Frenet equation (2.19) e˙ 1 = vκe2 implies that ω = vκ is the signed angular velocity of the tangent vector. Hence, 1 v ̺= = ∣κ∣ ∣ω∣

is the radius of curvature. The angular velocity of c and at the same time that of the tangent vector ¨ c to the hodograph is ϕ, ˙ and we confine our attention to the case C = r 2 ϕ˙ ≠ 0. Therefore, we obtain for the hodograph in the case of an arbitrary attraction law ∥¨ c∥ = a(r) the radius of curvature 2 ∥¨ c∥ a(r) r a(r) ̺h = = = . ∣ϕ∣ ˙ ∣ϕ∣ ˙ ∣C∣ This equation reveals immediately that ̺h is constant if, and only if, r 2 a(r) is constant, and this holds only if the magnitude of the attractive force, namely a(r), is proportional to r −2 .

b) ⇒ c): Let the circle k ′ with center P1′ and radius ̺h be the given hodograph. The origin O of the velocity diagram is assumed to be at a distance d to P1′ , and we start with the case d > 0. We use polar coordinates centered at P1′ such that O has the polar coordinates (π, d) (Figure 3.9).

c˙

e

k

̺h +

e P2′ P2

A′ d co

ϕ

c˙

k′

sϕ

P1′

r d ϕ

A

̺h O

P1 ¨ c

FIGURE 3.9. For a circular hodograph k ′ one component of c˙ immediately yields the polar equation of the orbit k. Let ϕ be the polar angle of any point P2′ ∈ k ′ . Then, the arrow pointing from O to P2′ represents the velocity vector c˙ of the corresponding point P2 of the orbit k. Since the acceleration vector ¨ of P2 acts along [P2 , P1 ], the tangent vector to k ′ is parallel to −c, the negative position c vector of the point P2 . For the orbit k, we now introduce polar coordinates with origin P1 such that the point A ∈ k corresponding to A′ = (0, ̺h ) of the hodograph k ′ gets the polar angle 0. Then, the polar angle ϕ of P2′ ∈ k ′ occurs also at the polar coordinates (r, ϕ) of the corresponding point P2 of the orbit k.

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Chapter 3: Differential Geometry

We learned from Figure 3.1 that c˙ can be decomposed into a radial component of signed length r ′ ϕ˙ and an orthogonal component of signed length r ϕ. ˙ The same decomposition can be performed on the hodograph (see Figure 3.9), and we obtain ̺h + d cos ϕ = r ϕ˙ = r

C C = , r2 r

where C = r 2 ϕ˙ still denotes twice the constant areal velocity. This yields immediately the polar equation of k C r= , ̺h + d cos ϕ which confirms that the orbit k is a conic with focal point P1 , with p = C/̺h and ε = d/̺h .10

In the excluded case d = 0, the vector c˙ is permanently orthogonal to ¨ c, and therefore, also to c, and consequently k is a circle. c) ⇒ b): Whenever a conic with polar equation (3.10) is traced with constant areal velocity, then, according to the computation on page 77, c˙ = r ′ ϕ˙ ( =

Cr ′ cos ϕ Cr − sin ϕ cos ϕ − sin ϕ ) + r ϕ˙ ( )= 2 ( )+ 2 ( )= sin ϕ cos ϕ sin ϕ cos ϕ r r

C C − sin ϕ 0 − sin ϕ ( )= [( ) + ( )] . ε cos ϕ p ε + cos ϕ p

This shows that the hodograph is a circle with radius ̺h =

10

∣C∣ . p

◾

Hamilton proved this step by revealing a direct relation between the power of O with respect to the circle k ′ and the Apollonian property of k.

85

3.2 Conics in planar differential geometry

3.2 Conics in planar differential geometry Arc length of an ellipse

The circumference l of a circle c of radius r > 0 can be computed with the simple formula l = 2πr.

The latter equation can be used as a definition of π. We can also use a parametrization of c like the standard trigonometric parametrization c(t) = r(cos t, sin t)

with t ∈ J = [0, 2π[.

The arc length l(c, J) of the curve c(t) in the interval J equals ˙ l(c, J) ∶= ∫ ∥c∥dt

(3.13)

J

where c˙ denotes the velocity vector, i.e., the derivative of the parametrization c(t) with respect to t. Thus, we find

●

l(c, J) = ∫

2π 0

r dt = 2rπ.

Exercise 3.2.1 Circumference of a circle.

Verify the above result by using the rational parametrization of the circle like the one given in (2.13) in the case a = b = r.

Now, we are interested in the circumference l of an ellipse with semimajor axis a and semiminor axis b, and thus, a > b. We use the trigonometric parametrization (2.12) of c(t) = (a cos t, b sin t)

with t ∈ J = [0, 2π[.

(3.14)

With (3.13), we find a formula for the circumference l l(c, J) = 4 ∫

π 2

0

√ a2 sin2 t + b2 cos2 t dt = 4a ∫

where ε=

√

π 2

0

√ 1 − ε2 sin2 t dt, (3.15)

b2 a2 is the numerical eccentricity. Note that 0 < ε < 1 for a > b and a > 0. In the last step, we have used the linear substitution t → π2 − t. The integral 1−

86

Chapter 3: Differential Geometry

in (3.15) is an elliptic integral of the second kind. Therefore, it is not possible to give a formula for the circumference of an ellipse (different from a circle) in terms of elementary functions. However, one can derive approximations from this elliptic integral. Approximations of the circumference of an ellipse

A rough approximation of the circumference of an ellipse with semimajor axis a and semiminor axis b is given by l ≈ π(a + b) which somehow imitates the analogous formula for the circle’s circumference. A much better approximation uses the quadratic mean value of a and b √ a2 + b 2 l ≈ 2π 2 and is not more than five per cent away from the actual circumference as long as a < 3b. Still better approximations are given by √ l ≈ π (3(a + b) − (3a + b)(a + 3b)) or by

l ≈ π(a + b) (1 +

3h √ ) 10 + 4 − 3h

with h = (a − b)2 /(a + b)2 . The latter two formulas stem from the Indian mathematician Srinivasa Ramanujan Iyengar (1887–1920). Using the numerical eccentricity ε, one can evaluate the exact formula (2k)!2 ε2k ⋅ ). 4 2k − 1 k=1 (2k ⋅ k!) ∞

l = 2bπ (1 − ∑

The series starts with

1 2 1⋅3⋅5 2 3 2bπ (1 − ( ) ε2 − ( ) ε − . . .) . 2 2⋅4⋅6

Another exact formula for the circumference l of an ellipse reads ∞

1

l = π(a + b) ∑ ( 2 )hk k=0 k

87

3.2 Conics in planar differential geometry

with h = (a − b)2 /(a + b)2 . The initial terms read

1 1 1 3 π(a + b) (1 + h + h2 + h + . . .) . 4 64 256

Curvature of conics and the construction of osculating circles In Example 6.4.4, we learn that at any point P of a conic c there exists a circle o being the image of c under a perspective collineation with center P and axis through P . The circle o shares the tangent (and of course the normal) with c, and we call it the osculating circle. In this section, we approach the osculating circle from the differential geometric view point. It yields the same circle. Vertices of an ellipse

According to (2.19) and (2.20), the curvature of a planar curve with the n parametrization c(t) = (x(t), y(t)) and t ∈ J can be computed as κ = av2 , ˙ is the speed and an is the length of the component of where v = ∥c∥ the acceleration vector ¨c orthogonal to the tangent in c(t), i.e., in the direction of the normal vector e2 = v1 (−y, ˙ x). ˙ This yields a formula for the signed curvature ˙ det(c(t), c¨(t)) κ(t) = , (3.16) 3 ˙ ∥c∥ cf. [27, 129]. The sign of κ depends on the orientation of c(t): A positive curvature means that the curve turns left, when traversing it with increasing parameter t. We apply the formula (3.16) to the parametrization of the ellipse c given in (2.12) and find for a > b κ(t) = √

ab

(a2 sin2 t + b2 cos2 t)3

.

(3.17)

In the case a = b = r we get κ(t) = 1r , i.e., the circle’s curvature is constant. Therefore, the center of curvature with position vector c⋆ = c + κ1 e2 is the center of a circle o which has the same curvature as c at P = c(t). Thus, o is called the osculating circle, and we shall learn that o is the best circular approximation of c at P . The first derivative of κ(t) with respect to its parameter t yields −3ab(a2 − b2 ) sin t cos t κ(t) ˙ =√ (a2 sin2 t + b2 cos2 t)5

88

Chapter 3: Differential Geometry

with zeros

t1 = 0, t2 =

π 3π , t3 = π, t4 = . 2 2

They correspond to the points V1 = (a, 0), V2 = (0, b), V3 = (−a, 0), V4 = (0, −b)

where the second derivatives of the curvature κ(t) of c satisfy κ ¨ (t1 ), κ ¨ (t3 ) < 0

and κ ¨ (t2 ), κ ¨ (t4 ) > 0.

This tells us that the points V1 and V3 are points of maximal curvature on the ellipse c, whereas V2 and V4 are points of minimal curvature on c. Note that the points V1 and V3 are the principal vertices, and V2 and V4 are the auxiliary vertices of the ellipse c (cf. Figure 3.10).

κ

̺

0

π 2

π

3π 2

t

0

π 2

π

3π 2

t

FIGURE 3.10. Left: Curvature plot of ellipses for different ratios of the semimajor and semiminor axes: 3 ∶ 2 (blue), 2 ∶ 1 (violet), and 6 ∶ 5 (magenta). Right: the respective plots of the curvature radii.

The respective curvatures κ and curvature radii ̺ = κ1 = κ3 =

a ⇐⇒ ̺1 = ̺2 = b2 b κ2 = κ4 = 2 ⇐⇒ ̺2 = ̺4 = a

b2 , a a2 , b

1 κ

are

(3.18)

compare with Theorem 2.1.5 on page 29. We can give a simple construction for the centers of curvature at the vertices of an ellipse c, as can be seen in Figure 3.11: The axes of the

89

3.2 Conics in planar differential geometry

ellipse c and the tangents at the vertices V1 and V2 confine a rectangle M V1 U V2 . We add the diagonal [V1 , V2 ] and draw the line perpendicular to the diagonal through the fourth corner U .

Corollary 3.2.1 The line through U and perpendicular to [V1 , V2 ] meets the principal and auxiliary axes in the centers V1⋆ and V2⋆ of curvature at the vertices V1 and V2 . V2

U

M V1⋆

V1

V2⋆ FIGURE 3.11. Construction of the curvature centers V1⋆ , V2⋆ at vertices V1 , V2 . Proof: It is easy to see that this construction yields the desired centers of curvature and the proper radii: Let M denote the center of the ellipse c. From Figure 3.11, we can read that a ∶ b = M V1 ∶ M V2 = b ∶ ̺1

and

a ∶ b = a ∶ ̺2

since the right-angled triangles M V1 V2 and V1 U V1⋆ are similar. The same holds true for the two right-angled triangles M V1 V2 and V2 V2⋆ U . It is easy to verify that the circles around Vi⋆ through Vi hyperosculate the ellipse (cf. Definition 6.4.1). A simple computation shows that the osculating circles at the vertices share only the vertices with c (cf. Exercise 2.1.1).

◾

We shall meet this construction again in Section 7.5, where we deal with the quadratic transformation of conjugate normals. Remark 3.2.1 We defined the vertices of conics in Section 2.1 as the points where a conic meets its lines of symmetry. This yields four points on the ellipse, two points on the hyperbola, and one point on the parabola. The only proper way of generalizing the notion of a vertex for curves without lines of symmetry is the following: Look for regular points on the curve where the curvature κ has a local maximum or minimum.

90

Chapter 3: Differential Geometry

Metric relations between curvature radii and tangential distances

Curves can also be defined as envelopes of one-parameter families of lines. If ϕ denotes the angle between the positive x-axis and the oriented normals of the family, then each line of the family is uniquely defined by the unit normal vector n = (cos ϕ, sin ϕ) and the oriented distance d(ϕ) to the origin of the coordinate system, such that dn is the position vector of the line’s pedal point with respect to the origin. The function d(ϕ) is called the support function of the curve. The curvature and the support function of an ellipse fulfill a simple relation: Theorem 3.2.1 For an ellipse with semiaxes lengths a and b, the curvature function κ and the support function d with respect to the center of the ellipse are related via d3 κ = 2 2. (3.19) a b Proof: We recall that x = a cos t and y = b sin t (cf. (2.12)) holds for any point on the ellipse x2 y 2 + = 1. a2 b2

(3.20)

After eliminating cos t and sin t from the curvature function given in (3.17), we arrive at an algebraic expression for the curvature: κ(x, y) = √

a4 b4

(b4 x2 + a4 y 2 )3

.

(3.21)

Via the velocity vector of the ellipse at some point P = (ξ, η), we obtain the Hesse normal form of the tangent at P as tP ∶

−b2 ξ x − a2 η y + a2 b2 = 0. √ b4 ξ 2 + a4 η2

(3.22)

If we insert x = y = 0 and relabel ξ → x and η → y, we obtain the signed distance of tP to the center of the ellipse, a2 b2 d(x, y) = √ (3.23) 4 b x2 + a4 y 2 Note that x and y have to fulfill (3.20). Combining (3.21) and (3.23) we can eliminate the nasty square root and obtain (3.19).

◾

●

Exercise 3.2.2 Analyze the curvature function of the ellipse given in (3.19) and show that the vertices of the ellipse appear where the support function is stationary.

91

3.2 Conics in planar differential geometry P

π 4

l1

l2 P

P1 ̺2

̺1

M

P2

P2⋆

P

P1⋆

P⋆

FIGURE 3.12. Left: The cube of the ratio l1 ∶ l2 equals the ratio ̺1 ∶ ̺2 of the curvature radii at the two contact points P1 and P2 . Right: For the particular point P , the osculating circle oP meets the ellipse c at the antipode P of P .

●

Exercise 3.2.3 Derive the analogous formula to (3.19) for hyperbolas.

A deeper result relating the curvature radii with metric values is due to the French mathematician Joseph Liouville (1809–1882) and given in Theorem 3.2.2 Let P be an exterior point of a conic c. The two tangents from P to c touch at P1 , P2 ≠ P . Let l1 = P1 P , l2 = P2 P , and let further −1 ̺1 = κ−1 1 , ̺2 = κ2 be the curvature radii of the conic at P1 , P2 . Then, the following holds ̺1 l1 3 =( ) . ̺2 l2 ●

Exercise 3.2.4 Give a straightforward proof of Theorem 3.2.2 by assuming P1 and P2 are arbitrary points on an ellipse / hyperbola / parabola with the parametrizations ellipse (a

1 − t2 2t ,b ), 1 + t2 1 + t2

hyperbola (a

1 + t2 2t ,b ), 1 − t2 1 − t2

parabola (2qt2 , 2qt)

with a, b, q > 0. Compute the intersection P of the tangents at P1 and P2 , the distances P1 P and P2 P , and the curvatures at the respective points on the conic.

In Figure 3.12 (left), the metric values mentioned in Theorem 3.2.2 are marked. Though Figure 3.12 shows only the case of an ellipse, Theorem 3.2.2 holds for hyperbolas and parabolas as well. Figure 3.12 (right) shows another curious result. There are four points P on the ellipse (corresponding to the parameter values t = ± π4 and t = ± 3π 4 in the trigonometric parametrization (2.12)) whose osculating circles

92

Chapter 3: Differential Geometry

meet the ellipse in a pair (P, P ) of opposite points. In this special case, the center P ⋆ of the osculating circle lies on the diameter line which is orthogonal to the diameter [P, P ]. ●

Exercise 3.2.5 Find a proof for the facts illustrated in Figure 3.12 (right).

●

Exercise 3.2.6 Vertices of a hyperbola.

Compute the radii of the osculating circles for the two vertices V1 = (a, 0) and V2 = (−a, 0) on the hyperbola x2 y 2 h ∶ 2 − 2 = 1 with ab ≠ 0 a b by using the two parametrizations h(t) = (±a cosh t, b sinh t) with t ∈ R and computing the curvature (cf. the curvature plot in Figure 3.13) κ=∓

(a2

ab 3

2

sinh t + b2 cosh2 t) 2

.

The radius of the osculating circles at V1 and V2 equals ̺1 = ̺2 =

κ

b2 . a

As shown in Figure 3.14

̺

0

t

0

t

FIGURE 3.13. Curvature plot and the curvature radius plot of the hyperbola for three different ratios of the semimajor and semiminor axes: 3 ∶ 2 (blue), 1 ∶ 1 (magenta), and 3 ∶ 4 (violet). (left), the center V1⋆ of the osculating (hyperosculating) circle at the hyperbola’s vertex V1 can be constructed in a simple way, provided that the vertex and the asymptotes are known.

●

Exercise 3.2.7 Vertex of a parabola.

A parabola p ∶ x2 = 2qy (with q > 0) can be parametrized as p(t) = (2qt, 2qt2 ) with t ∈ R. 1 Thus, the curvature radius function ̺ = κ reads ̺(t) = q(1 + 4t2 ) 2 . 3

Show that p has only one vertex V = (0, 0), which corresponds to t = 0. The radius of the osculating circle at V equals ̺0 = q.

In Section 4.2 (Theorem 4.2.3, page 155), we learn that the subnormals (the orthogonal projections of the normal’s segment between the curve and its principal axis onto the latter) of conics are linear functions in the

93

3.2 Conics in planar differential geometry

as

e ot pt ym

P t ̺0

axis V1

V1⋆

V

V⋆

̺0 ̺0

t′ h P′ V1⋆

FIGURE 3.14. Left: The construction of the center of the hyperbola’s osculating circle at the vertex V1 . Right: The radius ̺0 of the osculating circle at a parabola’s vertex V can be found if a line element, i.e., a point P ≠ V plus incident tangent, is known.

abscissa, i.e., the curve point’s distance from the auxiliary axis. The parabola is more special: The subnormal is constant, and thus, independent of the point on the parabola, as shown in Figure 3.14.

Evolute of an ellipse

What about the osculating circle oP for a generic point P = c(t0 ) of an ellipse c? Assume that o is such a circle with center P ⋆ (with coordinate vector c⋆ and radius R). Locally, that means, in a sufficiently small neighborhood of a generic point P = c(t0 ) on the ellipse c, the circle and the ellipse share the point, the tangent, and the curvature. These three conditions can be expressed algebraically: The first yields ⟨c(t) − c⋆ , c(t) − c⋆ ⟩∣t=t0 = R2 ,

i.e., a circle through c(t) (at any t) with the yet unknown center c⋆ and radius R. Note that both, center and radius are constant at a certain point P = c(t0 ). We differentiate once with respect to t and obtain ˙ ⟨c(t), c(t) − c⋆ ⟩∣t=t0 = 0

Ô⇒

⟨e1 , c(t) − c⋆ ⟩ = 0

(3.24)

where (e1 , e2 ) is the Frenet frame at P . The last equation tells us that the desired circle o is tangent to c at P = c(t0 ), for its radius vector c(t) − c⋆ is perpendicular to the tangent tP of c at P that aims in the direction ˙ c(t) or e1 . Thus, the center P ⋆ of o lies on the curve’s normal at P .

94

Chapter 3: Differential Geometry

We differentiate once again. With c˙ = vκe2 and ⟨e1 , e1 ⟩ = 1, we find which yields

v⟨κe2 , c(t) − c⋆ ⟩∣t=t0 + v = 0

(3.25)

1 c⋆ = c + e2 . (3.26) κ By virtue of (3.16), κ can be expressed in terms of the derivatives of the parametrization c(t) (with an arbitrary parameter t). We learned that c⋆ is the center of the osculating circle or the center of curvature. The term osculation is explained for pairs of conics in a completely different way in Definition 6.4.1 (see Section 6.3), although the two separate approaches lead to the same circle.11

If t runs through the entire interval J = [0, 2π[, then c(t) parametrizes the ellipse and c⋆ (t) parametrizes a curve called the evolute of the ellipse. The tangents of the evolute are the normals of the ellipse, which follows d ⋆ κ˙ from dt c = ve1 − κκ˙2 e2 − vκ κ e1 = − κ2 e2 . Figure 3.15 shows some ellipses together with their evolutes.

P

P⋆

FIGURE 3.15. Evolutes of different ellipses being the envelopes of the ellipses’ normals. The cusps of the evolute correspond to the vertices of the ellipse.

11

Two C r curves c and d are said to be in n-th order contact (with n ≤ r) at a certain point P if there exist parametrizations of both curves such that the derivatives of c and d at the point P agree up to order n. Sometimes this is called a C n -contact at P . Two C r curves are said to be in Gn -contact (geometric continuity of order n ≤ r) if the span of the first n derivative vectors of both curves agree at P . C n -contact always implies Gn -contact, the converse is not true. The projective differential geometric version deals with the first n + 1 derivative points, starting with c(t0 ) and d(u0 ) as the first derivatives.

95

3.2 Conics in planar differential geometry

We insert the parametrization (2.12) of the ellipse c into (3.26) and obtain a parametrization of the evolute as c⋆ (t) =

a 2 − b2 (b cos3 t, −a sin3 t) , ab

t ∈ J.

(3.27)

The evolute of an ellipse is an algebraic curve of degree six and class12 four. The parametric representation (3.27) fulfills the algebraic equation 2

with α =

a2 −b2 a

and β =

2

x 3 y 3 ( ) +( ) =1 α β

a2 −b2 b ,

(3.28)

equivalent to the polynomial equation

((a2 − b2 )2 − (a2 x2 + b2 y 2 )) − 27a2 b2 x2 y 2 (a2 − b2 )2 = 0. 3

The evolute of an ellipse has four cusps, which are the centers of curvature of the four vertices, see Figure 3.15. The scaling x y (x, y) ↦ (x′ , y ′ ) = ( , ) α β

transforms (3.28) into the astroid with the equation x′ 3 + y ′ 3 = 1. 2

2

For a generic point P on the ellipse c(t), there exists also an elementary construction of the center of curvature P ⋆ = c⋆ (t), cf. Figure 3.16 (left):

Lemma 3.2.2 Draw the tangent tP and the normal nP at P ∈ c. The normal intersects the principal axis in the point 1. The line parallel to tP through 1 meets the ellipse’s diameter through P in the point 2. The line [2, P ⋆ ] is parallel to the auxiliary axis of c, and P ⋆ is the center of curvature, which has to lie on the curve’s normal nP .

12

The class of a planar algebraic curve is, roughly speaking, the number of tangents that can be drawn from a generic point to the curve. In Section 7.5, we shall obtain the evolute of a conic as the image of the set of the conic’s tangents under a quadratic Cremona transformation. Then, it will be immediately clear why the class of the evolute is four.

96

Chapter 3: Differential Geometry tP

tP P

P

c

2 p

p

1

a

a

oP

nP P⋆

P⋆

c

M

2

nP

oP

1

FIGURE 3.16. Two constructions of the center of curvature for a point P on an ellipse c. The construction on the left-hand side also works if we interchange the roles of the axes. Proof: We can verify the construction shown in Figure 3.16 (left) by a simple computation. We start with the trigonometric parametrization of c given in (2.12) and set P = (a cos t, b sin t). For fixed t0 we find a point P on c. Then, the tangent tP and the normal nP of c at P are given by the Cartesian equations tP ∶ b x cos t + a y sin t = ab and nP ∶ − a x sin t + b y cos t = (b2 − a2 ) cos t sin t. The normal nP and the principal axis with equation y = 0 meet at 1 with coordinates ( a1 (a2 − b2 ) cos t, 0), and thus, the line l parallel to the tangent tP has the equation l ∶ ab x cos t+a2 y sin t = b(a2 −b2 ) cos2 t. The diameter through P is given by xb sin t−ya cos t = 0. Therefore, its intersection 2 with l has the same x-coordinate as P ⋆ xP ⋆ =

a2 − b2 cos3 t. a

Inserted into the normal’s equation, yields yP ⋆ . Thus, p⋆ (t) agrees with c⋆ from (3.27).

●

◾

Exercise 3.2.8 Alternative curvature construction.

In Figure 3.16 (right), we can see that there is a second construction that leads to the center P ⋆ of curvature for a generic point P on the ellipse c. Show that this construction leads to the same P ⋆ , i.e., it is equivalent to the first construction given above.

Theorem 3.2.3 When in the case of an ellipse or a hyperbola c the tangent tP and the normal nP at a generic point P ∈ c intersect the principal axis at points 2, 2′ and the auxiliary axis at 1, 1′ (see Figure 3.17), then the signed distances to P and the curvature center P ⋆ satisfy

●

P 1 ∶ P 2 = P ⋆ 1′ ∶ P ⋆ 2′ .

(3.29)

Exercise 3.2.9 Proof of Theorem 3.2.3.

Show that the construction of P ⋆ from Theorem 3.2.3 as shown in Figure 3.17 is equivalent to that given in Lemma 3.2.2.

97

3.2 Conics in planar differential geometry 1

tP

1′

c

c

P oP 2′

2

P

nP P⋆

2 P

2′

′

nP oP 1′

tP

P′

P⋆

1 FIGURE 3.17. An alternative constructive approach to the centers of curvature of ellipses and hyperbolas. P divides the segment 12 on tP at the same ratio as P ⋆ does with 1′ 2′ on nP .

●

Exercise 3.2.10 Evolute of an ellipse as the envelope of the ellipse’s normals.

Show that the evolute of an ellipse can be found as the envelope of its normals. Start with the equations of the normals nP ∶ − a x sin t + b y cos t = (b2 − a2 ) cos t sin t with t ∈ [0, 2π[ and intersect the lines nP with their derivatives n˙ P (with respect to t). The points nP ∩ n˙ P are precisely the points of the ellipse’s evolute.

●

Exercise 3.2.11 Evolutes of hyperbola and parabola as envelopes.

Find the evolutes of a hyperbola and a parabola as the envelopes of the normals.

The evolute of a parabola Assume that we are given a parabola p ∶ x2 = 2qy (with q ∈ R {0}), which can be parametized as p(t) = (2qt, 2qt2 ),

t ∈ R.

(3.30)

We insert (3.30) into (3.26) and find a parametrization of its evolute p⋆ (t) = q(−8t3 , 6t2 + 1)

(3.31)

which is even polynomial, and thus, it can be written as a Bézier curve (see Section 8.2). This curve is sometimes called a semicubical parabola. An example of a parabola’s evolute can be seen in Figure 3.18.

98

Chapter 3: Differential Geometry

o

P

p P⋆

p⋆

FIGURE 3.18. The evolute of a parabola as the envelope of its normals.

The curve given in (3.31) is algebraic, of degree three, and of class three, although it is the evolute of a parabola. Its only cusp is the point (q, 0), which is the center of the osculating circle at the vertex. An implicit equation of the parabola’s evolute from (3.31) reads 8(y − q)3 − 27qx2 = 0

which can be found by eliminating t from x = −8qt3 and y = 6qt2 + q.

●

Exercise 3.2.12 Center of curvature for a generic point on a parabola.

p

1

p

tP

tP oP

P

2

oP

P

l A

1

A

V

V⋆

nP d 2d P

⋆

P⋆

FIGURE 3.19. Two constructions of the center of curvature of parabolas. Assume P ∈ p is a generic point (not the vertex). We can find the center P ⋆ of p’s osculating circle at P in two different ways shown in Figure 3.19:

99

3.2 Conics in planar differential geometry

The construction shown on the left-hand side of Figure 3.19 is the analogue to the construction of the center of curvature of the ellipse (cf. Figure 3.16). The parabola’s normal nP at P meets the axis of the parabola at a point 1. There, we draw the parallel to the tangent tP . In the case of the ellipse, the point P has to be joined with the center of the curve. In the present case of the parabola, there is no center. However, there is a diameter through P , i.e., the line parallel to the parabola’s axis, which meets the aforementioned parallel to the tangent at the point 2, such that [2, P ⋆ ] is orthogonal to all diameters of p, especially to its axis. Prove this by direct computation.

Theorem 3.2.4 The normal nP at any point P of a parabola meets the parabola’s directrix at a point D. If P ⋆ is the parabola’s center of curvature at P , then the lengths of the segments DP and DP ⋆ fulfill

●

DP ∶ DP ⋆ = 1 ∶ 3. Exercise 3.2.13 Alternative approach to the center of curvature.

The construction in Figure 3.19, right, for the center P ⋆ of curvature of a parabola is also valid at the vertex (cf. Theorem 3.2.4). Give a proof of Theorem 3.2.4.

Evolute of a hyperbola h o h⋆ P

P⋆

FIGURE 3.20. A hyperbola h with its evolute h⋆ .

With the preparations from the previous sections, it is easily verified that the evolute of the hyperbola, parametrized by (±a cosh t, b sinh t),

t ∈ R,

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Chapter 3: Differential Geometry

admits the parametrization a2 + b2 (±b cosh3 t, ∓a sinh3 t) , ab

t ∈ R.

An implicit equation of the evolute of the hyperbola reads either 2

2

3 3 ax by ( 2 2) − ( 2 2) = 1 a +b a +b

or

(1 −

a2 x 2 − b2 y 2 (abxy)2 ) = 27 . (a2 + b2 )2 (a2 + b2 )4 3

Like the evolute of the ellipse, the hyperbola’s evolute is a sextic curve of class four with two real cusps being the centers of the osculating circles at the hyperbola’s vertices. Figure 3.20 shows a hyperbola with its evolute. The construction of the curvature center P ⋆ for a generic point P on the hyperbola is very similar to the case of the ellipse. In Figure 3.21, we have illustrated two equivalent constructions of P ⋆ . The construction mirrors that of the ellipse as shown in Figure 3.16.13 2

1

P M

P M

1

nP nP P⋆

tP o

2

P⋆

tP

h

o

h

FIGURE 3.21. The construction of the center of curvature of an ellipse shown in Figure 3.16 also applies to the hyperbola. Again, there are several equivalent ways to the center of curvature.

●

Exercise 3.2.14 A circle on three infinitesimally close points.

Another approach to the osculating circle of a C 2 -curve at a regular point P (that is not a point of inflection) is as follows: Assume P = c(t0 ) is a point on c, and let ε > 0 be sufficiently small such that the Taylor series of c(t) is arbitrarily close to c(t) at least in the interval 13

The constructions of the center of curvature shown in Figures 3.16 and 3.21 are sometimes ascribed to K.H. Schellbach, see [41, p. 72].

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3.2 Conics in planar differential geometry

J ∶= [t0 − ε, t0 + ε]. Then, there exists a circle o′ through the three points c(t0 − ε), c(t0 ), and c(t0 + ε) on c(J) (see Figure 3.22). If the circle o′ for ε → 0 degenerates into a line (the radius becomes infinitely large and the center disappears), then P is a point of inflection. This will not occur on conics. Show that the limit of the circle for ε → 0 is the osculating circle o (see Figure 3.22, left).

y o′

c′

o C′ c

p

C

B2

P+ B1 P−

y =m−q

c

x

P V

FIGURE 3.22. Left: The circumcircle o′ (with center C ′ ) of three sufficiently close points P − , P, P + ∈ c converges to the osculating circle o (with center C) at P if P − → P and P + → P provided that the discretization P − , P , P + comes from a regular C 2 -curve and no point of inflection occurs in this interval. Right: A circle c′ with double contact (at the points B1 and B2 ) converges to the hyperosculating osculating circle c at the parabola’s vertex V . The parabola p and the circle c are in third-order contact at V , i.e., they intersect with multiplicity four at V . The symmetric approach towards the hyperosculating circle at a vertex also works for ellipses and hyperbolas.

●

Exercise 3.2.15 Circle with intersection of multiplicity three or four.

When dealing with algebraic curves c, the osculating circle at a regular non-inflection point P can be defined as a circle whose intersection at P with c is at least of multiplicity three. A vertex V of an algebraic curve is a regular point where the curve and the osculating circle intersect at least with multiplicity four. The definition of a vertex of a generic curve as given on page 89 still holds and is equivalent. Conics are algebraic curves of degree two, and thus, this definition of a vertex also applies to conics. At the vertex of a conic (and any other algebraic curve) the osculating circle is hyperosculating. The case of a parabola shall demonstrate a way how to find a vertex with the help of the hyperosculating circle. We assume that p ∶ x2 = 2qy is the parabola. Let c ∶ x2 + (y − m)2 = r 2 be a circle, which is in double contact with p, see Figure 3.22 (right). Consequently, m, q, and r are related via q 2 + r 2 = 2qm, and thus, the y-coordinate of the two contact points B1 , B2

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Chapter 3: Differential Geometry

equals y = m − q. The x-coordinates are the solutions of the bi-quadratic equation x4 m + x2 (1 − ) + m2 − r 2 = 0. 4q 2 q

If the two points of contact fall in one (x = 0), then we have one point of intersection with multiplicity 2 ⋅ 2 = 4. This is the case if, and only if, m = q, i.e., the circle is centered at the point (0, q) and has radius r = q. The vertex is the point (0, 0) of contact. The parabola p and the circle c are in third-order contact at V , i.e., they intersect with multiplicity four at P . This verifies the results of Exercise 3.2.6.

●

Exercise 3.2.16 Osculating parabola.

An alternative to the approximation of a curve by an osculating circle could be the following: Let P = c(t0 ) be a regular point on a curve c. Now, there is a uniquely defined parabola p with P for its vertex and o for its osculating circle at the vertex P . How does the curvature κ0 = κ(t0 ) of c at P enter the equation of p?

●

Exercise 3.2.17 Centers of curvature for parabolas and hyperbolas.

Figure 3.23 shows a construction of the centers of curvature P ⋆ of an ellipse e, a parabola p, and a hyperbola h at a generic point P . In the case of a conic with center, i.e., a conic with two (real) focal points, the construction is as follows: Draw the normals to the focal rays at the focal points F1 and F2 . These normals meet the conic’s normal n at P in two points 1 and 2. Find the fourth harmonic point P ⋆ of P with respect to 1 and 2. The case of the parabola is simpler (see Figure 3.23, middle). Give a proof for these three constructions.

2

oP

1

P

P

L

F2

2

̺

P

⋆

oP

p

̺

t

n F1

t

N

P

e

n

oP

F1

n

F2

P⋆

F t

̺

1

P

h

⋆

FIGURE 3.23. Left and right: The center of curvature P ⋆ at the point P on a conic is the harmonic conjugate of P with respect to the points 1 and 2. The points 1 and 2 lie on the conic’s normal n such that the triangles P F1 1 and P F2 2 have right angles at the focal points F1 and F2 . Middle: The parabola differs somehow. Nevertheless, there is a similar but simpler construction.

The fourth common point

Assume that we are given the parabola p ∶ x2 = 2qy with q ∈ R {0}. Let P = (ξ, η) be a point on the parabola. According to Theorem 3.2.4 (see

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3.2 Conics in planar differential geometry

R oP ξ

ξ

ξ

p

ξ

P⋆ s

F

v A

P

3ξ 2

FIGURE 3.24. The linear construction of the remaining common point R of a parabola p and one of its osculating circles oP .

Section 3.2), we construct the center P ⋆ of the osculating circle oP . We know that oP and p intersect at P with multiplicity three since P is a generic point. However, there must be one further common point R which is a transversal intersection of oP and p. How do we find it? It is clear that R can be found by means of a linear construction: The osculating circle oP has the equation oP ∶ (x + 8qt3 )2 + (y − 6qt2 − q)2 = q 2 (1 + 4t2 )3

for its center is given in (3.31) provided that p is parametrized as in (3.30). The resultants14 of oP and p with respect to y and x are res(oP , p, y) = (6qt + x)(x − 2qtx)3 ,

14

res(oP , p, x) = (18qt2 − y)(2qt2 − y)3

The resultant of two polynomials with respect to a common variable is a polynomial defined by the coefficients of the given polynomials. Resultants can be used to eliminate variables, as is the case here. All common zeros of the two polynomials are zeros of the resultant. For a precise definition and properties of resultants we refer to [36].

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Chapter 3: Differential Geometry

whose zeros are the x- and y-coordinates of the common points P (with multiplicity three) and R: R = (−6qt, 18qt2 ) = (−3ξ, 9η).

The mapping (ξ, η) ↦ (−3ξ, 9η) which assigns to the point P of osculation the remaining intersection R of o and p is linear. Moreover, if t is considered to be an inhomogeneous coordinate on p, then P ↦ R is given by the mapping t ↦ −3t. This mapping is a projective mapping on the parabola (cf. Example 5.4.3 in Section 5.4, page 260). It is hyperbolic with the vertex p(0) as a fixed point (1, 0). The second fixed point does not matter when we speak about osculating circles: It is the ideal point p(∞) where no osculating circle is defined. Now, we can deduce a construction for the point R (cf. Figure 3.24): Draw a parallel to parabola’s axis s at distance 3P s on the opposite side of P . The triangle F AR has a right angle at A where A = (− 3ξ 2 , 0) is the point on the tangent v at p’s vertex with As = 12 Rs. In the case of an ellipse e with equation e∶

x2 y 2 + =1 a 2 b2

we can use the trigonometric parametrization P = (a cos t, b sin t) as given in (2.12). Let ct ∶= cos t and st ∶= sin t. Then, (3.27) yields the equation of e’s osculating circle at the points P oP ∶ (x −

a 2 − b2 3 a 2 − b2 3 1 3 ct ) + (y + st ) = 2 2 (a2 s2t + b2 c2t ) . a b a b 2

2

We compute the point R = e ∩ oP {P } in the same way as we did for the parabola. Therefore, we compute the resultants of e and oP , with respect to y and x, cancel out the constants and the obvious cubic factors corresponding to P . Thus, we find R = (act (4c2t − 3), −bst (4c2t − 1)).

Now, we rewrite ct (4c2t − 3) and st (4c2t − 1)) as follows:

4c3t − 3ct = c3t − ct (1 − s2t ) − 2ct s2t = c2t ct − s2t st = cos 3t,

st (4c2t − 1) = st (c2t − 1 + c2t + 2c2t ) = st c2t − s3t + 2st c2t = st c2t + s2t ct = sin 3t.

105

3.2 Conics in planar differential geometry e○ P○ e P M

t t

t t

R

P⋆

oP

R○

FIGURE 3.25. The central angle t corresponding to the point P of osculation is to be reoriented and multiplied by three in order to find the central angle of R. Note that the central angles are measured for the affine images of P and R on the principal circle e○ of the ellipse.

Thus, the point R can be written as R = (a cos 3t, −b sin 3t).

Figure 3.25 shows how to find the point R. The parameter t corresponding to the osculation point P equals the central angle on the principal circle e○ which is e’s image under a scaling of the y-coordinates (cf. Proclus’s construction in Figure 2.6. Section 2.1). We reverse the orientation of t and multiply it by three which gives the central angle of the point R○ ∈ e○ . The point R○ is the affine image of the desired point R. The construction given for the point R works even if t = 0, i.e., the point P is a vertex of e with P = R. Note that conversely there are three points P1 , P2 , and P3 with osculating circles through the same point R since t can be replaced by t± 23 π without changing R (see Figure 3.26). In Section 9.1, we shall learn that e is Steiner’s circumellipse of the triangle P1 P2 P3 . Consequently, we have:

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Chapter 3: Differential Geometry e

o3 P1

P2

o2 R P3

o1

FIGURE 3.26. Three osculating circles of e through R: The ellipse e is Steiner’s circumellipse of the triangle built by the three osculation points P1 , P2 , and P3 .

Theorem 3.2.5 There are always three different points of an ellipse e whose osculating circles share the remaining point of intersection R. Conversely, each point R ∈ e can serve as the remaining point. The case of hyperbolas is similar: ●

Exercise 3.2.18 Intersection of a hyperbola and an osculating circle.

2t Use either the rational parametrization (a 1+t , b 1−t 2 ) with t ∈ R or the parametrization in 1−t2 2

terms of hyperbolic functions (±a cosh u, b sinh u) with u ∈ R of the hyperbola h ∶ b2 x2 − a2 y 2 = a2 b2 and show that the osculating circle oP at P intersects h in the point R (Figure 3.27) with either (1 + 3t2 )(3 + t2 ) t4 + 14t2 + 1 R = (2bt , a(1 + t2 ) ) or R = (a cosh3 u, −b sinh3 u). 2 3 (t − 1) (t2 − 1)3

We conclude with a geometric interpretation of the result from Exc. 3.2.18. Assume AP is the area of the triangle M V P with M being the hyperbola’s center, V being its principal vertex, and P the point on the hyperbola. The edge from V to P is an arc of the hyperbola. We compute the area AP as a cosh t √ 1 x2 1 AP = a cosh t ⋅ b sinh t − ∫ b − 1 dx = abt. 2 2 a 2 a

Note that the function AP (t) is linear in t. Therefore, the area AR of the larger triangle M V R with R as the remaining point h ∩ oP {P } equals 1 3 AR = ab ⋅ 3t = abt. 2 2

107

3.2 Conics in planar differential geometry

Thus, we have

AP ∶ AR = 3 ∶ 1.

o P M

AP AR

V h

P⋆

R

FIGURE 3.27. The area AR of the curved triangle M V R equals 3AP with AP being the area of the curved triangle M V P .

Taylor parabolas and Ghys’s theorem There is a relatively new result dealing with the approximants of order n − 1 of a polynomial function of degree n. We shall consider here just the case n = 3, because the Taylor approximation in this case is of degree n − 1 = 2, i.e., a parabola. A theorem (cf. [52]) which is due to the French mathematician Étienne Ghys (born 1954) says: Theorem 3.2.6 The Taylor approximants of degree 2 to a real polynomial function of degree 3 are disjoint parabolas with the same ideal point. An illustration of Theorem 3.2.6 is given in Figure 3.28. The Taylor approximants of degree two are parabolas including the tangent at the point of inflection. In this particular point, the Taylor parabola degenerates into a straight line, namely the tangent to the graph of the polynomial function. Note that these parabolas share the ideal point, and thus, their axes are parallel. If we drop this condition, we find much more osculating parabolas. However, those mentioned in Theorem 3.2.6 form a family of

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Chapter 3: Differential Geometry

FIGURE 3.28. The degree two Taylor approximants to a cubic polynomial function are disjoint parabolas together with the tangent at the inflection point.

disjoint parabolas. Theorem 3.2.6 can be considered as an affine version of the Tait-Kneser theorem (cf. [52]).

The hyperosculating parabola The osculating circle o of a curve c at a point P is a good approximation of the curve, at least locally around P . At P , the osculating circle and the curve share just the point P , the tangent tP , and the curvature κ. However, we can find better approximating conics.

Assume c(s) is the arc length parametrization of a C 4 -curve c. Let P = c(t0 ) be a regular point that is not a point of inflection. We choose the Frenet frame at P as coordinate frame (see Figure 3.29). In this frame, a parabola p through P and tangent to tP can be parametrized by 1 1 p(t) = (t + a1 t2 , a2 t2 ) . 2 2

(3.32)

Obviously, the axis of p is parallel to (a1 , a2 ). Now, we determine a1 and a2 such that p and c do not only agree in P , tP , and the curvature κ at P . Moreover, the derivatives of the respective curvatures with respect ˙ to their arc lengths shall be equal. We compute p(t) = (1 + a1 t, a2 t) and ¨ (t) = (a1 , a2 ). With (3.16) the signed curvature κp of the parabola at p t = 0 and find a2 κp = 3/2 σ˙

109

3.2 Conics in planar differential geometry c p c nP

a nP P ⋆⋆

P⋆

axis

oP

a

oP

P⋆

A

V

P

tP

P

tP

FIGURE 3.29. Left: With respect to the Frenet frame of the curve c, the center P ⋆ of curvature has coordinates (0, ̺) and the evolute’s center of curvature P ⋆⋆ has coordinates (−̺̺, ˙ ̺). The point A on the affine normal a has the coordinates ( 13 ̺̺, ˙ ̺) (see (3.33)). Right: The hyperosculating parabola p at P shares the osculating circle oP with c. It is uniquely determined by the center P ⋆ of curvature, the line element (P, tP ), and the diameter a.

˙ At t = 0, we obtain σ(0) with σ = ∥p∥. ˙ = 1, σ ¨ (0) = −a1 , and κP (0) = a2 . The derivative κ′ =

dκ dσ

= κ/ ˙ σ˙ of the parabola’s curvature at t = 0 equals κ′ = −3a1 a2 .

By assumption, the curvature κc and its derivative κ′c of the given curve c at P satisfy κc = κp = a2

and κ′c = κ′p = −3a1 a2

p ′ with ̺ = κp −1 and ̺′ = (κ−1 p ) = − κ2p being the radius of curvature and its derivative. This yields

κ′

a1 = −

κ˙ p 1 = ̺̺ ˙ 3κp 3

1 and a2 = κp = . ̺

Therefore, the axis of the parabola is parallel to 1 a = ( ̺̺, ˙ ̺) . 3

(3.33)

The line l(t) = p + λ ⋅ a (with λ ∈ R) is a diameter of the hyperosculating parabola. This line is called the affine normal of c at P , since it is invariant

110

Chapter 3: Differential Geometry d

D ̺ 2

P

′

tP

oP P

P′

f

P

a

a ̺ 2

f

d

F ̺ 2

p

P⋆

p

FIGURE 3.30. Left: The center P ⋆ of curvature and the point P define a point on the directrix d (according to Theorem 3.2.4). The directrix d is perpendicular to any diameter (especially to a). Right: The focus F of the parabola is the reflection of P ′ in the tangent tP where P ′ is the orthogonal projection of P onto the directrix d.

under affine transformations, i.e., an affine mapping α applied to c sends this line to the affine normal of the image curve α(c). The construction of the hyperosculating parabola if the point P , the osculating circle oP , and the affine normal a (diameter through P ) are given, can be seen in Figure 3.30 (left): From Theorem 3.2.4, we know that P divides the segment on the parabola’s normal between the directrix d and the center P ⋆ of curvature with ratio 1 ∶ 2. Thus, we find a point D on the directrix d by adding a segment of length ̺2 = 12 P P ⋆ on the normal from P outside the parabola. The directrix d is orthogonal to all diameters of p and so it is orthogonal to a (note the left-hand side of Figure 3.30). The axis and the vertex of the (osculating) parabola p are found by first determining the focus F . For that purpose, we use the focal property of the parabola. Lines parallel to the axis of p (such as the affine normal a) are reflected in p. The reflections pass through the focus F . This gives a line through the yet unknown point F . Since P d = P P ′ = P F , we find the focus F . The point P ′ is the orthogonal projection of P onto the directrix d. Moreover, it is the reflection of F in the tangent tP (Figure 3.30, right). According to Definition 6.4.1 in Section 6.4, the osculating circle oP and the hyperosculating parabola p are related via a perspective collineation, with the affine normal as an axis and the center on it. Note that the

3.2 Conics in planar differential geometry

111

osculating circle oP osculates the curve c as well as the hyperosculating parabola at P . The hyperosculating parabola hyperosculates c at P . The affine normals of conics are their diameters, i.e., they are either parallel to each other (in the case of a parabola), or they are concurrent in the conic’s center. For curves different from conics, the totality of affine normals envelopes the affine evolute which can be seen as an affine counterpart of the evolute. The affine evolute of curve is invariant with respect to affine transformations.

The conic with local five-fold intersection The hyperosculating parabola p of a curve c at some (regular, non-inflection) point P is only one hyperosculating conic at P . In Definition 6.4.1 (cf. Section 6.4), we shall learn that there is a oneparameter family of conics hyperosculating p at P . Among these conics there is one conic f that approximates the curve c locally at P better than any other conic. In terms of algebraic geometry, f intersects c at P with multiplicity five. Sometimes, it is said that f and c are in fourth-order contact. This cannot be the case for p since it is only intersecting with multiplicity four. Note that four is the maximal multiplicity of intersection of two conics. The conic f with local five-fold intersection can be seen as a limit of a conic on five points on c which are sufficiently close together. If f is a conic with center, then its center is a point on the affine normal of c at P . In the following, we shall derive the center and an equation of f . Assume that the C 4 -curve c is parametrized by its arc length which is possible for any C 1 -curve, at least locally. Let further (e1 , e2 ) be the Frenet frame of c at P . Then, we use the Frenet equations e˙ 1 = κe2

and e˙ 2 = −κe1

where the dot indicates differentiation with respect to the arc length s. We can expand c at P in a Taylor series: ˙ c(s) = c(0) + sc(0) +

s2 s3 ˙ ¨(0) + c ¨(0) + . . . . c 2 6

We write down the expansion in the Frenet frame, i.e., the point P coincides with the origin of the coordinate system and the tangent tP and

112

Chapter 3: Differential Geometry a

a

P c

P⋆ e

e

oP

A

P

oP

f

c

P⋆

A

f

FIGURE 3.31. Comparison of the conic f with local five-fold intersection at P and the osculating circle oP . The evolute e and the affine evolute a are also shown. The point A ∈ a is the center of f . P ⋆ is the center of curvature, and thus, the center of the osculating circle oP . Left: P is not a vertex of c, i.e., a point with κ˙ = 0. Right: P is a vertex of c. In this case, the conic f and the curve intersect with multiplicity six at P .

normal nP aim in the direction e1 and e2 . Thus, we have κκ˙ κ4 − 3κ˙ 2 − 4κ¨ κ 5 κ2 ⎛s − s4 + s + . . .⎞ − s3 ⎟ ⎜ 6 8 120 ⎟. c(s) = ⎜ ⎟ ⎜ κ 2 κ˙ 3 κ ¨ − κ3 4 κ ¨˙ − 6κκ ˙ 2 5 s + s + . . .⎠ ⎝ − s + s + 2 6 24 120

(3.34)

The equation of a conic that touches c at P reads k ∶ Ax2 + 2Bxy + Cy 2 + 2Ey = 0,

with E ≠ 0.

We substitute the first and second coordinates of c(s) from (3.34) for x and y. This yields a polynomial condition on the coefficients A, B, C, and E of k in order to annihilate the equation of k. We extract the coefficients of powers of s. The first three, start at s2 , and read A + κE = 0,

3κB + κE ˙ = 0, 4κ2 A − 4κB ˙ − 3κ2 C + (κ3 − κ ¨ )E = 0.

Therefore, the conic k has fourth-order contact (or equivalently a local five-fold intersection) if, and only if, k ∶ 9x2 − 6̺xy ˙ + (9 + 2̺˙ 2 − 3¨ ̺)y 2 + 18̺y = 0.

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3.2 Conics in planar differential geometry

Figure 3.31 shows a conic with local five-fold intersection centered at A=(

−3̺̺ ˙ 9̺ , ) 2 ̺¨ − ̺˙ − 3 ̺¨ − ̺˙ 2 − 3

which is a point on the affine normal m at P . It is the point of contact between m and the affine evolute a. Remark 3.2.2 If we take just the linear and quadratic terms of the local expansion of c(s) from (3.34), we obtain as second order Taylor approximant the parametrization p(s) = (s,

κ 2 s ) (3.35) 2 which gives a parabola that osculates the given curve at c(0). The vertex of the parabola equals the point c(0), the parabola’s axis coincides with the curve’s normal. Later, in Section 3.3, we shall see how this can be generalized to the case of two-dimensional surfaces.

Mannheim curve, curvature diagram ̺ 4

2 3

5

1

6

0

κ

0

1

2

3

4

5

6

s

FIGURE 3.32. The curvature diagram (blue) and the Mannheim diagram (red) of the ellipse on the left-hand side. The two diagrams are drawn with different scales.

From conics, we can deduce further curves as we shall see in later sections. On the other hand, conics can define some curves in the plane: According to the fundamental theorem of curves in the Euclidean plane, the curvature function κ(s) depending on the arc length s defines a planar curve uniquely up to Euclidean motions. This means that we can define planar curves by prescribing a curvature function κ(s) ∶ I ⊂ R → R . The graph of the function κ(s) is usually referred to as the curvature diagram. Furthermore, with κ(s) we also know the curvature radius function ̺ = κ1 . This defines another diagram showing how the curvature

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Chapter 3: Differential Geometry

radius depends on the arc length s. Figure 3.32 shows both. The second diagram showing the graph (s, ̺(s)) is called the Mannheim diagram named after the French engineer and mathematician Amédée Mannheim (1831–1906). Sometimes, the diagram (s, κ(s)), or equivalently (s, ̺(s)) is called the Cesàro curve of a planar curve, named after the Italian mathematician Ernesto Cesàro (1859–1906). Let now the conics enter the scene. We shall interpret conics as the Mannheim diagram or the curvature diagram of curves in the Euclidean plane and ask for the curves defined in this way. We give just one example in detail in order to show how to find the curve c. For various other choices of diagrams showing a conic, we just present the results. The list provided in Table 3.1 is far from being complete.

̺

c s

FIGURE 3.33. An equilateral hyperbola (left) as a Mannheim diagram of a Cornu spiral (right).

What does the curve look like if the Mannheim diagram is an equilateral hyperbola given by ̺ = a s−1 with a ∈ R {0}?

It is well known that the curvature κ of a planar curve c and the slope angle ϕ of the tangent are related via κ=

dϕ . ds

115

3.2 Conics in planar differential geometry

Consequently, we have in this particular case s

ϕ = ∫ κ(σ)dσ = 0

s2 . 2a

Without loss of generality, we can assume the tangent at c(0) equals the x-axis of a Cartesian frame. Then, we can write the tangent vector depending on s as ˙ c(s) = (cos ϕ(s), sin ϕ(s)) = (cos

s2 s2 , sin ) . 2a 2a

Integrating once yields ∫0 c(σ)dσ ˙ = c(s) which in the present case gives s

⎛ σ2 σ2 ⎞ c(s) = ∫ cos dσ, ∫ sin dσ . 2a 2a ⎠ ⎝ s

0

s

0

The latter integrals are called Fresnel integrals (named after the French physicist Augustin-Jean Fresnel, 1788–1827). The parametrized curve we obtain is called a Clothoid or Cornu spiral after the French physicist Marie Alfred Cornu (1841–1902). It is used in curve design, especially for streets and railroad tracks in order to make smooth transitions (at least C 2 ) between differently curved profiles. So, we have seen that the Cornu spiral has an equilateral hyperbola for its Mannheim diagram. Figure 3.33 shows an example of a Cornu spiral and the corresponding equilateral hyperbola. Figure 3.34 shows some of the curves mentioned in Table 3.1. The epicycloid and the hypocycloid are trajectories of points on a circle rolling

FIGURE 3.34. From left to right: epicycloids, hypocycloids, a paracycloid, and a hypercycloid.

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outside or inside a fixed circle. The fixed circle is shown in Figure 3.34 (thin violet circle). Choosing various radii for the rolling circle (not displayed), we obtain different cycloids. The paracycloid and the hypercycloid can also be seen as traces of points undergoing superposed rotations. However, in these cases the circles are not real, i.e., at least the radii of the involved circles are complex numbers (different from real numbers) and still some points move on real curves. Finally, we collect pairs of conics being either Mannheim diagrams or curvature diagrams and the corresponding curves in Table 3.1. conic

equation

equilateral hyperbola equilateral hyperbola circle ellipse hyperbola hyperbola parabola

κ ̺

= as = as 2

logarithmic spiral

s +̺ =1 2

+

s2 a2

− as2 + 2

2

s a2

−

̺2 b2

=1

̺2 =1 b2 2 ̺ =1 b2

̺ = 2rs 2

corresponding curve Cornu spiral cycloid hypocycloid if a > b epicycloid if a > b hypercycloid paracycloid involute of a circle with radius r

TABLE 3.1. Mannheim curves / curvature diagrams and the thus defined curves.

Multifocal curves Ellipses can be defined by means of a constant sum of the two distances to fixed focal points as we have seen in Section 2.1. Replacing the sum by a difference leads to hyperbolas. There are many ways to generalize these distance properties: Sum of distances

Let x = (x, y) be the Cartesian coordinates of a point X in the Euclidean plane. Assume further that F1 , F2 , . . . Fn are n mutually distinct points in the Euclidean plane. Now, we can define an n-ellipse or a multifocal ellipse e with focal points Fi by n

e ∶= {X ∣ ∑ XFi = const.}. i=1

(3.36)

3.2 Conics in planar differential geometry

117

This definition includes the ellipses as we have seen in Section 2.1. If now n > 2, we obtain a new class of curves.

FIGURE 3.35. Multifocal ellipses: The black dots indicate the three focal points. The boundaries of the differently colored areas are multifocal ellipses.

In Section 2.1, we learned that the tangent and normal of an ellipse or a hyperbola at a point X are the bisectors of the focal rays. It is easy to see that the well-known construction of tangents (or normals) to ellipses and hyperbolas is a special case of the construction for the multifocal ellipses: The defining equation (3.36) for points X = (x, y) ∈ c can be written as n

E(x, y) ∶= ∑ XFi − d = 0. i=1

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Chapter 3: Differential Geometry

Now, we assume that fi = (fxi , fyi ) are the coordinates of the focal point Fi and compute the gradient of the function E. This yields gradE = (

n y − f yn x − f x1 x − f x n y − f y1 x−fi +. . . + , +. . . + )= ∑ . XF1 XFn XF1 XFn i=1 XFi

Since the gradient points in the direction of the normal n at X of the implicitly given curve (3.36), we have n

n=∑

1 (x−fi ). i=1 XFi

(3.37)

Thus, the normal vector n is the sum of properly oriented unit vectors pointing from the n focal points to the point X on the multifocal ellipse (Figure 3.36). So we can say that even the construction of tangents to ellipses is generalized. Note that the case of the ordinary ellipse is included.

F3

F3

n

F2

F1

F2

F1

t

FIGURE 3.36. Left: Among the various confocal trifocal ellipses we find curves passing through the focal points. Right: The tangent construction for multifocal ellipses generalizes that of ordinary ellipses: Equally long vectors pointing from the focal points towards the point on the curve sum up to a normal vector n of the curve.

An algebraic treatment of the multifocal ellipses is challenging. An implicit equation of a trifocal ellipse reads ((d2 − w12 − w22 − w32 ) − 4(w12 w22 + w22 w32 + w32 w12 )) −64d2 w12 w22 w32 = 0 (3.38) 2

2

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3.2 Conics in planar differential geometry

√ where wi = (x − fxi )2 + (y − fyi )2 (for i ∈ {1, 2, 3}) and is, therefore, of degree 8. However, this curve is of genus 9 and has the absolute points (0 ∶ 1 ∶ ±i) of Euclidean geometry for its singularities. These are four-fold points on e with δ-invariant15 6. The boundaries of the differently shaded areas shown in Figure 3.35 are only parts of the level sets of the algebraic variety described by (3.38). There are some exceptions: Choose d such that the sum w1 + w2 + w3 is minimized. Then, the real branch of e consists only of the Fermat point or Torricelli point X13 of the triangle F1 F2 F3 16 . If we choose d such that it equals the sum of the distances of one focal point Fi to the other two focal points Fj and Fk , then the multifocal ellipse passes through Fi since these points satisfy the distance relation. Product of distances

Instead of the constancy of the sum or difference of distances to focal points we can define a family of curves by the product n

∏ XFi = const.

(3.39)

i=1

For n = 2 and a slight modification of (3.39), these curves were probably first studied by the Greek geometer Perseus who lived around 150 BC. He named these curves spiric curves, for he found them as the planar intersections of a torus which was called a spire at that time (see Figure 3.37). The curves described by (3.39) (without modification) are called Cassini’s curves, named after the French-Italian astrologer, astronomer, engineer, and mathematician Giovanni Domenico Cassini (1625–1712). The curves of Cassini contain some isoptics and pedal curves (cf. Sections 9.2 and 9.4) of an ellipse, especially Booth’s lemniscate and Bernoulli’s lemniscate. The spiric curves are quartic curves. For arbitrary n, the curves defined by (3.39) are of degree 2n. The left-hand side of Figure 3.38 shows the graph surface with contour lines at constant product of three distances. The right-hand side of Figure 3.38 shows a top view of the surface. There, 15 16

Definitions and properties of the δ-invariant among others can be found in [97]. The Kimberling notation of triangle centers is explained in the footnote on page 413.

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FIGURE 3.37. Above: Spiric curves as planar intersections of a torus. Below: Some spiric curves.

the contour lines appear as the curves where the product of distances to three focal points is constant. Weighted sum or product of distances

Other generalizations are possible: such as the sets of all points with constant weighted sum of distances ∑bi=1 wi ⋅ XFi = const. or constant

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3.2 Conics in planar differential geometry

FIGURE 3.38. Left: Graph surface with contour lines being curves of constant product of distances. Right: The top view shows the curves of constant product of distances. The black spots indicate the focal points.

weighted sum of products Πbi=1 ⋅ XFi generalizations of conics.

ei

= const. which can also be seen as

Superellipses, superhyperbolas, superparabolas A further generalization of conics can be found by manipulating the equation of an ellipse or a hyperbola. In the standard equation of the ellipse, we replace the power 2 of the monomials by an arbitrary rational number n ≠ 0. This gives the equations x n y n ( ) + ( ) = 1, a b

n ∈ Q {0},

(3.40)

of a class of curves called Lamé curves, named after the French mathematician and physicist Gabriel Lamé (1795–1870). The Lamé curves are obviously symmetric with respect to the x- and yaxes, and thus, also with respect to the origin of the Cartesian coordinate system if n is even. For n = 2, we find ellipses among the Lamé curves. For n = 32 , we obtain the star-shaped evolutes of ellipses (3.28) in Section 3.2. So, we can say that even the evolute of an ellipse is a generalization of an ellipse since these two curves belong to the same class of curves, namely the Lamé curves.

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Chapter 3: Differential Geometry y y

y

x

x

x

FIGURE 3.39. Lamé curves for various n: Left: n = 2, 4, 6. Middle: n = 3, n = 5, n = 7. Right: n = 21 , n = 13 , n = 14 .

Some examples of Lamé curves are displayed in Figure 3.39. We observe that for n > 1 and n even the curves are getting more and more “rectangular”, i.e., the bounding rectangle with vertices (±a, ±b)T is the limit of these curves if n → ∞. The Danish mathematician and inventor Piet Hein (1905–1996) suggested to use Lamé curves with n = 25 (Figure 3.40) to make tables with round corners. Piet Hein called his special type of Lamé curve a superellipse. y

l e x

FIGURE 3.40. Piet Hein’s Lamé curve l (red) compared with the ellipse e (blue) having the same lengths of their axes. Hein’s superellipse comes closer to the corners of the common tangent rectangle of both curves.

Lamé curves with n ∈ Z {0, ±1, 2} cannot be parametrized by rational functions.

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3.2 Conics in planar differential geometry

The Lamé curves with 0 < n < 1 are star-shaped, as can be seen on the right-hand side of Figure 3.39. The limit of the Lamé curves for n → 0 is a pair of line segments. Sometimes the equations of Lamé curves are given in the form xn y n ∣ ∣ + ∣ ∣ = 1. a b This does not change the shape of the curves with even n, but forces the curves with odd n into the bounding rectangle. Of course, these curves are symmetric with respect to the x- and y-axes. Naturally, Lamé-like curves can also be obtained by replacing the + with a − in (3.40). We shall not discuss these curves here since this is far beyond the scope of our book.17 The curves with equation y

y

y

x

x x

(0 ∶ 0 ∶ 1)

(0 ∶ 0 ∶ 1)

(0 ∶ 0 ∶ 1) (1 ∶ 0 ∶ 0)

(1 ∶ 0 ∶ 0)

(1 ∶ 0 ∶ 0)

(0 ∶ 1 ∶ 0)

(0 ∶ 1 ∶ 0)

(0 ∶ 1 ∶ 0)

FIGURE 3.41. Some W -curves: affine images (above), projective images (below).

y = qxn ,

n ∈ Q {0}, q ∈ R {0}

(3.41)

are generalizations of parabolas in some sense. Like the usual parabola (n = 2), they touch the ideal line at the ideal point of the y-axis with homogeneous standard coordinates (0 ∶ 0 ∶ 1) (see Section 5.3). Furthermore, 17

For further details see [93].

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all these curves intersect the ideal line at this point with multiplicity n − 1 provided that n ∈ N and n ≥ 2. Felix Klein (German mathematician, 1849–1925) called these curves W -curves, because of their remarkable property involving a cross ratio18 described in Theorem 3.2.7. Apparently, some W -curves with different exponents n are collinear images of each √ 1 other, e.g., y = x3 and y = x 3 = 3 x can be mapped via a reflection x → y, y → x onto each other. Figure 3.41 shows some W -curves. The affine images together with the respective collinear images are displayed in order to illustrate the characteristic behavior of W -curves at infinity. The following theorem uses terms which will be explained in Section 5.3. Theorem 3.2.7 Let c be a W -curve (3.41). The tangent tP at P ∈ c, P ≠ (0, 0), intersects the base lines x0 = 0, x1 = 0, x2 = 0 of the projective standard frame in points B0 , B1 , B2 , respectively. Then, the cross ratio cr(B2 , B1 , B0 , P ) is the same for all curves with fixed n: cr(B2 , B1 , B0 , P ) = n.

(3.42)

Proof: Use the parametrization P = (1 ∶ t ∶ q −1 tn ) of c, compute the tangent tP , intersect it with the lines [Bi , Bj ] (with i ≠ j and i, j ∈ {0, 1, 2}), and compute the cross ratio.

◾

The W -curves are the orbits of points under the action of smooth oneparameter subgroups of the group of projective transformations in P2 (F). Therefore, logarithmic spirals and exponential curves also fall into the class of W -curves Finally, we shall point out that the evolute of a parabola is also a W curve. It is, up to a homothety and a translation, given by the equation 3 y = x 2 , and thus, it is one of the curves given in (3.41).

Offset curves of conics This section is devoted to the offsets of conics. The offset curve cd of a curve c in the (Euclidean) plane at distance d is defined as the set of all points (in the plane of c) which are at a fixed distance d ∈ R to the points of c. We can find the offset to a general C 1 -curve by simply attaching the normal vector de2 (t) (of length d) at c which gives a parametrization of 18

Felix Klein used the German word Wurf (throw) instead of cross ratio which is the reason for the notation ’W ’.

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3.2 Conics in planar differential geometry

the offset (as shown in Figure 3.42) cd (t) = c(t) + de2 (t).

(3.43)

The tangent td to the offset at Pd is parallel to the tangent of c at P independent of the choice of the value d ∈ R. Therefore, the one-parameter family of curves cd (with d ∈ R) is also called the family of parallel curves. Obviously, the offsets of a circle c are circles concentric with c. Circles are Q Q

P P

cd R c

⋆ P ⋆ =Q⋆ c

c−d

c

cd

FIGURE 3.42. Left: The curve c and its offset cd at distance d share normals and have parallel tangents. The offset d is measured along the common normals. Right: The family of offset curves of c (including c) share the evolute c⋆ as the envelope of the common normals.

the only conics whose non-trivial offset curves are conics. Since the offset d can be chosen with different signs, we obtain two branches of an offset curve. One corresponds to +d, the other one to −d. Points P and Q that lie on the same normal of c share the center P ⋆ = Q⋆ of curvature (Figure 3.42, right). The offset cd of a planar curve c can also be found as the envelope of all circles of radius d centered at c, see Figure 3.43, left. ◾

Example 3.2.1 Contour of a torus under an orthogonal projection.

Assume that the ellipse e(t) = (a cos t, b sin t) is the orthogonal projection of a circle with radius a. We compute the offset at distance d which can be considered the contour of a torus with major radius a and minor radius d. The torus can be generated in many different ways. For the moment, we consider the torus as the envelope of the one-parameter family of spheres with radius d centered at the spine curve, i.e., the circle with radius a. The contours of the spheres under orthogonal projection are circles with radius d and centers on e(t). Thus, the equation of all theses circles reads c ∶ (x − a cos t)2 + (y − b sin t)2 = d2

with t ∈ [0, 2π[.

(3.44)

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Chapter 3: Differential Geometry

FIGURE 3.43. Left: The two branches of the offset of an ellipse e form the envelope of a one-parameter family of circles. Right: The contour of a ring-shaped torus consist of an “inner” (red) and “outer” (cyan) branch. We compute the envelope of these circles by differentiating c with respect to t which gives c˙ ∶ a sin t(x − a cos t) − b cos t(y − b sin t) = 0.

(3.45)

Eliminating t from (3.44) and (3.45), we arrive at an implicit equation of the contour of the torus, and thus, of the offset curves of the ellipse e(t). With the abbreviations A = a2 + b2 ,

E = a2 y 2 + b2 x2 ,

Ω = x2 + y 2

this equation reads ed ∶ E 2 Ω2 + 4E 3 − 2(d2 + A)E 2 Ω + 2(a4 − A(a2 + d2 ))EΩ2 + 4a2 d2 (A − a2 )Ω3

(12a2 (a2 − A) − 10Ad2 + A2 + d4 )E 2 + (22a2 d2 (A − a2 ) + 4A(d2 + a2 )(d2 + b2 ))EΩ+ +Ω2 (a4 (a2 + 12d4 ) − 2Aa2 (a2 − 2d2 )(a2 − 3d2 ) + A2 (a4 − 10a2 d2 + 2d4 ))+

+2(a2 − b2 )((a2 + d2 )A2 − (5a4 − a2 d2 + 4d4 )A + 2a6 + 3a4 d2 + 2a2 d4 + d6 )E+

+((a4 − 4a2 d2 + d4 )A2 + (6a4 d2 + a2 d4 + d6 )A + a8 − 8a6 d2 + 3a4 d4 − 4a2 d6 )Ω −2a2 x2 (a2 − d2 )3 + (b2 − d2 )2 (a2 − d2 )2 (a2 − b2 )2 = 0.

The parallel curves of ellipses are algebraic curves of degree eight. Only in the case of a circle, i.e., a = b, the parallel curves degenerate into pairs of circles.

Consequently, we are able to construct points, tangents, and even osculating circles of a torus’s contour under an orthogonal projection. The offset curves of hyperbolas are also of degree eight. Only in case of the parabola, we can observe a reduction of the degree: ●

Exercise 3.2.19 Offset curves of a parabola.

Show that the offset curves pd of the parabola p ∶ x2 − 2qy = 0 at distance d are algebraic curves of degree six and have the equation pd ∶ 2P 2 (2Ω + P ) + 8d2 P Ω + 4d2 (d2 − 3q 2 − 4x2 )Ω − 4d2 (2d2 + 2q 2 − 3x2 )P +(d2 + q 2 − 2x2 )P 2 − d2 (4(d2 + q 2 )2 − 16x2 − 7x4 ) = 0

where P = x2 − 2qy and Ω = x2 + y 2 . Figure 3.44 (right) shows some offsets of a parabola.

Figure 3.44 shows some offsets to an ellipse and a parabola. Offset curves tend to have cusps which happens if at some point on the initial curve

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3.2 Conics in planar differential geometry

the curvature radius equals the offset d. Usually, these cusps are cusps of the first kind, i.e., like on the curve (t2 , t3 ) at (0, 0). Especially, if the offset d is chosen equal to the curvature radius at a vertex, then the cusps become cusps of the second kind, i.e., like on the curve (t3 , t4 ) at (0, 0).

FIGURE 3.44. The offset curves of an ellipse (left) and a parabola (right) have cusps. These cusps gather on the evolute of the original curve.

●

Exercise 3.2.20 Generalized offsets.

Now, that we know that ordinary offsets to conics are (in general) not conics, we still may ask ourselves: Is it possible to find a distance function d ∶ R → R that (applied on the conic’s normals) produces a conic as its graph? We start with the standard equations x2 y 2 ± = 1, a2 b2 of ellipses, hyperbolas, and parabolas. e, h ∶

p ∶ x2 = 2qy

with a, b, q ≠ 0

Show that for ellipses, hyperbolas, and parabolas the distance function d has to fulfill 8a4 b4 ̺ = (a2 ± b2 )3 d3 ,

8q 2 ̺ = d3 ,

where ̺ ∶ R → R is the curvature radius function of the conic (cf. [105, 120]). The thus defined generalized offsets are conics of the same affine type (see Figure 3.45). We shall meet them again in Exercise 6.4.3 as Frégier conics.

FIGURE 3.45. Generalized offsets of conics: The distance function applied on the conics normals equals a multiple of the cube root of the curvature radius.

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Chapter 3: Differential Geometry

3.3 Conics in differential geometry of surfaces Dupin indicatrix n

n ̺1

P

P

t1

t2 ̺1

̺2

t1

̺2

t2

FIGURE 3.46. The principal curvature radii ̺1 , ̺2 are the radii of the osculating circles of the normal sections in the principal directions t1 , t2 . Left: At an elliptic point P , both osculating circles are on the same side of the tangent plane. Right: At a hyperbolic point P , the osculating circles are on different sides of the tangent plane.

Conics also play an important role in the study of the local normal curvatures of a surface at a regular point P . For that, we assume that S is a surface in Euclidean three-space E3 given by a parametrization S ∶ U → E3 over some domain U ⊂ R2 . Let P = S(u0 , v0 ) be some point on the surface with (u0 , v0 ) ∈ U . We use a common notation for partial derivatives: Su =

∂S ∂S ∂2S ∂2S ∂2S , Sv = , Suu = , S = , S = . uv vv ∂u ∂v ∂u2 ∂u∂v ∂v 2

Points with Su × Sv ≠ 0 are called regular. In the following we consider only regular points on S where the first and second fundamental form are I

= E(u, v)u˙ u˙ + 2F (u, v)u˙ v˙ + G(u, v)v˙ v, ˙

II = L(u, v)u˙ u˙ + 2M (u, v)u˙ v˙ + N (u, v)v˙ v˙

where E(u, v) ∶= ⟨Su , Su ⟩, F (u, v) ∶= ⟨Su , Sv ⟩, G(u, v) ∶= ⟨Sv , Sv ⟩ are the coordinate functions of the first fundamental form, L(u, v) ∶= ⟨Suu , n⟩,

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3.3 Conics in differential geometry of surfaces

M (u, v) ∶= ⟨Suv , n⟩, N (u, v) ∶= ⟨Svv , n⟩ are the coordinate functions of the second fundamental form, and n ∶ U → S2 is the unit normal vector field on S parallel to Su × Sv . For any vector t = uS ˙ u + vS ˙ v in the tangent plane of S at the regular point P , the normal curvature, i.e., the curvature of all planar sections of S formed by planes through P spanned by the normal vector n and the tangent vector t, can be found via κn =

II . I

(3.46)

Obviously, at any point P , the normal curvature κn depends on u˙ ∶ v˙ only. In other words: All curves on S with the same tangent line at P have the same normal curvature there. The normal curvature can be positive, zero, or negative. Among the directions (u, ˙ v), ˙ we would like to find those which correspond to the maxima and minima of κn . For the sake of simplicity, we define F1 ∶= (

E F ) F G

and F2 ∶= (

The extremal value problem (u˙ v)F ˙ 2( straint (u˙ v)F ˙ 1(

L M

M ). N

u˙ ) Ð→ extr. under the conv˙

u˙ ) = 1 is solved by introducing the Lagrange function v˙ u˙ (u˙ v) ˙ (F2 − λF1 ) ( ) with the Lagrange multiplier λ. Differentiating v˙ the Lagrange function with respect to the vector (u, ˙ v) ˙ shows that the maxima or minima correspond to values of λ such that the 2 × 2-matrix F2 − λF1

or equivalently

(F2 ⋅ F−1 1 − λI2 )F1

becomes singular. Note that det F1 = ∥Su × Sv ∥2 > 0. The matrix F1 is positive definite since P is assumed to be regular, and so, the λs are eigenvalues of W ∶= F2 ⋅ F−1 1 . Since both matrices F1 and F2 are symmetric, F−1 and W are symmetric, too. The linear mapping 1 of the tangent plane of S at P onto itself described by W is usually referred to as the Weingarten map (after the German mathematician Julius

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Chapter 3: Differential Geometry

Weingarten, 1836–1910). Since W is symmetric, it can be diagonalized. Using the basis (t1 , t2 ) of W’s eigenvectors, this diagonal form is (

κ1 0 ) 0 κ2

with κ1 and κ2 are called the principal curvatures at P . Then, 1 1 H ∶= trW = (κ1 + κ2 ) 2 2

and K ∶= det W = κ1 κ2

are the mean curvature and the Gaussian curvature of S at P . The eigenvectors of W define the principal tangents t1 and t2 at P (Figure 3.46). For the unit tangent vector we have t = ut ˙ 1 + vt ˙ 2 with u˙ 2 + v˙ 2 = 1. In case κ1 = κ2 ≠ 0 and κ1 = κ2 = 0 the point P is called an umbilic or a flat point, respectively. In these cases κn is indpendent of the tangent vector at P . Written within the eigenvector basis of W (which is not uniquely defined if κ1 = κ2 ), the expression for the normal curvature from (3.46) simplifies to κn = κ1 u˙ u˙ + κ2 v˙ v. ˙ Without loss of generality, we may set (u, ˙ v) ˙ = (cos α, sin α), i.e., the tangent vector t encloses the angle α with the first principal tangent. Then, we have κn (α) = κ1 cos2 α + κ2 sin2 α

(3.47)

which is called Euler’s formula, due to Leonhard Euler (1707–1783). Now, we study the following diagram (Figure 3.47): We look for the √points Q on all (non asymptotic) surface tangents of S at P with P Q = ∣c ⋅ ̺n ∣ where ̺n = κ−1 n is the radius of curvature defined by the tangent [P, Q]. Here, c ∈ R {0} is an arbitrarily chosen fixed constant. The set i(c) of all points Q in the tangent plane to S at P is called the Dupin indicatrix (Charles Dupin, French mathematician, 1784–1873). The diagram i(c) shall be drawn in the Cartesian frame centered at P with the first and second principal tangent for its x- and y-axes. Comparing with (3.47), we can see that the x- and y-coordinates of Q, α, and ̺n are related via x cos α = √ ∣c ⋅ ̺n ∣

and

y sin α = √ , ∣c ⋅ ̺n ∣

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3.3 Conics in differential geometry of surfaces

and in the case κ1 κ2 ≠ 0, we thus have i(c) ∶

x2 y 2 + = ±c ̺1 ̺2

(3.48)

with an appropriate choice of the sign on the right-hand side. This is the equation of the Dupin indicatrix which defines up to two conics (including one singular type). t2

t2 t2

i(−c)

i(c) √ c⋅̺2

√ c⋅̺ n P

√ c⋅̺1

t

i(c)

t

P t1

i(c)

√ c⋅̺ n

√ c⋅̺1

P t1

√ c⋅̺ n

√ c⋅̺1

t

t1 i(c)

i(c) i(−c)

FIGURE 3.47. Dupin indicatrices at an elliptic (left), a parabolic (middle), and a hyperbolic point P (right).

The type of conic showing up as the Dupin indicatrix depends on the surface point P (which is still assumed to be regular). The point P is called elliptic, if κ1 κ2 > 0 and then i(c) is an ellipse provided that c⋅̺i > 0 for i ∈ {1, 2}. The point P is called hyperbolic, if κ1 κ2 < 0 and i(c) is a pair of conjugate hyperbolas. If one principal curvature is zero while the other one is not, then P is called a parabolic point. The case of a parabolic point P leads to a singular conic: Without loss of generality, we can assume that κ2 = 0. Then, the equation of the indicatrix changes to i(c) ∶ x2 = c ⋅ ̺1

√ and i(c) is the pair of lines x = ± c ⋅ ̺1 parallel to the second principal tangent. Note that c has to be chosen such that c ⋅ ̺1 > 0. Figure 3.47 shows the indicatrices of all regular types of surface points. In the case of a hyperbolic point P , there are two hyperbolas shown in Figure 3.47, depending on whether c is positive or negative. It makes sense to draw both curves: As can be seen in Figure 3.46, the surface changes the sides of the tangent plane at P . That means the surface intersects the tangent plane at P along curves that touch the asymptotic tangents which correspond to tangent vectors with κn = 0. These tangents are

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the asymptotes of the indicatrix i(c) at a hyperbolic point P and define the asymptotic directions at P . The integral curves of the two fields of asymptotic tangent vectors are the asymptotic lines. Any normal section of S in the asymptotic direction has a point of inflection at P . Assume that the surface tangent t at P is rotating about the surface normal n. There are two instances where t becomes an asymptote of i(c). In between the two asymptotes, the center of the osculating circle defined by t is on one side of the tangent plane. It changes to the other side of the tangent plane if we rotate t to the next sector. So, the two parts i(c) and i(−c) of the indicatrix at P allow us to clarify on which side of the tangent plane we have to find the center of the osculating circle. The two parts i(c) and i(−c) of the Dupin indicatrix at a hyperbolic point are said to form a pair of conjugate hyperbolas, cf. Section 8.1.

When treating indicatrices constructively, we choose c = ̺1 (or c = ̺2 ). Thus, one semiaxis of the indicatrix equals one principal curvature radius √ and we only have to construct ̺1 ̺2 , which is elementary. Two surface tangents are called conjugate if they are conjugate diameters of the indicatrix (cf. page 280), no matter if P is elliptic, parabolic, or hyperbolic. At an umbilical point, the indicatrix i(c) is a circle. There, any tangent vector is an eigenvector of W and the only eigenvalue κ1 = κ2 has algebraic multiplicity two. If the Weingarten map degenerates completely, i.e., W = 0 and then we call P a flatpoint. The Dupin indicatrix of a flatpoint is either empty or the ideal line of the tangent plane, depending on whether we have performed the projective closure of the tangent plane or not. Osculating paraboloid

In a sufficiently small neighborhood of a regular surface point P , any C 2 -surface S can be approximated by a quadratic function f (x, y) over the tangent plane. We choose a Cartesian coordinate system such that P equals the origin and such that the axes aim in the direction of the principal tangents. Then, f (x, y) = h11 (0, 0)x2 +2h12 (0, 0)xy +h22 (0, 0)y 2 , and obviously, the Dupin indicatrices to variable constants are the level sets of a paraboloid P with the equation f (x, y) = z. P shares P and the tangent plane at P with S. Moreover, the first and second fundamental forms of P and S agree at P . The paraboloid P is called the osculating (vertex) paraboloid, for P is P’s vertex. The osculating paraboloid is an

133

3.3 Conics in differential geometry of surfaces

elliptic or hyperbolic paraboloid if the point P is elliptic or hyperbolic. In P P P

P S

S P

P S

FIGURE 3.48. Osculating paraboloids with their level sets (from left to right): at an elliptic point, at a hyperbolic point, and at a parabolic point.

the case of a parabolic surface point P , the osculating paraboloid P is a parabolic cylinder touching the tangent plane at P along the asymptotic tangent. Figure 3.48 shows the three types of (regular) surface points with their respective osculating paraboloids. In Remark 3.2.2, we have derived the two-dimensional analogon (3.35) to the osculating paraboloid by simply cutting off the local expansion (3.34) of a curve. Details on the construction of hyperosculating parabolas of curves as well as the osculating paraboliods of surfaces can be found in [86].

Meusnier’s theorem and the osculating circle of an ellipse The following Theorem 3.3.1 is due to the French mathematician Jean Baptiste Marie Charles Meusnier de la Place (1754–1793): Theorem 3.3.1 Let S be a C 2 -surface in Euclidean three-space, let further P be a point on S and let t be a non asymptotic surface tangent of S at P . Then, all curves c on S that touch t at P have the same normal curvature κn . The osculating circles of all such curves c trace a sphere Σ if the osculating planes rotate about t. Proofs of this important theorem can be found in almost all textbooks on differential geometry, see e.g. [27, 129]. Figure 3.49 shows the consequences of Theorem 3.3.1, especially for the orthogonal projection of a circle c. Here, the surface S is the cylinder formed by the projecting lines through points of c. In Figure 3.49 (left), we can see that the orthogonal projection (in the direction p) of a circle c with radius a is an ellipse c′ . The osculating circle o of c′ at the auxiliary vertex P ′ has the center O ′ which is the intersection of c’s axis x with the

134

Chapter 3: Differential Geometry

x

p a ′′

c ϕ

P ′′

c

′

Σ

c c

O ′′ π ′′ C′ O′ a

P

rc ϕ r

rs ϕ

P′ r

Σ

C ′′

o

t P

o

o

FIGURE 3.49. Left: The hyperosculating circle o of the orthogonal projection of a circle c. Middle and right: The osculating circles of all regular surface curves through the line element (P, t) cover the sphere Σ.

principal plane π through p. The ellipse c′ (orthogonal projection of the circle c) has the semimajor axis length a and the semiminor axis length a cos ϕ. According to (3.18), the radius of o equals a/ cos ϕ which equals the length P ′ O ′ as can be read off from Figure 3.49 (left). Note that c′ and o are hyperosculating. That will be of importance in Section 4.4. In the middle of Figure 3.49, we can see that the centers of the osculating circles of all curves with the same normal curvature (at P in the direction t) form a circle. Thus, all these osculating circles fill a sphere Σ which is frequently called the Meusnier sphere. The right-hand side of Figure 3.49 shows these particular circles on Σ sharing the line element (P, t). ●

Exercise 3.3.1 Hyperosculating circles of a hyperbola.

Figure 3.50 shows a hyperbola as a planar section of a right cone together with the Meusnier spheres at the vertices. Figure out the construction of the centers of curvature at the vertices. By the same token, the two radii of curvature are equal. The tangent to h at the vertex Vi is the tangent of the parallel circle bi . Hence, the Meusnier sphere Σi at Vi corresponding to the tangent bi is the sphere centered on the cone’s axes and touches the cone Γ along the parallel circle bi . The osculating plane of h = ε ∩ Γ equals the plane ε and the axis of curvature of h at Vi equals the normal of ε through the center Mi of Σi . The pedal points of Mi on ε are h’s centers Vi⋆ of curvature. Since Vi are vertices, Vi⋆ are the centers of h’s hyperosculating circles at Vi . Figure 3.51 shows a side view such that the plane ǫ is projecting and appears as a straight line. According to the Theorem of Right Angles (see, e.g., [24, 71, 146]), in this view, the normals to the plane ε through the spheres’ centers Mi show up as the normals to the line ε′′ through the points Mi′′ . Consequently, the intersection point Vi′′ can immediately be read off as well as the radii of the hyperosculating circles.

135

3.3 Conics in differential geometry of surfaces

ε

h Σ1 V1⋆

Γ

o1

V1

n1

V2⋆

n2 V2

Σ2 o2

M2

M1 b1

b2

h

FIGURE 3.50. Meusnier’s theorem can be used to construct the centers of curvature V1⋆ , V2⋆ at the vertices V1 , V2 of a planar section h of a cone Γ. The plane ε of the conic h intersects the Meusnier spheres at V1 , V2 along the osculating circles of h. The construction shown in Figure 3.51 do change if the plane ε intersects the cone Γ along an ellipse of a parabola. Further, the construction allows us to infer that conics cannot have a point of inflection. The determination of curvature centers using Meusnier’s sphere is also suitable for constructing curvature centers P ⋆ of a point P on the development cv of a planar intersections c of the cone Γ. The development is the unfolding of the cone Γ into a plane, i.e., rolling it into a one of its tangent planes. In the language of differential geometry, the development is an isometric image of the cone and clearly planar for the cone’s Gaussian curvature is constantly equal to zero. The plane intersection c leaves a trace cv in the plane and the curvature of the plane curve cv equals the geodesic curvature κg of the corresponding space curve c. Hence, the center P ⋆ of curvature of the development cv is the intersection of c’s axis of curvature with the tangent plane τP at P . The Meusnier spheres are not to be confused with Dandelin spheres.

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Chapter 3: Differential Geometry

Σ1

Γ

M1′′ b′′1

V1⋆ ′′

ε′′ = h′′ V1′′ n′′1 n′′2

b′′2

V2′′

V2⋆ ′′ M2′′

Σ2 FIGURE 3.51. The implementation of the construction of the osculating circles of the planar intersection h using Meusnier’s theorem needs the notion of a suitable side view. In this side view, the intersection plane ε has to appear as a straight line which allows us to draw the surface normals and h’s axes of curvature as the normals of ε passing through the centers of the Meusnier spheres.

4 Euclidean 3-space C C

hc p

c

ec

C′

3c

2c 1c

γ

1 2

e p h

3

π

The central projection of circles with different radii may result in conics of any type. Depending on whether the projection cone C, i.e., the connection of the circles and the center C of the projection, avoids, touches, or intersects the vanishing plane, the image of the circle is an ellipse, a parabola, or a hyperbola.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2024 G. Glaeser et al., The Universe of Conics, https://doi.org/10.1007/978-3-662-70306-9_4

137

138

Chapter 4: Euclidean 3-space

4.1 Planar intersections of cones of revolution Conics are also called conic sections, and this name expresses directly that they are closely related to sections of a cone, more precisely, to planar sections of quadratic cones. This is the topic of the present chapter. We start with right cones C, i.e., with cones of revolution. Cones are always understood as ‘double cones’, i.e., they are not bounded by their apex. Furthermore, cones as well as spheres and cylinders are seen as surfaces and not as solids. Based on the Apollonian definition (Definition 2.1.1 on page 13), we still prefer proofs which hold for all three types of conics simultaneously. But now there is a difference to Chapter 2: The plane of the conic lies somewhere in the three-dimensional space E3 . Hence, we use Cartesian coordinates (x, y, z), and for the sake of simplicity we assume the z-axis in vertical position. In order to visualize 3D-geometry, we sometimes take benefit from elementary Descriptive Geometry and apply orthogonal projections, in particular those into the coordinate planes (see Figure 4.1): • The vertical projection P ↦ P ′ parallel to the z-axis generates the top view in the horizontal xy-plane. • The projection P ↦ P ′′ parallel to the x-axis generates the front view in the vertical yz-plane. • The projection P ↦ P ′′′ parallel to the y-axis generates the side view in the vertical xz-plane. In order to be precise, we strictly distinguish between preimages in the 3-space E3 and those obtained by a projection: The images are indicated by primes. We omit the primes only, when the preimage already lies in the image plane of the projection, and therefore, image and preimage are coinciding. In many cases, we use appropriate coordinate frames and corresponding views, so that, e.g., the cutting plane appears as a line. Then, we speak of an ‘edge view’ of the plane. Such particular views have the advantage that they show immediately what is significant.

Cones with a vertical axis We begin with the standard case: A cone of revolution (= right cone) with a vertical axis has to be intersected with a plane.

139

4.1 Planar intersections of cones of revolution z

z

side view

z

P"

view

y x

P"

P"’

y

P

top

x

P"’ front

iew

v side

front view

view

P’

y top view

P’

x FIGURE 4.1. Top view, front view, and side view.

Theorem 4.1.1 The curve c of intersection between a right cone C and a plane π is a conic, provided the plane π does not pass through the cone’s apex S and is not orthogonal to the cone’s axis s. When, under the assumption of a vertical axis s of C, α and β are the angles of inclination of π and of the generators of C, then the numerical eccentricity of c is ε = sin α/ sin β. Suppose, a sphere S touches the cone along a circle k and touches the plane π at a point F : Then, F is a focal point of c, and the line of intersection between π and the plane of k is the associated directrix l. The orthogonal projection onto π maps the cone’s axis s onto the principal axis of c. Proof: The axis s of the given right cone C is supposed to be vertical. We can ignore horizontal cutting planes as they intersect along circles. Therefore, we may assume that the cutting plane π is inclined at the angle α with 0 < α ≤ π2 . The generators of the cone C are inclined at the angle β, 0 < β < π2 . Figure 4.2 shows the front view, with the image plane passing through the cone’s axis s. The cutting plane π is in an edge-view. Let M be the center of a sphere S which is inscribed into the cone C and touches π in the point F . We call this sphere a Dandelin sphere, since the following proof dates back to G.P. Dandelin1 . S touches the cone along a circle k in a plane, which intersects the given cutting 1

Germinal Pierre Dandelin, 1794–1847, professor of mathematics in Liège and lieutenant of the Belgian army.

140

Chapter 4: Euclidean 3-space C

S

k

π ′′

E0

p

E ′′

M

l

′′

M

A

XE

S F c

′′

α

β X ′′

T

X

β s

X0

FIGURE 4.2. The planar sections of right cones are conics. plane π along a line l. In Figure 4.2 the line appears as a point l′′ , and the contour of S is the incircle of the triangle formed by π ′′ and the contours of the cone C.

In order to prove that the section c = C ∩π is a conic with focal point F and associated directrix l, we utilize what is depicted in Figure 4.2 on the right-hand side: Given a sphere S = (M ; r) and an exterior point X, for all tangents drawn from X to S, the points T of contact are at 2 2 the same distance to X. The square of this distance XT = XM − r 2 is the power of X w.r.t. S (compare with Figure 2.32 on page 51). Now, we choose a point X ∈ π ∩ C. There are two particular tangents to the Dandelin sphere S available through X: One touches at F , the other along the cone’s generator through X touches at a point E ∈ k. In the front view, we see the “true” distance XE after rotating [S, X] about the cone’s axis, until X comes to X0 and E to E0 on the right contour line of C. On the other hand, the front view shows already the “true” distance between X and the line l. Now, we apply the sine law to the triangle which is shaded in Figure 4.2 and obtain, that for all X ∈ π ∩ C XF ∶ Xl = XE ∶ Xl = X0 E0 ∶ X ′′ l′′ = sin α ∶ sin β = ε = const. The quotient ε = sin α/ sin β is the numerical eccentricity of the conic c = π ∩ C. Hence, for α < β we obtain an ellipse, for α = β a parabola, and for α > β a hyperbola (Figure 4.3).

◾

Figure 4.2 shows on the left contour line of C how the parameter p = ε ⋅ F l of the conic c can be found. We leave the proof to the reader. Another construction of p, based on Meusnier’s Theorem 3.3.1, is presented in Exercise 4.1.2 (Figure 4.9, left).

The proof given above can easily be adapted to the limiting case β = π/2. The limit of C is a right cylinder L (Figure 4.3, right). Again, we leave the details of the proof to the reader, and claim as follows:

141

4.1 Planar intersections of cones of revolution C

C

C

L π

S

A1

c′′ ellipse

π ′′ A2

S

π ′′ A

c′′ parabola

A2 π ′′

′′

S

γ A1 c′′

hyperbola

A1

c′′ ellipse

FIGURE 4.3. The type of the conic section c = π ∩ C depends on how the inclination of the cutting plane π relates to that of the cone’s generators.

FIGURE 4.4. Dandelin’s proof in the case of an ellipse: The intersecting plane is less inclined than the tangent planes of the cone (Exercise 4.1.1).

Theorem 4.1.2 Let L be a right cylinder and π a plane which is neither parallel nor orthogonal to the axis of the cylinder. Then, the curve c = L∩π is an ellipse with its minor axis orthogonal to the cylinder’s axis.

142

Chapter 4: Euclidean 3-space

In terms of the radius r of L and the angle γ between the cylinder’s axis and π, the semimajor and semiminor axes, and the eccentricities of c are, respectively, a = r/ sin γ,

b = r,

ε = cos γ,

e = εa = r cot γ.

FIGURE 4.5. Dandelin’s proof in the case of a hyperbola: The intersecting plane is steeper than the tangent planes of the cone (Exercise 4.1.1).

●

Exercise 4.1.1 Separate Dandelin proofs for ellipses and hyperbolas. Above, we presented a ‘Dandelin proof’ of Theorem 4.1.1, which works simultaneously for all types of conics. Since ellipses and hyperbolas have two focal points, we can apply the fundamental relation XF = XE (note Figure 4.2) two-times. In this way, we directly obtain that the related curves of intersection satisfy the standard definition of ellipses or hyperbolas. Formulate these proofs on the basis of Figures 4.4 and 4.5.

Projection of a conic section parallel to the cone’s axis Theorem 4.1.3 Let C be a right cone with a vertical axis s and with generators inclined under the angle β. If the conic c = C ∩ π lies in a

143

4.1 Planar intersections of cones of revolution z

π ′′

C p c

X ′′

X0′′

X1′′

π1′′

′′

′′

A

s′′ l′′

α

S

β

y X ′F

X ′l

′

S

A

F =s′

X0′

c′ X′

l

X1 FIGURE 4.6. The top view c′ of the conic section c is a conic with the focus S.

plane π which is inclined under the angle α, α < π2 , then the top view c′ of c is a conic with F = s′ as a focus and with the numerical eccentricity ε = tan α/ tan β. The horizontal plane π1 through the apex S intersects π in a line l, which appears in the top view as the directrix l associated with the focus F . Proof: As shown in Figure 4.6, we choose the apex S of C as the origin of our coordinate frame and the cone’s axis as z-axis. In the front view, the cutting plane π appears as a line π ′′ .

For each point X ∈ C ∩ π with the pedal point X1 in the xy-plane, we obtain by virtue of Figure 4.6 tan β =

XX1 X1 S

=

X ′′ X1′′ X1 S

=

X1 l tan α X1 S

,

hence

X′S X′l

=

X1 S X1 l

=

tan α = const. tan β

Contrary to the eccentricity ε = sin α/ sin β of the conic c itself, the top view c′ has the numerical eccentricity tan α/ tan β. This shows that c and c′ are of the same type. Here, we excluded hyperbolas in vertical planes (α = π2 ); their top view is a line, of course.

◾

Figure 4.7 shows a right cone with a vertical axis and different planar sections (compare also with the figure on page 11). All intersecting planes

144

Chapter 4: Euclidean 3-space

FIGURE 4.7. A cone of revolution, intersected by a pencil of planes. In the green area, all intersections are ellipses, in the yellow area, only hyperbolas occur. When the intersecting plane is parallel to a tangent plane of the cone, the intersection curve is a parabola (blue color).

pass through the same line in the horizontal plane through the apex. By virtue of Theorem 4.1.3, the top views of the conic sections share a focus and the associated directrix. Hence, the top views of these conics form exactly the same family which has been depicted in Figure 2.3 on page 15. Remark 4.1.1 Theorem 4.1.3 reveals that, once a cone C with a vertical axis s is fixed, all conics c′ with focus s′ (including circles with center s′ ) show up as top views of appropriate planar sections C ∩π. Hence, C induces a one-to-one correspondence between monofocal conics, i.e., conics sharing one focus, and planes π in E3 not passing through S.

145

4.1 Planar intersections of cones of revolution S

C A1

S

ε′′

C̃

c′′ s′′

̃ S

C A2

A′1 F1

C̃

̃ S F2

c′′

s′′

A′2

c′ FIGURE 4.8. The planar section c is symmetric w.r.t. its secondary axis s. Therefore, the cone C still passes through c after a 180○ -rotation about s (left), or after reflection in the plane ε which contains s and is orthogonal to the plane of c (right).

Ellipses and hyperbolas c′ have a second focus. Where does it come from? The conic c on the cone C has a second axis s of symmetry, which appears in the front view (Figure 4.8, left) as the midpoint between the vertices A1 , A2 on the contour lines of C.2 A rotation about this axis s through 180○ maps c onto itself, while the cone C is transformed into a second cone C̃ with vertical axis, which again passes through c. Therefore, the ̃ of C̃ coincides with the second focus. top view of the apex S There is a second symmetry of the conic c, which transforms the cone. Figure 4.8, right, shows how the symmetry of the conic c with respect to the plane ε through the minor axis s can be used to find another right cone C̃ passing through the same conic c. This procedure can be iterated. ●

Exercise 4.1.2 Parameter p of planar sections of cones.

Confirm for the conic section c = C ∩ π the construction of the parameter p, as demonstrated in Figure 4.9, left (compare with Figure 4.2). By virtue of Theorem 2.1.5, this construction

2

It is not so obvious that planar sections of a right cone have a second axis of symmetry. One might guess that at the principal vertex which is closer to the cone’s apex, the radius of curvature of the planar section is smaller than at the opposite vertex. It seems that even Dürer had this expectation (see Figure 1.10, left). Figure 3.50 shows for a hyperbola the centers of curvature at the two vertices. Of course, the radius of curvature is the same, though the respective constructions look quite different.

146

Chapter 4: Euclidean 3-space l2

S

C S

C

A A∗

α

π ′′

β α N β

F

c′′ α

c2

S

c′′1

p

β

εA F

AF / sin β

k ′′

N

k ′′

l′′

A∗1

β

F1 p1

′′ α l1 A1 =F2

FIGURE 4.9. Left: parameter p of the conic section c = C ∩ π. Right: universal proof of Theorem 4.2.1 (Exercise 4.1.3). also yields as a by-product the center of curvature A∗ for the vertex A of c. The latter can also be concluded by virtue of Meusnier’s Theorem 3.3.1. Hint: Note that p = (1 + ε)AF and ε = sin α/ sin β.

●

Exercise 4.1.3 Apollonian property of planar sections of cones.

Let a conic c1 be given with vertex A1 , focus F1 , and associated directrix l1 (Figure 4.9, right). Prove Theorem 4.2.1 simultaneously for all types of conics c1 by verifying that the apices S of right cones passing through c1 satisfy the Apollonian definition SA1 = ε2 ⋅ Sl2 , where l2 is the axis of the circle which osculates c1 at A1 . Hence, all points S lie on the focal conic c2 of c1 . The numerical eccentricities ε1 and ε2 of c1 and c2 are reciprocal. As parameter p2 of c2 we obtain p2 = p1 /ε1 = F1 l1 . Hint: We choose any sphere S, which touches the plane of c1 at the given focus F1 . Then, we determine a cone C, which touches S along a circle k in a plane passing through the given directrix l1 , such that one contour line of C contains the vertex A1 . By virtue of Theorem 4.1.1, in terms of the inclination angles α and β of the cone’s generators and the cutting plane (relative to the plane of k), the eccentricity of c1 is ε1 = sin α/ sin β. We apply the construction presented in Exercise 4.1.2 (Figure 4.9, left) and conclude, using the notation of Figure 4.9, right, that3 SN =

3

SA1 , sin β

Sl2 = SN sin α, hence SA1 =

1 Sl2 . ε1

By virtue of Theorem 3.3.1, the point N is the center of the Meusnier sphere corresponding to the horizontal tangent at A1 .

147

4.2 Pairs of focal conics, Dupin cyclides

4.2 Pairs of focal conics, Dupin cyclides We are going to report about a symmetric pairing of conics in E3 which has some remarkable geometric properties. We will meet these pairs again in volume 2 in connection with confocal quadrics in E3 . Definition 4.2.1 Given a conic c1 in E3 , the conic c2 is called focal conic of c1 if the two conics lie in orthogonal planes sharing the principal axis, and on this common axis each vertex of one conic is a focus of the other.

c2

c1

FIGURE 4.10. A pair of focal conics.

The focal conic of an ellipse with semimajor √ axis a and semiminor axis b is a√hyperbola with the semimajor axis a2 − b2 and the semiminor axis a2 − (a2 − b2 ) = b (see Figure 4.10), and vice versa. The focal conic of a parabola is a parabola with the same parameter; however, the two parabolas open to different sides. We do not speak of a pair of focal conics in the limiting case, consisting of a circle c1 and its axis c2 , though some of the following theorems are still valid in this case.

148

Chapter 4: Euclidean 3-space z

L

S

b

T2

ch

k ′′ S

T1

c′′e A2 F2

F1

M

y

A1

b c′h

S′

M A2

F2

F1

A1

y

ce x FIGURE 4.11. Which cones of revolution pass through a given ellipse ce ?

FIGURE 4.12. The cones of revolution which pass through a given ellipse have their apices on the focal hyperbola.

149

4.2 Pairs of focal conics, Dupin cyclides

Focal conics and cones of revolution Theorem 4.2.1 For any given conic c1 in E3 , the locus of apices S of right cones through c1 , is the focal conic c2 (up to the focal points of c1 ). The axis of the cone through c1 with apex S ∈ c2 is tangent to c2 at S. z ch

F1

A1

F1

A2

M S ′′

A1

M

A2

y

F2 c′h

T1 S

y

F2

c′′e

ce

k′

S T2

tS x

FIGURE 4.13. Which cones of revolution pass through a given hyperbola ch ? Proof: We discuss the three types of conics separately, since the universal proof of Theorem 4.2.1 is a little bit tricky and not as instructive (see Exercise 4.1.3 and Figure 4.9, right). Case 1: c1 is an ellipse ce in the horizontal xy-plane with axes a, b and the y-axis as principal axis (see top view and front view in Figure 4.11). By virtue of Theorem 4.1.1, it is necessary that any right cone passing through c1 has its axis in the vertical plane through the principal axis, i.e., in the yz-plane. Furthermore, there must be a sphere S inscribed into this cone, which touches the xy-plane at one focus F1 of ce . So, let us choose such a sphere S with radius r and inspect the front view in Figure 4.11. The second tangents drawn from the vertices A1 and A2 of ce to the contour of S intersect at a point S — except the case where these tangents are parallel, i.e., the sphere S has the radius r = b. The lines [S, A1 ] and [S, A2 ] are the contour of a right cone, which touches S along a circle k. In fact, this cone already passes through the ellipse ce , since by virtue of Theorem 4.1.1 the curve of intersection with the xy-plane has the principal vertices A1 , A2 and the focus F1 . Thus, it is identical with ce . Let Ti for i = 1, 2 denote the point of contact between [S, Ai ] and S. Then, we identify in the front view of Figure 4.13 some equal distances: A1 T1 = A1 F1 ,

A2 T2 = A2 F1 ,

In the depicted case r < b we conclude that

ST1 = ST2 .

SA2 − SA1 = ST2 + T2 A2 − ST1 − T1 A1 = A2 F1 − A1 F1 = F1 F2 = const.

150

Chapter 4: Euclidean 3-space

FIGURE 4.14. All cones of revolution passing through a given hyperbola have their apices on the focal ellipse. In the case r > b the point S lies closer to the second focus F2 . Therefore, we use the second sphere: It is inscribed into the same cone but touches the xy-plane at the focus F2 , and it has a radius smaller than b. Hence, we obtain a similar result, but under SA2 < SA1 . So, in all cases the requested locus of apices satisfies the standard definition of a hyperbola ch in the yz-plane with the foci A1 and A2 (see also Figure 4.12). The asymptotes of ch are axes of right cylinders L passing through ce (Figure 4.11). Case 2: Let the given conic c1 be a hyperbola ch in the yz-plane with the vertices A1 and A2 . Then, the apices of the right cones passing through ch lie in the xy-plane. Figure 4.13 shows from the top and front view only one half. The rest can be obtained by reflection in the y-axis. We follow the same strategy like in case 1: We specify a sphere S, which is in contact with the yz-plane at the focus F1 of ch , and inspect the top view. The tangents drawn from the vertices A1 and A2 to the contour of S (contact points T1 and T2 ) intersect at a possible apex S which obviously satisfies the following conditions: A1 S + A2 S = A1 T1 + ST1 + A2 T2 − ST2 = A1 F1 + A2 F1 = F1 F2 = const. This characterizes the focal ellipse ce of the given hyperbola ch (see also Figure 4.14). Case 3: Figure 4.15 shows that all right cones C passing through a given parabola c1 have their apex S on the focal parabola c2 , since SA = ST1 + T1 A = ST2 + T2 l = Sl.

◾

We conclude this section with some additional theorems on focal conics. Theorem 4.2.2 Given a pair of focal conics (c1 , c2 ), each arbitrary line segment P1 P2 with Pi ∈ ci for i = 1, 2 defines a line segment Q1 Q2 on the principal axis such that P1 P2 = Q1 Q2 holds and that the point Qi is a vertex of a further conic di which is confocal with ci and passes through Pi (see Figure 4.16).

151

4.2 Pairs of focal conics, Dupin cyclides

l

T2

C S

k ′′ tS

S

T1 c′′1

F

A

c1 c2 c′2

A

F

S′ FIGURE 4.15. Which cones C of revolution pass through a given parabola c1 ? Remark 4.2.1 In volume 2, we shall learn that this theorem is a special case of Ivory’s theorem in three dimensions. It will be useful for proving thread constructions of quadrics (note [109, Section 7.3]). Proof: We start with two parabolas, one in the xy-plane and the other in the yz-plane. We adjust the standard parametrization of parabolas from (2.17) and set (Figure 4.16) P1 = (2pu, 2pu2 , 0),

P2 = (0,

p − 2pv2 , 2pv) , 2

(u, v) ∈ R2 .

This yields P1 P2

2

= =

4p2 u2 + (2pu2 −

p 2

+ 2pv2 ) + 4p2 v2 2

2p u + 2p v + 4p u + 4p2 v4 + 8p2 u2 v2 + 2 2

2 2

2 4

Now, we verify that P1 P2 = ∣p∣ [2(u2 + v2 ) +

p2 4

= p2 [2(u2 + v2 ) + 12 ]

1 ] = ∣y1 − y2 + p∣ = Q1 Q2 , 2

2

(4.1)

where y1 = 2pu2 and y2 = p2 − 2pv2 are the respective y-coordinates of P1 and P2 . We define Q1 = (0, y1 + p2 , 0) and Q2 = (0, y2 − p2 , 0). The parabola through P1 , which is confocal with c1 has the vertex Q1 . This follows for v = 0 from (4.1) since the directrix of this parabola has the y-coordinate y1 + 2pu2 + p2 . Similarily, we can confirm that Q2 is the vertex of the confocal parabola through C2 . When c1 is assumed as an ellipse and c2 as its focal hyperbola, then we refer to the coordinate frame of Figure 4.11 with the y−axis as the common principal axis and the origin at the

152

Chapter 4: Euclidean 3-space z

c2 P2 d2 p/2

Q2

P1 P2 A

p/2

F

c1

x

d1

Q1 y

P1

FIGURE 4.16. P1 P2 = Q1 Q2 for points P1 and P2 on a pair of focal parabolas. common center M . We recall the parametrization of ellipses given in (2.12) and of hyperbolas given in (2.14). Thus, we can set up the points P1 ∈ c1 and P2 ∈ c2 as √ P1 = (b sin u, a cos u, 0), P2 = (0, ± a2 − b2 cosh v, b sinh v), u ∈ [0, 2π), v ∈ R,

where the upper sign at the y-coordinate of P2 defines the right branch and the lower sign the left branch of the hyperbola c2 . Hence, we compute √ 2 2 P1 P2 = b2 sin2 u + (a2 − b2 ) cosh v ∓ 2a a2 − b2 cos u cosh v + a2 cos2 u + b2 sinh2 v √ = (a2 − b2 ) cos 2 u ∓ 2a a2 − b2 cos u cosh v + a2 cosh2 v. √ With e = a2 − b2 , we obtain e a P1 P2 = ∣e cos u ∓ a cosh v∣ = ∣ y1 − y2 ∣ , (4.2) a e where yi denotes the y-coordinate of the point Pi for i = 1, 2. The point Q1 = (0, ae y1 , 0) is a vertex of the hyperbola which is confocal with ce and passes through P1 . This can be verified by direct computation. But we can also recall the particular case of Ivory’s theorem which is depicted in Figure 2.24: The scaling (x, y, 0) ↦ (0, ae y, 0) maps the ellipse ce onto the segment bounded by its focal points while each point remains on the same hyperbola of the confocal family. In the same way we can prove that Q2 = (0, ae y2 , 0) is a vertex of the ellipse in the yz-plane which passes through P2 and is confocal with the hyperbola c2 .

◾

Figure 4.16 shows the equal distances in the case of parabolas. The case of an ellipse and its focal hyperbola is displayed in Figure 4.17, however with a slightly modified notation. Here, Theorem 4.2.2 gives rise to a generalization of the standard definitions of ellipses and hyperbolas, which might explain the choice of the name ‘focal’ hyperbola or ellipse.

153

4.2 Pairs of focal conics, Dupin cyclides

ch

E d2 S2 S1

T1 A1

d1 X ′′

S1 X c′h

T2

F2 S2 X

F1 A1

A2 c′′e

F1

F2

S1′

U

S2′ A2 ce

X FIGURE 4.17. Generalized gardener’s construction: For points S1 , S2 fixed on different branches of the focal hyperbola ch we have S1 X + S2 X = const. for all X ∈ ce .

Corollary 4.2.1 Given an ellipse ce , let S1 and S2 be points chosen on different branches of the focal hyperbola ch of ce . Then, for all points X ∈ ce , the sum S1 X + S2 X is constant. Conversely, given any two points R1 , R2 ∈ ce , for all points Y ∈ ch the difference ∣R1 Y − R2 Y ∣ is constant. Proof: With Theorem 4.2.2, we recognize in Figure 4.17 equal distances Si X = Ti U for i = 1, 2. For S1 on the left branch and S2 on the right branch of the hyperbola ch , we obtain by (4.2) S1 X = a cosh v1 + e cos u,

S2 X = a cosh v2 − e cos u.

The sum S1 X + S2 X = T1 T2 is independent of the choice of X ∈ ce .

On the other hand, for the two points R1 , R2 ∈ ce with corresponding parameter values u1 and u2 we get for all Y ∈ ch : ∣R1 Y − R2 Y ∣ = ∣e(cos u1 − cos u2 )∣. By the same token, such a constant difference can also be identified at a pair of focal parabolas.

◾

The constant sum S1 X + S2 X = a(cosh v1 + cosh v2 ) gives rise to a generalized gardener’s construction (Figure 4.17). If we fix the endpoints of a string of length a(cosh v1 + cosh v2 ) at S1 and S2 on ch and pull the string taut at a point X in the xy-plane, then X will trace the ellipse. Or, in other words: Let us imagine the two curves as wires and choose a rubber

154

Chapter 4: Euclidean 3-space

S2 ch S1

v2 ϕ

ϕ tX

X

ce v

v1

FIGURE 4.18. The constant length of the string from S1 via X to S2 is also a consequence of congruent angles between S1 X or S2 X and the tangent tX .

band instead of the string. Now, we fix one endpoint at S1 ∈ ch , thread the band the ellipse ce , and span it at the point S2 on the other branch of ch . Surprisingly, the point X ∈ ce will not tend to a particular point on ce where the total length attends its minimum, since for all X we get the same length. This means, the point X can move on the ellipse without changing the tension along the band (Figure 4.18). This phenomenon is a consequence of the congruent angles between the tangent tX to ce at X and the segments XS1 and XS2 . Both are lying on a right cone with apex X and axis tX . Similar to the proof of Graves’s theorem (Theorem 2.2.5, see Figure 2.29) we can conclude that the sum of distances S1 X + S2 X must be constant, since the velocity vector v of X has equal components v1 and v2 in direction of XS1 and XS2 , but with different signs. What happens, when the point X, which pulls the string between S1 and S2 taut, is not kept within the xy-plane? Then, in each plane through the line [S1 , S2 ] the point X traces an ellipse with the foci S1 and S2 . The rotation about [S1 , S2 ] generates an ellipsoid E of revolution, which is depicted in the front view of Figure 4.17. This ellipsoid E intersects the xy-plane exactly along the ellipse ce .

155

4.2 Pairs of focal conics, Dupin cyclides

Corollary 4.2.2 Let (c1 , c2 ) be a pair of focal conics. Then, for all common secants [X1 , X2 ] with Xi ∈ ci , i = 1, 2, the connecting planes with the tangents ti at Xi to ci are orthogonal. Proof: This is trivial when X2 is a focus of c1 since t2 is orthogonal to the plane of c1 . Otherwise, by virtue of Theorem 4.2.1, there is a right cone C passing through c1 , with apex X2 and axis t2 . The plane connecting [X1 , X2 ] with t2 is a diameter plane of C, while the connection with t1 is tangent to C along the generator [X1 , X2 ].

◾

Remark 4.2.2 It can be proved (see vol. 2) that orthogonal projections map a pair (c1 , c2 ) of focal conics onto confocal conics (see, e.g., Figures 4.16 and 4.18), when edge views of one of the conics’ planes are excluded. By virtue of Corollary 4.2.2, we can already confirm that the views of c1 and c2 under an orthogonal projection must intersect orthogonally. This follows, since each point X of intersection is the image of any secant X1 X2 , and the tangents at X are edge views of the orthogonal planes mentioned in the corollary above.

Theorem 4.2.3 Let the x-axis of a Cartesian coordinate frame be an axis of symmetry of a conic c. If the normal line to c at a point P = (x, y) ∈ c intersects the x-axis at Pn = (xn , 0), then xn is a linear function of x. For parabolas, the difference xn − x is constant. The point Pn is the center of a circle k which touches c twice: at P and at its reflection in the x-axis. The squared radius r 2 of k is a quadratic function of x.

Proof: By virtue of Theorem 2.1.2, conics are irreducible curves of degree two. When the x-axis is an axis of symmetry of the conic c, then in the general equation (2.11) the variable y shows up only with exponent two. Hence, we may set up c ∶ F (x, y) = a11 x2 + a22 y 2 + 2a1 x + a = 0,

where a22 ≠ 0, since otherwise c would be reducible. The gradient grad(F ) = (

∂F ∂F , ) = (Fx , Fy ) = (2a11 x + 2a1 , 2a22 y) ∂x ∂y

at the point P = (x, y) ∈ c is orthogonal to c, since for each parametrization (x(t), y(t)) of c the substitution F (x(t), y(t)) is the zero-function in t, and therefore, Fx x˙ + Fy y˙ = 0. We set up the point Pn as x 2a11 x + 2a1 xn ( ) + λ( )=( ). y 2a22 y 0

This yields for points P with y ≠ 0 λ=

−1 2a22

and

xn = x −

a11 a1 x− , a22 a22

which is the stated linear function. The squared distance P Pn = (xn − x)2 + y 2 is a quadratic function of x. The signed distance xn − x is called the subnormal of point P . It is constant if, and only if, a11 = 0, hence, by virtue of Theorem 2.1.2, the conic c is a parabola. 2

156

Chapter 4: Euclidean 3-space

S C P ′′ c

′′

k

′′

π ′′

x

S

x

Pn c

k P

FIGURE 4.19. For the conic c = C ∩ π, the subnormal xn − x of P = (x, y) is a linear function of x. If the given axis of symmetry is the principal axis, then Theorem 4.2.3 can also be concluded, when c is given as a planar section of a right cone C (see Figure 4.19). We choose a sphere S which is inscribed into C and intersects the plane π along a circle k. This circle with center Pn has a double contact with c. The points P of contact belong to the circle of contact between S and C. Obviously, the point Pn is the point where the normal line at P intersects the axis of symmetry. In Figure 4.19, the front view shows the x-coordinates of P and of Pn . Suppose, the inscribed sphere S varies. Then, the point P ′′ is mapped onto Pn by a sequence of three parallel projections from lines to lines. This causes the stated linearity of the function xn (x).

◾

●

Exercise 4.2.1 A flexible bipartite framework.

For any pair (c1 , c2 ) of focal conics, let points X1 , . . . , Xn ∈ c1 and Y1 , . . . , Ym be the knots of a bipartite bar-and-joint framework with m, n ≥ 2 such that no knot lies on the common principal axis of the conics. Why can all knots move along their conics such that all mutual distances Xi Yj remain unchanged? Hint: By virtue of (4.1), in the case of two parabolas the distance Xi Yj depends only on the y-coordinates of the points Xi and Yj . Choose a sufficiently small k > 0 and add k to the y-coordinates of all Xi , while k is subtracted from the y-coordinates of all knots Yj . Due to (4.2), a similar method works for ellipses and hyperbolas (see [150]).

4.2 Pairs of focal conics, Dupin cyclides

157

Dupin cyclides

FIGURE 4.20. Dupin cyclides D are channel surfaces (cf. Remark 4.2.3) in two ways. The spine curves are focal conics. The circles of contact with the enveloped spheres form an orthogonal net.

We introduced non-parabolic Dupin cyclides on page 37 in the following way. Given two circles g1 and g2 with different centers, the envelope of a one-parameter family of spheres, whose equators h contact g1 and g2 , is called a Dupin cyclide, provided, the type of contacts between the equators and the given circles is either everywhere the same or changes simultaneously. We can reformulate the restrictions on the kinds of contact by using signed radii for all involved circles. In accordance with (2.22), the equation M1 M2 = ∣r1 − r2 ∣ characterizes the contact between the circles (M1 ; r1 ) and (M2 ; r2 ). In case of an interior contact the signs of r1 and r2 are equal, while in the case of an exterior contact the signs differ. This convention, which will also be used for spheres in 3-space, allows us to define Dupin cyclides more precisely. Definition 4.2.2 Let two circles gi = (Fi ; ri ), i = 1, 2 with signed radii r1 and r2 be given in the same plane, where F1 ≠ F2 and r1 ≠ r2 . Then,

158

Chapter 4: Euclidean 3-space

the envelope of the family of spheres, whose equators h = (X; ̺) satisfy XFi = ∣ri − ̺∣ for i = 1, 2 , is a (non-parabolic) Dupin cyclide D, provided the set of equators is not empty.

All equators h are in contact with g1 and g2 . For variable ̺ ∈ R, the kind of contact can only change at ̺ = 0, and then this happens for g1 and g2 simultaneously. This is exactly what we wanted to achieve. A simultaneous change of signs for r1 and r2 has no effect on the cyclide D. In order to obtain also parabolic cyclides, we have to replace in Definition 4.2.2 the circle g2 by a line l such that Xl is the signed distance from point X to l, i.e., positive in one half-plane of l and negative in the other. Then, the circles h = (X; ̺) satisfying XF1 = ∣r1 − ̺∣ and ̺ = Xl are the equators of a parabolic Dupin cyclide (see Figure 2.20). Theorem 4.2.4 Each Dupin cyclide D is the envelope of two oneparameter families of spheres. The centers of the spheres form a pair of focal conics (c1 , c2 ). All normal lines of D are common secants of (c1 , c2 ) (Figure 4.20). Each sphere of one family touches all spheres of the other family. For each family, the planes of the circles of contact have a common intersection line. Along each of these circles, there is a right cone tangent to the cyclide. The apices of these cones are located on one of the mentioned lines. When one family is replaced by the other, the two lines change their roles. The circles of contact constitute an orthogonal net on D. Remark 4.2.3 The envelope of a one-parameter family of spheres is called a channel surface. The centers of the spheres form the associated spine curve. Dupin cyclides, together with the torus, are the only surfaces which are channel surfaces in two different ways. Another approach to Dupin cyclides via pentaspheric coordinates can be found in [82, p. 53ff]. Proof: We proceed in six steps and treat only the non-parabolic case. A similar proof for parabolic Dupin cyclides is left to the reader as Exercise 4.2.2. 1) Let two circles g1 = (F1 ; r1 ) and g2 = (F2 ; r2 ) with F1 ≠ F2 and r1 ≠ r2 be given in a horizontal plane. The two radii are signed, and these signs remain fixed. We show first that the circles h = (X; ̺) satisfying XFi = ∣ri − ̺∣ have their centers on a conic c1 with focal points F1 and F2 (Figure 4.21).

To this end, we use the following transformation: We subtract from all involved circles the signed radius r1 of g1 , while the centers remain fixed. This preserves each mutual contact, but transforms the circles h into circles which pass through F1 and contact the circle (F2 ; r2 − r1 ). The rest follows from Corollary 2.2.2.

2) The sphere S with equator h = (X; ̺) touches the enveloped surface along a circle k. This circle is defined as the limit of the circle of intersection between S and a neighboring sphere,

159

4.2 Pairs of focal conics, Dupin cyclides tY

c2 q D

Y

′′

l

′′

p′′

c′′1 F1

A1

A2

F2 k2

k1

g1

K1

h D′

X

S h1

c1

K2

c′2 U1

U2

A1

F1

k

′

g2 F2 Q =q ′

C′

p

tX V1

V2

A2

h2

P P0

FIGURE 4.21. Top and front view of the Dupin cyclide D which envelopes the spheres with equators h being tangent to two given circles g1 and g2 . when the latter tends towards S within the family of spheres. The plane of k is vertical. Furthermore, the axis tX of k is tangent to the conic c1 at X. We already know two points K1 , K2 ∈ k, the respective points of contact of the equator h and the given circles g1 and g2 .

The lines connecting the center X of the sphere S with points of the circle k ⊂ S are orthogonal to S and to the envelope D. These lines form a right cone with apex X and axis tX . Since the centers F1 and F2 of the given circles are focal points of the conic c1 , the comparison with Figures 4.11, 4.13, and 4.15 reveals that this cone passes through the focal conic c2 of c1 . Hence, all normals of the Dupin cyclide D are common secants of the pair of focal conics (c1 , c2 ).

3) There is a right cone C, which touches S and D along k. Let P be its apex. The power 2 P K1 of P w.r.t. the equator h = (X; ̺) is the same as with respect to g1 and g2 . Hence, for all X ∈ c1 , the apex P is a point of the horizontal radical axis p of the given circles g1 and g2 . 4) The common diameter [F1 , F2 ] of the given circles is the principal axis of c1 . At the principal vertices A1 , A2 of c1 , the radii of the corresponding equators h1 , h2 attain extremal values ̺1 , ̺2 , respectively. [F1 , F2 ] is also an axis of symmetry of the Dupin cyclide D.

The tangents to h at K1 and K2 are symmetric with respect to the axis tX of the cone C (Figure 4.21). The line [K1 , K2 ] intersects g2 at a second point, and there the tangent to

160

Chapter 4: Euclidean 3-space

g2 is parallel to the tangent to g1 at K1 . Therefore, the point Q of intersection between the common diameter [F1 , F2 ] and the plane of k is the center of a similarity which maps g1 onto g2 . Thus, the point Q is independent of the choice of X ∈ c1 . For all circles k, their planes share the vertical line q through Q.

Let Ui and Vi for i = 1, 2 denote the points of intersection between [F1 , F2 ] and gi . Then, the point Q satisfies the equation QU1 ∶ QV1 = QV2 ∶ QU2 . This implies that QU1 ⋅QU2 = QV1 ⋅QV2 . 2 Hence, the power of Q with respect to h1 , i.e., QA1 − ̺21 = QU1 ⋅ QV2 , is the same as that with respect to h2 . The vertical line q through Q is the radical axis of the two circles k1 , k2 of contact with the centers A1 and A2 at the principal vertices of c1 . In the same way, it follows that the point P0 of intersection between the common diameter and the line p is the center of a similarity with h1 ↦ h2 and k1 ↦ k2 . 5) We choose the principal axis [F1 , F2 ] of c1 as y-axis and the secondary axis as x-axis of a Cartesian coordinate frame (like in Figure 4.11). By virtue of Theorem 4.2.2, the distance between any two points X ∈ c1 and Y ∈ c2 depends only on the respective y-coordinates y1 of X and y2 of Y . If c1 is an ellipse with semimajor axis a and numerical eccentricity e, then, according to (4.2), XY = ∣ ae y1 − ae y2 ∣. Let all points X ∈ c1 and Y ∈ c2 be centers of spheres with respective signed radii rX =

e y1 + r 0 , a

rY =

a y2 + r 0 e

(4.3)

with any real constant r0 . This defines two families of spheres, one centered on c1 and the other centered on c2 . Because of XY = ∣̺1 − ̺2 ∣, any two spheres (X; rX ) and (Y ; rY ) taken from different families are in contact. The point of contact lies on the connecting line [X, Y ]. In particular, all spheres centered on c1 contact the spheres with centers F1 and F2 , i.e., where y2 = ±e. Their equators in the xy-plane have the radii rY = r0 ± a.

Conversely, if these two equators with centers F1 , F2 and respective signed radii r1 = r0 +a and r2 = r0 −a are given, we obtain r0 = (r1 +r2 )/2 and a = ∣r1 −r2 ∣/2. According to Definition 4.2.2, the envelope of both families of spheres is the same Dupin cyclide D, which has been studied above. By the same token, when the constant r0 changes, D is replaced by an offset surface. 6) The surface normals of the Dupin cyclide passing through a fixed point Y ∈ c2 form a cone of revolution. Its curve of intersection with the sphere (Y ; rY ) is a circle l, which is at the same time the circle of contact between this sphere and D. For Y → ∞ the sphere becomes a plane. There are two such planes passing through p. The front view in Figure 4.21 shows them in an edge view. All spheres (X; rX ) of the first family contact these two planes, which reveals again that the radius rX is a linear function of the y-coordinate y1 of X.

Let T ∈ D on the normal line [X, Y ] be the point of contact between the two spheres (X; rX ) and (Y ; rY ). There are two circles of D passing through T : The circle k of contact between (X; rX ) and D lies in a plane through the vertical line q; the circle l of contact between (Y ; rY ) and D lies in a plane through the line p (Figure 4.22).

The tangents to k and l at T span the tangent plane of D, which is orthogonal to [X, Y ]. The tangent to l must intersect p and passes, therefore, through the apex P of the right cone C, which touches D along k (compare with Figure 4.21). The tangent to l at T is, therefore, orthogonal to k. The circles of contact of both families of spheres form an orthogonal net on D.4

◾

4

In fact, the circles of contact between the spheres of both families and the Dupin cyclide D are the curvature lines of D.

161

4.2 Pairs of focal conics, Dupin cyclides Q l q T X Q0

g1

P0 g2

p

k

P Y

FIGURE 4.22. The circles of contact between the Dupin cyclide and its two families of spheres constitute an orthogonal net. Remark 4.2.4 According to the proof given above, the point Q has the same power with respect to the two circles h1 and h2 (Figure 4.21). Therefore, Q is the center of an inversion (see Section 7.5) which maps h1 onto itself, as well as h2 . This inversion exchanges g1 with g2 , while all circles h remain fixed. Consequently, also in space there is an inversion with center Q mapping D onto itself, since all spheres of the first family remain fixed. Similarily, the point P0 is the center of an inversion which maps each sphere of the second family onto itself. By the way, Dupin cyclides can also be obtained by applying a 3D inversion onto a torus, a right cone, or a right cylinder.

●

Exercise 4.2.2 Parabolic Dupin cyclides.

Modify the proof of Theorem 4.2.4, as provided above, for the case of a parabolic Dupin cyclide.

●

Exercise 4.2.3 Equation of Dupin cyclides.

Prove that a Dupin cyclide D with an ellipse c1 and its focal hyperbola c2 as spine curves satisfies in the coordinate frame of Figure 4.11 the equation (x2 + y 2 + z 2 + b2 − r02 ) − 4b2 x2 − 4(ay + er0 )2 = 0 √ of degree four with e = a2 − b2 and any constant r0 ∈ R, when, by virtue of (4.3), the enveloping spheres have the radii r1 = ae y1 + r0 and r2 = ae y2 − r0 . 2

Hint: The spheres centered on c1 satisfy the equation (x − b sin u)2 + (y − a cos u)2 + z 2 = (e cos u + r0 )2 ,

with u ∈ [0, 2π[ as family parameter. Because of (a2 − e2 ) cos 2 u + b2 sin2 u = b2 , we can rewrite it as F (x, y, z, u) ∶= x2 + y 2 + z 2 − 2bx sin u − 2 cos u(ay + er0 ) + b2 − r02 = 0 .

162

Chapter 4: Euclidean 3-space

The circles of contact with the enveloped cyclide D satisfy, beside F (x, y, z, u) = 0, also ∂ F (x, y, z, u) = 0, which leads to 2bx cos u − 2 sin u(ay + er0 ) = 0. The stated equation of ∂u D is follows from the elimination of u from F (x, y, z, u) = 0 and its partial derivative with respect to u. We obtain the same equation when starting with the spheres centered on the focal hyperbola c2 . This gives x2 + y 2 + z 2 ∓ 2 cosh v(ey + ar0 ) − 2bz sinh v − b2 − r02 = 0 , and for the planes through the circles of contact 2 sinh v(ey + ar0 ) ∓ 2bz cosh v = 0 .

Hence, Dupin cyclides with an ellipse and hyperbola as spine curves are of degree four, while parabolic cylides are cubic surfaces. The latter can be proved in a similar way.

4.3 Perspective images of conics

163

4.3 Perspective images of conics The main goal of this section is to prove that not only circles, but all conics have the property that their perspective images are again conics, provided the carrier plane of the conic does not pass through the center of the projection.

Quadratic cones In analogy to the definition of curves of degree two on page 20, we call in the Euclidean 3-space E3 the set Q of points whose Cartesian coordinates (x, y, z) satisfy a given quadratic equation b11 x2 + 2b12 xy + 2b13 xz + . . . + b33 z 2 + 2b1 x + 2b2 y + 2b3 z + b = 0

(4.4)

a surface of degree two. Again, the degree is invariant under changes (x, y, z) ↦ (x′ , y ′ , z ′ ) of the Cartesian coordinate system, since the new coordinates are linear functions of the initial coordinates, and vice versa. The surface with the equation (4.4) is called reducible if the polynomial on the left-hand side splits into two linear polynomials. Then, the surface Q consists of two planes or only of one. Similarily to the planar case, a polynomial can be irreducible over R, but reducible over C, and the point set Q can be empty, e.g., in the case x2 + y 2 + z 2 + 1 = 0.

A surface Q of degree two is called a quadratic cone C if there exists exactly one point S ∈ C such that with each point P ∈ C {S} all points of the connecting line [S, P ] belong to C. The point S is called the apex of C. A quadratic cone must be irreducible, since otherwise the apex would not be unique. Let (4.4) be the equation of a quadratic cone C with its apex S at the origin. Then, the polynomial F (x, y, z) on the left-hand side satisfies F (0, 0, 0) = b = 0. Furthermore, P ∈ C {0}, i.e., F (xP , yP , zP ) = 0, must imply F (λxP , λyP , λzP ) = λ2 (b11 x2P + . . . + b33 zP2 ) + 2λ(b1 xP + b2 yP + b3 zP ) = 0

for all λ ∈ R. This is only possible, when for all P ∈ C in the equation above the quadratic term and the linear term vanish. Being a quadratic polynomial, F (x, y, z) must be homogeneous of degree two, and therefore, b1 = b2 = b3 = 0.

The lemma below focuses on two important properties of quadratic cones.

164

Chapter 4: Euclidean 3-space

Lemma 4.3.1 The lines in E3 connecting a given point S with the points of a conic, which is not coplanar with S, form a quadratic cone. Conversely, a quadratic cone intersects all planes not passing through the apex along conics. Proof: We choose a Cartesian coordinate frame with the origin at the point S such that the given conic lies in the plane z = 1. If the conic satisfies the equation a11 x2 + 2a12 xy + a22 y 2 + 2a1 x + 2a2 y + a = 0, then

a11 x2 + 2a12 xy + a22 y 2 + 2a1 xz + 2a2 yz + az 2 = 0

is the equation of a quadratic cone containing the given conics. In the reverse direction, this confirms the second statement of Lemma 4.3.1, since without loss of generality the intersecting plane can be specified with the equation z = 1.

◾

Remark 4.3.1 As a consequence of Theorem 2.1.2, each irreducible homogeneous quadratic polynomial with any zero other than (0, 0, 0) defines a quadratic cone as its set of zeros.

By virtue of the principal axes transformation which will be provided in volume 2, there is always an appropriate coordinate system such that the corresponding homogeneous equation of the quadratic cone with apex S = (0, 0, 0) contains only squares; all terms with mixed products xy, yz, and yz vanish. Since this equation must admit other zeros than the origin, quadratic cones have the standard equation (in a suitably adjusted coordinate frame) C∶

x2 y 2 + − z 2 = 0 with a 2 b2

a ≥ b > 0.

(4.5)

It turns out that among the planar sections of quadratic cones there are always circles. We can prove this by checking the intersection between the cone C in (4.5) and the sphere S ∶ x2 + y 2 + (z − 1 − a2 )2 = a2 (1 + a2 )

(Figure 4.23). We eliminate x2 from the two equations and obtain (a2 − b2 )y 2 − b2 (1 + a2 )(z − 1)2 = 0.

This is the equation of a pair of planes through the point (0, 0, 1), which lies in the interior of S. Hence, the two planes which are displayed in red color in Figure 4.23, intersect S along two circles, which also belong to

165

4.3 Perspective images of conics C S z

z z=1

x

S ′′

x

S S ′′′

y

S′

y FIGURE 4.23. Each quadratic cone has circular sections and (at least) three planes of symmetry.

C. All planes which are parallel to one of these two planes, intersect the quadratic cone C along circles. We call them circular sections of C. This means also that each quadratic cone can be defined as the connection of a circle with a non-coplanar apex S. Therefore, quadratic cones other than right cones can also be called oblique circular cones.

Central projection of conics The basic term in this subsection is the central projection. It is defined by an image plane π and a center C ∉ π (Figure 4.24). The pedal point of C with respect to π is called principal point H; the distance d = CH is the principal distance. The image X c of any point X other than C is the point of intersection between the ‘line of sight’ [C, X] and the image plane π. The image of any scene is called (central or linear) perspective. It can happen that [C, X] is parallel to π. Then, we define the image X c as a point at infinity. In this case, X lies in the vanishing plane π v which passes through C and is parallel to π. On the other hand, points at infinity can have a finite image: Figure 4.25 shows that the images of lines parallel to the coordinate axes have common points Xuc , Yuc , Zuc , respectively, whose preimages are at infinity (for the extension of the Euclidean plane by points at infinity see Section 5.1).

166

Chapter 4: Euclidean 3-space va ni sh in g πv

im pl a

ag

π

ne

y

e

pl a

ne

y′

X

Xc z

C

H

x

x′ d

FIGURE 4.24. Central projection X ↦ X c with center C, principal point H, and camera frame (x, y, z).

Theorem 4.3.1 Central projections map conics c onto conics cc , provided the plane of c does not pass through the projection center C. Depending on the number 0, 1, or 2 of intersection points between c and the vanishing plane, the image cc is an ellipse, a parabola, or a hyperbola. Proof: This follows directly from Lemma 4.3.1: The cone connecting the projection center with the conic c is quadratic, and this cone intersects the image plane π along a conic cc . Exactly the points of intersection between c and the vanishing plane are mapped onto the points at infinity of the conic cc .

◾

Of course, Theorem 4.3.1 is also valid for circles. Figure 4.26 shows the perspective of a packing of congruent circles over a square grid. The circles depicted in Figure 4.27 form a parabolic pencil. They are located in the xy-plane and contact the y-axis at the origin O. All types of conics can be found among the images. Figure 4.25 shows a perspective of circle packings in all three coordinate planes. The points Xuc , Yuc , and Zuc are the images of the ideal points of the coordinate axes. The central projection is the geometric idealization of the photographic mapping with C as the focal center of the lenses, with π as the plane carrying the film or the CCD sensor, and with d as focal length. However,

167

4.3 Perspective images of conics

z

Yuc

Xuc

x y

Zuc FIGURE 4.25. Perspective of unit circle packings in the coordinate planes.

one has to note that the central projection is defined for all points X ≠ C in space, while the photographic mapping depicts only points inside a four-sided pyramid with apex C in one half-space of the vanishing plane. In Figure 4.27, some of the circles are mapped onto hyperbolas; their second branch would not be visible under a photographic mapping. It does not matter that — contrary to Figure 4.24 — at the photographic mapping the image plane does not belong to this half-space. As demonstrated in Figure 4.28, the plane π shows the same image as its reflection π in the center C. There is just one difference: The negative plane π shows the image upside down.

168

Chapter 4: Euclidean 3-space

FIGURE 4.26. Perspective of a circle packing over a square grid.

Each photographic mapping defines an associated coordinate system in space, the camera frame (x, y, z) (Figure 4.24). Its origin is placed at the center C. The principal ray of sight pointing from C into the visible halfspace is the z-axis. The horizontal and vertical directions of the image sensor define the x and y-axis, spanning the vanishing plane π v . When the principal point H is specified as the origin of a coordinate frame (x′ , y ′ ) in the image plane π, then the photographic mapping and the associated central projection obey the matrix equation x ⎛ x ⎞ d 1 0 0 ⎛ ⎞ x′ ⎜ y ⎟ ↦ ( ′ )= ( )⎜ y ⎟. y z 0 1 0 ⎝ ⎠ ⎝ z ⎠ z

Now, we bring this in a more general form: We replace the camera frame by arbitrary world coordinates (x, y, z), namely ⎛ x ⎞ ⎛ a ⎞ ⎛ x ⎞ ⎜ y ⎟ = ⎜ b ⎟ + R⎜ y ⎟, ⎝ z ⎠ ⎝ c ⎠ ⎝ z ⎠

169

4.3 Perspective images of conics z

O

x y

FIGURE 4.27. Perspective of a parabolic pencil of circles. ne ga

π

tiv e

pl a

va ni s πv hing ne

im pl a

π

ag

e

ne

pl a

ne

CP Xc

H C

H

CP Xc

d

CP X

d c

FIGURE 4.28. Central projection X ↦ X onto the negative plane π.

170

Chapter 4: Euclidean 3-space

where R is an orthogonal 3×3-matrix. A similar coordinate transformation (x′ , y ′ ) → (x′ , y ′ ) can be performed in the image plane π. This reveals that finally the coordinates x′ and y ′ of X c can be expressed as rational functions of the world coordinates (x, y, z) of the preimage X, such that the numerator and the denominator are linear in x, y, and z, and the denominators in x′ and y ′ are equal. These formulas can be used to transform any rational parametrization of a conic c onto a rational parametrization of the image cc . C

H C

F1c

u

F1

F2

π uc

S

πv FIGURE 4.29. The perspective contour uc of a sphere S is a conic uc . The points F1 , F2 ∈ S are projected onto the focal points of uc .

In order to obtain the perspective of any sphere S, we draw the tangents from the center C to the sphere S. They form a right cone C, provided the projection center C lies in the exterior of S. The cone C touches S along a circle u and intersects the image plane π in a conic uc , the contour of S, whose principal axis passes through the principal point H. Figure 4.29 shows the orthogonal projection of the scene onto a plane through C, H, and the sphere’s center. When S intersects the vanishing plane π v along a circle, then uc is a hyperbola. In the case of contact between S and π v the contour uc is a parabola. We obtain an ellipse uc under S ∩ π v = ∅ (note Figure 4.30).

By virtue of Theorem 4.1.1, the focal points of the contour uc = C ∩ π are the points of contact of the image plane π and the Dandelin spheres. Each Dandelin sphere can be transformed onto the given sphere S by a

171

4.3 Perspective images of conics

similarity with center C. This similarity sends the corresponding focal point onto a point on S, whose tangent plane is parallel to π. Therefore, the projections of the endpoints of the diameter F1 F2 orthogonal to the image plane π are the focal points of the contour uc (Figure 4.29).

H

p

e h

FIGURE 4.30. Ellipse e, parabola p and hyperbola h as contours of spheres.

Similar results are valid for parallel projections, which are the limiting cases of central projections, when the center C tends to infinity: Conics are projected onto conics of the same type, and the contour of a sphere is an ellipse. We will get a deeper insight in chapter 8 which treats affine geometry. We speak of a linear image, whenever a scene in E3 is mapped onto a plane by a central or parallel projection.

172

Chapter 4: Euclidean 3-space

4.4 Spatial interpretation of conic constructions In this section, we shall learn that some constructions of conics and related to conics can be solved by simply interpreting the given planar figures as some projection of a geometric object in a three-dimensional space and solving the transferred problem there. The projection of the solution to the spatial problem yields the solution to the initial problem. Unfortunately, there is no general concept for this method. Some interpretations are obvious, some seem to be tricky. The method is restricted to special problems: Any given configuration of points, lines, and conics that can be interpreted as a linear image of points and conics on an auxiliary quadric (surface of degree two, including cones and cylinders) or lines tangent to the auxiliary surface can be treated that way. The major advantage of this method is that it allows us to find the solutions without any knowledge from Projective Geometry and is, therefore, accessible to everybody who is familiar with Descriptive and Constructive Geometry in three dimensions. In the following we shall solve some problems in order to show to which problems the technique of spatial interpretation applies to.

Ellipses on three points and two parallel tangents U1

s

A′

A

U1′ =s′ 1′

1

e

p′

p B

e′ =∆′

M′

M B

′

B′

2′

2 C U2

A′

C′ t

U2′ =t′

C′

FIGURE 4.31. The ellipses on three points A, B, C and two parallel tangents s, t are orthogonal projections of four planar intersections of a cylinder of revolution.

173

4.4 Spatial interpretation of conic constructions

How to find ellipses on three points A, B, C tangent to two tangents s, t in admissible position? Figure 4.31 shows the initial situation: We are given three points A, B, C and two parallel tangents s, t. We assume that none of the given points is incident with a given line. Further, A, B, and C shall not be collinear and lie entirely in the strip bounded by s and t. This is what we call admissible position in this case. The key idea is the following: The two parallel tangents s and t can be viewed as the contour of a cylinder ∆ of revolution under an orthogonal projection onto some plane parallel to the cylinder’s axis. If s and t are displayed in a side-view which is the result of an orthogonal projection in the direction of s and t, then any generator of the cylinder ∆ appears as a point. Consequently, the cylinder ∆ shows up as a circle ∆′ whose radius is half the distance between s and t. The points A, B, C are now considered to be images of points on ∆. Therefore, they can be seen in the side-view as points A′ , B ′ , C ′ on the circle e′ = ∆′ , see Figure 4.31. A e

A′

e′ =∆′

axis

s

B′

B C′ C U2

t

U2 =t′

FIGURE 4.32. There is a perspective affine mapping between the circle e′ and the requested ellipse e. Pairs (A, A′ ) of corresponding points are joined by mutually parallel fixed lines of the affine mapping. The axis of the affine mapping is the locus of points that have coinciding images in both orthogonal projections.

Note that there are two choices for A′ ∈ ∆′ : on the left or on the right-hand side (Figure 4.31), i.e., on the back or on the front side of the cylinder ∆. This is also the case for the preimages of B and C. We choose the particular configuration of A′ , B ′ , and C ′ shown in Figure 4.31. The pair (A, A′ ) determines a point P in Euclidean three-space which is located

174

Chapter 4: Euclidean 3-space

on the cylinder ∆. The same holds true for (B, B ′ ) and (C, C ′ ). We call these points Q and R.

The points P , Q, and R span a plane that meets ∆ along an ellipse e. The side-view e′ of e agrees with the side-view ∆′ of ∆. In order to complete the conic e, i.e., the solution to the initial problem, one has to construct at least the points U1 and U2 of contact of e with s and t. The latter points are the contour points on e, and therefore, they are located on the principal line5 with the side view p′ (cf. Figure 4.31). Axes and vertices can be found in many ways. A constructive way is displayed and explained in Figure 9.7. We skip the details of the construction here, since many CAD-systems have tools to construct ellipses from various pieces. On the other hand, there is a perspective affine mapping that maps e′ onto e (Figure 4.32). It can be used to complete the drawing.

FIGURE 4.33. Eight ellipses on the cylinder ∆: Any two equally colored ellipses map to the same ellipse under a certain orthogonal projection.

More important is the fact that there are, in general, four solutions. This is clearly seen when we consider the following: As mentioned above, there are two choices for the side-view of a given point. In Figure 4.31, the different choices are distinguished by bars on top of the symbol. Thus, there are eight planes that can be defined on these points and we have eight intersections with ∆, see Figure 4.33. However, there are four pairs 5

In Descriptive Geometry, a principal line is a line that is parallel to the image plane of a projection. Similarly, a principal plane is parallel to the image plane of any projection.

175

4.4 Spatial interpretation of conic constructions

of planes that result in coinciding images: For example the planes ε ∶= [P, Q, R] and ε ∶= [P , Q, R] are symmetric with respect to the principal plane through ∆’s axis. Thus, the two curves ε ∩ ∆ and ε ∩ ∆ are mapped to the same curve under the orthogonal projection that yields the initial figure (Figure 4.31, left). There exists no ellipse on three points touching s and t if at least one of the given points lies outside the strip bounded by s and t. The choice of the auxiliary surface for obtaining a spatial preimage should be made such that constructions become as simple as possible. In the present example, we could also take an elliptic cylinder (different from a cylinder of revolution), but this would not simplify the construction. Only theoretically, it would be the same way of solving the problem, and it would yield the same four ellipses. B′

B 1

1′

A

a

A′

U1

B′

A′ U1′ 2′

2

U2′

U2

b

∆′

∆′ C

C′

C′

FIGURE 4.34. Hyperbolas on three points A, B, C and two parallel tangents s, t may be found as an orthogonal projection of planar intersections of a hyperbolic cylinder (cf. Exercise 4.4.1). The fat hyperbola on the left-hand side corresponds to the particular choice of A′ , B ′ , C ′ in the side view on the right-hand side.

●

Exercise 4.4.1 Hyperbolas on three points and two parallel tangents. Discuss the type and number of solutions to the example displayed in Figure 4.31 depending on the position of A, B, C relative to s and t.

●

Exercise 4.4.2 Modify the configuration of given points and lines from Example 4.4.1: Assume that the points A, B, C lie outside of the strip bounded by the parallel lines s and t. Hint: In this case, the auxiliary cylinder is a hyperbolic cylinder, i.e., its base is a hyperbola. A short version of the construction may look as shown in Figure 4.34.

176

Chapter 4: Euclidean 3-space

●

Exercise 4.4.3 Find all parabolas on three points and a given tangent. Use a parabolic cylinder as the auxiliary surface. An example is shown in Figure 4.38.

U s

c′

b′

c A 2

S a t

U′

b

1

2′

B′

B l

A′

S′ a′

1′ s =t′

C′

′

C

V

l′

V′

FIGURE 4.35. The ellipse l is one of four conics passing through A, B, C touching the lines s and t. It is the orthogonal projection of a planar intersection of a cone or revolution.

Conics on three points and a pair of intersecting tangents What happens if s and t are intersecting in a proper point? Then, we cannot expect to find conics of a certain affine type interpolating the given points and lines. From the constructive point of view, we have to replace the auxiliary cylinder by a quadratic cone Γ. For the sake of simplicity, Γ should be chosen as a cone of revolution. Again the two given lines s and t are interpreted as the contour lines (see Figure 4.35) and the conics (ellipses, hyperbolas, and, in some cases, even parabolas) are found as the orthogonal projections of planar intersections of the cone Γ with its axis parallel to the image plane. Figure 4.35 shows the given lines s, t together with the points A, B, C. To the given configuration of points and lines (interpreted as an orthogonal projection of a cone of revolution), we attach another orthogonal view in the direction of the cone’s axis. In the attached view, we can see the parallel circles a, b, c on the cone Γ carrying the points A′ , B ′ , C ′ that map to A, B, C. Like in the previous cases, there are two possible choices for A′ , B ′ , and C ′ . For the moment, we treat one certain choice.

177

4.4 Spatial interpretation of conic constructions

Our goal is to find two further points of a solution l. Then, the conic l is well-defined (or even over-determined). Here, we pay attention to the contour points U and V which are located on a principal line. Once the contour points (points of contact of a solution l with s and t) are found, the solution l is over-determined by two line elements and three further points. As mentioned before, there are several different ways to find axes and vertices of the solution l: the approach via Constructive Geometry, or via Projective Geometry, or even with purely planar and elementary techniques like the perspective affinity joining the two orthogonal projections. We skip the details here.

s A

B

B

′

A′

C t

C′

FIGURE 4.36. All four solutions to the problem given in Figure 4.35 are orthogonal projections of eight planar sections of the cone Γ. Curves with the same color on the right-hand side map to the same curve on the left-hand side.

Figure 4.36 shows all four conics that touch the two intersecting lines s and t and pass through the three points A, B, C. Any solution is the image of two conics which are displayed in the same color in Figure 4.36. In a similar way, i.e., using cylinders and cones as auxiliary surfaces, we can solve the following problems: For the construction of conics on two points and three tangents, we interpret two out of the three tangents as the contour lines of a cone or cylinder (if two of these lines happen to be parallel). For construction purposes it is much simpler to assume that the contour is that of a surface of revolution. The third line is then considered as a tangent of the auxiliary surface, and the two given points shall lie on the surface.

178

Chapter 4: Euclidean 3-space

●

Exercise 4.4.4 Conics on line elements, points, and tangents. Apply the method of spatial interpretation to the following configuration of given elements. Discuss the number of solutions depending on the position of lines and points relative to each other: Figure 4.37 shall give an idea how possible configurations and solutions may look like. 1. Find all conics on one line element (A, a) plus two tangents b, c, and one point B. 2. Find all conics on one line element (A, a) plus two points B, C, and one tangent b.

l1

l2

s1

l B

a

A

b

A l1

s2

l2 a l1

c

C

B

b

l2 FIGURE 4.37. Conics determined by given line elements, points or lines as mentioned in Exercise 4.4.4: Left: Given one line element (A, a), two tangents b, c, and one point B. Right: Given one line element (A, a), two points B, C, and one tangent b. In both cases, some solutions are degenerate: a repeated l (left-hand side) and a pair (s1 , s2 ) of lines (right-hand side). These lines meet the requirements of solutions at least algebraically.

The example of ellipses tangent to two parallel lines given at the very beginning of this chapter (see Figure 4.31) is somehow special. A priori, the affine type of the conics showing up as solutions is known. Another special case in the sense of affine geometry would be the following variation: Find all parabolas on one tangent (proper line) and three (proper and non-collinear) points. In this case, the given line is interpreted as the contour line of a parabolic cylinder. The solutions to the initial problem are orthogonal projections of planar intersections of the cylinder. The image plane is parallel to the cylinder’s generators. An example is displayed in Figure 4.38. The construction is indicated and reduced to the construction of the contact point T1 of one particular solution l1 .

179

4.4 Spatial interpretation of conic constructions C′

l1

C l4

∆′ =l1′ =l2′ =l3′ =l4′

l2

B′

B

1′

1 l3

A′ A T1′

t

T1

FIGURE 4.38. A parabolic cylinder ∆ serves as an auxiliary surface when we have to find all parabolas on three points A, B, C, and a given tangent t.

Conics on three points and touching a given conic twice There is a second group of problems that can be solved by means of spatial interpretation. However, there may be configurations of points that may not yield a single solution. We only deal with those cases that show (real) solutions and start with a circle c and three points A, B, C in the interior of c (Figure 4.39, right). We are looking for all conics that pass through the three given points and touch c twice. Consequently, we interpret c as the contour of a sphere Σ (indeed any regular quadric of revolution with the same contour would also be a possible choice provided that its points are mapped to the interior of c). The points A, B, and C are the orthogonal projections of points P , Q, R on Σ. (Again, there are two possible choices of P as preimages of A, and similar for Q and R.) The plane ε spanned by P , Q, and R meets Σ in a circle k1 whose orthogonal projection gives one solution l1 (see Figure 4.39, left). Note that the solution l1 touches c twice while the preimage k1 meets c transversely. Again, the situation in three-dimensional space shows eight conics that map to only four solutions. This is due to the symmetry of the auxiliary surface Σ with respect to the carrier plane of c. The red solution l4 shown in Figure 4.39 (on the right) is not a wrong solution. The double contact happens in a pair of complex conjugate points joined by a real line. Figure 4.40 shows that a one-sheeted hyperboloid of revolution can serve as an auxiliary surface if the three given points are exterior points of c. Again, one solution seems to be wrong. However, like in Figure 4.39, the

180

Chapter 4: Euclidean 3-space

l3

R

Σ k4

c k1 P

Q l4

k2

l1

l2

B

k3 c

P

C

A

Q

R FIGURE 4.39. Left: Eight circles on the sphere that map to four ellipses touching the contour circle twice. Right: Four ellipses l1 , l2 , l3 , l4 are in double contact with the circle c and pass through three points A, B, C. These ellipses are obtained from the eight ellipses on the left by an orthogonal projection.

seemingly wrong solution is in double contact with c at a pair of complex conjugate points. ● Exercise 4.4.5 What to do if c is not a circle? The technique of spatial interpretation is not restricted to the case of a circle. If c happens to be an ellipse, the auxiliary surface may no longer be a surface of revolution. However, in such cases it is useful to interpret c as the contour of an ellipsoid (which may still be one of revolution) or as the contour of a one-sheeted hyperboloid (which cannot be of revolution in this case). An ellipsoid has to be chosen if the given points A, B, and C are interior points of c. In case of exterior points, we have to use the one-sheeted hyperboloid. ●

Exercise 4.4.6 What to do if c is a parabola? If there is a parabola c to be touched twice by conics on three points A, B, and C, there are again two cases to be distinguished: If the given points A, B, and C are interior points, we interpret c as the contour of an elliptic paraboloid (which could be, for the sake of simplicity, a paraboloid of revolution). In the case of three exterior points, we view c as the contour of a hyperbolic paraboloid (which could be an orthogonal one). The last case is illustrated in Figure 4.41. Figure 4.42 shows the four conics l1 , l2 , l3 , and l4 on A, B, C that touch the parabola c. None of the four conics can be an ellipse. Each solution is the orthogonal projection of a planar intersetion of a hyperbolic paraboloid which carries only parabolas and hyperbolas.

Note that all examples given in this section have one property in common: There are no real solutions if the given conic c separates one of the given

181

4.4 Spatial interpretation of conic constructions

C

c

B

A

FIGURE 4.40. Left: Eight conics (ellipses) on a one-sheeted hyperboloid that map to four ellipses touching the contour circle c twice (cf. Exercise 4.4.5). Right: The four ellipses on A, B, C in double contact with c are obtained from the eight ellipses on the left by an orthogonal projection to the carrier plane of c. Note that the nearly circular solution seems to be wrong, for it doesn’t touch c. However, from the view point of algebraic geometry, it is a solution.

points from the others. A mixture of interior and exterior points of c cannot be interpolated by a conic that touches a given conic twice. In the case of lines and line elements, the given points have to lie in the same wedge enclosed by two given tangents in order to allow for real solutions.

Hyperosculating conics on two points touching a given conic How to find conics which hyperosculate a given conic c and interpolate two given points A and B. Hyperosculation can be viewed as a limiting case of double contact: Let the two contact points converge towards each other, and the double contact becomes one contact but with contact order three, i.e., algebraically speaking, the two curves intersect at one point with multiplicity four. Another important fact follows from Theorem 3.3.1: Assume (P, t) is a line element on a sphere Σ. As already shown in Figure 3.49, the planes through t intersect Σ along circles which share the line element (P, t). The orthogonal projection onto the plane of the

182

Chapter 4: Euclidean 3-space

FIGURE 4.41. The spatial interpretation (cf. Exercise 4.4.6) shows eight conics (parabolas and hyperbolas) on a hyperbolic paraboloid meeting the contour (cyan) transversely. Any pair of conics with the same color map onto one conic under the orthogonal projection onto the contour’s plane.

great circle u through (P, t) yields ellipses with the common hyperosculating circle u. In Definition 6.4.1 (see Section 6.3), we shall give a different, but nevertheless equivalent definition of hyperosculation. Figure 4.43 shows the two conics that hyperosculate a circle c and pass through two given points A and B. As in the aforementioned cases, we cannot expect real solutions if one of the given points is an interior point and the other one is an outer point of c. Again the (affine) type of the auxiliary surface depends on the given conic c and on whether the given points are interior or exterior points. For a circle c the simplest auxiliary surface is a surface of revolution: a sphere (or an ellipsoid) if A and B are interior points of c, a one-sheeted hyperboloid otherwise. If c is a hyperbola, we use a hyperboloid, and if c is a parabola, we choose a paraboloid as auxiliary surface. Figure 4.44 shows the situation in three-space, i.e., the result of the spatial interpretation.

183

4.4 Spatial interpretation of conic constructions l2 l1

l4

l3 l4 c

C

l2

A B

l1

l3

FIGURE 4.42. Conics in double contact with a parabola (cf. Exercise 4.4.6). The orange solution l4 is in double contact with c at a pair of complex conjugate points.

Here, the simple case of conics that osculate a circle c is treated. The auxiliary surface is a sphere. The key idea is to interpret the two given points A and B as points on the auxiliary quadric Φ which again leaves two possibilities for each point (Figure 4.44). The line s = [P, Q] connecting the preimages of A and B meets the carrier (principal) plane ε of c in a point T . The two tangents from T to c touch c at the contact points H1 and H2 of the conics l1 and l2 we are looking for. Note that the position of T relative to c is responsible for the number (or existence) of real solutions. The choice of s = [P, Q], i.e., one point on either side of ε, results in a point S = s ∩ ε in the interior of c. Consequently, there are no real tangents from S to c. Therefore, the contact points as well as the conics interpolating A and B and hyperosculating c are not real. ●

Exercise 4.4.7 Parabolas or hyperbolas on two points which hyperosculate a given conic. Replace the circle from above with an ellipse, a parabola, or a hyperbola and look for all conics on two points that hyperosculate the given conic c. What could possible auxiliary surfaces look like depending on the affine type of the conic c?

184

Chapter 4: Euclidean 3-space

H2

T c

B

l1

l2 A

H1 FIGURE 4.43. Spatial interpretation also helps to find hyperosculating conics interpolating two additional points. Note that the line [A, B] is concurrent with the two tangents at the hyperosculation points H1 , H2 .

Conics on three points with a given focus Conics on three points A, B, C and a given focus F can also be found via spatial interpretation. We consider the three given points as the top views of three points on a cone Γ of revolution with vertical axis. The common focal point F is the top view of Γ’s apex (cf. Theorem 4.1.3). Figure 4.45 shows the construction of the conic on three points A′ , B ′ , C ′ , and prescribed focus F ′ . (The initial points got primes, because they are interpreted as the top view of three points in E3 .) The front view (points labeled with a double prime) shows the straight contour of the cone and the three points A′′ , B ′′ , C ′′ . The right-side view (triple primes) attached to the top view is chosen such that the plane ε = [A, B, C] is in edge-view and can be seen as a straight line ε′′′ . The right-side view l′′′ = ε′′′ of the particular solution l coincides with the image of l. Vertices and the center (if at all present) can be found in the side view. Figure 4.45 shows the constructive approach to one particular solution l. Once a choice is made whether the corresponding points A, B, C in three-space are on the “upper” or “lower” half of the cone Γ, the plane ε is determined. The intersection l = ε∩Γ maps to a conic that passes through A′ , B ′ , C ′ and has F ′ for a focal point. Figure 4.46 shows all solutions to the initial problem. As in some examples before, there are eight planes determined by these two times three points in Euclidean three-space.

185

4.4 Spatial interpretation of conic constructions Σ P k2 ε

k1 l2 A

Q H2

l1 B

H1 k1

c

s

Q P

k2 T

FIGURE 4.44. Spatial interpretation also helps to find hyperosculating conics interpolating two additional points: The dotted lines are the solutions to the initial problem. The dashed lines are the symmetric copies of the two circles on the upper hemisphere.

These planes can be arranged in four pairs of planes (ε, ε) such that ε and ε are symmetric with respect to the horizontal plane through the apex of Γ. Therefore, the curves ε ∩ Γ and ε ∩ Γ have coinciding top views and we find four different solutions. Example 7.2.2 in Section 7.2 shows a different way to find conics on three points with a common focal point by using the polarity with regard to a circle centered at F . Once the preimages of A, B, C on Γ are specified on different half-cones, the corresponding solutions are hyperbolas with A, B, C on different branches. Hence, there is only one solution out of the four that gives a continuous transition between the given points along an ellipse, or a parabola, or one branch of a hyperbola. This is the solution which was mentioned in the preface on page vi concerning the dwarf planet Ceres.

186

Chapter 4: Euclidean 3-space Γ ′′′

′′′

S

′′′

1

l ′′′ = ε ′′′

A ′′′ C

′′′ =

3

Γ

′′′

b

S ′′ ′′′ =

A′′

′′′

M

H ′′

4

C ′′

l′′

′′′

B

B ′′ A′

′′′

4

2

B

1′

H′

′

′

M

′

F ′ =S ′

C′

b′

l′ 3′ 2′ FIGURE 4.45. Any conic on A′ , B ′ , C ′ with focus F ′ is a top view of a planar section of a cone Γ of revolution. The focus F ′ is the top view of Γ’s apex S. The constructive solution for one of the in total four solutions is shown. A

B

F C

FIGURE 4.46. The top view of a point on Γ may correspond to two different points in Euclidean three-space. Thus, there are eight different planar intersections which lead to four different conics with the focus F and passing through A, B, C.

5 Projective Geometry

Projective Geometry is the proper frame work for understanding the geometry of conics. It differs from Euclidean Geometry and allows us to treat points and lines in a unifying way: There is no difference between points at infinity and proper points. The lines in a projective plane are closed like any conic, and the line at infinity is a line like any other. The above surface was discovered in 1901 by Werner Boy and is an immersion of the real projective plane into threespace. Thus, Boy answered a question raised by David Hilbert whether the projective plane can be embedded into a three-dimensional space or not.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2024 G. Glaeser et al., The Universe of Conics, https://doi.org/10.1007/978-3-662-70306-9_5

187

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Chapter 5: Projective Geometry

5.1 Projective planes Models of projective planes In this section, we shall briefly describe projective planes. Therefore, we give a short overview of the axiomatic approach to projective planes. We also describe the vector space model of a projective plane. 10

11

20 21 19 6

2 3

18 3

4

5

16 2 7

6

15

9 5

4

17

8 1

12

1

7

14 13

13

8 12 11 10 9

FIGURE 5.1. Models of finite projective planes. Left: The plane of order three has thirteen points {1, 2, . . . , 13} and as many lines (drawn in different colors). Right: The plane of order four has twenty-one points and lines.

From the incidence geometric point of view, a projective plane consists of a set P of points and a set L of lines. Lines are always considered as sets of points. Sometimes we emphasize this fact by calling them range of points. Closely related to ranges of points are pencils of lines which are defined as the set of all lines through a certain point, say V . The point V is frequently called the vertex or carrier of the pencil. Briefly, we say pencil V . Figure 5.1 shows two examples of finite projectve planes. Each range of points has a certain color in order to emphasize that certain points belong to this set. If for some point P ∈ P and some line l ∈ L the relation P ∈ l holds, we say that the point P is contained in the line l, or P is incident with l, or lies on l, or l goes through P , or l passes through P . Any set of points on

5.1 Projective planes

189

the same line is called collinear. Any set of lines through the same point is called concurrent.

What makes the pair (P, L) a projective plane P2 ? It is a list of axioms which is as follows:

(A1) Any two different points P ≠ Q can be joined by a unique line l = [P, Q], the line connecting P and Q. (A2) Any two different lines l ≠ m intersect in one point S = l ∩ m, the point of intersection of l and m. (A3) There exists a quadrangle, i.e., four points no three of which are collinear. ′ Two projective planes P2 = (P, L) and P2 = (P ′ , L′ ) are called isomorphic if there exists a bijective mapping κ ∶ P → P ′ and L → L′ such that P ∈ l holds if, and only if, κ(P ) ∈ κ(l) for all P ∈ P and all l ∈ L. Sometimes, we say that (P, L) and (P ′ , L′ ) are models of the same abstract projective plane. From the axioms (A1) and (A2), we can deduce: Any two different lines meet in precisely one point, otherwise, if they met in two different points, these lines would be equal by (A1). Further, we can infer that any line carries at least three points, or through any point, there are at least three lines. A range of points carries as many points as there are lines in a pencil. Moreover, the sets P and L are equipotent, i.e., there exists a bijective mapping between P and L. A set of four lines with the property that no three of them are concurrent, i.e., no three have a point in common, is also called a quadrangle. ◾

Example 5.1.1 Some projective planes.

Projective minimal plane. The fact that there exists a quadrangle together with the unique point of intersection of any two different lines allows us to infer that a projective plane has at least seven points: the points A, B, C, D of the quadrangle together with the three diagonal points [A, B] ∩ [C, D], [A, C] ∩ [B, D], [A, D] ∩ [B, C]. Actually, there exists a projective plane with exactly seven points. It is called the Fano plane, named after the Italian mathematician Gino Fano (1871–1952). An example of the “minimal projective plane” can be seen in Figure 5.6. For more details, see Example 5.1.2. The projectively closed real plane. All the drawings we produce take place in the real plane E2 . There, we observe that some lines are not intersecting, see Figure 5.2. Usually, we call such lines parallel. Parallelity of lines in a plane is an equivalence relation.1 Therefore, we can 1

A relation on a set S is a subset R of the Cartesian product S × S, and therefore, R is the union of ordered pairs of elements of S. With x ∼ y we indicate that (x, y) ∈ R or x is related to y. An equivalence relation R satisfies the following conditions for all a, b, c ∈ S: (1) a ∼ a

190

Chapter 5: Projective Geometry real projective plane M

L

M n

Q

m

L P l k

FIGURE 5.2. A scene from the real projective plane: The unique line l connecting P and Q intersects the parallel line k in the ideal point L. The parallels m and n share the ideal point M . (In drawings, we indicate ideal points by arrows. Note that arrows in opposite directions define the same ideal point.) speak of classes of parallel lines. In order to perform the projective closure of the real plane, we add a unique ideal point or point at infinity to each class of parallel lines. The set of all ideal points is called ideal line or line at infinity. The thus extended plane is called the real projective plane and we denote it by P2 (R).

The real projective plane is, indeed, a projective plane: The axiom (A1) is fulfilled since any point P in the real projective plane can be joined with any other point. In practice: To join P with an ideal point means to draw a parallel to some line through P (cf. figure 5.2). Further, the axiom (A2) is fulfilled since any two lines are either intersecting or parallel. If two lines are parallel, then they intersect in the common ideal point. Finally, there is no doubt about the existence of a quadrangle.

The bundle model of a projective plane. From the real plane P2 (R), we can easily create a further model of a projective plane. Let O be a point which is not contained in P2 (R). Then, we project the points and lines in the projective plane from O. This means that we connect the points and lines of P2 (R) with the point O. Of course, points at infinity yield lines parallel to P2 (R). Any point P ∈ P2 (R) is now “blown up” to a line [O, P ] through O; and any line l ∈ P2 (R) becomes a plane [O, l] through O. The points and lines of the real projective plane are in a one-to-one correspondence with the lines and planes through O. We call this the bundle model of the real projective plane.2 The point O is sometimes called the carrier of the bundle. Figure 5.3 shows points, lines, and a quadrangle in the bundle model of a projective plane. (reflexive), (2) a ∼ b implies b ∼ a (symmetric), and (3) a ∼ b and b ∼ c together imply a ∼ c (transitive). These three conditions are completely independent. 2 The name bundle is a translation of the German word Bündel. The word Bündel is used for linear two- or more-parameter families of geometric objects. The manifold of lines through a common point in a three-dimensional space is called a star. Hence, one could also speak of the star model of a projective plane.

191

5.1 Projective planes

O

O

po int

line

R

P Q pro jec tive pl ane

pro jec tive pl ane

FIGURE 5.3. The bundle in E3 is a projective plane isomorphic to the real projective plane. Left: The “lines” and “points” are represented by planes and lines through the vertex O of the bundle. Right: A “quadrangle” of the projective plane is a four-sided pyramid with apex O.

It is not hard to verify that the bundle also serves as a model of a projective plane: For any two points p and q in the bundle (lines through O), there is a unique line in the bundle (plane through O) joining these two, and thus, (A1) is fulfilled. Any two lines in the bundle (planes through O) intersect in a point (line through O) which fits to axiom (A2). Four planes through O constitute a quadrangle in the bundle model provided that no three of these planes are coaxial (have a common line). So, the existence of quadrangles (as requested in (A3)) is guaranteed, too.

Σ

Σ

P D

m S

A l

Q

C B

FIGURE 5.4. Left: Two lines l and m in the spherical model intersect at a point S. Right: The lines of a quadrangle in the spherical model are great circles of the sphere Σ.

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Chapter 5: Projective Geometry

The sphere model of a projective plane. From the bundle model, we can create a further model of a projective plane. We take a sphere Σ centered at the carrier O of the bundle. It means no restriction to assume that the radius of Σ equals one. Any line p in the bundle intersects Σ in a pair of opposite or antipodal points. Thus, the points in the sphere model are the pairs of opposite or antipodal points on Σ. The lines in the bundle are planes through the carrier O. Since O is the center of Σ, any plane of the bundle intersects Σ along a great circle which represents a line in this model. The quadrangles in this model appear as the intersection of a four-sided pyramid with apex O with the sphere Σ, see Figure 5.4. The vector space model of a projective plane. Let F be an arbitrary field3 and F3 a threedimensional vector space over F. It can be shown that the set P of one-dimensional subspaces together with the set L of two-dimensional subspaces defines a projective plane. We call it the vector space model of a projective plane and denote it by P2 (F). Points P are defined as non-zero vectors p ∈ F3 . Instead of P = {λp∣λ ∈ F} we write P = pF or briefly even P = p.

Later, we shall use a vector space in order to find an algebraic model of a projective plane. The bundle model of a projective plane is the geometric realization of a vector space model, cf. page 205.

A very important concept of Projective Geometry is the Principle of Duality. Any valid theorem that is formulated in terms of points, lines, connection, intersection, and incidence remains true if we interchange the words points and lines, intersection and connection, but keep incidence between points and lines. This “new” theorem is called the dual theorem. The fact that dualizing valid theorems produces new valid theorems in Projective Geometry was possibly first observed by the French mathematician Joseph Diaz Gergonne (1771–1859). In order to make the geometry of a projective plane more meaty, one has to add further axioms: The axiom of Fano, as given below, is added to the list of axioms if the projective plane should display a property of quadrangles that is known from P2 (R):

3

A (non-empty) set F with an operation + is called a (commutative) group if for all x, y, z ∈ F: (1) x + y ∈ F and x + y = y + x for commutativity, (2) there is a unique element 0 (zero element), such that x + 0 = 0 + x = x, (3) there is a unique element −x (additive inverse) such that x + (−x) = 0, (4) and x + (y + z) = (x + y) + z (associativity), and additionally, for commutativity x + y = y + x. A (non-empty) set F is called a (commutative) field if there are two different operations, say + and ⋅, acting on F such that F together with + is a commutative group and F {0} together with ⋅ fulfills for all x, y, z ∈ F: (1) x ⋅ y ∈ F. (2) There is a unique element 1 ≠ 0 (neutral element) such that x ⋅ 1 = 1 ⋅ x = x, (3) there is a unique element x−1 (left multiplicative inverse element) such that x−1 ⋅ x = 1, (4) x ⋅ (y + z) = x ⋅ y + x ⋅ z and (x+y)⋅z = x⋅z +y ⋅z (distributive laws), and additionally, for commutative fields x⋅y = y ⋅x ∈ F.

193

5.1 Projective planes Q D R

Q

Q P

D

C

C

A

B

Q

C P

B

R

A

B

A

P

Q C

R

P

C

B

P

R C P

D

R A

P

Q

Q

D

B

D D

C C

Q

R

P A

R D

A

C

D

Q

P B

D P

A

R A

B P

P

C

C B

Q

FIGURE 5.5. Quadrangles in the real projective plane: The diagonal points P , Q, and R, are never collinear.

(FA) The three diagonal points of a quadrangle, i.e., the points of intersection of opposite sides of the quadrangle are never collinear. In Figure 5.5, we can see different appearances of quadrangles in the real projective plane. It does not matter if vertices or diagonal points are ideal points (lie at infinity) or not. The three diagonal points are never collinear. So, in the real projective plane the axiom (FA) is fulfilled. The dual of a quadrangle is called a quadrilateral. It is a set of four lines such that no three of them are concurrent. A quadrilateral defines six vertices and three diagonals. The axiom (FA) implies that the diagonals of a quadrilateral are never concurrent. ◾

Example 5.1.2 The Fano plane. We shall continue the first item of Example 5.1.1.

Synthetic approach. In this example, we use the notation given in Figure 5.6. According to the axiom (A3), there exists a quadrangle P1 P2 P3 P7 in the Fano plane together with the diagonal points P4 = [P2 , P3 ]∩[P1 , P7 ], P5 = [P3 , P1 ]∩[P2 , P7 ], and P6 = [P1 , P2 ]∩[P3 , P7 ]. The axiom (A1) says that there is a unique line l7 joining P4 and P5 . Further, l7 has to have a unique point P6 of intersection with [P1 , P2 ].

Analytic approach. We identify the points of the Fano plane with the following vectors, cf. Figure 5.6 (right): P1 = (1, 0, 0), P2 = (0, 1, 0), P3 = (0, 0, 1), P4 = (0, 1, 1), P5 = (1, 0, 1),

P6 = (1, 1, 0), P7 = (1, 1, 1).

We use 0 and 1 as the elements of the field Z2 , i.e., they satisfy the following rules: 0 + 0 = 1 + 1 = 0, 0 + 1 = 1 + 0 = 1, 1 ⋅ 1 = 1, 0 ⋅ 0 = 0 ⋅ 1 = 1 ⋅ 0 = 0. Now, we apply the usual addition of vectors. By the latter rules, we find that three points are collinear if, and only if, any two of the corresponding vectors sum up to the third. This allows us to prove all collinearities shown in Figure 5.6. The fact that the diagonal points of a quadrangle are collinear is equivalent to

194

Chapter 5: Projective Geometry (0, 0, 1)

P3

l5

l2

l4

(1, 1, 1)

)

(1, 0

P4 P7

1, 1

l6

(0 ,

P5

, 1)

l3

l7 P1

l1

P6

P2

(1, 0, 0)

(1, 1, 0)

(0, 1, 0)

FIGURE 5.6. Left: The Fano plane with its seven points P1 , . . . , P7 and its seven lines l1 , . . . , l7 . In the Fano plane, the axiom of Fano is not valid. Right: The analytic model of the Fano plane. the fact that the characteristic4 of the field, upon which the algebraic model is built, equals two. The coordinatization of a projective plane will be described on page 205.

The axiom (FA) is not valid. The two equivalent models of the Fano plane given above show the characteristic property of this projective plane: The three diagonal points P4 , P5 , and P6 of the quadrangle P1 P2 P3 P7 are collinear, and thus, the Fano axiom (FA) is not fulfilled in the minimal projective plane. It can be shown that P2 (F) is a Fano plane if, and only if, the characteristic of the field F equals 2.

In this book, we mainly concentrate on Fano planes without mention. Only in exceptional cases, we will assume that (FA) does not hold. For the axiomatic approach to Projective Geometry, the axiom of Desargues (named after the French architect and mathematician Gérard Desargues (1591–1661)) and the axiom of Pappus are very important. Both axioms are usually stated as theorems which are, indeed, theorems in the real projective plane. A proof of the latter will be given on page 202. Desargues’s axiom reads: (DE) If two triangles are perspective to a point, then their sides are perspective to a line (note Figure 5.7). 4

The characteristic of a field F is denoted by charF. The symbol charF = n means that the n-fold sum of the unit element 1 equals 0, i.e., 1 + 1 + . . . + 1 = 0 and n ∈ N {0} is the smallest ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ number with this property.

n times

195

5.1 Projective planes p A1 A2 C1 C2 P B2

B1

FIGURE 5.7. A Desargues configuration: The triangles A1 B1 C1 and A2 B2 C2 are perspective. The point P is the perspector. The sides are perspective with the perspectrix p.

The axiom (DE) together with the axioms (A1) – (A3) implies its dual which is exactly its converse. The configuration described in the axiom/theorem of Desargues is frequently called a Desargues configuration or Desargues figure. The point to which the triangles are perspective is called the perspector. The line where assigned sides of the triangles intersect is called the perspectrix. Figure 5.7 shows such a configuration including the perspector and the perspectrix. It is self-dual, since each line (in the configuration) carries three points of the configuration, and through each point (of the configuration) there pass exactly three lines. We call a projective plane a Desarguesian plane if the axiom of Desargues holds. In fact, (DE) holds in any projective plane P2 (F).

The Desargues configuration is a (103 , 103 ) configuration. This means that there are 10 points and each one is incident with three lines. The second pair tells us that there are 10 line each carrying three points. Since the Desargues figure is self-dual, this configuration is usually simply denoted by (103 ). There are ten further (103 ) configurations which are not isomorphic to the Desargues configuration (cf. [61]). Figure 5.8 shows three examples of (103 ) configurations which are not isomorphic to Desargues’s configuration.

The axiom of Desargues guarantees the existence of perspective collineations (cf. page 246). Furthermore, it can be proved that in any Desarguesian plane coordinates from a field F can be introduced such that

196

Chapter 5: Projective Geometry

FIGURE 5.8. Three more (103 ) configurations which are not isomorphic to the Desargues configuration.

there is an isomorphism to a projective plane P2 (F) over a field F which needs not be commutative (see [23]). C2 P

p

C2

B2

C1

C1

P C1 B 1

B2 B1 A1

A2 P

C2 B1

P P B2

A2 P

A1

P A1

A2

FIGURE 5.9. Some versions of Desargues configurations in the (projectively extended) Euclidean plane and their relations to elementary transformations: translation (left), central similarity (middle), shearing (right).

Figure 5.9 shows special forms of Desargues figures in the real projective plane and their relations to elementary transformations. A translation is defined by a Desargues figure with an ideal point P as the perspector and the ideal line as perspectrix. The two triangles in the middle of Figure 5.9 are similar, i.e., corresponding sides are parallel and the lengths of corresponding sides have a constant ratio. Since this is a Desargues figure, corresponding points are collinear with the perspector P , and thus, P is the center of a central similarity. The perspectrix p coincides with the ideal line. On the right-hand side of Figure 5.9, we see a shearing or shear transformation relating the two triangles of the Desargues figure. The perspectrix

197

5.1 Projective planes

p is a proper line and the perspector P is the ideal point of the perspectrix. Finally, we formulate the axiom of Pappus. Pappus of Alexandria (290–350) was a Greek mathematician who left us an eight-volumes book on geometry and mathematics. The following axiom was originally formulated as a theorem. It is closely related to projectivities and to conics. It states: (PP) Assume that 1, 3, 5 are three points on a line l, and let further 2, 4, 6 be three points on another line m. Then, the points A ∶= [1, 2] ∩ [4, 5],

are collinear.

B ∶= [2, 3] ∩ [5, 6],

C ∶= [3, 4] ∩ [6, 1]

The points and lines described in (PP) form a Pappus figure of Pappus configuration. It is a self dual configuration of type (93 ). Later, we shall see that (PP) also holds for six points on a conic. So, the union of the two lines mentioned in (PP) can be seen as a conic. 6

m

M 6

2

2

4 4

p

C

A

B 1 1

5

P c

b 3

a 5

l

3

L

FIGURE 5.10. Pappus’s theorem (left) and its dual form (right).

Figure 5.10 illustrates the axiom (PP). It further displays the dual version of Pappus’s theorem which is usually ascribed to Charles Julien Brianchon (1783–1864) and is related to Pappus’s theorem in a natural way. We call a projective plane a Pappian plane if the axiom of Pappus is valid. The axiom (PP) is stronger than (DE). A classical result which is due to Gerhard Hessenberg (1874–1925) states that (A1) – (A3) and (PP)

198

Chapter 5: Projective Geometry

imply (DE). Hence, each Pappian projective plane is also isomorphic to a projective plane P2 (F). Furthermore, it can be proved that the projective plane P2 (F) is Pappian if, and only if, F is commutative. This means that the axiomatic approach to Pappian projective planes is equivalent to the analytic approach via coordinates which come from a commutative field. In the sequel, we will use both, because each of them, the analytic approach as well as the so-called synthetic approach, has its advantages. If not stated otherwise, the projective planes we are dealing with are always Pappian.

199

5.2 Perspectivities, projectivities

5.2 Perspectivities, projectivities C

C1

P

C2 l

m

M x

m L

l

l

X

a

FIGURE 5.11. The basic mappings in Projective Geometry (as described in C = m with perspector C. Middle: the Definition 5.2.1): Left: the perspectivity l ∧ = perspectivity l ∧ P between a range of points and a pencil of lines. Right: the a = C2 between two pencils of lines with perspectrix a. perspectivity C1 ∧

The basic mappings in Projective Geometry are perspectivities. These mappings are one-to-one and onto, and they relate ranges of points with pencils of lines, pencils with pencils, and ranges with ranges. The simplest among these mappings (as shown in Figure 5.11, in the middle) relates a line l and a pencil of lines with carrier P ∉ l such that corresponding elements X ∈ l and x ∋ P are incident. Here are the fundamental notions:

Definition 5.2.1 A perspectivity α ∶ l → m with the center C ∉ l, m sends L ∈ l to M ∈ m whenever L, M , and C are collinear. We use the symbol C = m. l∧ The dual version is also called a perspectivity and the symbol =T S∧ a

means that corresponding elements are lines s ∋ S and t ∋ T sharing a point on an axis a. We also call the pencil of lines through P perspective to the range l (P ∉ l) of points if for each line x through P the corresponding point X ∈ l is incident with x. In this case, we write = P l∧

or

= l. P∧

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Chapter 5: Projective Geometry

Figure 5.11 illustrates the contents of Definition 5.2.1. In the middle, we = P . The inverse mapping is symbolized by P ∧ = l. see the basic mapping l ∧ The left-hand side shows the action of the perspectivity α ∶ l → m which can be viewed as the composition of two perspectivities of the basic type. The right-hand side shows a perspectivity between two pencils of lines. The vertices of the pencils are the points C1 and C2 , while the axis of the perspectivity is the line a. C2

M N

c2

b′ n

A =A′

B′

C′

a=a

m

c′ b c1

B A

′

C

B C

c a

C1

l

b

c

L

FIGURE 5.12. The reduction of projectivities. Left: A projectivity between two lines l and m given by A ↦ A′ , B ↦ B ′ , C ↦ C ′ can be realized as the comC1 C2 = n ∧ = m. Right: The dual version, the position of only two perspectivities: l ∧ projectivity between two pencils of lines with vertices L and M given by a ↦ a′ , c1 c2 =N ∧ = M. b ↦ b′ , c ↦ c′ , can be realized as a composition of two perspectivities L ∧

Perspectivities between ranges/pencils are uniquely defined by prescribing two pairs of corresponding elements. In general, the composition of two perspectivities is no longer a perspectivity: Definition 5.2.2 The composition of finitely many perspectivities between ranges of points or pencils of lines is called a projectivity. In order to indicate that two ranges of points l and m are joined by the α projectivity α (different from a perspectivity) we write l − ∧ m. The symbol (A, B, C) − ∧(A′ , B ′ , C ′ ) means that A, B, and C are mapped to A′ , B ′ , and C ′ , respectively, via a projectivity.

201

5.2 Perspectivities, projectivities

The projectivities act transitively on ordered triplets of collinear points (or concurrent lines), i.e., there is a projectivity for prescribed three mutually distinct collinear points (or concurrent lines) and their images. In Pappian planes, i.e., in projective planes where Pappus’s theorem (cf. page 202) is valid, this projectivity is even uniquely determined, i.e., projectivities act sharply transitive on ordered triples of collinear points and pencils of concurrent lines. Hence, the following “fundamental theorem” ((FT) in brief) is valid: Theorem 5.2.1 (FT) A projectivity α ∶ l → m in a Pappian projective plane is uniquely defined by prescribing three pairs (Ai , A′i ) of points with α(Ai ) = A′i for all i ∈ {1, 2, 3, }.

In any projective plane, (PP) is equivalent to (FT), for details, see [23, 69].

The chains of subsequent perspectivities can be arbitrarily long, but only finitely many perspectivities are allowed. However, in any Pappian plane, there is a possibility to shorten chains of perspectivities with the help of the following reduction theorem: Theorem 5.2.2 In projective planes with (FT), any projectivity between two different ranges l and m of points can be realized as the product of at most two perspectivities between ranges. A projectivity α ∶ l → m whith l ≠ m is a perspectivity if, and only if, the point l ∩ m is mapped onto itself. Proof: Assume A, B, C are three different points on a line l, and A′ , B ′ , C ′ are three different points on a line m. Without loss of generality, we can suppose A ∉ m and A′ ∉ l. By (FT), there exists exactly one projectivity α ∶ l → m with α(A) = A′ , α(B) = B ′ , and α(C) = C ′ . In order to show that α can be decomposed into two perspectivities between ranges of points, we choose a line n ≠ m, n ≠ [A, A′ ] through A′ and further a point C1 on the line [A, A′ ] with

=1 n C1 ≠ A, A′ as can be seen in Figure 5.12 (left). Now, we have a first perspectivity β1 ∶= l ∧ ′ that maps A, B, C to A = A , B ∶= [C1 , B] ∩ n, C ∶= [C1 , C] ∩ n. Because of the choice of C1 and n, the points B, B, B ′ , C ′ form a quadrangle, and the point C2 ∶= [B, B ′ ] ∩ [C, C ′ ] is different from all other points and can, therefore, serve as the center of a second perspectivity C

=2 m that maps A = A′ , B, C to the points A′ , B ′ , C ′ , respectively. Thus, α = β2 ○ β1 . β2 ∶ n ∧ (FT) guarantees that for any suitable choice of n and C1 the product of the two corresponding perspectivities remains the same. C

The case of a projective mapping between two pencils of lines can be treated in the same way by simply dualizing the construction as illustrated in Figure 5.12 (right).

If α(l ∩ m) = l ∩ m, then we may set C = C ′ = l ∩ m. Now, A, A′ , B, B ′ is a quadrangle and for =1 m. C1 = [A, A′ ] ∩ [B, B ′ ] we get α as the perspectivity l ∧ C

◾

202

Chapter 5: Projective Geometry

The statements of Theorem 5.2.2 are also valid in their dual formulation (illustrated in Figure 5.12). In the case l = m the minimal number of perspectivities is two only if there is a point which is mapped onto itself. Otherwise, we need at least three perspectivities. Later, we shall use the following result: Theorem 5.2.3 For any four collinear points A, B, C, D there exists a projectivity with the following property A ↦ B, B ↦ A, C ↦ D, D ↦ C. Proof: Let m ≠ l be a line through D and Z ∉ {l ∪ m}. Now, we define the points A = [Z, A]∩ m, B = [Z, B] ∩ m, C = [C, Z] ∩ m, and B ′ = [A, B] ∩ [C, Z], as displayed in Figure 5.13. The existence of m and Z and all further points is guaranteed by the axiom (A3).

Z m

A B′

B

C

A

l

B

C

D

FIGURE 5.13. The proof of Theorem 5.2.3. Then, we follow the chain of perspectivities =m∧ = [Z, C] ∧ =l l∧ Z

A

B

with A ↦ A ↦ Z ↦ B, B ↦ B ↦ B ′ ↦ A, C ↦ C ↦ C ↦ D, and D ↦ D ↦ C ↦ C and the proof is finished.

◾

Pappus’s theorem and the axis of a projectivity A useful result on projectivities between lines depends on whether the theorem of Pappus is valid in the underlying projective plane or not. In the real and complex projective plane, Pappus’s theorem holds because the underlying fields R and C are commutative. However, there are some projective planes which are not Pappian, e.g., the projective plane P2 (H)

203

5.2 Perspectivities, projectivities ′

C

′

A

=4

′

B

=2

m

=6

′

′

Y

m

′

′

C

X

B

′

A

pα

A

C pα

B

A=1

B =3

A l

C =5

B

X

Y

l

C

FIGURE 5.14. The proof of Theorem 5.2.4: the construction of the axis of a projectivity (left), the important property of the axis (right).

where H denotes the Hamiltonian quaternions. When dealing with conics, we always assume that the axioms of Pappus and FANO are valid. The original version of the axiom (PP) (cf. page 197) is in close relation to projectivities between two different ranges of points. We show Theorem 5.2.4 Let α ∶ l → m with l ≠ m be a projectivity between two ranges of points in a Pappian projective plane. For any two pairs (X, X ′ ) and (Y, Y ′ ) of corresponding points with X, Y ∈ l, X ≠ Y , and X ′ = α(X), Y ′ = α(Y ), the lines [X, Y ′ ] and [X ′ , Y ] meet on a fixed line pα called the axis of the projectivity.

In the case of a projectivity α between two different pencils of lines, we can find a center Pα of the projectivity. Proof: Let α ∶ l → m be given by A ↦ A′ , B ↦ B ′ , C ↦ C ′ with A, B, C ∈ l and A′ , B ′ , C ′ ∈ m where C ∉ m and C ′ ∉ l, cf. Figure 5.14. According to Definition 5.2.2, the projectivity α should decompose into a finite sequence of perspectivities. = pα ∧ =m In fact, when we set A = [1, 2]∩[4, 5], B = [2, 3]∩[5, 6], and pα = [A, B] we have α ∶ l ∧ because A ↦ A ↦ A′ , B ↦ B ↦ B ′ , and C ↦ C = [C, C ′ ] ∩ pα ↦ C ′ . C′

C

For any two points X, Y ∈ l {C} we find X ′ = α(X) by running through the chain of perspectivities (fig 5.14, right). Thereby, we find that [X, C ′ ] ∩ [X ′ , C] and [Y, C ′ ] ∩ [Y ′ , C] are points of pα . Now, we label the points in the way Y ′ = 1, C = 2, X ′ = 3, Y = 4, C ′ = 5, X = 6 and conclude by (PP) also [X, Y ′ ] ∩ [X ′ , Y ] ∈ pα .

◾

204

Chapter 5: Projective Geometry

A projectivity may not always map points from one line to another line. When we deal with conics, we are sometimes concerned with the case of projectivities on a line or in a pencil. This will be of importance in Section 7.3, especially in Theorem 7.4.1. For more clarity, we shall analyze some examples on page 221. As a consequence of the Fundamental Theorem 5.2.1, we can say: If a projectivity α ∶ l → l has three mutually distinct fixed points, i.e., points F with the property α(F ) = F , then α is the identity mapping idl on l. The mapping idl has only fixed points.

205

5.3 Coordinatization

5.3 Coordinatization In Projective Geometry, there is a need for a suitable coordinatization that allows us to describe the objects and transformations in a simple way. Homogeneous coordinates turned out to be the right choice. This kind of coordinates goes back to August Ferdinand Möbius (1790– 1868), a German mathematician and astronomer. In his famous book Der barycentrische Calcul (cf. [98]), he introduced barycentric coordinates as a method to assign coordinates to points in a plane with respect to a base of three points. In the following, we show how to assign homogeneous coordinates to points which are given in Cartesian coordinates. Thus, we perform the projective extension of the Euclidean plane E2 analytically and reveal in a constructive way its isomorphy to P2 (R). Before that, we give a very short summary on affine and Cartesian coordinates.

Inhomogeneous and homogeneous coordinates on a line

Let us first explain the term affine coordinates of a point X on a line l. We fix two different points O and E on l, as can be seen in Figure 5.15. The point O is called the origin and E is called the unit point. (It means no restriction to assume that O is on the left-hand side of E as shown in Figure 5.15.) We assign the coordinates 0 and 1 to the points O and E. If X is a point on the line l = [O, E], then its inhomogeneous coordinate x is defined as the affine ratio x ∶= OX ∶ OE =∶ ar(O, E, X)

(5.1)

provided the distances are signed: The coordinate x is negative if we find X left from O; and it is positive, if it is right from O. If the point X lies between O and E, then 0 < x < 1.

Now, we turn to the case where X is a point in a plane and use two coordinate axes, i.e., a pair of non-parallel lines. One of them shall be the x-axis, the other one shall be the y-axis. Though it does not really matter, we may assume that the x-axis is “horizontal” and the y-axis is “vertical” as illustrated in Figure 5.15. Their point of intersection O is called origin of the coordinate system. On the x-axis, we pick a point Ex ≠ O; similarly, we choose Ey ≠ O on the y-axis. These two points are usually referred to as the unit points on the

206

Chapter 5: Projective Geometry y

P

Py Ey 0 O

1 E

x X

x O

Ex

Px

FIGURE 5.15. Affine coordinates on a line and in the plane.

coordinate axes and the lengths OEx and OEy are called the unit lengths (on the axes and on all lines parallel to the axes).

The coordinates (xP , yP ) of a point P in the plane are now found in the following way: Draw two lines through P , one parallel to each axis. The respective intersections with the coordinate axes shall be labeled with Px and Py . Then, the coordinates xP and yP are the coordinates of Px and Py on the lines [O, Ex ] and [O, Ey ], i.e., xP = ar(O, Ex , Px ) and yP = ar(O, Ey , Py ).

(5.2)

The generalization to affine coordinates in n-dimensional (n > 3) space is straightforward. Note that the x-axis and the y-axis are not necessarily orthogonal. The coordinates xP and yP are defined as ratios in (5.2). The pair (xP , yP ) is called affine coordinates of the point P , in general. If the two axes enclose a right angle and the two unit lengths are equal, then we speak of Cartesian coordinates named after the French mathematician, philosopher, and writer René Descartes (1596–1650).

Homogeneous coordinates

Affine coordinates are no longer useful in the projective extension P2 (R) of E2 since points at infinity would get coordinates (∞, ∞) and could not be distinguished. Therefore, we introduce homogeneous coordinates. The previously mentioned coordinates are called inhomogeneous just in order to emphasize the difference. We start with homogeneous coordinates of points on a line l. We assume that a projective line, i.e., a line in a projective plane is embedded as the line with the equation x0 = 1 in the x1 x0 -plane, as can be seen in

207

5.3 Coordinatization

P =(p0 ∶p1 )

(0, 0)

(1 ∶0)

x

b1 =(1, 0)

1∶ (0 ∶

0)

v2 1,

,v v0

=( v

,p

(p 0

(1 ∶0 ∶0)

)

b0 =(1, 0, 0)

1)

b0 =(0, 1)

(0 ∶1)

V =(v0 ∶v1 ∶v2 )

(0, 0, 0) 0)

(0 ∶ 0 ∶

y 1)

b2 = ( 0 , 0 ,1

) = b1 FIGURE 5.16. Homogeneous coordinates on a line and in a plane.

x0 =1

1, (0 ,

Figure 5.16. Any vector (p0 , p1 ) ∈ R2 {(0, 0)} starting from (0, 0) aims at a point on the line l. For p0 ≠ 0 this point on l is finite and has the coordinates (1, p1 p−1 0 ) considered as a point in the x1 x0 -plane. However, the vector p = (p0 , p1 ) points in the same direction as any of its scalar (non-zero) multiple does. We say that p represents the point P on the line, but also any scalar multiple of p can serve as a representative of P . So, only the ratio p0 ∶ p1 matters and we say that (p0 ∶ p1 ) are the homogeneous coordinates of P on l and write briefly P = (p0 ∶ p1 ). We call p a coordinate vector of P and use the notation P = pR in order to express that we have chosen some representative of P . More precisely, pR = {p ⋅ λ ∣ λ ∈ R}

is a one-dimensional subspace of R2 . Note that (λp0 ∶ λp1 ) are the homogeneous coordinates of the same point for any λ ∈ R {0}. On the other hand, any vector (p0 , p1 ) ≠ (0, 0) defines a unique point on l. The vector (0, 0) must be excluded since it does not define a line through (0, 0) and is not directed to a point on l.

Because of the special choice of the coordinate system in the x1 x0 -plane, the line l is parallel to the x1 -axis. Thus, the vector b1 = (0, 1) points in the direction of l. Therefore, b1 = (0, 1) is a representative of the ideal point or point at infinity of l. The homogeneous coordinates of l’s point at infinity are thus (0 ∶ 1).

The key idea of the coordinatization with homogeneous coordinates of points on a line can now easily be carried over to a homogeneous coordinatization of the points in the real projective plane. All the vectors v in

208

Chapter 5: Projective Geometry

the three-dimensional vector space R3 are triplets (v0 , v1 , v2 ) of numbers taken from R. For practical reasons, we assume that the projective plane is embedded as the plane x0 = 1 into the space E3 . In order to support our imagination, the x0 -axis shall point upwards as shown in Figure 5.16, on the right-hand side. (0 ∶0 ∶1)

(1 ∶1 ∶1) (1 ∶0 ∶0)

(0 ∶1 ∶1)

(1 ∶ 1 ∶ 0) (1 ∶ 2 ∶ 0) (1 ∶ 3 ∶ 0)

(0 ∶1 ∶0)

FIGURE 5.17. The fundamental triangle and a projective scale on one side.

Any point P in the plane x0 = 1 has coordinates (1, px , py ). However, it is not only the vector v = (1, px , py ) emanating from (0, 0, 0) that points to P . Any scalar multiple (λ, λpx , λpy ) points in the same direction and the line spanned by this vector intersects the plane x0 = 1 in the same point P . As in the previous case, we shorten notations by writing P = pR as abbreviation for {p ⋅ λ ∣ λ ∈ R}. Once again, only the ratio (1 ∶ px ∶ py ) matters, and thus, we define (1 ∶ px ∶ py ) as the homogeneous coordinates of P . As indicated in Figure 5.16, there is an affine coordinate system in the plane x0 = 1. The axes are labeled with x and y. Thus, (px , py ) are the inhomogeneous coordinates of P . On the other hand, we can extract the homogeneous coordinates of a point from the vector (v0 , v1 , v2 ): We simply write (v0 ∶ v1 ∶ v2 ) in order to emphasize that these are homogeneous coordinates and only the ratio of these three values matters. If v0 ≠ 0, we can easily switch to a homogeneous coordinate representation of the point V = vR determined by v with (v0 ∶ v1 ∶ v2 ) = (1 ∶ v1 v0−1 ∶ v2 v0−1 ). There is one major advantage of homogeneous coordinates: Ideal points (points at infinity) can be described by homogeneous coordinates. The

209

5.3 Coordinatization

vectors (0, f1 , f2 ) are parallel to the plane x0 = 1. To any of these vectors, we find a one-parameter family of parallels in the plane x0 = 1. Thus, (0 ∶ f1 ∶ f2 ) are the homogeneous coordinates of an ideal point in the plane x0 = 1. The vector f = (0, f1 , f2 ) is a representative of the common ideal point of all lines parallel to the vector (f1 , f2 ) ∈ R2 , considered as a vector in the plane x0 = 1. We call the triangle with vertices (1 ∶ 0 ∶ 0), (0 ∶ 1 ∶ 0), (0 ∶ 0 ∶ 1) the fundamental triangle of one particular coordinatization, (cf. Figure 5.17). The point (1 ∶ 1 ∶ 1) is called unit point. We will see that any four points that form a quadrangle can be chosen as the vertices of a fundamental triangle plus the unit point. We know that instead of the standard basis {b1 , b2 , b3 } of R3 any three linearly independent vectors a, b, c can be used as a basis of a coordinatization. This leads to another system of homogeneous coordinates in P2 (R) with the fundamental points A = aR, B = bR, C = cR, and the unit point E = (a + b + c)R which is never collinear with any two of the points A, B, C. When in particular a, b, c are orthonormal in E3 , we call the coordinates homogeneous Cartesian coordinates. If we replace the fundamental points A, B, C by any other three noncollinear points A′ , B ′ , C ′ , then we replace the basis {a, b, c} by another basis, say {a′ , b′ , c′ }. From Linear Algebra, we know that the coordinates (x0 , x1 , x2 ) are then replaced by the coordinates (x′0 , x′1 , x′2 ) which are given in matrix form by ′ ⎛ x0 ⎞ ⎛ x0 ⎞ ⎜ x′1 ⎟ = T ⎜ x1 ⎟ . ⎝ x′2 ⎠ ⎝ x2 ⎠

Therein, T is a regular 3 × 3-matrix with entries from R. The columns of T are the ‘new’ coordinates of the previous basis vectors a, b, c. Because of the homogeneity of the coordinates, the matrix T is unique only up to a non-zero real factor. When using the matrix notation, it makes sense to write the homogeneous coordinates as columns. Therefore, from now on we insert the symbol T for ‘transposed’ when we write the coordinates of a column vectors in a line. Because of (DE) it is easy to see that all this works in the same way when R is replaced by any commutative field F, e.g., by C.

210

Chapter 5: Projective Geometry

Computing with homogeneous coordinates How to make calculations with homogeneous coordinates? Two points P and Q are represented by their homogeneous coordinates (p0 ∶ p1 ∶ p2 ) and (q0 ∶ q1 ∶ q2 ) or, equivalently, P = pF and Q = qF. In the following, coordinates and scalars are taken from any commutative field F. The points X on the line l = [P, Q] spanned by P and Q can be parametrized with the help of homogeneous parameters (λ ∶ µ). For the point X = xF, we have x(λ, µ) = {p ⋅ λ + q ⋅ µ ∣ (λ, µ)T ∈ F2 {(0, 0)T }}. Proportional pairs (λ ∶ µ) represent the same point on l. Sometimes, we shall write pF + qF instead of {p ⋅ λ + q ⋅ µ ∣ (λ, µ)T ∈ F2 {(0, 0)T }}. Note that pF + qF ≠ (p + q)F.

For variable (λ, µ) ∈ F2 the linear combination x(λ, µ) is a twodimensional subspace of the vector space F3 . Thus, it can also be described by a linear equation of the form ⟨l, x⟩ ∶= l0 x0 + l1 x1 + l2 x2 = 0,

(l0 , l1 , l2 ) ≠ (0, 0, 0).

(5.3)

We call (l0 ∶ l1 ∶ l2 ) the homogeneous coordinates of the line l since scaling the equation of l in (5.3) does not change its solutions. As we have done with points, we shall also write l = lF and call l a representative of l.5 Note that the incidence between the point xF and the line lF is expressed by ⟨l, x⟩ = 0, which in E3 would mean the orthogonality l–x between vectors l and x. Let X = xF with x = (x0 , x1 , x2 )T be a point on l = [P, Q], where P = pF and Q = qF (Figure 5.18). Then, the vectors p, q, and x are linearly dependent. Consequently, we have det(p, q, x) = 0 which is again an equation of l. The latter determinant can be written in a different form as det(p, q, x) = ⟨p × q, x⟩,

5

In the case of a non-commutative field F, there is a difference between left and right multiplication. Therefore, we have to distinguish between a left and a right vector space on F. Consequently, lines should then be written as Fl if points are written as pF.

211

5.3 Coordinatization l =p×q

x0

q

y Q l

P

p j. pro

ne pla

x

x1 FIGURE 5.18. The line l = [P, Q] through P = pF and Q = qF has the representative l = p × q which satisfies ⟨l, p⟩ = ⟨l, q⟩ = 0.

where ⟨a, b⟩ = a1 b1 +a2 b2 +a3 b3 is the canonical scalar product of vectors in F3 and a × b = (a2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − a2 b1 ) is the induced exterior product of vectors in F3 . Thus, a coordinate vector l of the line l = [P, Q] can be found via l = p × q. (5.4)

Replacing p and q by non-zero multiples means changing the representatives of P and Q. The cross product of p and q then changes to a multiple of l and yields another representative of the same line. Linearly dependent vectors p and q describe the same point and p× q = 0 expresses the fact that there is no unique line joining a point with itself. Obviously, in homogeneous Cartesian coordinates, the ideal line or line at infinity spanned by two (different) points (0 ∶ f1 ∶ f2 ) and (0 ∶ g1 ∶ g2 ) with f1 g2 − f2 g1 ≠ 0 has the homogeneous coordinates (1 ∶ 0 ∶ 0) and has, thus, an equation of the form x0 = 0. In the dual version, assume that two lines l and m are given by the respective homogeneous coordinates l and m (Figure 5.19). A coordinate vector s of the point S of intersection is given by s = l × m.

(5.5)

This is clear since S is incident with both lines, and thus, we have ⟨l, s⟩ = 0 and ⟨m, s⟩ = 0,

and consequently, s is a non-trivial scalar multiple of l × m provided that l and m are linearly independent. Linearly dependent vectors l and m

212

Chapter 5: Projective Geometry

l

y

x0 m

s=

S

j. pro

l m

l×

m

ne pla

x x1

FIGURE 5.19. The point S = sF of intersection of two lines l = lF and m = mF can be found as s = l × m.

are coordinates of the same line, and consequently, there is more than a single common point. This statement is expressed by l × m = 0. Lemma 5.3.1 For any two different points P = pF and Q = qF in P2 (F), the connecting line is [P, Q] = (p × q)F. For any two different lines l = lF and m = mF, the point of intersection is (l × m)F.

Any line l in a projective plane is spanned by two points P and Q. Any point X = xF on l can also be described by the two homogeneous coordinates (x0 , x1 )T ∈ F2 {(0, 0)T } on the line l, if there are representatives p and q such that p ⋅ x0 + q ⋅ x1 = x.

This also holds for lines in a pencil: Any point S = sF is the carrier of a pencil of lines spanned by two different lines l = lF and m = mF through S. If u = uF is any line through S, then there are representatives l and m such that l ⋅ u0 + m ⋅ u1 = u. Then, (u0 ∶ u1 ) are the homogeneous coordinates of the line u = uF in the pencil about S.

Affine ratios, cross ratios, and projective frames In E2 , the inhomogeneous coordinate x of a point X on a line l = [0, E] with respect to the coordinate system with origin O and unit point E has been defined as the affine ratio x = ar(O, E, X) as given in (5.1). The name affine ratio indicates that ar(O, E, X) is invariant under affine transformations, i.e., transformations that map any point X with inhomogeneous coordinate vector x to x′ = Ax + a. Here, A is a regular 2 × 2

213

5.3 Coordinatization

matrix and a ∈ R2 is a vector.6 The invariance of ar(O, E, X) under affine transformations is easily checked: If ar(O, E, X) = λ = OX/OE, then OE ⋅ λ = OX, and equivalently, (e − o) ⋅ λ = x − o with o, e, and x being the inhomogeneous coordinate vectors of O, E, and X. Thus, x can be expressed in terms of o and e by x = o ⋅ (1 − λ) + e ⋅ λ

which is a so-called affine combination. Now, we apply the affine transformation to O, E, X which gives o′ = a, e′ = Ae + a, and x′ = Ae ⋅ λ + a. Then, we compute o′ ⋅ (1 − λ) + e′ ⋅ λ = a ⋅ (1 − λ) + (Ae + a) ⋅ λ = Ae ⋅ λ + a = x′

which shows that the affine images O ′ , E ′ , and X ′ define the same affine ratio as O, E, and X, i.e., ar(O, E, X) = ar(O ′ , E ′ , X ′ ). Now, we extend E2 to the real projective plane P2 (R). Let O, U , and E be three points on a line l. We assign homogeneous coordinates to these points on the line l by setting O = (1 ∶ 0) and U = (0 ∶ 1). Without loss of generality, we can assume that E = (1 ∶ 1). This is possible since the homogeneous coordinate vectors of three collinear points are always linearly dependent. However, sometimes the homogeneous representatives o, u of O, U have to be rescaled in order to fulfill o + u = e. If X with representative x is a further point that is collinear with O, U , and E, then it has homogeneous coordinates (x0 ∶ x1 ) if x = ox0 + ux1 . All this works in P2 (F) with an arbitrary (preferably commutative) field F in the same way.

We define the cross ratio cr(A, B, C, D) ∈ F ∪ {∞} of four collinear points (on a line l) A = aF, B = bF, C = cF, and D = dF with A ≠ D and B ≠ C in P2 (F) by det(a, c) ⋅ det(b, d) cr(A, B, C, D) ∶= . (5.6) det(a, d) ⋅ det(b, c)

If A= D or B = C and no other two points coincide, we set cr(A, B, C, D)= ∞. Rescaling of vectors, i.e., changing the representatives of points, does not alter the cross ratio. When we deal with projectivities (cf. page 221) 6

The definition of an affine transformation in the n-dimensional space Fn over any field F is analogous: Take some regular n × n matrix A and some vector a ∈ Fn .

214

Chapter 5: Projective Geometry

and projective collineations (cf. page 246), we shall see that the cross ratio is invariant under projective transformations. Thus, it is defined in a useful and proper way. We observe that special values of cr(A, B, C, D) appear if D attains special positions with respect to the points A, B, C, e.g., cr(A, B, C, A) = ∞, cr(A, B, C, B) = 0, and cr(A, B, C, C) = 1. Note that 0 and 1 always exist independently of the underlying field F.

The base points of our coordinatization on l, O = (1 ∶ 0) and U = (0 ∶ 1) together with E = (1 ∶ 1) and X = (x0 ∶ x1 ) have the cross ratio cr(O, U, E, X) = xx01 . We call F ∶= (O, U ; E) a projective frame or projective coordinate system on a line. The cross ratio x0 x−1 1 formed by O, U , E, and X can be interpreted as a coordinate of X in the given projective frame. In this sense, cr(A, B, C, D), as defined in (5.6) can be viewed as projective coordinate of D with respect to the frame (A, B; C).

When O, U , E are replaced with three other points, then the basis {o, u} of the two-dimensional subspace is replaced by another basis {o′ , u′ }. We know from Linear Algebra that the coordinates (x0 ∶ x1 ) for points X on the line l are replaced with (x′0 ∶ x′1 ) satisfying (

x′0 x0 ) ′ ) = T( x1 x1

with a regular 2× 2-matrix T ∈ F2×2 which is unique only up to a non-zero scalar from F. Now, it is easy to prove that for the new coordinates a′ , b′ , c′ , d′ of the four points A, B, C, D the formula (5.6) delivers the same result since det(a′ , c′ ) = det T ⋅ det(a, c) and det T ≠ 0.

Any four ordered collinear points define a unique cross ratio. On the other hand, the cross ratio does not uniquely define a quadruple of collinear points. However, it is surprising that the 24 permutations of four collinear points yield only six different cross ratios: If δ = cr(A, B, C, D), then the five other values of the cross ratio are 1 − δ, δ−1 , (1 − δ)−1 , δ(δ − 1)−1 , and (δ − 1)δ−1 . For example, cr(B, A, C, D) = cr(A, B, C, D)−1 , due to (5.6). Another consequence of (5.6) is the product rule for cross ratios concerning five collinear points A, B, C, D, E: cr(A, B, C, D) = cr(A, E, C, D) ⋅ cr(E, B, C, D).

(5.7)

215

5.3 Coordinatization

Suppose that in the real projective plane P2 (R) the point U is the ideal point of the line [O, E]. Then, for any finite point X = xR, we have −1 x = ox0 + ux1 = x−1 0 (o + x0 x1 u) with u = o − e. Therefore, we have −1 x−1 = cr(U, O, E, X). 0 x1 = ar(O, E; X) = cr(O, U, E, X)

(5.8)

Let A, B, C, D be four finite points on the line [O, E]. Then, by virtue of (5.8) and (5.7), we find cr(A, B, C, D) = cr(A, U, C, D) ⋅ cr(U, B, C, D) =

ar(B, C, D) . ar(A, C, D)

(5.9)

Finally, it should be said that cross ratios can also be defined for quadruples of concurrent lines. It is easy to prove that the cross ratio of four concurrent lines a, b, c, d equals the cross ratio of the four points l ∩ a, l ∩ b, l ∩ c, l ∩ d for any line l which is not incident with the common point of a, b, c, and d. With the help of the cross ratio formula (5.6), one can show that the cross ratio of four concurrent lines a, b, c, and d equals cr(a, b, c, d) =

sin( m2 + n2 , i.e., there are no real tangents from P ’s center to c. The conic d is a hyperbola if R2 < m2 + n2 , or equivalently, there are two real tangents from P ’s center to c.

If R2 = m2 +n2 , then c contains the center of p and d is a parabola. The angle ϕ enclosed by the axis of the parabola and the x-axis of the coordinate system is still given by tan 2ϕ = √ 2mn . 2 2 m +n

Note that the center of p is a focal point of the polar transform (cf. Def. 7.1.5).

This classification of the polar image of a conic remains valid if the hyperbolic polarity πP from Example 7.2.1 is replaced by an elliptic polarity since an elliptic polarity also has a center. The center is just the pole of the line at infinity. The results of this section can also can be used in order to solve some conic problems by means of a graphical construction. ◾

Example 7.2.2 Find conics on points or lines with a given focus.

The generic case. Assume that we are given three non-collinear points P1 , P2 , P3 and a further point F . We are looking for all conics passing through P1 , P2 , P3 and having F for a focus. Applying a polarity to a conic interchanges points and lines. A focus F , for example, is the meet of two isotropic tangents. When polarized w.r.t. the circle f with the center F , they map to the points of contact, i.e., the absolute points of Euclidean geometry. The given points map to three lines: π(Pi ) = pi , i ∈ {1, 2, 3}. By assumption, the given points form a triangle, and therefore, their polars pi are the side lines of a triangle (see the light grey triangle in Figure 7.18). The conics we are looking for, are now transformed to conics ei (with i ∈ {1, 2, 3, 4}) that touch the side lines pi of the triangle and pass through the absolute points of Euclidean geometry. Thus, the conics ei are the incircle together with the excircles of the triangle. Applying the polarity once again yields the four solutions li . In general, there are four conics on three points and a given focus. These four conics are either four hyperbolas or three hyperbolas plus an ellipse or parabola depending on whether the

304

Chapter 7: Polarities and pencils

e3

l3

l2

e2 p1

p3

l3

l1

e1

P1

l1

P2

p2 l4 P3

l2

F

e4

f

FIGURE 7.18. The construction of conics on three points P1 , P2 , P3 with a given focus F is equivalent to the construction of all circles tangent to three lines. The four solutions l1 , . . . , l4 are the polar images of the four tri-tangent circles e1 , . . . , e4 of the triangle all of whose sides lines p1 , p2 , p3 are the polars of P1 , P2 , P3 w.r.t. any circle f centered at F .

given focus F is an interior point of one of the tritangent circles of the lines pi or lies on one of them. F cannot be the interior point of more than one circle ei because the tritangent circles ei of the lines pi are the incircle together with the excircles of the triangle built by the lines p1 , p2 , p3 . These circles are known to be disjoint. Homofocal parabolas. We try a variation of the previous example. We are looking for all parabolas on two points P1 , P2 and a given focus F (Figure 7.19). Again, we apply the polarity w.r.t. to a circle f centered at F : The given points P1 , P2 are mapped to lines p1 , p2 and possible solutions (which should pass through P1 and P2 ) are, thus, transformed into conics tangent to p1 and p2 . The focus considered as the intersection of a pair of isotropic tangents to the solutions maps to the absolute points of Euclidean geometry, and therefore, the polar transforms of the desired conics are circles tangent to p1 , p2 . Further, the ideal line is mapped to F , because the center of f and the ideal line correspond to each other in the polarity.

305

7.2 Definition of a conic according to von Staudt

p1

P1 l2 p2

P2

l1

M2

F

e1

f

e2

M1

FIGURE 7.19. The two parabolas l1 and l2 on two points with a common focus are found in the following way: Find the circles e1 and e2 which are tangent to the polar lines p1 , p2 of P1 , P2 w.r.t. f . Then, apply the polarity w.r.t. f to e1 , e2 . Hence, we are looking for the circles tangent to p1 , p2 passing through F . In the generic case, there are two solutions e1 and e2 , as shown in Figure 7.19. Their polar transforms are the parabolas l1 and l2 .

306

Chapter 7: Polarities and pencils

7.3 Pencils of conics Pencils of conics are special one-parameter families of conics. The projective classification of the pencils of conics leads to five different types. All these types can be classified by means of the set of singular conics included in the pencil. The computation as well as the construction of common points of two conics out of this pencil can be reduced to the case where one of these curves of degree two is reducible. This is interesting at least from the theoretical point of view, since in general it is not possible to find the common points of two conics by ruler and compass. In this section, the term ‘conic’ means a curve of degree two, no matter if it is irreducible (regular) or reducible (singular).

An introductory example y

√ x0= 3 2

a

y0 =

2 ay x =

2a

√ 3 22 2a

a y2 =2

x

ax

FIGURE 7.20. The Delian cube duplication problem: The volume of the two cubes on the left have ratio 2 ∶ 1. How can one construct the scaling factor of the edge length?

Let us look at the following classical problem. A cube of side length a = 1 is to be duplicated, i.e., the volume shall be doubled. The question is: What is the side length of the new cube with twice the volume? This problem is usually called the cube duplication problem. The name Delian problem was also given to this question: It is reported that the citizens of the Greek island Delos once asked the oracle how to defeat

307

7.3 Pencils of conics

the plague. The oracle wanted the Delians to build a new altar of twice the volume of the old altar. Unfortunately, the altar was a regular cube, and so, the Delians had to duplicate a cube. The ancient Egyptians and Indians also knew this type of mathematical problem. √ In principle, one has to find or construct 3 2 in order to solve the Delian problem. Sad to say, but ruler and compass are not sufficient to construct the cube root of two, as shown in 1837 by the French mathematician Pierre Laurent Wantzel (1814–1848). √ However, the Greek geometers discovered a way to construct 3 2 using two parabolas, as shown in Figure 7.20. Assume that p1 ∶ y 2 = 2ax

and p2 ∶ ay = x2

(7.11)

are the two parabolas. Obviously, (7.11) is a system of two quadratic equations in the unknowns x and y. The solutions of (7.11) are √ √ 3 3 (x, y) = (0, 0), (x, y) = ( 2a, 4a). (7.12)

The first solution is of no√importance for the Delian problem. The second one with a = 1 gives x = 3 2, precisely that what the Delians were looking for. If we allow a tool that draws parabolas, then the problem is solvable in a constructive way. We have solved a system of two quadratic equations in two variables. An equivalent problem is to search for all points common to two curves of degree two (cf. Section 6.3). In this particular case, we could extract the solutions by mere elementary operations. However, this will not be sufficient when it comes to the more general case with two quadratic equations. On the other hand, how do the configurations of two conics look like? It can easily be imagined that there may occur more than two common points, see Figure 7.21.

FIGURE 7.21. Some configurations of pairs of conics with different numbers of common points.

308

Chapter 7: Polarities and pencils

Pencils of conics in the projective plane In the following we assume that the field F is algebraically closed, i.e., any non-constant polynomial with coefficients in F has at least one root in F. This allows us to generalize and extend the case of a pair of circles which is somehow special: Any two circles share the circle points / absolute points (but only after the projective closure and complex extension of the Euclidean plane). Their radical axis l joins the two common proper points S1 and S2 whether they are real or not. I

I

ω d

c

l S1

S2

FIGURE 7.22. A complex and projective view on two circles: There are four common points spanning three pairs of lines.

The four common points of the two “circles” c and d, as illustrated in Figure 7.22, can be thread up by three pairs of lines, one real pair (magenta), two of them consisting of a pair of complex lines (dashed in cyan and violet): The real pair is (l m) with l = [I, I] and m = [S1 , S2 ]). The two complex pairs are ([I, S1 ], [I , S2 ]) and ([I, S2 ], [I, S1 ]). These pairs of lines can be viewed as singular conics each sharing the same four points with c and d. Consequently, it is natural to treat pairs of conics in the projective plane. Whenever we deal with equations and coordinates, we prefer the homogeneous representation and extend the underlying field F properly. As we shall see, a purely synthetic approach to pencils of conics is also possible.

Definition of a pencil of conics

From Section 5.1, we already know that in P2 (F) a conic is uniquely defined by prescribing five points such that no three of them are collinear.

309

7.3 Pencils of conics

On the other hand, five lines considered as the tangents of a conic also define a unique conic if no three of these lines are concurrent. What happens if we remove one point or line? In the following, we use Definition 7.3.1 The family of all regular conics through the vertices of a quadrangle B1 B2 B3 B4 is called a pencil (of conics) of the first kind. The points Bi with i ∈ {1, 2, 3, 4} are called base points of the pencil. Some conics in a pencil of the first kind are displayed in Figure 7.23. How many conics are in this pencil? If we choose an arbitrary point P which is not collinear with any two of the given base points, then we know that there is a unique conic through B1 , . . . , B4 , and P . So, it seems that there are as many conics in the pencil as there are points in the plane (besides the one which are collinear with the base points). However, there is only one degree of freedom in a pencil of conics, and the pencil is a fibration of the projective plane. This can easily be seen with (6.13) by writing down a conic’s equation in terms of homogeneous coordinates a00 x20 + 2a01 x0 x1 + . . . + a22 x22 = 0. Inserting the homogeneous coordinates of the four given base points, we obtain a system of four linear equations in the six unknown coefficients aij with i, j ∈ {0, 1, 2}. The solution of this system is a two-dimensional subspace of F6 provided that the rank of the coefficient matrix is 4. Hence, there are as many conics in a pencil as there are points on a projective line. t1

t2

B4 B3 B1

B1 B2

B2

B2

B3 B1

t1

FIGURE 7.23. Definition of pencils of conics: from left to right; the first three kinds.

310

Chapter 7: Polarities and pencils

There are some other types of pencils of conics which differ from the viewpoint of synthetic geometry but not from the algebraic point of view: Definition 7.3.2 Assume that B1 , B2 , and B3 are the vertices of a triangle and t1 is a line through B1 that does neither contain B2 nor B3 . Then, the family of (regular) conics through B1 , B2 , B3 that touch t1 at B1 is called a pencil of the second kind. The pencil of the second kind can be viewed as a limiting form of a pencil of the first kind: One base point of a pencil of the first kind, say B4 , has moved infinitely close to B1 . In the limit, the two points become one point plus the tangent there. An example of a pencil of the second kind can be seen in Figure 7.23 (middle). Naturally, the second type of pencil contains a one-parameter family of conics as is the case for the pencil of the first kind. The transition from the pencil of the first kind to that of the second kind gives a rough idea of how to proceed in order to find the remaining kinds of pencils of conics. Assume that we have two such limiting procedures: Let B4 move towards B1 , and B3 move towards B2 . Then, we obtain: Definition 7.3.3 Assume B1 and B2 are two (different) points, and t1 and t2 are two straight lines such that B1 ∈ t1 , B2 ∈ t2 , and either ti is different from the line [B1 , B2 ]. The family of regular conics through B1 and B2 that touch t1 at B1 and t2 at B2 is called a pencil of the third kind. The pencil of the third kind consists of doubly touching conics and is a one-parameter family of conics. Some of the conics in a pencil of the third kind can be seen in Figure 7.23 (right). There are two further types of pencils. We can modify a pencil of the second kind as described in Definition 7.3.1 and assume that the third base point B3 is moving towards the point B1 . Algebraically speaking, we have three points infinitely close together, and therefore, any two conics in this pencil intersect at B3 with multiplicity three. However, there is a more geometric way to define new pencils of conics. For that purpose, we use Definition 6.4.1 and recall the perspective collineation α between any two conics in the pencil.

311

7.3 Pencils of conics

From the definition of osculation and hyperosculation, it is clear that the family of conics which are osculating a given conic k at B1 and share a further point B2 with k is one-parametric: We can vary the center C of α on t1 . It is also obvious that the family of conics hyperosculating a given conic k at a point B1 is one-parametric: On a line [B1 , K] with K ∈ k through the center B1 of α ∶ k ↦ l, we have one degree of freedom for the choice of the α-image L of K. Note that K is not allowed to lie on t1 (Figure 6.22). So we have the natural

Definition 7.3.4 The family of regular conics osculating a given conic k at a point B1 and passing through a further point B2 ∈ k is called pencil of the fourth kind. The family of conics hyperosculating a given conic k at a point B1 is called a pencil of the fifth kind.

Figure 7.24 shows some conics in a pencil of the fourth and fifth kind. The base points and some special conics in the pencil are highlighted.

t1

t1

B1 B1 B2 c

c

FIGURE 7.24. Pencils of conics in P2 (R): Left: The pencil of the fourth kind with base points B1 , B2 , and the base tangent t1 consists of all conics that osculate a conic c at B1 . Right: The pencil of the fifth kind consists of all conics that hyperosculate c at B1 . In both cases, c is also an element of the pencil.

312

Chapter 7: Polarities and pencils

Analytic representation

The analytic representation of pencils of conics in P2 (F) enables us to extend the previously defined pencils to singular conics. The following result is necessary for the characterization of singular conics by means of the rank of the coefficient matrix. Lemma 7.3.1 In P2 (F) with charF ≠ 2, the curves of degree two with the equation xT Kx = 0 where K ∈ F3×3 with KT = K is 1. a conic or the empty set if, and only if, rkK = 3, 2. a repeated line if, and only if, rkK = 1, 3. a pair of lines or a single point if, and only if, rkK = 2. Proof: We have already learned that a regular symmetric matrix defines a polarity where the set of self-conjugate points is either empty or a conic. In the case of det K = 0 the singular points S = sF are solving the system of homogeneous linear equations Ks = 0. Singular points have the property that with any other point P = pF of the curve, i.e., with pT Kp = 0, all points of the line [S, P ] = (λs + µp)F with (λ, µ) ∈ F2 {(0, 0)}, satisfy the quadratic equation. In the case of rkK = 1 there is a line of singular points. If there exists another point p satisfying the quadratic equation, all points in the plane satisfy this equation and K = 0.

◾

We assume that two conics in P2 (F) have the following equations k ∶ xT Kx = 0 and l ∶ xT Lx = 0

with symmetric matrices K, L ∈ F3×3 . Then, we can state:

Theorem 7.3.1 If the pencil of conics of any kind is spanned by the conics k ∶ xT Kx = 0 and l ∶ xT Lx = 0, then all conics of the pencil are included in the family of curves satisfying any non-trivial linear combination κxT Kx + λxT Kx = xT (κK + λL)x = 0,

(κ, λ) ∈ F2 {(0, 0)}.

(7.13)

Proof: The coordinates of each base point Bi annihilate the equations of k and l, and therefore, also each linear combination. Thus, all conics defined by (7.13) pass through the base points. If any two points P , Q are conjugate w.r.t. k and l, then they are conjugate w.r.t. each curve given in (7.13), since pT Kq = pT Lq = 0 implies pT (κK + λL)q = 0. If, therefore, k and l share the line element (Bi , ti ), then each P ∈ t1 is conjugate to Bi w.r.t. k and l. Hence, all other conics in (7.13) contain this line element.

If k and l are hyperosculating at B1 , then each P ∈ t1 has the same polar line w.r.t. k and l. Consequently, all other conics in (7.13) hyperosculate k and l at B1 .

If finally k and l span a pencil of the fourth kind, then l share (B1 , t1 ) and another point B2 ∉ t1 with k such that no other point or the tangent at B2 is common to k and l. This condition holds for all other conics in the pencil as well as for each conic in (7.13).

313

7.3 Pencils of conics

Hence, for all pencils of any kind, each included conic satisfies also (7.13). Conversely, if any regular conic satisfying (7.13) is given, choose any point Q different from the base points. The pencil as well as the linear family (7.13) send just one conic through this point Q, and the two conics must coincide. Thus, the set of regular conics in (7.13) is a pencil.

◾

Note that either conic in the pencil can be seen as a point in a fivedimensional space. Then, a pencil of conics is a line in this space. The pencils of conics defined and described by (7.13) are linear pencils. In [63], the exponential pencil spanned by the above mentioned conics k ∶ xT Kx = 0 and l ∶ xT Lx = 0 is defined as ̃ ⋅ (K ̃ −1 ⋅ L) ̃ t−1 x = 0, c(t) ∶ xT L

t∈R

(7.14)

̃ = exp(K) and L ̃ = exp(L) are the matrix exponentials of K and where K L. It is a matter of elementary computations to show that the coefficient matrix in (7.14) is the exponential of the coefficient matrix of the linear pencil. The thus defined pencils allow for a different classification of the linear pencils. Moreover, these families of conics are closed under conjugation and dualization. The latter is definitely not true for the linear pencils. The result of Theorem 7.3.1 is a good reason to extend the Definitions 7.3.1, 7.3.2, 7.3.3, and 7.3.4 as given below. Definition 7.3.5 The set of regular or singular conics which satisfy any linear combination of the equations of two different conics k and l, is called a pencil of conics. In this sense, we extend all kinds of pencils of conics by the respective singular conics. ◾

Example 7.3.1 Conics of a pencil through certain points.

In P2 (R), we are looking for the conic p through P = (− 12 , 2) in the pencil of conics spanned by k ∶ 2y 2 − 3x − 5 = 0 and l ∶ 5x2 − 2y 2 + 3x = 0. The equations of all conics in the pencil are given by κ(2y 2 − 3x − 5) + λ(5x2 − 2y 2 + 3x) = 0 with κ ∶ λ ≠ 0 ∶ 0.

If a certain curve from that family runs through P , then the ratio κ ∶ λ ≠ 0 ∶ 0 is to be determined such that the latter equation is fulfilled. Therefore, we substitute the first and second coordinates of P for x and y and find 3κ − 11λ = 0

⇐⇒

κ ∶ λ = 11 ∶ 6.

The conic p through P (as displayed in Figure 7.25) has the equation p ∶ 6x2 + 2y 2 − 3x − 11 = 0.

314

Chapter 7: Polarities and pencils y l

P

k

1

p x

1

FIGURE 7.25. Conics of a pencil and the conic p (blue) through P .

We can look for special types of conics in a given pencil. We show this by means of examples: ◾

Example 7.3.2 Parabolas in a pencil of conics.

l p2 k p1 y

1 1

x

FIGURE 7.26. The parabolas p1 and p2 in the pencil of conics spanned by k and l, cf. Example 7.3.2.

315

7.3 Pencils of conics

In the projectively extended Euclidean plane let two conics k and l be given by their equations k ∶ 9x2 + 9y 2 − 6x − 52y − 40 = 0, l ∶ 83x2 + 333y 2 − 222x − 1674y + 1480 = 0. We homogenize the equations and write the pencil in the form given in (7.13). The intersection with the ideal line u is given by x0 = 0. In the matrix representation of (7.13), we have to remove the first column and the first row, and we arrive at the quadratic form (9κ + 83λ)x2 + (9κ + 333λ)y 2 = 0.

The zeros of this quadratic form correspond to the ideal points of the conics in the pencil. In order to find parabolas, we have to determine κ ∶ λ such that the conics touch the ideal line. In other words: We have to look for double solutions. The existence of a double solution of a quadratic form is equivalent to the vanishing of the determinant of the coefficient matrix. In our example, we find κ ∶ λ = −83 ∶ 9 and κ ∶ λ = −37 ∶ 1.

Now, the equations κ ⋅ k + λ ⋅ l = 0 are the equations of the two parabolas in the pencil and read p1 ∶ 9y 2 − 43y − 6x + 40 = 0

and

p2 ∶ x2 − y = 0.

Figure 7.26 shows four conics in the pencil: k, l and the two parabolas p1 and p2 . From this example, we learn that there are at most two parabolas in a generic pencil of conics in P2 (R). This fits to Theorem 7.4.1, as we shall see later. However, there exist pencils of conics which contain only parabolas.

◾

Example 7.3.3 Circles in a pencil of conics.

We would like to find the circles in a pencil of conics if there are any. The pencil in question shall be spanned by two circles k and l with their Cartesian equations k ∶ 3x2 + yx + 6y 2 − 14y − 12 = 0

and

l ∶ 3x2 + 4yx + 15y 2 − 38y − 12 = 0.

Now, we have to find κ ∶ λ in (7.13) such that the coefficient matrix of the quadratic form in x ∶ y is a scalar multiple of the 2 × 2 unit matrix I2 . Therefore, we extract the matrices of the quadratic forms by removing the first rows and columns of the coefficient matrices of the homogeneous equations and arrive at K=(

6 1

1 ) 12

and

L=(

3 2

2 ) 15

where K is multiplied by 2. From κK + λL = αI2 with α ∈ R we find 6κ + 3λ = 12κ + 15λ, κ + 2λ = 0, κ + 2λ = 0 which is a system of three homogeneous linear equations in two unknowns, namely κ and λ. Normally, we cannot expect a non-trivial solution. Since K and L are symmetric, the third equation equals the second one and can, thus, be canceled. Since the first equation simplifies to κ + 2λ = 0, it is equivalent to the second, and we end with one homogeneous equation with solutions κ ∶ λ = −2 ∶ 1. Consequently, there is one circle c in the pencil. It has the equation c ∶ − 2 ⋅ (6x2 + 2yx + 12y 2 − 28y − 24) + 1 ⋅ (3x2 + 4yx + 15y 2 − 38y − 12) = = −9x2 − 9y 2 + 18y + 36 = −9(x2 + y 2 − 2y − 4) = 0.

Figure 7.27 shows the √ conics k and l as well as the circle c. The circle c is centered at (0, 1), and the radius equals 5.

316

Chapter 7: Polarities and pencils c y

k 1 l

1 x

FIGURE 7.27. There is only one circle in the pencil of conics spanned by k and l (cf. Example 7.3.3).

Later, in Sec. 7.4 (p. 347), we shall see that there exist pencils of conics that contain only circles (including straight lines as degenerate circles). These pencils will be termed pencils of circles.

Singular conics in a pencil In the pencil of conics given in (7.13), we always find singular conics, i.e., degenerate conics. In the sequel we discuss the pencils of all five kinds. Following Lemma 7.3.1, a necessary and sufficient condition for a conic to be singular is the singularity of the coefficient matrix. The singularity of the 3 × 3-matrix is equivalent to det(κK + λL) = 0.

(7.15)

From that we infer that there are at most three singular conics in the pencil since this determinant is a cubic form in κ ∶ λ, i.e., a homogeneous cubic polynomial. We shall emphasize that we are not interested in the trivial solution κ ∶ λ = 0 ∶ 0 for it corresponds to the zero matrix which is not the coefficient matrix of a conic. It turns out that the singular conics can serve as a basis of a pencil in three types. As we shall see, there are two kinds of pencils of conics that cannot be spanned by choosing only singular conics. In the following, we assume that K and L are regular. Later we shall see what happens if we drop this assumption.

317

7.3 Pencils of conics

Remark 7.3.1 For any root κ ∶ λ of (7.13) the quotient κ/λ can be seen as generalized eigenvalue of K w.r.t. L. We will meet these eigenvalues again in (9.25) in Section 9.5. Case 1 - conics through the vertices of a quadrangle

s2 B B B B B B333333333333 B B

B B B B B B4444444444444 B B s3 s1

l

B B B111111111 B B B B 111 B B B B B B2222222222222 B B

k FIGURE 7.28. The first kind of a pencil of conics: Left: some conics, the four real base points B1 , . . . , B4 , and the singular conics s1 , s2 , and s3 . Right: the common polar triangle (blue).

First, we are dealing with the case of three different roots of (7.15). Any of these roots has multiplicity one, and thus, we have three different singular conics s1 , s2 , s3 in the pencil. The singular conics s1 , s2 , and s3 are pairs of lines through four common points B1 , B2 , B3 , and B4 . This is a pencil of conics of the first kind. All conics in the pencil, including the singular ones, share the four base points B1 , . . . , B4 . Figure 7.28 shows some conics from a pencil of the first kind. The three singular conics are pairs of lines displayed in pink. The diagonal triangle of the quadrilateral of base points B1 , . . . , B4 is the common polar triangle of all regular conics in the pencil. Figure 7.28 (right) also shows this polar triangle. More than one pencil of the first kind is involved in the following result, called the three-conics-theorem: Theorem 7.3.2 Assume that three conics c1 , c2 , c3 share two points S1 and S2 . Then, the lines connecting the remaining two intersections of either pair of conics are concurrent.

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Chapter 7: Polarities and pencils

Figure 7.29 shows the three conics mentioned in Theorem 7.3.2 as well as its dual counterpart. l

c3 c1

s2 c3 S1 s1 c2 c2 S2 c1 FIGURE 7.29. Left: Three conics with two common points S1 , S2 and the three concurrent chords. Right: In the dual version, the intersections of the pairs of remaining common tangents are collinear.

●

Exercise 7.3.1 The three-conics-theorem. Give a proof of Theorem 7.3.2. Why don’t we have to give a proof for the dual statement? What happens if S1 and S2 are the absolute points of Euclidean geometry? Show that Theorem 7.3.2 (together with its dual version) is still valid even if some conics degenerate into pairs of lines.

Closely related to Theorem 7.3.2 is the following strange result, the fourconics-theorem which can be found together with Theorem 7.3.3 in [45]: Theorem 7.3.3 If two points of intersection of each pair of three conics c1 , c2 , and c3 lie on a conic c, then the lines joining the remaining two intersections of each pair are concurrent. ●

Exercise 7.3.2 The four-conics-theorem. Give an analytic proof of Theorem 7.3.3 by assuming c1 and c2 are two different conics on (1 ∶ 0 ∶ 0), (0 ∶ 1 ∶ 0), (0 ∶ 0 ∶ 1), (1 ∶ 1; 1).

Figure 7.30 illustrates Theorem 7.3.3 and its dual counterpart. ●

Exercise 7.3.3 Give a precise formulation of the dual version of Theorem 7.3.3. It is illustrated in Figure 7.30 (right).

319

7.3 Pencils of conics c1 c2 c c3

FIGURE 7.30. Left: the four conics theorem in its original version. Right: the dual version of the four conics theorem. Cases 2 and 3 - conics in contact

t1 ⊂s1

l

s2

s2

k

B3 s 1

B2

B1

t1 ⊂s1

B1

t2 ⊂s1

s2

B2

l

k

FIGURE 7.31. Left: pencil of the second kind with base points B1 , B2 , B3 and singular conics s1 and s2 . Right: pencil of the third kind with base points B1 , B2 and singular conics s1 and s2 .

If the cubic form given in (7.15) has two different zeros with respective multiplicities one and two, we find precisely two singular conics s1 and s2 in the pencil. The singular conic s1 that corresponds to the double root may be a pair of lines or a repeated line whereas the singular conic s2 corresponding to the simple root is always a pair of distinct lines. In the first case, we have

320

Chapter 7: Polarities and pencils

a pencil of the second kind. All conics in the pencil pass through three different points one of which has multiplicity two. In other words: Any pair of conics in the pencil of the second kind share two different points B2 , B3 , and a line element (B1 , t1 ). Figure 7.31 shows some of the conics in a pencil of the second kind besides the singular conics. Now, there is no common polar triangle to all the conics in the pencil. Only B1 and the point [B2 , B3 ] ∩ t1 have equal polars w.r.t. all conics in the pencil.

If the singular conic s1 corresponding to the double root of (7.15) turns out to be a repeated line (sometimes called a double line), we have a pencil of the third kind. Now, there are only two different base points, say B1 and B2 , each of multiplicity two. Thus, the conics in the pencil of the third kind touch each other at B1 and B2 . The line s1 = [B1 , B2 ] joins these contact points. The singular conic s2 is the union of the tangents t1 at B1 and t2 at B2 . The right-hand side of Figure 7.31 shows some conics in a pencil of doubly touching conics as well as the respective singular conics. Case 4 and 5 - osculating and hyperosculating conics

l

k

B2 B

s

k

l

B1 s

s

FIGURE 7.32. Pencils of conics: Left: pencil of the fourth kind with base points B1 , B2 and singular conic s. Right: pencil of the fifth kind with base point B and singular conic s.

Finally, the cubic form given in (7.15) may have one root with multiplicity three. In this case, there is only one singular conic s in the pencil.

7.3 Pencils of conics

321

Depending on whether s is a pair of lines (rk(κK + λL) = 2) or a repeated line (rk(κK + λL) = 1), we have a pencil of the fourth or fifth kind.

In the case of a pencil of the fourth kind, all conics of the pencil share two points B1 and B2 . At B1 they have a common tangent t which is one component of the singular conic s. The second component of s is the line joining B1 and B2 . Any pair of conics in the pencil intersects at the point B1 with multiplicity three and at B2 with multiplicity one. Therefore, any two conics of the pencil of the fourth kind osculate each other. Sometimes we say that the pencil of the fourth kind is a pencil of osculating conics. On the left-hand side of Figure 7.32 we can see some conics of a pencil of the fourth kind. The only parabola in the depicted pencil is shown in red, and the common osculating circle is shown in orange. If the one and only singular conic s in the pencil is a repeated line, then there is only one base point B. Any two conics in the pencil of the fifth kind intersect at B with multiplicity four. This is also the case for the singular conics, and therefore, s is the common tangent to all conics in the pencil. Any two conics in the pencil intersect with multiplicity four at B, i.e., they are hyperosculating at B, and thus, the pencil of the fifth kind is also called pencil of hyperosculating conics. In a pencil of the fifth kind, we can find a circle only if B is a vertex of some conic in the pencil. However, there is always a hyperosculating parabola at any point B on any regular conic c. Note that the pencils of the fourth and fifth kind cannot be spanned by singular conics exclusively.

Different appearances of pencils of conics in P2 (R) Depending on whether the base points of a pencil of conics in P2 (R) are real or not, whether they are proper (finite) or not, we see different versions of pencils of conics. We shall describe some of them here in order to make the reader familiar with this fact and in order to enable the reader to recognize different pencils of conics as such at hand of some images. We do not aim at a complete list of different cases. Case 1 - Four common points, the generic case

Consider a pencil of the first kind with two base points, say B3 and B4 , at infinity. From the viewpoint of Euclidean geometry, the conics in this pencil are hyperbolas if B3 and B4 are real points. If B3 and B4 are com-

322

Chapter 7: Polarities and pencils

B B111 B1 B222 B

B2

FIGURE 7.33. Some versions of pencils of the first kind in P2 (R). Left: Two real ideal base points and two real proper base points. Middle: A pair of complex conjugate ideal base points together with a real proper pair. The absolute points of Euclidean geometry are the two ideal base points, and thus, the conics are circles. Right: Two pairs of complex conjugate base points.

plex conjugate, then the conics in the pencil are ellipses. In Figure 7.33 (left), we can see the conics of a pencil of the first kind with a pair of proper real base points and a pair of ideal base points. In this particular example, we have B3 = (0 ∶ 1 ∶ 0) and B4 = (0 ∶ 0 ∶ 1). The remaining base points are B1 = (1 ∶ 1 ∶ 1) and B2 = (1 ∶ −1 ∶ −1). Note that all conics in this pencil are equilateral hyperbolas with parallel asymptotes. This pencil can be spanned by k ∶ xy − 1 = 0 and

l ∶ xy − x + y − 1 = 0.

Note that these curves are frequently considered to be the graphs of rational functions. The middle of Figure 7.33 shows an example of a pencil of the first kind with two real proper base points and a pair of complex conjugate ideal base points. The proper points could be placed at B1 = (1 ∶ 0 ∶ 1)

and B2 = (1 ∶ 0 ∶ −1)

whereas the ideal points could be chosen as absolute points of Euclidean geometry, i.e., B3 = (0 ∶ 1 ∶ i) and B4 = (0 ∶ 1 ∶ −i).

Because of the latter choice, the regular conics in this pencil are circles. Replacing B1 and B2 with a complex conjugate pair of points, we find a pencil of conics which is still of the first kind and contains only circles.

323

7.3 Pencils of conics

The right-hand side of Figure 7.33 shows some of the circles in this pencil. We shall treat pencils of circles in more detail in Section 7.4 (cf. page 347). ◾

Example 7.3.4 A pencil of equilateral hyperbolas. We assume that the four base points of a pencil of the first kind are the vertices A, B, C of a triangle ∆ in the Euclidean plane together with ∆’s orthocenter O as shown in Figure 7.34 (left). As a matter of fact, the three

B′ B C C O

k

k

A

S

C′ O′

h

O A

B

A′

FIGURE 7.34. Left: The pencil of conics with base points A, B, C, and O contains three pairs of lines and only equilateral hyperbolas. Kiepert’s hyperbola k passes through the centroid of ∆. Right: The orthocenter of any triangle whose vertices are chosen on an equilateral hyperbola h is also located on h. singular conics in this pencil are the three pairs of lines ([A, B], [C, O]), ([B, C], [A, O]), and ([C, A], [B, O]), i.e., any side line of ∆ together with the altitude through the opposite vertex. Any of these pairs can be viewed as a limiting case of an equilateral hyperbola with principal axis equal to zero. Any regular conic in this pencil is an equilateral hyperbola. This can be seen as follows: We impose a Cartesian frame on ∆ such that A = (p, 0), B = (q, 0), and C = (0, r) with p ≠ q and p, q, r ≠ 0. Then, O = (0, −pqr −1 ) and the equation of the conics in the pencil are linear combinations of the equations of any two singular conics: xyλ + (qr − rx − qy)(qx − ry − pq)µ = 0

with (λ, µ) ≠ (0, 0). The ideal points of all conics in this pencil are real and belong to orthogonal directions √ v1,2 = (µ(q 2 − r 2 ) − λ ± µ2 (q 2 + r 2 )2 + 2λµ(r 2 − q 2 ) + λ2 , 2qrµ)

since ⟨v1 , v2 ⟩ = 0.

Among the hyperbolas shown in Figure 7.34 (left), we find the Kiepert hyperbola h (named after the German mathematician Friedrich Wilhelm August Ludwig Kiepert (1846– 1934)). The hyperbola h also contains the centroid S of the base triangle ∆.

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Chapter 7: Polarities and pencils

From Example 7.3.4, we can deduce: Theorem 7.3.4 Let A, B, C three non-collinear points chosen on an equilateral hyperbola h. Then, the orthocenter of the triangle ABC is a point on h. The contents of Theorem 7.3.4 are illustrated in Figure 7.34 (right) for two different triangles ABC and A′ B ′ C ′ with vertices on one equilateral hyperbola h. Case 2 - One point of contact

In the case of a pencil of conics sharing one line element and two arbitrary points, the different appearances in P2 (R) can be distinguished by the relative position of the line element and the ideal line and the reality of the two further base points. tttt33333

B1

B2 B B B33333 B

FIGURE 7.35. Some versions of pencils of the first kind in P2 (R). Left: A pencil containing only parabolas which could also be considered the graphs of all quadratic functions y = a(x2 − 1) with zeros −1 and 1; a ∈ R. Right: A pencil of equilateral hyperbolas. These are the graphs of rational functions of the form x with a ∈ R. y = 1−ax

Let us consider the following two examples which are illustrated in Figure 7.35. The pencil spanned by k ∶ x2 − y − 1 = 0 and l ∶ x2 + y − 1 = 0

7.3 Pencils of conics

325

is a pencil of the second kind, consists of parabolas only, and has the three real base points B1 = (1 ∶ −1 ∶ 0), B2 = (1 ∶ 1 ∶ 0), B3 = (0 ∶ 0 ∶ 1).

At the ideal point B3 , all the conics in the pencil share the tangent x0 = 0 which is the ideal line. The singular conics in the pencil are two pairs of lines: a pair of parallels and the union of the ideal line and the line joining B1 and B2 . Another variant of a pencil of the second kind can be seen on the righthand side of Figure 7.35. We have chosen the base points B1 = (0 ∶ 1 ∶ 0), B2 = (0 ∶ 0 ∶ 1), B3 = (1 ∶ 0 ∶ 0),

and the line x − y = 0 for the common tangent of the curves at B3 . Therefore, we see a pencil of equilateral hyperbolas all of whose orthogonal pairs of asymptotes are parallel and the curves in the pencil touch at B3 . In this case, the singular conics are xy = 0 and the union of the ideal line with the line x = y. Note that the left and right image in Figure 7.35 are collinear copies of each other, even from the viewpoint of real geometry. Case 3 - double contact

The graphs of the quadratic functions y = ax2 with variable a ∈ R {0} are parabolas which constitute a pencil of conics of the third kind with a proper base point B1 = (1 ∶ 0 ∶ 0) and an ideal base point B2 = (0 ∶ 0 ∶ 1). The image, as displayed in Figure 7.36 (left), shows some of the curves and the singular conics in the pencil. As the ideal line is a part of one of the singular conics, the singular conic consisting of the common tangents at the base points can only be seen partly. The repeated line, i.e., the connection of B1 and B2 , is drawn as a double line. Next, we consider B1 = (1 ∶ 1 ∶ 0) and B2 = (1 ∶ −1 ∶ 0) to be the base points of a pencil of the third kind. Further, we let (0 ∶ 0 ∶ 1) be the common point of the common tangents of all conics at B1 and B2 . Now, the pencil of the third kind contains ellipses and hyperbolas as well (cf. Figure 7.36). The ellipses fill the strip in between the parallel tangents, whereas the hyperbolas cover the exterior of the strip. In this particular example in P2 (R), we have chosen the repeated line as the perpendicular to the tangents at B1 and B2 .

326

Chapter 7: Polarities and pencils

FIGURE 7.36. Some versions of pencils of the third kind. Left: The graphs of the family of functions y = ax2 touch at B1 = (1 ∶ 0 ∶ 0) and at the ideal point B2 = (0 ∶ 0 ∶ 1). Right: Ellipses (blue) and hyperbolas (violet) in a pencil of the third kind.

FIGURE 7.37. Some particular versions of pencils of the third kind. Left: Concentric circles form a pencil of doubly touching conics. Right: Concentric equilateral hyperbolas are concentric circles in pseudo-Euclidean geometry.

The base points of the pencil of the third kind can be chosen on the ideal line (Figure 7.37). The concentric circles displayed on the left form a pencil of conics of the third kind. The base points form a complex

327

7.3 Pencils of conics

conjugate pair, B1 = (0 ∶ 1 ∶ i) and B2 = (0 ∶ 1 ∶ −i),

and since these points are the absolute points of Euclidean geometry, the regular conics in the pencil are Euclidean circles. Thus, the repeated line in the pencil is the real ideal line which is spanned by B1 and B2 . Note that the concentric circles touch at B1 and B2 along complex tangents meeting at the common center. The pencil shown in Figure 7.37 (right) has the real base points B1 = (0 ∶ 1 ∶ 1) and B2 = (0 ∶ 1 ∶ −1).

It consists, therefore, of equilateral hyperbolas. The singular conics in the pencil are the repeated ideal line and the asymptotes of the hyperbolas. In pseudo-Euclidean geometry (see Section 10.2), we would interpret these as concentric circles, since, in the standard model, the absolute points can be chosen as B1 and B2 . Case 4 - osculating conics

A pencil of osculating conics can, for example, be spanned by k ∶ x2 − x − y = 0 and l ∶ x2 + x − y = 0.

These parabolas intersect at B1 = (1 ∶ 0 ∶ 0) with multiplicity one and at B2 = (0 ∶ 0 ∶ 1) with multiplicity three. Therefore, these two curves osculate at B2 , and consequently, any two conics in the pencil osculate at B2 . The ideal line is the common tangent to all conics in the pencil. There is only one singular conic in the pencil. It is a pair of lines consisting of the ideal line and the line [B1 , B2 ]. Some conics of this pencil are shown in Figure 7.38 (left). Another example of a pencil of conics of the fourth kind is given by the one-parameter family of conics with the equations 1 + λ − y = 0 with λ ∈ R. x Again, the point of osculation lies at (0 ∶ 0 ∶ 1). The conics in this pencil are equilateral hyperbolas sharing the asymptote x = 0. The complementary family of asymptotes is given by y = λ. Figure 7.38 (right) shows some of

328

Chapter 7: Polarities and pencils

FIGURE 7.38. Some versions of pencils of osculating conics: Left: The parabolas x2 + λ ⋅ x − y = 0 with λ ∈ R osculate at (0 ∶ 0 ∶ 1) and meet at the proper point (1 ∶ 0 ∶ 0). Right: The hyperbolas x1 + λ − y = 0 with λ ∈ R also osculate at (0 ∶ 0 ∶ 1) and share the ideal point (0 ∶ 1 ∶ 0).

the conics in the pencil. In both pencils, the singular conics are a pair of lines containing the common tangent at the point of osculation and the ideal lines. In the displayed image, we see only “one half” of the singular conics. Both examples show a further specialty: Any two parabolas/hyperbolas in the respective pencil differ only by a translation. This can easily be seen by direct computation, i.e., by letting x ↦ x′ = x + a, y ↦ y ′ = y + b. The equation x2 + λ ⋅ x − y = 0 of a parabola in the pencil changes to 2 x′ + x′ (2a + λ) − y + a2 + λ ⋅ a − b = 0. Obviously, only the coefficient of x changes. Since this coefficient is a linear function in the first parameter a of the translation, it can achieve any real value if λ does. However, we also have the possibility to argue without any computation. According to Definition 6.4.1, any pair of regular conics in the pencil of osculating conics is related via a perspective collineation. Assume k and l are (regular conics) taken from the pencil. They shall osculate at B2 = (0 ∶ 0 ∶ 1). Then, there exists a perspective collineation α with center B2 and axis [B1 , B2 ] with α(k) = l. The perspective collineation α is a translation for the center B2 is an ideal point and the axis is the ideal line. In the case of the hyperbolas, the translation fixes the lines parallel to x = 0.

329

7.3 Pencils of conics Case 5 - hyperosculating pair, intersection with multiplicity four

Finally, we shall look at special examples of pencils of the fifth kind. First, let us assume that λ⋅x+

1 − y = 0 with λ ∈ R x

are the equations of the conics in the pencil. Once again, we can also view these curves as the graphs of rational functions. Therefore, we have a pencil of hyperbolas. The left-hand side of Figure 7.39 shows some curves of the pencil. They are hyperosculating at B = (0 ∶ 0 ∶ 1). The only singular conic in the pencil is the repeated line x = 0.

Another simple example in the projectively extended Euclidean plane is the family of parabolas given by x2 + λ − y = 0 with λ ∈ R as shown on the right side of Figure 7.39. Note that these parabolas only differ by a translation in the direction of the common axes. All the curves in the pencil are congruent.

FIGURE 7.39. Some versions of pencils of hyperosculating conics: Left: A pencil of hyperbolas hyperosculating at the common ideal point. Right: Coaxial and congruent parabolas are hyperosculating at the common ideal point.

330

Chapter 7: Polarities and pencils

Singular pencils A pencil of conics in the sense of Definition 7.3.5 (page 313) can be spanned by two singular curves in the pencil. However, spanning a pencil by two singular conics, i.e., degenerate curves of degree two, may lead to a pencil that contains only singular conics. We call a pencil of conics singular if all conics in the pencil are singular. Figure 7.40 shows the bases of three types of singular pencils. The pencil P (λ, µ) = λx2 + µy 2 = 0 is s2

s2

s1

s2

s1

s1

FIGURE 7.40. Singular pencils of conics spanned by three different pairs of singular conics.

spanned by a pair of distinct repeated lines (Figure 7.40, left) and contains pairs of real lines if λµ > 0 and pairs of complex conjugate lines if λµ > 0. Changing the base by letting P (λ, µ) = λx2 + µ(x2 − y 2 ) = 0 or P (λ, µ) = λ(x2 − y 2 ) + µxy = 0 gives rise to further examples of singular pencils (cf. Figure 7.40, middle and right). ●

Exercise 7.3.4 Classification of singular pencils. Try to classify pencils of singular conics from the viewpoint of Projective Geometry. Are there distinctions to be made if the analytic model is based on a finite field F with charF ≠ 0? Is it possible to find pencils that are spanned by singular conics containing a finite number of non-singular pencils?

Ranges - The dual counter parts of pencils of conics According to the principle of duality, any statement and any configuration of geometric objects in Projective Geometry has a dual version. A range of conics of the first kind is the set of all conics tangent to four lines that form a quadrilateral, i.e., no three of them are collinear. Note that in this particular case conics (of the range) are considered as a set of lines. We have called these objects dual conics in Section 7.1. Since line elements (lines with incident points) are self-dual, line elements are part of the base for ranges of conics of the second and third kind. A range of conics of the third kind is the family of conics that share two line elements (A, a) and (B, b) in admissible position which means that

331

7.3 Pencils of conics

A ∉ b and B ∉ a. The regular conics in this range belong also to a pencil of the third kind. However, the singular conics are different. The dual of a pair of lines is a pair of pencils of lines. The range of dual conics of the second kind consists of all conics that share a line element (A, a) and two further tangents b and c with A ∉ b, c and b ∩ c ∉ a. The range of the fourth kind comprises the family of all (dual) conics osculating a given conic c at a line t ∈ c and sharing a further tangent u ≠ t. The range of the fifth kind consists of all (dual) conics hyperosculating a given conic c at a line t ∈ c. Again, the regular conics belong also to a pencil of the fifth kind. However, the included singular dual conic is a repeated pencil of lines. l3

l4

l2

l1

FIGURE 7.41. The three types of ranges of conics that differ from their dual counter parts: Left: A ranges of the first kind consists of all conics tangent to a quadrilateral. Middle: A ranges of the second kind consists of all conics with a common line element and two further common tangents. Right: The conics in a range of the fourth kind share an osculating element (black parabola with a certain point of osculation) and one further tangent.

The synthetic treatment of ranges of conics does not differ too much from the treatment of pencils. Every construction is to be dualized. The same holds true for the analytical treatment: Point coordinates are replaced with line coordinates. When we visualize the ranges of conics, we should be aware of the fact that the dual conic is a family of lines being the family of tangents of what we call a conic (note Figure 1.5). The images in Figure 7.41 show conics as sets of points which are just the envelopes of the dual conics. In Figure 7.41, we have displayed those three types of ranges that differ from their dual counter parts. The pencils of third and fifth kind are self-dual,

332

Chapter 7: Polarities and pencils

i.e., the base consisting either of a pair of line elements or a conic to be hyperosculated at some point stays the same when we apply a duality. In the case of a range of the first, second, or fourth kind, the base changes to a quadrilateral consisting of tour independent lines l1 , l2 , l3 , l4 , a line element (l1 , P ) plus two tangents l2 , l3 ≠ l1 (not through P ), and a dual conic c to be osculated (at some line and point) and a further tangent l. ●

Exercise 7.3.5 Conics with four given tangents

Prove the following characterization: The point F is a focus of a conic out of a range of the first kind if the pedal points of the tangents w.r.t. F lie on a circle or are aligned.

Confocal conics span a range Prominent examples of ranges are the families of confocal conics. In the case of central conics, they share the isotropic tangents through the focal points (Definition 7.1.5). Therefore, this is a range of the first kind which contains the pairs of line pencils with the two real focal points as carriers as well as that with the two complex conjugate focal points as singular curves. The third singular curve is the set of isotropic lines. Confocal parabolas share the focal point and the axis. They define a range of the second kind, and the singular curves are accordingly. We are going to prove a classical result which generalizes the theorem mentioned in Exercise 2.2.12 and depicted in Figure 2.35. Theorem 7.3.2 If the tangents drawn from any two points A1 , B1 of a conic c1 to a confocal conic c0 form a quadrilateral, then through each other pair of opposite vertices (Ai , Bi ), i = 2, 3 , passes a conic ci of the confocal range through c0 and c1 . The quadrilateral is incircular, i.e., it has an incircle. This theorem has already been published by M. Chasles [29, p. 841] and later by W. Böhm2 in [18, p. 221]. The same theorem was studied in [89], [121] and [2]. In [76], the authors reproved it in a differential geometric way. Below, we present and prove a projective generalization 2

Wolfgang Böhm (1928–2018), German mathematician, founder of the journal ‘Computer Aided Geometric Design’.

7.3 Pencils of conics

333

of this statement which at the same time expresses properties of Poncelet grids (Figures 7.44 and 9.32).

Theorem 7.3.3 Let c0 be any conic and A1 , B1 two points such that the tangents t1 , . . . , t4 drawn from A1 and B1 to c0 form a quadrilateral. Its remaining pairs of opposite vertices are denoted by (Ai , Bi ), i = 2, 3 (Figure 7.42.

(i) For any conic c1 passing through A1 and B1 there exist conics ci through Ai and Bi such that ci belongs to the range Rc spanned by c0 and c1 . (ii) If Rc includes pairs of line pencils with carriers (Ej , Fj ), j = 1, 2, . . ., then there exist conics dj tangent to t1 , . . . , t4 and passing through Ej and Fj . (iii) The tangents at Ai and Bi to ci for i = 1, 2, 3 , as well as the tangents at Ej and Fj to dj for j = 1, . . . , 3 meet at a common point T . (iv) This result holds also in the limiting case t1 = t2 , where the chord A1 B1 of c1 contacts c0 at B2 and coincides with two of the four tangents t1 , . . . , t4 . Then, all conics dj touch c0 at B2 and are tangent to t3 and t4 .

Proof: Note that in this proof the term ‘conic’ stands for regular conics, seen as set of their tangent lines, as well as for pairs of line pencils and for single line pencils with multiplicity two. Expressed in terms of homogeneous line coordinates, the corresponding quadratic forms have the ranks 3, 2, or 1, respectively. The term net denotes a two-parameter linear system of dual curves of degree 2. It is spanned by three conics provided that they are not contained in a range. Within any net, conics and ranges can be seen as points and lines of a projective plane. Any two ranges in a net must have a conic in common (compare with [30, Théorèmes I – IV]). The conics being tangent to t1 , . . . , t4 define a range Rt , which includes for i = 1, 2, 3 the pairs of line pencils (Ai , Bi ) as well as the initial conic c0 . On the other hand, c0 and c1 span a range Rc , which contains the pairs of line pencils (Ej , Fj ). Since both ranges share the conic c0 , they span a net N of conics.

The pair (A1 , B1 ) of line pencils spans together with c1 the range of conics sharing the points A1 , B1 and the tangents there, which meet at point T . This range, which also belongs to N , contains the rank-1 conic with carrier T . Now, each pair of line pencils (Ai , Bi ), i = 2, 3 , spans with the pencil T again a range within N . This range shares with the range Rc a conic ci passing through Ai and Bi with respective tangent lines through T . A similar argument holds for the pair of line pencils (Ej , Fj ) which proves the existence of a conic dj ∈ Rc through Ej and Fj with tangent lines passing through T . All these conclusions remain valid in the case (iv), when Rt consists of conics which touch c0 at B2 and are tangent to t3 and t4 .

◾

334

Chapter 7: Polarities and pencils A2 t4 t3

A1 T

t1

c1

d2

A3 c3

B2 d1

E1

B1

t2 F1

B3 p

c0 s c2

FIGURE 7.42. Theorem 7.3.3 in the particular case of confocal conics c0 and c1 .

Obviously, Theorem 7.3.2 treats the particular case of Theorem 7.3.3 where the given conics c0 and c1 are confocal. This is shown in Figure 7.42. Now, the real focal points and the absolute circle points serve as pairs of points (Ej , Fj ), j = 1, 2, as mentioned in (ii) and (iii). The latter correspond to the incircle d2 of the quadrilateral t1 . . . t4 . This circle has the center T (see also Figures 7.44 and 9.53). Remark 7.3.2 1) With each net of conics two particular algebraic curves are associated (see [41, Sect. 77]). One is the Hessian curve of degree three named after L.O. Hesse3 . It is the envelope of all lines l with an indeterminate pole w.r.t. a (singular) conic included in the net. This means that such a line l either connects the carriers of the pairs of line pencils as rank-2 conic or it passes through the carrier of a rank-1-conic. However, the locus of the carriers of the included line pencils is called Cayley’s curve of the net. In general, it is of degree six. 2) In the particular case depicted in Figure 7.42, Cayley’s curve contains the strophoid s through the points of contact of all tangents drawn from T to any conic in Rc (compare with Figure 7.65). The Hessian consists of the line pencil T and of Chasles’s parabola p of T w.r.t. the confocal range Rc . The latter results from the fact that with each line l the Hessian of N 3

Ludwig Otto Hesse, German mathematician, 1811–1874.

335

7.3 Pencils of conics Rt

Rc c1

A3 B3 T

c2

A2 B2

c3

A1 B1 E1 F1

d2 c0

d1

E2 F2

FIGURE 7.43. A projective plane representing the net N containing the ranges Rc and Rt and the conics ci , i = 0, 1, 2, . . . and dj , j = 1, 2, . . ., pairs of line pencils and the rank-1 conic with carrier T . B3

c3 A1 c1 A2

c2 c0

T

A3

B1

B2 FIGURE 7.44. A portion of a Poncelet grid with incircles according to iterated applications of Theorem 7.3.2. includes also the line l′ which is conjugate to l w.r.t. all conics of N which assign to l a unique pole. Since in our example N contains the range Rc of confocal conics, the line l′ corresponds to l in the quadratic transformation of conjugate normals (see page 368). Consequently, the lines l′ envelop Chasles’s parabola of T w.r.t. c0 or any other conic of Rc (Figure 7.42). 3) Nets of conics which include a rank-1-conic, are characterized by a reducible Hessian curve. If only one rank-1-conic is included, then the Hessian splits into a regular conic and this singular line pencil (note Chasles’s parabola p and the pencil with the carrier T in Figure 7.42).

336

Chapter 7: Polarities and pencils

7.4 Desargues’s involution theorem First, we formulate the theorem of Gérard Desargues (first given in 1639) in the most general form. Afterwards, we treat the five cases of pencils of conics separately. In Example 7.3.2, we have determined the parabolas in a pencil of conics. Therefore, we have homogenized the equations of the conics in the pencil. Then, the intersection with the ideal line which are the solutions of a quadratic equation were depending on the homogeneous parameters (κ, λ) in the pencil. Consequently, we had to determine (κ, λ) ≠ (0, 0) such that the quadratic equation has a double root. The geometric background is laid down in Desargues’s involution theorem: Theorem 7.4.1 If the points S1 , S2 are the intersection points of a curve k of any given pencil of conics and a given line l, then S1 , S2 are corresponding in an involution δl , provided that l does neither pass through a base point of the pencil nor is a component of any curve in the pencil. Proof: Let A = (aik ) and B = (bik ) be the coefficient matrices of the two conics c and d which span the pencil, and let x0 = 0 be the the equation of the line l in P2 (F). The points X = (0 ∶ x1 ∶ x2 ) and Y = (0 ∶ y1 ∶ y2 ) are conjugate w.r.t. c if, and only if, a11 x1 y1 + a12 (x1 y2 + x2 y1 ) + a22 x2 y2 = 0.

(7.16)

The pairs (X, Y ) constitute an involution if the determinant a11 a22 − a212 ≠ 0. Otherwise l is a tangent or, in the case a11 = a12 = a22 = 0 a component of l. There is an analogous bilinear form characterizing the conjugate position of X and Y w.r.t. d. If (P, Q) is a pair of conjugate points w.r.t. c and d, then it is conjugate w.r.t. all conics in the pencil. Consequently, if fixed points S1 and S2 of such an involution of conjugate points exist, they are harmonic w.r.t. P and Q. Hence, the fixed points are assigned pairs of an involution on l.

In the case of an algebraically closed field F the proof is already done. However, in any other case, the pair (P, Q) does not necessarily exist. Therefore, we give a second proof which is valid for all comutative fields F with charF ≠ 2.

Let us project the points X = (0 ∶ x1 ∶ x2 ) on l from the center A = (1 ∶ 0 ∶ 0) onto the “standard conic” s ∶ x0 x2 − x21 = 0. Then, X is mapped to X ′ = (x21 ∶ x1 x2 ∶ x22 ). The points X, Y which are conjugate w.r.t. c satisfying (7.16) are projected to points X ′ , Y ′ ∈ s which are collinear with Ic = (a22 ∶ −a12 ∶ a11 ). This is either the center of an involution on s or the image T ′ of the point T of contact between c and l, if l is tangent to c. The verification of this statement is left to the reader as an exercise. In the same way, the second conic d defines a point Id = (b22 ∶ −b12 ∶ b11 ). The corresponding centers Ik of all other conics k in the pencil are represented as linear combinations. Hence, Ik , Ic , and Id are collinear. Before confirming that Ic ≠ Id let us finish the reasoning: If k intersects the line l at S1 and S2 , their projections S1′ , S2′ ∈ s are fixed points of the involution on s with center Ik . Hence, S1′ and S2′ lie on the polar line pk of Ik w.r.t. s. While Ik traverses

337

7.4 Desargues’s involution theorem

the line [Ic , Id ], the polars pk pass through a fixed point I. If I ∈ s, then there is a point of intersection B ∈ l with B ′ = I which belongs to all conics in the pencil, and l would pass through any base point, but this was excluded. Therefore, all pairs (S1′ , S2′ ) are aligned with a point I ∉ s. They are corresponding in an involution on s which, after projection from s back to l confirms the statement.

Finally, we have to exclude the case Ic = Id : Suppose α(a22 , −a12 , a11 ) + β(b22 , −b12 , b11 ) = (0, 0, 0) for any (α, β) ≠ (0, 0). Then, the conic with coefficient matrix αA + βB has the equation (αa00 + βb00 )2 x20 + 2(αa01 + βb01 )x0 x1 + 2(αa02 + βb02 )x0 x2 = 0. The conic is reducible and contains l as a component. This has been excluded as well.

◾

In order to get a deeper insight into Desargues’s involution theorem, we add synthetic proofs for the different kinds of pencils.

The most simple case - pencils of the first kind Proof: 1. Assume P1 P2 P3 P4 is the quadrangle of base points of the given pencil and c is a conic of the pencil, i.e., Pi ∈ c for all i ∈ {1, 2, 3, 4}. Let further l be a line which does not contain any Pi . Let S and S ′ be common points of c and l (see Figure 7.45).

Then, the points Q ∶= [P2 , P4 ] ∩ l, R ∶= [P2 , P3 ] ∩ l, S, and S ′ are four different points. Let α be the projectivity from the pencil P1 to the pencil P2 that generates c. If π1 is the perspectivity from the pencil P1 to l, and similar, π2 is the perspectivity from the pencil P2 to l, then π2−1 ○ α ○ π1 ∶ g → g is a projectivity with [P1 , P3 ] ∩ l =∶ A ↦ R,

[P1 , P4 ] ∩ l =∶ B ↦ Q,

S ↦ S,

S′ ↦ S′.

Now, there exists an involutive projectivity η ∶ l → l with Q ↦ R ↦ Q and S ↦ S ′ ↦ S. The

P1

P2

c P3 R

A

P4 S

S

′

l

Q B

FIGURE 7.45. The Desargues involution for a pencil of the first kind on a line l, as stated in Theorem 7.4.1. composition δl ∶= η ○ π2 ○ α ○ π1−1 maps A ↦ Q, B ↦ R, S ↦ S ′ , S ′ ↦ S. Since S ≠ S ′ , δl is involutive. Since A ≠ B, Q ≠ R, A ≠ R, B ≠ Q, the involution δl is uniquely defined by A ↦ Q and B ↦ R. This means that the quadrangle P1 P2 P3 P4 of base points of the pencil of the first kind uniquely determines δl and δl does not depend on the choice of the conic c. 2. If δl is the involutive projectivity determined by A ↦ Q and B ↦ R, then η ○ δl ∶ l → l maps A ↦ R and B ↦ Q and fixes S and S ′ . Consequently, the projectivity α−1 ∶= π2−1 ○η○δl ○π1 from the pencil P1 to the pencil P2 is identical with the projectivity α which generates the conic c,

338

Chapter 7: Polarities and pencils

because α−1 ([P1 , P3 ]) = [P2 , P3 ], α−1 ([P1 , P4 ]) = [P2 , P4 ], and α−1 ([P1 , S]) = [P2 , S]. Hence, the point S ′ is also a point of c since α−1 ([P1 , S ′ ]) = [P2 , S ′ ].

◾

In the case of a pencil of conics of the first kind, the Desargues involution δl is determined by the quadrangle P1 P2 P3 P4 of base points. Also in the case of a pencil of the second or third kind, the base points and base lines determine the involutive projectivity δl on any line l that is in an admissible position w.r.t. the base points (see Figure 7.46), and the proof given above remains valid. P1 =P3

P1 P3 P4 A′ l

C

P1 =P3 P2 =P4

P2 P2 C ′A

P4 B′ B A

l

A′

B′ B

A=A′

B′

l

B

FIGURE 7.46. Assigned pairs of points of the Desargues involution δl for pencils of the first kind (left), the second kind (middle), and the third kind (right).

●

Exercise 7.4.1 Three corresponding pairs in the Desargues involution. Show the following: The three pairs of opposite side lines in a quadrangle P1 P2 P3 P4 in a projective Pappian Fano plane intersect any line l which does not pass through a vertex of the quadrangle in three pairs of corresponding points of an involutive projectivity on l. =1 [P2 , P3 ] ∧ =4 l. Hint: Study the action of l ∧ P

P

If a line l contains one or even two diagonal points of the quadrangle P1 P2 P3 P4 of base points, then these points are fixed points of δl . This is clear from the proof of Theorem 7.4.1. The pencils of the second determines fixed points of δl on special lines. In the case of a pencil of the third kind, one fixed point is known from the beginning. It is the point A = A′ (in Figure 7.46) that comes as the point of intersection of l with the line joining the only two base points of this type of pencil. For fixed points of the Desargues involution δl which are not determined by the special choice of l there is a very important meaning. We can show: Theorem 7.4.2 Any fixed point of the Desargues involution δl on the line l is either the point of contact of l with a regular conic out of the pencil or a singular point of a singular conic included in the pencil.

339

7.4 Desargues’s involution theorem

Proof: Let F be a fixed point of δl which is not located on any side of P1 P2 P3 P4 . In the case of a pencil of the first kind, the base points P1 , P2 , P3 , and P4 together with the point F determine a unique conic c. The line l is tangent and contains only the point F : Otherwise, l would also carry δl (F ) ≠ F (according to Theorem 7.4.1) which contradicts δ(F ) = F . If F lies on some side of P1 P2 P3 P4 , then it must be a diagonal point. Conversely, if c and l are in contact at a point X with X ≠ δl (X), then, according to Theorem 7.4.1, c goes through δl (X) which is not possible for a tangent of the conic c.

◾

◾

Example 7.4.1 Applications of Desargues’s theorem. Desargues’s involution theorem is useful for the construction of conics from certain given configurations of lines, points, and conics in the projective extension of E2 . In this example we shall give a list of possible problems. This list is far from being complete. Parabolas on four points. We continue Example 7.3.2 and look for the parabolas in a pencil of the first kind. A parabola is a conic in the real plane that touches the ideal line ω. Therefore, we have to find the fixed points A1 and A2 of the Desargues involution δω , if there are some.

Z s

̃ 3

3

pδs

P3 3′ 3

′

1

P2

A1

2 2

p2

F1

̃ 1

F2 A2

1 1′

̃ 2

A1 P4

3′

2′

1′

P1

2

p1 1

A2

2

3

FIGURE 7.47. How to find the parabolas in a pencil of the first kind? The six lines through the base points P1 , P2 , P3 , P4 have six ideal points which comprise three pairs of corresponding points in the Desargues involution δω on the ideal line ω. The projection of ω to the conic s from Z ∈ s yields an involutive projectivity on s whose fixed points F1 and F2 correspond to the fixed points of δω . The lines [Z, F1 ] and [Z, F2 ] are parallel to the axes of the two wanted parabolas p1 and p2 . Assume we are given the four base points P1 , P2 , P3 , and P4 of a pencil of the first kind, as shown in Figure 7.47. We are looking for the parabolas in the pencil, i.e., we want to find the parabolas through the points P1 , P2 , P3 , and P4 , if there are some. A parabola is a conic that touches the ideal line ω, and thus, we use Theorem 7.4.2 which says that the fixed points of δω belong to those conics that touch ω.

340

Chapter 7: Polarities and pencils

In order to handle the involution δω , we project the ideal points 1, 2, 3 (the ideal points of the lines [P1 , P2 ], [P1 , P3 ], [P1 , P4 ]) as well as the ideal points 1′ , 2′ , 3′ (the ideal points of the opposite lines [P3 , P4 ], [P2 , P4 ], [P2 , P3 ]) to an arbitrary conic s from some of its points Z. Note that the ideal points of opposite sides in the quadrangle of base points are assigned points in δω . For the sake of simplicity, we have chosen a circle s. This yields a projectivity δs on s with (̃ 1, 1), (̃ 2, 2), and (̃ 3, 3) for its assigned pairs. The projectivity δs is involutive since δω is. Thus, already two pairs of assigned points determine δs . A priori, we do not know which pairs are useful for the construction of the fixed points. Therefore, we have projected all points onto s. Now, Theorem 6.4.4 comes into play: The axis pδs carries the points [̃ 1, 2] ∩ [1, ̃ 2], [̃ 1, 3] ∩ [1, ̃ 3], [̃ 2, 3] ∩ [2, ̃ 3].

The fixed point F1 and F2 of δs are the points of intersection of s and pδs . Finally, we project the fixed points of δs to that of δω by simply connecting Z with F1 and F2 . The latter lines are parallel to the axes of the parabolas p1 and p2 in the pencil. In Figure 7.47, the parabolas p1 and p2 are shown though we do not describe all constructions in order to find the vertices and the precise position of the axes.

◾

Example 7.4.2 Conics in a pencil of the second kind touching a line l. The line l does not pass through any of the base points P1 , P2 , and P3 . The base points together with the tangent t1 at P1 determines the Desargues involution δl on l as shown in Figure 7.46. Thus, we obtain the following pairs of assigned points [P1 , P2 ] ∩ l =∶ 1 ↦ 1′ ∶= [P1 , P3 ] ∩ l,

[P2 , P3 ] ∩ l =∶ 2 ↦ 2′ ∶= t1 ∩ l.

Since δl is involutive, we have δ(1′ ) = 1, and thus, we have at least three pairs of assigned points which are sufficient according to Theorem 5.2.1. According to Theorem 7.4.2, the fixed points of δl are the points of contact of conics in the pencil with the line l.

We choose a conic s and a point Z ∈ s, project the points 1, 1′ , 2, 2′ from Z to s. With 1′ = 3 and 1 = 3′ we can apply Theorem 6.4.4 and find the fixed points F1 and F2 of the projectivity δl lifted to s (as in the previous example). The fixed points are projected to the points C1 and C2 where the conics c1 and c2 touch l, see Figure 7.48.

◾

Example 7.4.3 Conics on three points and two tangents. Assume that three non-collinear points P1 , P2 , P3 and two lines l and m are given, as shown in Figure 7.49. The lines l and m shall not pass through any of the given points. Now, we are looking for conics through P1 , P2 , and P3 that touch both l and m. If a conic c1 is a solution of the given problem, then it touches l and m at the points C1 and C2 . Thus, c1 and the pair (l, m) span a pencil of the third kind with the two line elements (C1 , l) and (C2 , m) for its base. Let g12 ∶= [P1 , P2 ] be the line through the given points P1 and P2 . According to Theorem 7.4.2, the line g12 (which is not passing through a base point) intersects the conics as well as the lines l and m in pairs of assigned points in the Desargues involution δ12 on g12 . The assigned pairs are l ∩ g12 ↦ m ∩ g12 , P1 ↦ P2 and δ12 is uniquely determined.

A fixed point F12 of δ12 is the point of intersection of g12 with a possible chord [C1 , C2 ]. So, the connection of two fixed points of the Desargues involutions on different lines, e.g., g12 ∶= [P1 , P2 ] and g13 ∶= [P1 , P3 ], is a chord of a possible conic c1 and intersects l and m in the contact points C1 and C2 . As we can see in Figure 7.49, the six fixed points on the sides of the triangle gather on four lines, e.g., the fixed points F12 , F13 , and F23 are collinear. (The proof of this fact is left as an

341

7.4 Desargues’s involution theorem 2′

1

2 F2

pδl

F1

l Z

{

c1

C1 2 1′

t1 c2 P1

P2

1′ 2′

C2 P3

s 1

FIGURE 7.48. The conics c1 , c2 in a pencil of the second kind with base points P1 , P2 , P3 and base line t1 are uniquely determined if the contact points C1 , C2 with the line l are known. These contact points are the fixed points of the Desargues involution δl on l. The involution δl is projected from the center Z to the Steiner conic s where we find the fixed points. (The image points are labeled in the same way.) The axis pδl intersects s at the fixed points F1 , F2 which are mapped to the desired contact points C1 , C2 via the projection from Z to l. exercise to the reader.) Consequently, there are four chords which are the double lines in the pencils of the third kind hidden in this configuration of conics and lines. Therefore, there are up to four (real) solutions.

◾

Example 7.4.4 Conics that touch a given conic twice. In this example, we shall treat conics on three points P1 , P2 , P3 that touch a given conic c twice. Like in the previous item, we are concerned with pencils of doubly touching conics. In the previous example, two lines l and m were given. However, this pair of lines can be seen as a degenerate conic. Again, we aim at a determination of the points of contact of the given conic c and the conics we are looking for. Just as in the example before, the Desargues involutions on the sides of P1 P2 P3 are defined by their intersections with c and the pair of given points on it, e.g., in the case of the line [P1 , P2 ], the involution δ12 is determined by the following two pairs of corresponding points P1 ↦ P2 and S1 ↦ S2 , where S1 and S2 are the common points of c and [P1 , P2 ], if they exist.

The fixed points of the three Desargues involutions, and thus, the points of contact of c and the solutions give rise to the same configuration of points and lines as in the previous example, cf. Figure 7.49. The left-hand side of Figure 7.50 shows the configuration of given points, fixed points of Desargues involutions, and points of contact.

◾

Example 7.4.5 A conic on three real points and a pair of complex conjugate points. How to find a conic c on a pair (K, K) of complex conjugate points and three real points A, B, C?

342

Chapter 7: Polarities and pencils

given points

c1 F12 C2 F23 F13

fixed points points of contact

C1 g12

g13 P1 P2

l

g23

m

FIGURE 7.49. Conics on three points and two tangents: There is a Desargues involution on each side of P1 P2 P3 . Triples of fixed points are collinear and there are four such lines which meet the tangents l and m at points where one of the solutions touches. The pair (K, K) of complex conjugate points can be given by its real join l and by the elliptic involution with the fixed points K and K. If there exists a solution c, then the given involution is the involution α of points conjugate w.r.t. c. Applying a perspective collineation κ with l as its vanishing line and the Laguerre point Lα for its center to A, B, C results in three points, say A′ , B ′ , C ′ . The collineation α maps the pair (K, K) to the pair of absolute points of Euclidean geometry. Thus, the κ image c′ of c is the unique circle on A′ , B ′ , C ′ . The κ−1 -image of c′ gives the conic we are looking for.

In Section 4.4, we have seen a different approach to these kind of conic problems. However, the constructive methods and possibilities apply to many more kinds of constructive problems compared to the spatial interpretation. Desargues’s involution theorem for pencils of the fourth and fifth kind

In the previous section, we have learned that the base figure of a pencil of conics, i.e., the base points together with base tangents, determines the Desargues involution. In the case of a pencil of the fourth or fifth kind, the base figure contains a non-degenerate conic. However, there is still an involution determined on any line, that neither contains a base point,

343

7.4 Desargues’s involution theorem F12

P1

c

F13

P2

P1

P3

C2 c

C1

P3

P2

F23

FIGURE 7.50. Left: The conic c, the three given points P1 , P2 , P3 , the fixed points of the Desargues involutions on the sides of P1 P2 P3 , and the points of contact of the solutions. Two points of contact of one solution are always collinear with one fixed point of each of the three involutions. Right: Four conics on the same three points in double contact with c.

nor coincides with a base line, nor passes through a common pole of the conics in the pencil. We state and show: Corollary 7.4.1 Let (P1 , t1 , P2 , c) be the base of a pencil of conics osculating the conic c at P1 with the common tangent t1 at P1 and passing through P2 . Then, all conics in the pencil meet any line l that is neither passing through P1 nor through P2 in corresponding pairs of an involutive projectivity. In case of a pencil of the fifth kind with hyperosculation at P1 , the tangent t1 meets l in a fixed point of the involution. Proof: According to Definition 6.4.1, any conic d in the pencil defined by the base (P1 , t1 , P2 , c) can be mapped onto c by an elation η with center P1 and axis a ∶= [P1 , P2 ]. Assume now that a line l with P1 , P2 ∈/ l) carries two different points A and B of a such conic d in the pencil (cf. Figure 7.51, left-hand side). The line l = [A, B] meets the axis a of the collineation η in a point C. Also the η-image of l meets a in C. Further, the points A′ = η(A) and B ′ = η(B) are points on η(d) = c. Since C /∈ c, it is the center of an involutive projectivity α ∶ c → c that = P1 ∧ = c in order to define the interchanges A′ and B ′ . We use the projective mapping β ∶ l ∧ projective mapping δl ∶= β −1 ○ α ○ β that acts on the line l. It is involutive, for it sends A to B and B to A. It is independent of the choice of the conic d in the pencil. Since η sends A to A′ , η is uniquely defined.

344

Chapter 7: Polarities and pencils P1

d

t1

t1

a

A

c

A P2

B′

l

A

d ′

l′

′

C

P1

=a

C

l′ B ′

A l

B c

B

FIGURE 7.51. Pencils of conics osculating each other at a common point also define a Desargues involution on a line in an admissible position. In case of a pencil of the fifth kind, the base consists of (P1 , t1 , c) where (P1 , t1 ) is a line element of c. The axis of η ∶ d → c now coincides with t1 (Figure 7.51, right). Everything else of the previous proof remains valid.

◾

◾

Example 7.4.6 Parabolas that osculate a hyperbola. Figure 7.52 shows a hyperbola h with the line element (P1 , t1 ) and a further point P2 . We are looking for all parabolas that osculate h at P1 and pass through P2 .

In order to determine the parabolas, we are searching for their points at infinity, i.e., the direction of the axis. The base of the pencil of osculating conics consist of the line element (P1 , t1 ), the point P2 , and the conic h. Now, we look at the Desargues involution on the line at infinity: We find the two ideal points 1 and 1′ of h as a pair of corresponding points. A further pair comes from the singular conic in the pencil. The tangent t1 at P1 has the ideal point 2′ which corresponds to the ideal point 2 of the line [P1 , P2 ]. In this particular example, we can simplify the construction of the fixed elements of the involution by using h as the Steiner conic (cf. Example 6.4.8). Furthermore, we use P1 as the center of the perspectivity that maps the range of points on the ideal line to the points on h. In Figure 7.52, the images =1 h are labeled 1, 2, 1′ , and 2′ . The construction of fixed of points under the perspectivity ω ∧ points F 1 and F 2 follows the advice given in Example 6.4.8. In Figure 7.52, the center of the involution on h is the ideal point I = 2. The dashed lines [P1 , F 1 ] and [P1 , F 2 ] join P1 with the ideal points being fixed points of the Desargues involution on the line at infinity, and thus, they are parallel to the axes of the parabolas p1 , p2 we were looking for. P

◾

Example 7.4.7 Conics that hyperosculate a given conic and pass through two given points. Figure 7.53 shows a parabola p and two further points A and B. We are looking for all conics that hyperosculate p somewhere and contain A and B if there are some. The conics in a pencil of the fifth kind intersect a generic line l in pairs of corresponding points in an involution (see Corollary 7.4.1). A fixed point F of this involutive projective mapping on l is either a point of contact of l with a conic in the pencil or a point on the tangent at the point H of hyperosculation.

Therefore, we determine the fixed points of the involution on l = [A, B]. The points A and B form a pair of corresponding points as well as the two common points 1 and 2 of l and p do. As a matter of fact, one fixed point, say F2 , lies in the interior of p. The other fixed point F1 is an outer point of p. From F1 we can draw two real tangents to p that meet the given

345

7.4 Desargues’s involution theorem I =2

1′ =1′

1′ =1′

h I =2

F2

p2

P2 =2

F2

F1 p1

2′ I =2

1 =1

P1 =2′ t1

1 =1

F1 FIGURE 7.52. How to find the parabolas p1 and p2 (if there are some) that osculate the hyperbola h at P1 and share a further point P2 with h: According to Theorem 7.4.1, the Desargues involution on the line at infinity is uniquely determined. The hyperbola serves as Steiner conic and the fixed points of the involution determine the axes of the desired parabolas.

conic at potential points H1 and H2 of hyperosculation. The conics c1 and c2 appearing as solutions to the present problem, can be completed by applying the elations with centers H1 / H2 and the respective axes [F1 , H1 ] / [F1 , H2 ] to the given conic p (see Definition 6.4.1).

This problem can also be solved by means of spatial interpretation as explained in Section 4.4.

Desargues’s involution for ranges of conics

The principle of duality allows us to formulate the dual version of Desargues’s involution Theorem 7.4.1. It reads: Corollary 7.4.2 The tangents drawn from a generic point P to the conics of a range of any kind constitute corresponding pairs of an involutive projective mapping in the pencil P . The fixed lines in this involution are tangents of conics in the pencil that pass through P .

346

Chapter 7: Polarities and pencils l

c2

2

p

B c1 F2

H2

H1 A 1

F1 FIGURE 7.53. The conics c1 and c2 hyperosculate p at H1 and H2 , respectively. According to Theorem 7.4.1, the tangents to p at the latter points pass through the fixed points of the Desargues involution involution on l.

The conics mentioned in Theorem 7.4.2 are to be considered as dual conics, i.e., the set of tangents of a conic. Figure 7.54 shows how the base of a range of conics defines the Desargues involution in the pencil of lines around a generic point. The depicted figures are obtained by dualizing the respective configurations shown in Figures 7.45 and 7.51. ●

Exercise 7.4.2 A parabola tangent to three lines. Show the following result: The orthocenter O of each triangle built by three finite tangents of a parabola p lies on p’s directrix. Hint: Check Desargues’s involution at O.

◾

Example 7.4.8 Conics on two points and three tangents. Dual to Example 7.4.4, we briefly describe how to find conics on three given tangents a, b, c and two points P , Q. Figure 7.55 shows an example. Now, we look at the Desargues involutions in the pencils of lines with vertices a ∩ b, b ∩ c, and c ∩ a. Assume f1 is a fixed line of δa∩b and let f2 be a fixed line of δb∩c , cf. Figure 7.55. For example, the involution δa∩b is defined by a ↦ b,

[a ∩ b, P ] ↦ [a ∩ b, Q].

Dualizing the figure from the previous example, we find that T1 = f1 ∩ f2 is the common point of two tangents of a solution conic s. Consequently, we obtain a configuration of six fixed lines where three out of them pass through one such point Ti . There are four points Ti , and thus, there are four real solutions provided that all three Desargues involutions are hyperbolic.

347

7.4 Desargues’s involution theorem l3

1′

P1 l1 l2

l1 1

2

2′ 1′ P

l1

l2 1′

l1

l1

P1

1′

f

1

2

f P

2′

2 2′

P2

P1

l3 P

l2 l4

P1

1

′ 1 1

P l2

1

P

FIGURE 7.54. Desargues’s involution in pencils of lines about a generic point P induced by the bases of ranges of conics of the first to fifth kind (from top-left to bottom-right): (1, 1′ ) an (2, 2′ ) are pairs of corresponding points, f indicates a fixed line. Note, that the bases of ranges of the fourth and fifth kind contain a dual regular conic.

Pencils of circles Circles are special conics in many ways. From the viewpoint of geometry in the complex extended real projective plane, a circle is a conic through the absolute points I = (0 ∶ 1 ∶ i) and I = (0 ∶ 1 ∶ −i) of Euclidean geometry (see Example 6.4.8 on page 266). It is natural to distinguish pencils of circles by the number and type of proper base points. Thus, we have the following cases (cf. Figure 7.56): 1. circles on two real points - elliptic pencil, 2. circles on a pair of complex conjugate points - hyperbolic pencil, 3. circles on to a common line element - parabolic pencil, and 4. concentric circles. According to this list, the elliptic and the hyperbolic pencil of circles are pencils of conics of the first kind. The parabolic pencil is a pencil

348

Chapter 7: Polarities and pencils

T ³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹·4¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ

T2

f2

T1 c P T3

c

s a

Q

b

P Q

f1

b

a

FIGURE 7.55. Conics on two points and three tangents. Left: The fixed lines of the Desargues involution in the pencils about a ∩ b, b ∩ c, and c ∩ a meet in the common points Ti (with i ∈ {1, 2, 3, 4}) of the tangents at P and Q of the desired conics. Right: In this case all three Desargues involutions are hyperbolic, and thus, we have four real solutions.

of conics of the second kind. Concentric circles form a pencil of conics of the third kind, because any two of the circles touch at the absolute points of Euclidean geometry. We shall see very soon where the names of the pencils originate. Thus, pencils of circles are pencils of conics whose bases contain the points I and I.

FIGURE 7.56. The four types of pencils of circles: an elliptic, a hyperbolic, a parabolic pencil, and a pencil of concentric circles.

A pencil of circles is an object of equiform geometry, i.e., the fourparametric group of transformations in the Euclidean plane generated by Euclidean motions and uniform scalings.

349

7.4 Desargues’s involution theorem The elliptic pencil - two real base points

Since the scale does not matter, it means no restriction to assume that the two common points B1 and B2 of all circles in the pencil are B1 = (0, 1)

and B2 = (0, −1).

Thus, the Cartesian equations of all circles in this elliptic pencil read x2 − 2xt + y 2 = 1 with t ∈ R.

(7.17)

It is clear and easily verified that in this pencil of conics (of the first kind), there are the following singular conics: [I, I ] ∪ [B1 , B2 ],

[I, B1 ] ∪ [I, B2 ],

[I, B2 ] ∪ [I, B1 ].

The first is a pair of real lines; the remaining pairs are pairs of conjugate complex lines. The line [B1 , B2 ] is the radical axis of any pair of circles in the pencil. The centers of all circles lie on the bisector of the segment B1 B2 . We call this bisector the axis of the pencil. The axis is a line of symmetry of the pencil as well as a common line of symmetry for all circles in the pencil. B1

l

l

m

n

a 2

1

2

′

1

′

m a

2 1

F1

1

′

3

F2 3

2′

′

B2 FIGURE 7.57. Left: An elliptic pencil is characterized by the elliptic involution on the axis a. The two real base points B1 and B2 are the Laguerre points of the Desargues involution on the axis a. Right: A hyperbolic pencil induces a hyperbolic involution on the line a of symmetry. The centers F1 and F2 of the two null circles in the pencil are the fixed points of the Desargues involution on the axis a. For any two circles in the hyperbolic pencil we can say that one circle is either completely inside or completely outside the other.

We choose two circles l and m of the elliptic pencil (see Figure 7.57). According to Thales’s theorem (cf. 6.3.3), we can see the intersections

350

Chapter 7: Polarities and pencils

1, 1′ and 2, 2′ of l and m with the axis a at right angles from either of the two base points. However, the choice of l and m does not matter. Thus, the pair of intersections of any circle with the axis of the pencil is seen at right angles from the base points. Therefore, B1 and B2 are the Laguerre points of the elliptic involution δa on the axis a interchanging the intersection points of the circles in the pencil (see Example 6.4.8 in Section 6.4, page 266). We can summarize: Theorem 7.4.3 In the projective extension of E2 , the circles in an elliptic pencil meet the axis of the pencil in pairs of corresponding points in an elliptic involution with the two real base points for its Laguerre points. The hyperbolic pencil - no real base point

Again we make use of the fact that pencils of circles are objects of equiform geometry. Since the actual size and position is not of importance, we may choose the pair of complex conjugate base points as B1 = (0, i)

and B2 = (0, −i).

The regular circles in the hyperbolic pencil have the equations x2 − 2tx + y 2 = −1 with t ∈ R {−1, 1}

(7.18)

and the three singular conics in this pencil of conics of the first kind equal that in the previous case. The singular curves (x ± 1)2 + y 2 = 0 which are obtained for t = ±1 are called null circles. They split into two isotropic lines through their real centers F1 = (1, 0) and F2 = (−1, 0), respectively. These points are fixed under the Desargues involution δa on the axis a. the axis a ∶ y = 0 in two points A circle √k(t) from the pencil meets √ 1 = (t − t2 − 1, 0) and 1′ = (t + t2 − 1, 0), see Figure 7.57. Exactly for the values t = 1 and t = −1, these two points coincide and we denote these double points by F1 = (1, 0) and F2 = (−1, 0) which can be considered as null circles, i.e., circles with radius zero. With (5.6), we can easily verify cr(1, 1′ , F1 , F2 ) = −1

independent of t ∈ R {−1, 1}. Since the characteristic cross ratio of a hyperbolic involution equals −1 (cf. page 225), we can say:

351

7.4 Desargues’s involution theorem

Theorem 7.4.4 The circles in a hyperbolic pencil of circles meet the axis a in pairs of corresponding points in a hyperbolic involution whose fixed points are the centers of the two included null circles. ●

Exercise 7.4.3 Orthocircles of a hyperbolic pencil.

F2

Show that any circle (7.18) in the

F1

FIGURE 7.58. The Thales circle on the segment F1 F2 of fixed points of the hyperbolic involution in the hyperbolic pencil of circles is an orthocircle of all circles in the pencil. hyperbolic pencil meets √ the Thales circle at right angles (see Figure 7.58). Hint: The radius of a circle (7.18) equals t2 − 1, the Thales circle has radius 1 and is centered at (0, 0), the distance of the centers equals t. Now, apply Pythagoras’s theorem.

●

Exercise 7.4.4 Pencils of orthogonal circles. We are given a hyperbolic pencil and an elliptic pencil of circles (see Figure 7.59) with the equations: h∶ e∶

x2 − 2tx + y 2 + 1 = 0 x2 + y 2 − 2uy − 1 = 0

with t ∈ R {−1, 1}, with u ∈ R.

Show that for any (t, u) the circle h meets the circle e at right angles. Further, the base points of the elliptic pencil are the fixed points of the involution induced by the hyperbolic pencil.

The results from Exercises 7.4.3 and 7.4.4 can be summarized: Theorem 7.4.5 For any hyperbolic pencil of circles there exists an elliptic pencil of circles such that any circle of one family meets all circles of the other family at right angles. The axes of the pencils are orthogonal. The centers of the null circles of the hyperbolic pencil are the real base points of the elliptic pencil.

352

Chapter 7: Polarities and pencils

FIGURE 7.59. A hyperbolic pencil h and an elliptic pencil e of circles forming and orthogonal net of circles. This two-parameter family of circles is known as Apollonian circles.

The two pencils of circles from Theorem 7.4.5 are sometimes also called conjugate to each other. Apollonian circles

The two one-parameter families of circles displayed in Figure 7.59 are frequently called Apollonian circles, due to Apollonius of Perga. However, there is an elementary description of the circles in both pencils, the elliptic and the hyperbolic one. Let q ∈ R {1} be any constant, and look at the set of points c(q) = {X∣XF1 ∶ XF2 = q}.

(7.19)

Since F1 = (−1, 0), F2 = (1, 0), and X = (x, y), the set c(q) is described by the equation (x2 + y 2 + 1)(1 − q 2 ) + 2x(1 + q 2 ) = 0. (7.20) √ Now, substitute q = t+1 t−1 and (7.20) changes to (7.18). Consequently, the Apollonius circles defined by (7.19) form a hyperbolic pencil of circles as long as t ∈ R {−1, 1}.

Any circle of the elliptic pencil within the Apollonius circles is the locus of points where the segment F1 F2 between the two base points is seen at a

353

7.4 Desargues’s involution theorem F F111111 F

F F222222 F FIGURE 7.60. The family of isoptic curves of the line segment F1 F2 consists of the circles of the elliptic pencil with base points F1 and F2 . Any isoptic curve is a pair of circular arcs. Only in case of an optic angle equal to π2 it is a unique circle: the Thales circle on F1 F2 .

constant angle: on the arc from F1 to F2 at an angle of, say, ϕ and on the complementary arc at an angle of π − ϕ, according to the theorem of the angle of circumference. As can be seen in Figure 7.60, the isoptic curve of the segment F1 F2 , i.e., the locus of points from which the segment can be seen at constant angle ϕ, consists of parts of two circles and has a two-fold symmetry (see also Section 9.2). The parabolic pencil - a common line element

The parabolic pencil of circles consists of all circles that share a line element. Though pencils of circle are objects of equiform geometry, there is only one parabolic pencil up to Euclidean motions. The parabolic pencil of circles is self-similar, i.e., applying a similarity to the pencil may interchange the circles in the pencil but leaves the entire pencil unchanged. It is easy to check that the circles l ∶ x2 − 2tx + y 2 = 0 with t ∈ R

share the point O = (0, 0) and touch the line x = 0 there, and thus, they form a parabolic pencil. In particular, this is true for the point shaped circle corresponding to t = 0 which should rather be viewed as the pair of isotropic lines through O. The circles in the parabolic pencil m ∶ x2 + y 2 − 2uy = 0 with u ∈ R

354

Chapter 7: Polarities and pencils

intersect all circles l at right angles, for all m touch y = 0 at O. Figure 7.61 shows the two parabolic pencils that form a rectangular grid of circles. So, the pencil conjugate to a parabolic pencil of circles is again parabolic.

FIGURE 7.61. Left: A parabolic pencil of circles. Right: Concentric circles.

355

7.4 Desargues’s involution theorem Concentric circles

Finally, the pencil of concentric circles builds an orthogonal net together with the common diameters of all circles. This pencil is invariant under rotations about the common center, for it is invariant under reflections about all common diameters. On any diameter, the points of intersection of the circles can be arranged in pairs of corresponding points in a hyperbolic involution, i.e., the reflection in the common center, see Figure 7.61. Bipolar coordinates

We can introduce curvilinear coordinates (u, v) in the plane E2 such that the parameter lines u = const. and v = const. form the orthogonal net consisting of the circles of an elliptic pencil and the conjugate hyperbolic pencil. To this end, we choose points F1 = (e, 0) and F2 = (−e, 0), e ∈ R {0}, as the real base points. The circles of the two pencils satisfy the respective equations l ∶ x2 + y 2 − 2uy − e2 = 0 and m ∶ x2 + y 2 − 2vx + e2 = 0.

It is a matter of elementary computations to find the common points S1 and S2 of l and m. The coordinates of S1 and S2 are algebraic expressions in terms of u and v. After a parameter substitution (u, v) → (U, V ) with u=

e(1 − U 2 ) 2U

and v =

e(V 2 + 1) 2V

at least one of the points S1 and S2 can be given in terms of rational coordinate functions as (

eV (1 + U 2 ) eU (V 2 − 1) , ). U2 + V 2 U2 + V 2

This is a rational parametrization of the Euclidean plane such that the curves U = const. or V = const. are circles of an elliptic and a hyperbolic pencil forming an orthogonal net. Coordinates with these parameter lines are called bipolar. Usually, bipolar coordinates are expressed in terms of angles and the computations involve hyperbolic and ordinary trigonometric functions.

356

Chapter 7: Polarities and pencils

7.5 Quadratic Cremona transformations In this section, we shall deal with the quadratic birational mappings, frequently called Cremona transformations (after Antonio Luigi Gaudenzio Giuseppe Cremona, 1830–1903, Italian mathematician, structural engineer, and politician). Unlike collinear transformations, quadratic transformations are defined by homogeneous quadratic coordinate functions. Any such transformation is called birational if its inverse can also be given by homogeneous polynomial coordinate functions. Quadratic birational transformations are important objects in algebraic geometry. According to Max Noether (German mathematician, 1844– 1912), any planar algebraic curve can be transformed into a planar algebraic curve with only ordinary singularities by a finite sequence of quadratic birational transformations. In this sense the quadratic transformations are the fundamental building blocks and generate the Cremona group. First, we classify the quadratic Cremona transformations and study the three basic types. Then, we pay attention to some special mappings in P2 (R) such as the inversion (in conics in general and especially in circles), the transformations of doubly conjugate points or doubly conjugate lines, the transformation of conjugate normals, Hirst’s inversion, and the pedal transformation.

Classification of birational quadratic mappings Let

Φ ∶ P ∶= P2 (F) → P′ = P2 (F) with xF = (x0 ∶ x1 ∶ x2 ) ↦ (g0 (x) ∶ g1 (x) ∶ g2 (x))

be a rational mapping with homogeneous polynomials gi (x0 , x1 , x2 ), i = 1, 2, 3, of degree N without any non-trivial common divisor. After excluding the points of the exceptional set E ∶= {xF ∣ g0 (x) = g1 (x) = g2 (x) = 0},

i.e., the so-called base points or exceptional points of Φ, we obtain the mapping ϕ ∶ P E → P′ with xF ↦ x′ F = (g0 (x) ∶ g1 (x) ∶ g2 (x)) .

For the moment, we assume that the two involved projective planes P and P′ are different, but nevertheless isomorphic, and equipped with their own coordinate frames.

357

7.5 Quadratic Cremona transformations

The preimage of any line l′ ∶ u0 x′0 + u1 x′1 + u2 x′2 = 0 in P′ with (u0 ∶ u1 ∶ u2 ) ∈ F3 {(0, 0, 0)} is the cycle4 of degree N , given by ϕ−1 (l′ ) ∶ u0 g0 (x) + u1 g1 (x) + u2 g2 (x) = 0.

These cycles form a linear two-parameter family of curves, called the net associated with Φ. The net is spanned by the three cycles g0 (x) = 0,

g1 (x) = 0,

g2 (x) = 0,

where the cycle gi = 0 is the preimage of the line x′i = 0 in terms of the homogeneous coordinates chosen in P′ . The base points of the net are the common points of the base curves, and therefore, they agree with base points of Φ in the exceptional set E, as defined above. In case of a birational mapping, any image point x′ F (off an exceptional set E ′ in P′ ) has to have exactly one preimage. Since x′ F can be considered as the intersection of two different lines l′ and m′ , the two curves ϕ−1 (l′ ) and ϕ−1 (m′ ) have to intersect in precisely one point xF ∈ P E. This has to be true for any point in P′ E ′ . It is not possible that two curves of the net are in contact since there exists one, and only one, curve of the net passing through a given line element in general position. The quadratic transformations are obtained if N = 2. Thus, we are looking for all the nets of conics with the additional property that any two curves ϕ−1 (l′ ) and ϕ−1 (m′ ) intersect in exactly one point which is (in general) different from all base points. Furthermore, it is necessary that any point in P E can be found as a remaining point of intersection between two curves from the net.

The two curves ϕ−1 (l′ ) and ϕ−1 (m′ ) span a pencil of conics. Depending on the type of pencil (cf. Section 7.3), these curves share 4, 3, 2, or 1 point. One of them is the point xF, while the remaining ones must be contained in the exceptional set E. Since a contact of ϕ−1 (l′ ) and ϕ−1 (m′ ) at xF is not allowed, a pencil of doubly touching conics (third kind) and a pencil of hyperosculating conics (fifth kind) cannot occur. Thus, there are three 4

We distinguish between a cycle and a curve: An algebraic curve in the projective plane is the set of points whose homogeneous coordinates satisfy an irreducible homogeneous polynomial equation. A cycle is the union of a finite number of algebraic curves. Thus, the equation of a cycle is the product of a finite number of irreducible homogeneous polynomials. Some of the factors may even have a multiplicity greater than one.

358

Chapter 7: Polarities and pencils

types of associated nets left, provided, the field F is algebraically closed, like C: 1. The three base points P1 , P2 , P3 are the vertices of a triangle. 2. There are two different base points P1 , P2 . The curves of the net share a line element (P1 , t1 ) and the point P2 ∉ t1 . 3. There is only one base point P1 . The curves of the net osculate each other at P1 .

Note that in P2 (R) two of the three base points in type 1 can be complex conjugate. g2

P2

P2

P2 g1

g2

g0 P0

P1 g1

g0 P0

P1 g1

g0

P0

P1 g2

FIGURE 7.62. The three curves g0 , g1 , and g2 span the associated net of conics. From left to right: the three types 1, 2, and 3. Type 1: Three different base points

For the sake of simplicity, we choose the three base points of the quadratic transformation as the base points of the projective frame. Thus, P1 = (1 ∶ 0 ∶ 0), P2 = (0 ∶ 1 ∶ 0), and P3 = (0 ∶ 0 ∶ 1). Now, the associated net is spanned by g0 ∶ x1 x2 = 0,

g1 ∶ x2 x0 = 0,

g2 ∶ x0 x1 = 0.

Adapting the frame in P′ such that these pairs of lines are exactly the images of the coordinate lines x′0 = 0, x′1 = 0, and x′2 = 0, we get the coordinate functions of the transformation Φ: Φ ∶ (x0 ∶ x1 ∶ x2 ) ↦ (x′0 ∶ x′1 ∶ x′2 ) = (λ0 x1 x2 ∶ λ1 x2 x0 ∶ λ2 x0 x1 ).

(7.21)

with coefficients λi ∈ F {0}. Choosing the image point of (1 ∶ 1 ∶ 1) as the unit point of the frame in P′ , we achieve λ0 = λ1 = λ2 = 1. Type 2: Two different base points

Again, we choose a proper coordinate system and let P1 = (1 ∶ 0 ∶ 0), t1 ∶ X2 = 0, and P2 = (0 ∶ 0 ∶ 1). Then, the associated net is spanned by

359

7.5 Quadratic Cremona transformations

the curves

g0 ∶ x1 x2 = 0,

g1 ∶ x0 x2 = 0,

g2 ∶ x21 = 0,

because these three curves pass through all the base points and touch t1 at P1 , and they are linearly independet. (For tangents to a singular conic see page 277.) With the suitable choice of the unit point, we arrive at the normal form for the second type of quadratic Cremona transformation: Φ ∶ (x0 ∶ x1 ∶ x2 ) ↦ (x′0 ∶ x′1 ∶ x′2 ) = (x1 x2 ∶ x0 x2 ∶ x21 ).

(7.22)

Type 3: A single base point

We choose the regular conic g1 ∶ x21 − x0 x2 = 0. It shall be osculated by all other conics of the associated net at P1 = (1 ∶ 0 ∶ 0). The singular conic g0 ∶ x1 x2 = 0 (a pair of lines) and g1 span a pencil of osculating conics. The repeated line g2 ∶ x22 = 0 and g1 span a pencil of hyperosculating conics. Therefore, g0 , g1 , and g2 span the associated net of the quadratic transformation with a single base point. After choosing an appropriate unit point in P′ , the coordinate functions of the transformation read Φ ∶ (x0 ∶ x1 ∶ x2 ) ↦ (x′0 ∶ x′1 ∶ x′2 ) = (x1 x2 ∶ x21 − x0 x2 ∶ x22 ).

(7.23)

The conics and pairs of lines, labeled with g0 , g1 , and g2 which span the associated net of conics for all the three types of birational quadratic mappings, are illustrated in Figure 7.62. We can summarize in the following theorem: Theorem 7.5.1 In projective planes P2 (F) over an algebraically closed field F there are three types of quadratic Cremona transformations Φ ∶ P → P′ . Provided an appropriate choice of coordinate frames in both projective planes P and P′ , the coordinate functions of the three types are given by (7.21) with λ0 = λ1 = λ2 = 1, (7.22), and (7.23). The respective inverse transformations read ϕ−1 ∶ (x′0 ∶ x′1 ∶ x′2 ) ↦ (x0 ∶ x1 ∶ x2 ) = (x′1 x′2 ∶ x′2 x′0 ∶ x′0 x′1 ), ϕ−1 ∶ (x′0 ∶ x′1 ∶ x′2 ) ↦ (x0 ∶ x1 ∶ x2 ) = (x′1 x′2 ∶ x′0 x′2 ∶ x′0 ), 2

−1

ϕ

∶

(x′0

∶

x′1

∶

x′2 )

↦ (x0 ∶ x1 ∶ x2 ) =

(x′0 2

− x′1 x′2

∶

x′0 x′2

∶

x′2 2 ).

(7.24)

360

Chapter 7: Polarities and pencils

Proof: We have to show that the so far necessary conditions are sufficient.

In the case of type 1, we use x′0 x0 = x0 x1 x2 = x′1 x1 = x′2 x2 and find the system x′0 x0

= 0,

−x′1 x1 −x′2 x2

x′0 x0

=0

of linear equations which can be solved for (x0 ∶ x1 ∶ x2 ).

A similar procedure for type 2, yields x′0 x0 = x0 x1 x2 = x′1 x1 and x′0 x1 = x21 x2 = x′2 x2 , and consequently, the system of linear equations x′0 x0

= 0,

−x′1 x1 −x′2 x2

x′0 x1

= 0.

This system of equations can also be solved for (x0 , x1 , x2 ).

In the case of type 3, we have x′0 x2 = x1 x22 = x′2 x1 and x′0 x1 − x′s x0 = x21 x2 − x22 x0 = (x21 − x0 x2 )x2 = x′1 x2 . Thus, we find the system of linear equations x′2 x0

−x′2 x1

+x′0 x2

−x′0 x1

+x′1 x2

= 0, = 0.

The solutions of the latter system are the coordinate functions of the inverse mapping.

◾

As an immediate consequence of the proof of Theorem 7.5.1, we have: Theorem 7.5.2 Each of the birational quadratic transformations listed in Theorem 7.5.1 can be determined by two bilinear forms, depending on the type of transformation: • Type 1: x0 x′0 − x1 x′1 = x′0 x0 − x2 x′2 = 0, • Type 2: • Type 3:

x0 x′0 − x1 x′1 = x′1 x0 − x2 x′2 = 0,

x2 x′0 − x1 x′2 = x0 x′2 − x1 x′0 + x2 x′1 = 0.

Proof: We have to distinguish between the three types of transformations. The coordinate frames in P and P′ are (P0 , P1 , P2 ; P ) and (P0′ , P1′ , P2′ ; P ′ ), where P and P ′ are correponding under ϕ. • Type 1: The bilinear form x′1 x1 − x′2 x2 = 0 is equivalent to (x′1 ∶ x′2 ) = (x2 ∶ x1 ). This is the analytic representation of a projective mapping π0 from the pencil of lines about P0 to the pencil of lines about P0′ (see Theorem 5.4.1). The mapping π0 acts as follows: π 0 ∶ L P0 − ∧ LP ′

0

with

′ ′ ′ ⎧ ⎪ ⎪ l2 = [P0 , P1 ] ∶ x2 = 0 ↦ π0 (l2 ) = [P0 , P2 ] ∶ x1 = 0, ⎨ ′ ′ ′ ⎪ ⎪ ⎩ l1 = [P0 , P2 ] ∶ x1 = 0 ↦ π0 (l1 ) = [P0 , P1 ] ∶ x2 = 0.

We can do the same with x0 x′0 − x2 x′2 = 0 and find (x′0 ∶ x′2 ) = (x2 ∶ x0 ) which gives the analytic representation of a projective mapping β from the pencil about P1 to the pencil about P1′ : π 1 ∶ L P1 − ∧ LP ′

1

with

′ ′ ′ ⎧ ⎪ ⎪ m1 = [P1 , P0 ] ∶ x2 = 0 ↦ π1 (m1 ) = [P1 , P2 ] ∶ x0 = 0, ⎨ ′ ′ ′ ⎪ ⎪ ⎩ m2 = [P1 , P2 ] ∶ x0 = 0 ↦ π1 (m2 ) = [P1 , P0 ] ∶ x2 = 0.

361

7.5 Quadratic Cremona transformations

Finally, there exists a third projective mapping π2 from the pencil about P2 to the pencil about P2′ with [P2 , P0 ] ↦ [P2′ , P1′ ], [P2 , P1 ] ↦ [P2′ , P0′ ]. The points X ∉ E can be transformed by mapping [P0 , X] and [P1 , X] to their respective images through P0′ and P1′ . The image point is X ′ = π0 ([P0 , X]) ∩ π1 ([P1 , X]). Obviously, X ′ is also incident with π2 ([P2 , X]. Note the difference between the constructions of the image point under a birational quadratic mapping and under a collinear transformation. In case of a birational quadratic mapping of type 1, the line [P0 , P1 ] is not mapped to [P0′ , P1′ ], neither under π0 nor under π1 . All points X ∈ [P0 , P1 ] {P0 , P1 } are mapped to the same point P2′ = π0 (l1 ) ∩ π1 (m1 ). Thus, P2′ is an exceptional point for ϕ−1 , since its preimage is not uniquely determined. With Ẽ = [P0 , P1 ] ∪ [P1 , P2 ] ∪ [P2 , P0 ] and Ẽ′ = [P0′ , P1′ ] ∪ [P1′ , P2′ ] ∪ [P2′ , P0′ ], the restriction ϕ ̃ of ϕ as a mapping P Ẽ → P′ Ẽ′ is one-to-one and onto. It can be written in terms of homogeneous coordinates as 1 1 1 ϕ ̃ ∶ P Ẽ → P′ Ẽ′ , (x0 ∶ x1 ∶ x2 ) ↦ (x′0 ∶ x′1 ∶ x′2 ) = ( ∶ ∶ ), x0 x1 x2 ̃ since x0 x1 x2 ≠ 0 for any xF ∉ E.

• Type 2: In this case there are only two projective mappings: π0 from the pencil about P0 to the pencil about P0′ , and π1 from the pencil about P1 to the pencil about P1′ . The analytic representations of π0 and π1 read π0 ∶ (x′0 ∶ x′2 ) = (x2 ∶ x1 )

π1 ∶ (x′0 ∶ x′1 ) = (x1 ∶ x0 ).

and

Again, π0 ([P0 , P1 ]) ≠ [P0′ , P1′ ] holds true, since x1 = 0 causes x′2 = 0. Unlike in the previous case, π1 ([P1 , P0 ]) = [P0′ , P1′ ], because x1 = 0 implies x′0 = 0.

• Type 3: In this case, there is only one projective mapping π0 left, acting from the pencil about P0 to the pencil about P0′ . Its analytical representation also follows from the coordinate functions of the birational quadratic mapping and reads π0 ∶ (x′0 ∶ x′2 ) = (x1 ∶ x2 ).

The second bilinear form expresses the fact that the ϕ-image x′ F of the point xF is a point on the polar line of xF with regard to the correlation

since

′ ⎛ u0 ⎞ ⎛ 0 κ ∶ ⎜ u′1 ⎟ = ⎜ 0 ⎝ u′ ⎠ ⎝ 1 2

−1 0 0

0 ⎞ ⎛ x0 ⎞ 1 ⎟ ⎜ x1 ⎟ 0 ⎠ ⎝ x2 ⎠

u′0 x′0 + u′1 x′1 + u′2 x′2 = −x1 x′0 + x2 x′1 + x0 x′2 = 0. With a cyclic shift of the coordinates (y0′ ∶ y1′ ∶ y2′ ) ∶= (x′2 ∶ x′0 ∶ x′1 ), the latter two bilinear forms (Theorem 7.5.2) change to x2 y1′ − x1 y0′ = x0 y0′ − x1 y1′ + x2 y2′ = 0. The correlation κ is now replaced by ′ ⎛ v0 ⎞ ⎛ 1 ̃ κ ∶ ⎜ v1′ ⎟ = ⎜ 0 ⎝ v′ ⎠ ⎝ 0 2

0 −1 0

0 ⎞ ⎛ x0 ⎞ 0 ⎟ ⎜ x1 ⎟ . 1 ⎠ ⎝ x2 ⎠

Now, the coordinate matrix of ̃ κ is symmetric. Identifying the coordinate frame of the preimage with the new coordinate frame in the image plane, then ̃ κ is the polarity w.r.t. the conic p ∶ y0′ 2 − y1′ 2 + y2′ 2 = 0. The points ϕ(X) is the intersection of π0 ([P0 , X]) with the polar line of X with regard to p.

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Chapter 7: Polarities and pencils

◾ The following theorem will be useful, when we deal with special quadratic transformations. A proof can be found in books [25, 34]: Theorem 7.5.3 Under a birational quadratic transformation of type 1 any algebraic curve c of degree n is transformed to an algebraic curve of degree 2n, in general. If the exceptional points P0 , P1 , P2 of the transformation are points of c with respective multiplicities µ(Pi ) = mi for i = 0, 1, 2, then the degree of the (proper) image curve equals 2n − m0 − m1 − m2 . The multiplicities of the exceptional points P0′ , P1′ , P2′ on c′ in the image plane P′ equal n − m1 − m2 , n − m2 − m0 , and n − m0 − m1 , respectively (Figure 7.63). Moreover, if R ≠ P1 , P2 is a common point of c and [P1 , P2 ] and R is r-fold on c, then the line π0 ([P0 , R]) is an r-fold tangent of c′ at P0′ . The contents of Theorem 7.5.3 will be of importance when we deal with special quadratic transformations, e.g., the inversion which maps circles to circles. c′

P2=P2′

P2=P2′ c

c′

P0=P0′

c

P1=P1′ P0=P0′

P1=P1′

FIGURE 7.63. Quadratic Cremona transformation Φ ∶ P → P′ = P with canonical identification of base points. Left: A conic c touches the base line [P0 , P2 ], meets the line [P1 , P2 ] in two real points, and has no real point on [P0 , P1 ]. Thus, the image curve c′ has a cusp of the first kind at P1′ = P1 , a double point with two real tangents at P0′ = P0 , and an isolated double point at P2′ = P2 . Right: The conic c goes through the base point P2 . Thus, the image curve splits into a cubic c′ , being the proper image, and the line [P0′ , P1′ ] = [P0 , P1 ]. Since c meets [P0 , P1 ] in two real points, c′ has a double point with two real tangents at P2′ = P2 .

363

7.5 Quadratic Cremona transformations

Similar results hold for the transformations of type 2 and 3. For more details, we refer again to [25, 34]. In the beginning of the present section, we defined quadratic transformations by prescribing three homogeneous quadratic polynomials as the coordinate functions of the transformation. The geometric approach via pencils of conics showed that (from the viewpoint of Projective Geometry) there are only three types of birational quadratic transformation. However, is not possible to prescribe three arbitrarily chosen quadratic forms in order to define a quadratic birational transformation. The coefficients of the three underlying forms have to fulfill certain algebraic conditions in order to make the transformation invertible, and thus, birational (cf. [47]). The geometric condition is simply that the three conics obtained from the analytic representation of the transformation share three points (algebraically counted).

Special quadratic Cremona transformations Transformation of doubly conjugate points

Assume we are given two different curves of degree two, c ∶ xT Ax = 0 and d ∶ xT Bx = 0, in any projective plane P2 (F). Herein, A = (aik ) and B = (bik ) with i, k ∈ {0, 1, 2} are symmetric matrices in F3×3 . Let further P denote the pencil spanned by these two curves. The pencil should contain conics in order to exclude cases where both c and d are singular and share a line. We are going to show that the pairs (P, P ′ ) of conjugate points w.r.t. both curves c and d, are related in a birational quadratic transformation Φ depending on the type of the pencil. So these pairs of points are doubly conjugate. The resulting transformation Φ is involutive, i.e., ϕ = ϕ−1 , since conjugacy is a symmetric relation (cf. Theorem 7.1.2). Two points X and X ′ are conjugate w.r.t. c and d if their homogeneous coordinates (x0 ∶ x1 ∶ x2 ) and (x′0 ∶ x′1 ∶ x′2 ) fulfill the two bilinear equations 2

′ ∑ aik xi xk = 0 and

i,k=0

2

′ ∑ bik xi xk = 0.

i,k=0

364

Chapter 7: Polarities and pencils

They can be considered as linear homogeneous equations in the unknowns (x′0 ∶ x′1 ∶ x′2 ) which yields 2

∑ai0 xi x0 ′

i=0 2

2

+ ∑ ai1 xi x′1 i=0 2

2

+ ∑ ai2 xi x′2 i=0 2

= 0,

′ ′ ′ ∑ bj0 xj x0 + ∑ bj1 xj x1 + ∑ bj2 xj x2 = 0.

j=0

j=0

j=0

The solutions are, up to a common factor, x′0 = x′1 = x′2 =

2

∑ (ai1 bj2 − ai2 bj1 )xi xj ,

i,j=0 2

∑ (ai2 bj0 − ai0 bj2 )xi xj ,

(7.25)

i,j=0 2

∑ (ai0 bj1 − ai1 bj0 )xi xj .

i,j=0

Obviously, we have found a rational mapping ϕ ∶ (x0 ∶ x1 ∶ x2 ) ↦ (x′0 ∶ x′1 ∶ x′2 ). It is of degree two, provided that the quadratic forms on the right-hand side of (7.25) do not have a non-constant common divisor. By construction, ϕ is involutive, and therefore, it is birational. The mapping ϕ does not change if we replace the curves c and d with other curves from the pencil P because the conditions for conjugacy are linear combinations of the homogeneous equations above. In order to make Φ a birational quadratic transformation, the existence of one, two, or three base points is necessary. These are points where Φ does not produce a unique image point, i.e., in this case we are looking for points whose polar lines w.r.t. c and d agree or are even undetermined. If P is a pencil of the third kind (two line elements) or a pencil of the fifth kind (pencil of hyperosculating conics), then there are infinitely many such exceptional points. In the first case, these are the points on the chord common to all conics in the pencil. In the second case, the points on the common tangent at the hyperosculation point play this particular role. It is easy to show that the quadratic forms on the right-hand side of (7.25) have a common linear factor if, e.g., c is a repeated line. In this case, Φ becomes a collineation. Consequently, the only cases that are left, originate from a pencil P of the first, second, or fourth kind. According to Theorem 7.5.1, the trans-

7.5 Quadratic Cremona transformations

365

formation Φ is then of type 1, 2, or 3. This type of quadratic Cremona transformation is called transformation of doubly conjugate points. The fixed points of this mapping are the self-conjugate points of c and d, i.e., the base points of the pencil P. The projective mappings π0 , π1 , π2 in the pencils about the base points are involutive and their fixed lines pass through these fixed points. Now, we can state: Theorem 7.5.4 Let c and d be two curves of degree two out of a pencil P in P2 (F) which is either of the first, or the second, or the fourth kind.

1. The transformation Φ of doubly conjugate points with respect to c and d is an involutive quadratic birational transformation of type 1, or 2, or 3 if the pencil P is of type 1, or 2, or 4. The mapping Φ remains unchanged if c and d are replaced by other curves from P. 2. The exceptional points of Φ and its inverse are exactly those points whose polar lines with regard to c and d coincide. In the case of a pencil of the first kind, these are the vertices of the common polar triangle. The fixed points of Φ are the base points of the pencil P. In the case of a quadratic transformation of the first type, the projective mappings in the pencils about the exceptional points are involutive projectivities whose fixed lines are the lines joining the base points of Φ with the base points of P.

3. The poles of lines l /∋ P0 , P1 , P2 with regard to all conics in P are contained in a conic l′ that passes through all exceptional points of Φ. If P is of the first kind with base points B1 , B2 , B3 , and B4 , then l′ contains on each line [Bi , Bj ] the fourth harmonic point L′ij to Lij ∶= [Bi , Bj ] ∩ l with respect to Bi and Bj , i.e., H(Bi , Bj , Lij , L′ij ). Proof: The only thing that is left to show is the third part of the theorem. The locus of all image points P ′ of points P ∈ l is an irreducible curve of degree two (cf. Theorem 7.5.3). The image l′ passes through all exceptional points. On the other hand, the image point P ′ can be found as the intersection of the polar lines of P with regard to c and d which form pencils with the poles Lc and Ld of l w.r.t. c and d as carriers, when P traverses l. Thus, l′ is generated by a projective mapping from the pencil Lc to the pencil Ld , Lc (πc (P )) − ∧ l(P ) − ∧ Ld (πd (P )), and l′ is passing through the pencils’ vertices. Here, and in the following, πc and πd denote the polarities with regard to c and d. Since any two different curves from P yield the same quadratic Cremona transformation, and thus, the same image curve l′ , it has to carry the poles of l with regard to all conics in the pencil. This is what has been stated.

366

Chapter 7: Polarities and pencils

If P is a point on a fixed line of the involution π0 , e.g., P ∈ [B1 , B2 ], then the conjugate point P ′ which lies on the polars πc (P ) and πd (P ), is the fourth harmonic point of P with respect to [B1 , B2 ], since B1 , B2 ∈ c, d.

◾

It is worth to be noted that in the special case of the projectively closed Euclidean plane, the line l can be chosen as the line at infinity and the common polar triangle can be chosen as a finite triangle. Then, l′ carries the centers of all conics in the pencil P, the vertices of the diagonal triangle of the base points of P, and furthermore the midpoints of all sides of the base quadrangle of P. This conic l′ is called the nine-point conic. This conic appears also in Section 9.1, cf. Corollary 9.1.1. Doubly conjugate lines

The principle of duality (see Section 5.1, especially on page 192) guarantees that the results from this section so far have valid dual counter parts. The dual version of the transformation of doubly conjugate points is called the transformation of doubly conjugate lines. We denote this transformation by Φ⋆ . Φ⋆ acts on the set of lines in P2 (F) as follows: Let c⋆ and d⋆ be two (dual) conics from a range F of the first, or the second, or the fourth kind. Pairs (l, l′ ) of lines are doubly conjugate if l′ ∈ πc⋆ (l) and l′ ∈ πd⋆ (l).

Again, πc⋆ and πd⋆ denote the polar systems with regard to c⋆ and d⋆ . Moreover, the lines l and l′ are doubly conjugate with respect to any two different curves out of F. The exceptional lines of Φ⋆ are those lines whose poles with regard to c⋆ and d⋆ coincide. In the case of a range of the first kind, these are the lines p0 , p1 , p2 of the polar triangle common to all conics of F. The fixed lines of Φ⋆ are the base lines of the range F. The points of intersection of corresponding lines l and l′ with the exceptional lines are corresponding in involutive projective mappings π0 , π1 , π2 acting on the base lines. The fixed points of these involutions are incident with the fixed lines of Φ⋆ , i.e., they are the points of intersection of tangents common to c⋆ and d⋆ . Any two out of the three involutions are sufficient to determine the quadratic transformation Φ⋆ . A pencil of lines about P0 on the exceptional line p0 is mapped to a pencil of lines about π0 (P0 ) ∈ p0 , since the pencil that corresponds to p0 with

7.5 Quadratic Cremona transformations

367

vertex p1 ∩ p2 splits off. The points p0 ∩ p1 and p0 ∩ p2 are corresponding in this involution π0 , as well. On the other hand, pencils of lines whose vertices are not in p0 ∪ p1 ∪ p2 are transformed to irreducible curves of class two containing the lines p0 , p1 , p2 .

Orthogonal conjugate lines

In the projectively extended Euclidean plane, we can study a special case of doubly conjugate lines. Assume that c⋆ and d⋆ are a pair of confocal conics with a center C. Thus, the four common tangents t1 , t2 , t3 , t4 of c⋆ and d⋆ are isotropic lines, i.e., two of them pass through A1 = (0 ∶ 1 ∶ i) while the others pass through A2 = (0 ∶ 1 ∶ −i). The three diagonal lines p0 , p1 , p2 of the base quadrilateral, i.e., the axes of c⋆ and d⋆ together with the line at infinity, form the exceptional set E ⋆ = p0 ∪ p1 ∪ p2 of Φ⋆ . The quadratic Cremona transformation Φ⋆ defined by c⋆ and d⋆ induces an involution π2 on the ideal line ω ∶ x0 = 0 leaving the absolute points A1 and A2 of Euclidean geometry fixed (cf. Example 6.4.8 on page 266). Therefore, any line l ∉ E ⋆ is orthogonal to its image l′ = Φ⋆ (l). Φ⋆ is called transformation of conjugate normals with respect to any conic in the range E of confocal conics spanned by c⋆ and d⋆ . Note that one conic is sufficient to determine this range. The involutions π0 and π1 on the axes are called the focal involutions. In the case of a conic c with center C, in both focal involutions the center C corresponds to the ideal point. The fixed points of the focal involutions π1 and π2 are the focal points of c and agree with those from Definition 7.1.5 in Section 7.1. Each vertex V of c is mapped to its center of curvature V ⋆ under the focal involution. This holds true for ellipses as well as hyperbolas as can be seen in Figure 7.64: In the case of the ellipse (left-hand side), we apply Φ⋆ to the line [V1 , V2 ] joining both vertices. The conjugate normal passes through E (pole of [V1 , V2 ] w.r.t. c) and is orthogonal to [V1 , V2 ]. We know this construction from Corollary 3.2.1 (Figure 3.11, Section 3.2), where the proof was based on the comparison of similar right-angled triangles. The construction of the centers of curvature at the vertices of the hyperbola shown on the right-hand side in Figure 7.64 is equivalent to the construction from Exercise 3.2.6 (cf. Figure 3.14). If Φ⋆ is defined by a parabola, then the transformation of conjugate normals is a quadratic

368

Chapter 7: Polarities and pencils V2 E

c

c C F2

V1⋆ F1

l

E

V1

l′

C V1 F2

F1

V1⋆

V2⋆ FIGURE 7.64. Centers of curvature at vertices: ellipse (left), hyperbola (right).

Cremona transformation of type 2. The focal involution on the parabola’s axis is the reflection in the focal point. There is a deeper reason for the fact that the focal involutions send vertices to the corresponding centers of curvature, as shown below. Conjugate normals and the evolute of a conic

The transformation Φ⋆ of conjugate normals with respect to the conic c maps the set of c’s tangents to the set c’s normals. From Section 3.2, we know that the envelope c⋆ of normals of a curve is its evolute. According to Theorem 7.5.3, the evolute of a conic is a curve of class four. This mirrors the results from Section 3.2. The axes of c and the ideal line ω are double tangents of the evolute. The vertices of c and the contact points of the respective normals with c⋆ , i.e., the centers of curvature at the vertices, are corresponding in the related focal involutions. Evolutes of ellipses, parabolas, and hyperbolas are displayed in Figures 3.15, 3.18, and 3.20. In the case of an ellipse and a hyperbola, we have learned in Exercise 3.2.8 that the construction of centers of curvature is equivalent to the transfer of an affine ratio which is based on (3.29). Since the quadratic transformation Φ⋆ of conjugate normals preserves cross-ratios, on the tangent tP the cross-ratio of P ∈ c together with the points on the exceptional lines is equal to the cross-ratio of the respective image points on the image line nP . One of the four points in question on tP and its corresponding point

369

7.5 Quadratic Cremona transformations

on nP are at infinity. Hence, by virtue of (5.8), the cross-ratio reduces to an affine ratio.

l′

P

2

c1

t1

p1

l

p0

F2 M

F1

1

c

t2

π1 (2)

p

h

P1⋆

s

π0 (1)

c2

FIGURE 7.65. The transformation of orthogonal conjugate lines l ↦ l′ w.r.t. the conic c maps the pencil P onto tangents of Chasles’s parabola p. The polarity in c sends p onto the Apollonius hyperbola h which intersects c at the pedal points w.r.t. P . These pedal points belong also to the strophoid s which is the pedal curve of p w.r.t. P .

◾

Example 7.5.1 Chasles’s parabola and the Apollonian hyperbola. The transformation Φ⋆ of conjugate normals w.r.t. the ellipse or hyperbola c maps the pencil of lines through a point P outside the exceptional lines onto the tangents of a conic p, which contacts all three exceptional lines (Figure 7.65). Therefore, p is a parabola which is named after the French mathematician Michel Chasles, 1793–1880. On each axis, the contact point with p corresponds to the pedal point w.r.t. P in the related focal involution. The parabola p remains the same when c is replaced by any other conic from the confocal family, e.g., by the ellipse c1 and the hyperbola c2 passing through P . Therefore, p contacts the corresponding tangent lines t1 and t2 which bisect the angle < ) F1 P F2 . The points of contact of these lines with p are the centers P1⋆ , P2⋆ of curvature of P w.r.t. c1 and c2 . It can be shown that the center P1⋆ of curvature of c1 is the pole of the tangent t1 to c1 with respect to c2 (Figure 7.65), and vice versa. The directrix of p (orthoptic curve of p) passes through P and the center M of c. Common tangents of p and c contact c at the pedal points of normals drawn from P onto the conic c. When p is polarized in c, we obtain an equilateral hyperbola h = πc (p) which contains P , the center M , the ideal points of the axes, and the at most four pedal points of normals on c w.r.t.

370

Chapter 7: Polarities and pencils

P . It is the Apollonian hyperbola which we will meet again in Section 9.3. The pedal points in question are also located on the strophoid s which already has been depicted in Figure 2.27.

Inversions In this section, we shall look at inversions from the projective point of view. This gives a more general concept and shows that the inversion from page 7.1.2 is just a special case and its properties are clear and obvious. Projective Inversions

Let c be a conic in P2 (F) and let further C ∈ P2 (F) be some point. Two points X and X ′ in P2 (F) are said to be inverse if (1) (2)

X, X ′ , C are collinear and X, X ′ are conjugate with regard to c.

(7.26)

The mapping X ↦ X ′ is involutive, provided there exists a unique image point X ′ of X. We call such a mapping a (projective) inversion. The point C is called the center of the inversion, and the points of c remain fixed. Without loss of generality, we may assume that c ∶ x0 x2 − x21 = 0 and

C = P1 = (0 ∶ 1 ∶ 0) if C ∉ c, and C = P0 = (1 ∶ 0 ∶ 0) if C ∈ c.

Then, in both cases we have two bilinear conditions. The first condition equals the polar form of c x0 x′2 + x2 x′0 − 2x1 x′1 = 0,

and, depending on whether C ∉ c or C ∈ c, the second means (x0 ∶ x2 ) = (x′0 ∶ x′2 ) (x1 ∶ x2 ) = (x′1 ∶ x′2 )

⇐⇒ ⇐⇒

x0 x′2 − x2 x′0 = 0 x1 x′2 − x2 x′1 = 0

if C ∉ c, if C ∈ c.

Solving the systems of linear equations yields the coordinate functions of the respective transformations (x′0 ∶ x′1 ∶ x′1 ) = (x0 x1 ∶ x0 x2 ∶ x1 x2 ) if C ∉ c, (x′0 ∶ x′1 ∶ x′1 ) = (2x21 − x0 x2 ∶ x1 x2 ∶ x22 ) if C ∈ c.

371

7.5 Quadratic Cremona transformations

Hence, this transformation is involutive, and therefore, birational.

Under the assumption C ∉ c, the exceptional points of the inversion are P1 = C and the points P0 , P2 of contact of the tangents drawn from C to c if they exist. The projective mapping π0 ∶ P1 − ∧ P1 is the identity mapping. From the bilinear forms we can deduce the analytic description (x′0 ∶ x′1 ) = (x2 ∶ x1 ) of the projective mapping π1 ∶ P0 − ∧ P2 , and we have π2 = π1−1 . At this transformation, two base points are interchanged: P0 ↦ P2 and P2 ↦ P0 . The conic c is generated by π1 (or π2 ) according to Definition 6.1.1, since all the points of c are fixed.

X

X

P2 X

P2

′

c

c P0

P1 =C

X′ P0 =C

P1

FIGURE 7.66. Projective inversions with the center C, either not on c (left) or on c (right).

We construct the image X ′ of X using the quadrangle on c which contains the points P0 and P2 and has X as a diagonal point (Figure 7.66). Since one of the remaining diagonal points lies on [P0 , P2 ] which is the polar line of P1 w.r.t. c, the opposite side of the diagonal triangle passes through P1 and, consequently, the third diagonal point coincides with X ′ . If C ∈ c, then C is the only exceptional point and π0 is the identity mapping. Therefore, this involutive quadratic transformation is of type 3. Summarizing this, we can say: Theorem 7.5.5 The projective inversion in c with center C is an involutive quadratic (birational) transformation either of type 1 or of type 3, depending on whether C ∉ c or C ∈ c.

372

Chapter 7: Polarities and pencils

In the first case, the base triangle consists of the center C of inversion and the points T1 , T2 of contact of the two tangents drawn from C to c. The projective mapping in the pencil about C is the identity mapping. The conic c is generated by the projectivities π1 = π2−1 ∶ T1 − ∧ T2 . Any point on c {T1 , T2 } is left fixed. In the case C ∈ c, C is the only base point. Inversion in a circle

Assume that c is a circle, and the center C of inversion coincides with the circle’s center. Then, the projective inversion in c with center C is called the inversion or reflection in the circle c which we have seen in Example 7.1.1 on page 281. The exceptional points of the inversion are C and the absolute points I = (0 ∶ 1 ∶ i) and I = (0 ∶ 1 ∶ −i) of Euclidean geometry. If the radius of c equals r, then any pair (P, P ′ ) of inverse points satisfies CP ⋅ CP ′ = r 2 .

According to Theorem 7.5.3, any line l (not incident with a base point) is mapped under the inversion to an irreducible curve l′ of degree two passing through all three exceptional base points C, I, and I. Thus, l′ is a circle through C (see Figure 7.67). P l c

c P′ C =A

C p

l′

FIGURE 7.67. Left: The inversion in a circle c with center C (of the inversion) maps a point P to P ′ on P ’s polar line with regard to c. Right: A straight line l is mapped to a circle l′ through C.

A curve k of degree two that does not contain any of the base points is mapped to a quartic curve k ′ with double points at the base points C,

373

7.5 Quadratic Cremona transformations

I, and I. Curves with double points at the absolute points of Euclidean geometry are called bicircular. If k is a circle with C ∉ k, then the isotropic lines [C, I] and [C, I ] split off, and the proper image curve k ′ is a circle. Therefore, the inversion maps circles to circles, provided that lines are counted as circles. The center of a circle is in general not mapped to the center of the image circle. ◾

Example 7.5.2 Center of the inverse circle.

1. In order to derive the coordinate representation of the inversion in terms of Cartesian coordinates, we assume c ∶ x2 + y 2 = r 2 and C = (0, 0). Let further X = (ξ, η) ≠ (0, 0) be an arbitrary point. Then, the polar lines of X with regard to c has the equation pX ∶ xξ + yη = r 2 , and the line [X, C] has the equation xη − yξ = 0. Computing [X, C]∩ pX , we obtain the coordinate functions of the inversion as (ξ, η) ↦ (

r2 ξ r2 η , 2 ). 2 + η ξ + η2

ξ2

(7.27)

2. The image of a circle k in general position is a circle k ′ , as we have deduced above. But where is the center of k ′ ? We may assume that k ∶ (x − d)2 + y 2 = R2 is the circle to be inverted in c. This means no restriction because the coordinate system can always be rotated about C such that the center of k lies on the x-axis. Inserting the coordinate functions of the inversion (and using x and y instead of ξ and η), we find r 2 − 2dr 2 x + (d2 − R2 )(x2 + y 2 ) = 0

which is the equation of the image circle. The center can be found by completing to full squares and equals dr 2 ( 2 , 0) . d − R2 Figure 7.68 shows that there is a simple construction of the center of the inverse circle.

Circles that intersect c at right angles are fixed as a whole, but not pointwise since in this case r 2 equals the power of C with respect to l. Quadratic Cremona transformations (indeed any rational mapping) preserve the contact order of curves, provided, the contact takes place outside the exceptional set. To be precise: If the multiplicity µP (c, d) of two curves c, d at such a common point P is m, then the intersection multiplicity µP ′ (c′ , d′ ) at P ′ of the transformed curves c′ and d′ also equals m. In the special case m = 2 we obtain that curves in contact are mapped to curves in contact. Therefore, these mappings are so-called contact transformations. If a curve c ∈ P E and a circle (or line) l are in second or third order contact at P , i.e., m = 3 or m = 4, then c′ and l′ are in second or third order contact at P ′ , and l′ is again a circle or line. Therefore, the inversion maps osculating circles of c to such of c′ . Moreover, vertices of c, i.e., points with stationary curvature, are mapped to vertices of c′ .

374

Chapter 7: Polarities and pencils c k k′ C

Ck′

Ck

FIGURE 7.68. How to find the center Ck′ of the inverse k ′ of a circle k ? First, invert C in k, and second, invert the obtained point Ck in c.

The inversion in a circle is conformal but not orientation preserving. The inversion in c (with center C) maps the tangent t to a curve k at P to a circle t′ that touches the image curve k ′ at P ′ . The two angles that are enclosed by t′ and [C, P ] have equal measures but different orientations and the angle at C is congruent to the angle −a2 . Then, the affine transformations α ∶ X = (x, y) ↦ X ′ = (x′ , y ′ ) = (λx, y −

k ) with c → c′ yield the new expressions 2

c∶ kF (X) = (λ2 − 1)x2 − 2ky = 0,

c′ ∶

λ2 − 1 ′ 2 x − 2ky ′ − k 2 = 0. λ2

Lemma 8.1.4 Let α be an affine transformation between two confocal conics of the same type in E2 as defined before. If A′ and B ′ are the respective α-images of any two points A, B in the plane, then A′ B − AB ′ = k [F (A) − F (B)] 2

2

with the function F according to (8.15) or (8.16).

(8.17)

392

Chapter 8: Affine Geometry

Proof: Due to (8.13) follows for A = (x, y) and B = (ξ, η) 2

A′ B − AB ′

2

= [λ2 x2 − 2λxξ + ξ 2 + µ2 y 2 − 2µyη + η2 ]

− [x2 − 2λxξ + λ2 ξ 2 + y 2 − 2µyη + µ2 η2 ]

= (λ2 − 1)x2 + (µ2 − 1)y 2 + (1 − λ2 )ξ 2 + (1 − µ2 )η2 = A′ B − AB ′ = k [F (A) − F (B)] , 2

2

which confirms the claim for central conics. A similar computation proves the stated equation for parabolas.

◾

m P1 P1′

c c

′

P4′

M′ M

P4

O P3′

P2′

P3

P2 FIGURE 8.10. The affine transformation α which sends the conic c to the confocal conic c′ , transforms the points P1 , . . . , P4 ∈ c on a circle m with center M ′ to points P1′ , . . . , P4′ ∈ c′ on a circle centered at the preimage M of M ′ .

Here are a few consequences:

(i) For any two points P, Q ∈ c holds F (P ) = F (Q) = 1. Thus, Lemma 8.1.4 confirms again Ivory’s statement P Q′ = P ′ Q (compare with Theorem 2.2.3 and see Figure 8.11). There are two other relations of ‘Ivory type’: (ii) For the center O which remains fixed under α, and P, Q ∈ c we have 2 2 2 2 OP − OP ′ = OQ − OQ′ = −k because of F (O) = 0, hence OP ′ + OQ = OP + OQ′ . 2

2

2

2

393

8.1 Conjugate diameters of ellipses and hyperbolas

ÐÐ→ Ð→ ÐÐ→ This implies for the involved vectors because of ∥OQ − OP ′ ∥2 = ∥P ′ Q∥2 = ÐÐ→ ÐÐ→ Ð→ ∥P Q′ ∥2 = ∥OQ′ − OP ∥2 for the scalar product ÐÐ→ Ð→ Ð→ ÐÐ→ ⟨ OP ′ , OQ ⟩ = ⟨ OP , OQ′ ⟩.

(iii) Suppose that P1 , P2 , . . . ∈ c belong to a circle m centered at M ′ (see 2 2 Figure 8.10). Then, from Lemma 8.1.4 follows M ′ Pi −M Pi′ = k(F (M )− 1) independent of i. This means that α transforms the points Pi ∈ c ∩ m to points of a circle centered at the α-preimage M of M ′ . In other words, the pencil spanned by c and m is mapped to a pencil which contains again a circle, this time with the center M . (iv) Suppose that the circle m is the osculating circle of c at P . Then, m and c span a pencil of osculating conics and we can state: If M ′ is the center of curvature of c at P , then M is the center of curvature of c′ at P ′ . Consequently, the evolute of the α-image c′ is the α-preimage of the evolute of c. P R c P′

R′ c′

Q′ F1

O

Q

F2

FIGURE 8.11. Particular case of Ivory’s theorem in E2 with P Q′ = P ′ Q, P R′ = P ′ R, and consequently QR = Q′ P + P R′ .

We conclude with a particular case of an Ivory quadrangle in E2 . Referring to Figure 8.11, if Q ∈ c lies on the tangent to c′ at P ′ , then P ∈ c lies on the tangent to c′ at Q′ . A verification of this statement which holds for central conics (8.15) and parabolas (8.16) and is left to the readers. This statement can be generalized to a result that has already been cited by W. Böhm in [17, p. 153]. But most probably, it is of earlier origin (note [7, p. 118]). As stated in Theorem 10.2.1, it is also valid in the Minkowski plane and the hyperbolic plane.

394

Chapter 8: Affine Geometry

Theorem 8.1.4 Each Ivory quadrangle in the Euclidean plane E2 has diagonal lines that are tangent to the same curve of the confocal range. P c Q

P′

′

c

c0

Q′

FIGURE 8.12. The two diagonal lines in the Ivory quadrangle P P ′ Q′ Q contact the same confocal conic c0 . Proof: Our proof is based on homogeneous coordinates as introduced in Section 5.3. We begin with the case of central conics (Figure 8.12).

Suppose that P = (1 ∶ x ∶ y) and Q = (1 ∶ ξ ∶ η) are two points of the conic c as given in (8.15), while P ′ = (1 ∶ λx ∶ µy) and Q′ = (1 ∶ λξ ∶ µη) are their α-images on the conic c′ . Then, by (5.4), the diagonal lines [P, Q′ ] and [P ′ , Q] have the respective homogeneous line coordinates ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ µxη − λyξ ⎞ ⎜ x ⎟ × ⎜ λξ ⎟ = ⎜ y − µη ⎟, ⎝ y ⎠ ⎝ µη ⎠ ⎝ ⎠ λξ − x

⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ λxη − µyξ ⎞ ⎜ λx ⎟ × ⎜ ξ ⎟ = ⎜ µy − η ⎟. ⎝ µy ⎠ ⎝ η ⎠ ⎝ ⎠ ξ − λx

These two lines contact the confocal conic c0 with the elliptic coordinate l, i.e., the line coordinates satisfy the dual equation of c0 if, and only if, −(µxη − λyξ)2 + (a2 + l)(y − µη)2 + (b2 + l)(λξ − x)2 = 0 and −(λxη − µyξ)2 + (a2 + l)(µy − η)2 + (b2 + l)(ξ − λx)2 = 0.

We show the equivalence of these two conditions by verifying that the difference of the lefthand sides vanishes. This difference equals (λ2 − µ2 )(x2 η2 − y 2 ξ 2 ) + (a2 + l)(1 − µ2 )(y 2 − η2 ) + (b2 + l)(1 − λ2 )(x2 − ξ 2 ).

From F (P ) = F (Q) = 1 in (8.15) follows

y 2 − η2 x2 − ξ 2 = − , a2 b2

y2 x2 = 1 − , b2 a2

η2 ξ2 = 1 − . b2 a2

Moreover, we have 1 − λ2 = −k/a2 and 1 − µ2 = −k/b2 . Thus, the difference reduces to

b2 (a2 + k) − a2 (b2 + k) 2 2 k k (x η − y 2 ξ 2 ) − 2 (a2 + l)(y 2 − η2 ) − 2 (b2 + l)(x2 − ξ 2 ) a2 b2 b a b2 − a2 x2 − ξ 2 = k 2 2 (x2 η2 − y 2 ξ 2 ) + k(a2 + l − b2 − l) a b a2 2 k(b2 − a2 ) 2 ξ2 x = [x (1 − 2 ) − ξ 2 (1 − 2 ) − (x2 − ξ 2 )] = 0. a2 a a

8.1 Conjugate diameters of ellipses and hyperbolas

395

In the case of parabolas c, c′ from (8.16) we proceed similarly. This time, the image points have the coordinates P ′ = (1 ∶ λx ∶ y − k2 ) and Q′ = (1 ∶ λξ ∶ η − k2 ). The lines [P, Q′ ] and [P ′ , Q] contact the confocal parabola c0 with the dual equation (a2 + l)u21 + lu22 − 2u0 u2 = 0 if, and only if, (a2 + l)(y − η +

(a + l)(η − y + 2

k ) 2

2

k ) 2

2

+ l(λξ − x)2 − 2(xη −

k 2

x − λyξ)(λξ − x) = 0 and

+ l(λx − ξ) − 2(yξ −

k 2

ξ − λxη)(λx − ξ) = 0.

2

Again, straightforward computation reveals that the difference of the left-hand sides in these equations, namely 2k(a2 + l)(y − η) +

kl 2 2k (ξ − x2 ) + 2 (yξ 2 − x2 η) − k(x2 − ξ 2 ) a2 a

vanishes due to F (P ) = F (Q) = 0 with F defined in (8.16).

◾

The Theorem 8.1.4 can be used to reprove a theorem published 2017 by I. Izmestiev and S. Tabachnikov in [76]. For this purpose we introduce the following notation. Definition 8.1.2 A billiard is the trajectory of a mass-point within a planar domain called the billiard table with ideal physical reflections in the boundary. At the same token, billiards in ellipses will be the topic of Section 9.6. Remark 8.1.2 Sometimes in the literature, a ‘billiard’ is understood as the boundary of the billiard table, while the paths of mass-points are called billiard trajectories or, in the periodic case, simply periodics (compare, e.g., [141] and [115]).

Theorem 8.1.5 Given a range of confocal conics, let P P ′ Q′ Q be an Ivory quadrangle with diagonal lines [P, Q′ ] and [P ′ , Q] tangent to the confocal conic c0 . If the contact points with c0 are no interior points of the segments P Q′ and P ′ Q, then there exist infinitely many inscribed quadrangles A1 . . . A4 with tangents of c0 as sides lines, and they all are periodic billiards of equal lengths in the Ivory quadrangle (Figure 8.13). Remark 8.1.3 It is noteworthy, that in Ivory quadrangles the inscribed billiards offer a continuous transition from one diagonal to the other while the perimeter remains constant. Proof: Given the Ivory quadrangle P P ′ Q′ Q (note Figure 8.14), let (k, l), (k ′ , l), (k ′ , l′ ), and (k, l′ ) be the respective elliptic coordinates of the vertices P , P ′ , Q′ , and Q. When a point D1 moves from P along the diagonal P Q′ towards Q′ , then its second elliptic coordinate varies continuously from l to l′ . The same holds for D2 when moving along P ′ Q from P ′

396

Chapter 8: Affine Geometry c

c

P

P

A1

A1 A2 c A2

Q

′

Q

′

c P′ A3 c0

A4 Q

c0

A4

′

P′

A3 Q′

FIGURE 8.13. Left: In the Ivory quadrangle P P ′ Q′ Q with diagonals tangent to c0 all billiards A1 A2 A3 A4 with sides that contact c0 are closing and their perimeters equal 2 P Q′ . Right: As a counter example, if for a diagonal P Q′ or P ′ Q the contact point with c0 is an interior point, then ‘inscribed’ quadrangles A1 A2 A3 A4 exceed the border.

to Q. Therefore, for each D1 ∈ P Q′ exists a unique point D2 ∈ P ′ Q with the same second elliptic coordinate. Similarly, there exists a unique D4 ∈ QP ′ sharing with D1 the first elliptic coordinate. We can complete D2 D1 D4 by the point D3 to a smaller Ivory quadrangle. Since the diagonal line [D2 , D4 ] = [P ′ , Q] contacts c0 , the same must hold for the other diagonal line which implies that D3 ∈ P Q′ . Continuity guarantees that the tangent [P, Q′ ] from D1 to c0 does not switch over to the other tangent line. Let c and h be the two confocal conics through P and c1 and h1 those through D1 . The affine transformations c1 → c and h1 → h, which again are unique due to continuity, send D1 to vertices A1 ∈ c and A2 ∈ h, respectively (Figure 8.14). The Ivory quadrangle A1 P A2 D1 has one diagonal line tangent to c0 . Hence, also [A1 , A2 ] contacts c0 . Similarly, we obtain from D3 the points A3 on the curved side P ′ Q′ and A4 ∈ Q′ Q with [A3 , A4 ] tangent to c0 . Moreover, the Ivory quadrangles A2 P ′ A3 D4 and A1 D2 A4 Q show that finally all four sides of the quadrangle A1 A2 A3 A4 are tangents of c0 .3 From focal properties (note Theorem 2.2.4) or from Desargues’s involution theorem follows that A1 A2 A3 A4 is a four-sided billiard in the Ivory quadrangle P P ′ Q′ Q (Figure 8.13, left). Finally, Ivory’s theorem reveals (see Figure 8.14) that A 1 A 2 + A 2 A 3 + A 3 A 4 + A 4 A 1 = P D1 + P ′ D4 + D3 Q ′ + D2 Q

= (P Q′ − D1 D3 ) + (P ′ Q + D2 D4 ) = 2P Q′ . Hence, all these billiards have the same length. 3

◾

The existence of periodic billiards in the Ivory quadrangle can also be proved with the aid of a canonical parametrization (Theorem 9.6.10), as mentioned on page 506.

397

8.1 Conjugate diameters of ellipses and hyperbolas P

A1 c

A2 D1 Q

D4 D2 c′

h′ D3

P′

A4 A3 h

Q

′

c1

h1

FIGURE 8.14. The billiard A1 A2 A3 A4 in the Ivory quadrangle P P ′ Q′ Q along with the two diagonals.

Principal axes transform within the affine setting

Now, we want to treat conics in the affine plane A2 (F) in terms of affine coordinates. We start with the homogeneous equation x ∶ xT Ax = 0 with x = (x0 , x1 , x2 ) and A ∈ F3×3 . We switch to affine coordinates be setting (x0 ∶ x1 ∶ x2 ) = (1 ∶ x ∶ y). Thus, we obtain c ∶ a11 x2 + 2a12 xy + a22 y 2 + 2a01 x + 2a02 y + a00 = 0

(8.18)

with coefficients aij ∈ F and i, j ∈ {0, 1, 2}. This time, we allow A ∈ F3×3 to be singular. Let the vector x stand for x = (x, y) ∈ F2 .

Now, the equation of c can also be written as the sum of a quadratic form in x and y, a linear from in x and y, and a constant, as xT Ax + 2aT x + a00 = 0

(8.19)

x = x′ + m

(8.20)

with the vector a = (a01 , a02 ) and the matrix A = (

a11 a12 ) ∈ F2×2 . a12 a22 In a first step, we apply a translation to c such that the linear form in (8.19) becomes the zero-form. For that purpose we let with m ∈ R2 to be determined. We insert (8.20) into (8.19) and find x′ Ax′ + 2(mT A+aT )x′ + mT Am + 2aT m + a00 = 0, T

(8.21)

398

Chapter 8: Affine Geometry

since mT A + aT = (AT m + a)T and A T = A. Only in the cases where A is regular, the vector m can be chosen such that the linear form in (8.21) vanishes. At this point we have to distinguish between two cases: A is either regular or singular: 1. A is regular. Then, we have m = −A −1 a

(8.22)

which represents the center. (8.21) changes to F (x) = x′ Ax′ + α = 0 T

(8.23)

where α = aT A−1 a − 2aT A −1 a + a00 . It remains to discuss the cases α = 0 or α = −1 since otherwise we can divide the equation by −α.

Though A ∈ F2×2 is symmetric, it will in general not be a diagonal matrix. However, it is easy to diagonalize A by an appropriate choice of our affine coordinate frame. We only have to pay attention to the fact that A defines an involution of conjugate diameters as well as that of conjugate ideal points. Let E1 = e1 = (1, 0) and E2 = e2 = (0, 1) be the unit points. Then, we can express the entries of A as T T T a11 = eT 1 Ae1 , a12 = e1 Ae2 , a21 = e2 Ae1 , a22 = e2 Ae2 .

This reveals that A has diagonal form if, and only if, the base vectors point into conjugate directions. So, we can choose one basis vector e′1 such that a′11 = e′1 T Ae′1 ≠ 0. Then, we have e′1 T Ae′2 = 0 for the second basis vector. If F = R, we can still scale e′1 and e′2 in an appropriate 1 ′ way. In the case a′11 > 0 we let e′′1 = √a1′ e′1 , otherwise e′′1 = √−a ′ e1 . Thus, the new entry a′′11 can be +1 or −1. Similarily, we obtain a′′22 = +1, or a′′22 = −1, or a′′22 = 0. The latter case only arises if A is singular. 11

11

2. In the case det A = 0, we have no solution for m. But after changing T to the new basis {e′1 , e′2 } with e′2 Ae′2 = 0, we get (x + a01 )2 + 2a02 y + 2 ′ a00 − a01 = 0. We set x = x + a01 , and under the assumption a02 ≠ 0, we rescale e2 in order to obtain a02 = 1, and eliminate the constant by a translation along the y-axis. If a02 = 0, the rescaling of e1 reduces the constant either to 0, −1, +1.

Theorem 8.1.5 In the real affine plane A2 (R) there is an affine coordinate frame such that the equation of any regular or singular conic can be

399

8.1 Conjugate diameters of ellipses and hyperbolas

reduced to one of the following standard equations: x2 + y 2 − 1 = 0, x2 − y 2 − 1 = 0, x2 + y 2 + 1 = 0, x2 + 2y = 0, x2 + y 2 = 0, x2 − y 2 = 0, x2 + 1 = 0, x2 − 1 = 0, x2 = 0.

Remark 8.1.4 A general theory of the principal-axis transformation of surfaces of degree 2 in the Euclidean n-space and a list of normal forms can be found, e.g., in [109, Section 3.1].

●

Exercise 8.1.4 Center of a conic and the critical points of quadratic functions.

Assume f (x, y) = a11 x2 + 2a12 xy + a22 y 2 + 2a01 x + 2a02 y + a00 is a quadratic function over R2 . Show that the maxima/minima of f are found at the zeros of the gradient gradf = 2Ax + 2a, −1 or equivalently, at xm = −A a.

10 (−2, −3)T

43.603○

36.869○ 1 O

1

1

(1, 4)T

O

(5, 2)T 10

61.928○

1

O FIGURE 8.15. The conics from Exercise 8.1.5

●

Exercise 8.1.5 The following equations describe conics in the Euclidean plane (cf. Figure 8.15). Find out the type and all metrical invariants, such as centers (if there is one), the lengths of axes, and the angle enclosed by the principal axis and the x-axis of the coordinate system. In case of a parabola, find the vertex. Hint: The eigenvectors of the coefficient matrix A of the quadratic part deliver a basis of an orthonormal frame centered either at the center or the vertex of the conic depending on whether it is an ellipse/hyperbola or a parabola. 1. −38x2 + 168xy − 87y 2 − 656x + 858y − 1968 = 0,

2. 10656x2 − 5880xy + 10369y 2 + 2208x − 77072y + 31936 = 0, 3. 64x2 + 240xy + 225y 2 − 2650x − 1284y + 10918 = 0.

400

Chapter 8: Affine Geometry

8.2 Conics are rational quadratic Bézier curves Bézier curves and the algorithm of de Casteljau We shall describe briefly and without going into too much detail what a Bézier curve is. We restrict ourselves to quadratic Bézier curves since conics allow rational quadratic parametrizations. B1

B1 t t 1−

t

B2 B0

1− t t B02

B11

t

1−

B01

B11

t

B2 B0

t

t

B01

1−

1−

t

B1

B2 B0

FIGURE 8.16. Repeated affine combinations of points reduces the number of control points in each step of de Casteljau’s algorithm. Finally, the algorithm terminates if the last polygon consists of just one point.

Let B0 = b0 , B1 = b0 , and B2 = b2 be three different non-collinear points in an affine plane A2 (F), e.g., in the Euclidean plane. We call these points control points and the open polygon is called a control polygon. Now, by choosing a value t ∈ [0, 1], we construct points B01 and B11 in between the given control points as affine combinations b10 = (1 − t)b0 + tb1 and b11 = (1 − t)b1 + tb2 . Once again, we can introduce a new “in between point” B02 as b20 = (1 − t)b10 + tb11

with the same fixed t ∈ [0, 1].

In the first step, we defined two new points from three given points. In the second step, we gained one new point from two points. This procedure is called algorithm of de Casteljau due to the French physicist and mathematician Paul de Casteljau (1930–2022). (It can easily be guessed how it works with an arbitrary number of initial points.) Figure 8.16 shows how to construct the point B02 according to de Casteljau. It is obvious that the b20 depends on t which plays the role of an affine ratio, and if t runs through the entire interval [0, 1], then b20 is the para-

401

8.2 Conics are rational quadratic Bézier curves

metrization of a quadratic Bézier curve4 . We obtain this parametrization in full length by inserting the affine combinations for b10 and b11 . This leads to b20 (t) = (1 − t)2 b0 + 2t(1 − t)b1 + t2 b2 . (8.24)

Equation (8.24) parametrizes a parabola if t ∈ R. The part of the Bézier curve that is parametrized over the unit interval [0, 1] is shown in Figure 8.17 (left). The three quadratic functions f0 = (1 − t)2 ,

f1 = 2t(1 − t),

f 2 = t2

(8.25)

are the well-known Bernstein polynomials of degree 2 which constitute a basis in the space of univariate polynomials of degree 2. Now, we have: Lemma 8.2.1 Any quadratic Bézier curve is a parabola. Any parabola can be parametrized as a quadratic Bézier curve. Proof: In order to show that (8.24) parametrizes a parabola, we assume (without loss of

y

B1

B2 B0

B2 =(b, c)

x B0 =(0, 0)

B1 =(a, 0)

FIGURE 8.17. Left: A parabola as a quadratic Bézier curve with control points B0 , B1 , and B2 . Right: An appropriately chosen coordinate system, cf. proof of Corollary 8.2.1. generality) that b0 = (0, 0)T , b1 = (a, 0)T , and b2 = (b, c)T with a ≠ 0, a ≠ b, b ≠ 0, c ≠ 0 (see Figure 8.17). Then, we compute an implicit equation of the curve by eliminating the parameter t from x = 2at(1 − t) + bt2 and y = ct2 and find (cx + (2a − b)y)2 − 4a2 cy = 0 which is a parabola with axis direction (2a − b, c)T .

For the second part, we refer to Example 6.3.1 in Section 6.3. Here, we have shown that any conic allows a rational parametrization. For a parabola we choose the center of the stereographic projection (that gives the rational parametrization) as the one and only ideal point of the parabola. Then, the parametrization becomes polynomial.

◾

4

Pierre Étienne Bézier (1910–1999) was a French engineer.

402

Chapter 8: Affine Geometry

One important property of Bézier curves is the affine invariance of their representation. Since B02 (t) is a composition of affine combinations, the affine image of any Bézier curve is the Bézier curve defined by the transformed control polygon.

Conics with center The quadratic Bézier representation can only be used for the parametrization of parabolas (cf. Lemma 8.2.1). What about the other conics? From Section 2.1 (and also from Section 6.3), we already know that conics admit rational parametrizations. Thus, we can use the concept of rational Bézier curves. A rational quadratic Bézier curve can be written in the form 2

c(t) = (∑ wi fi (t)) i=0

−1

2

⋅ ∑ wi bi fi (t)

(8.26)

i=0

where bi are the coordinate vectors of the control points, fi (t) are the Bernstein polynomials (8.25) of degree 2, and wi ∈ R are weights. Increasing weights of any control point, pulls the curve towards this point. With appropriately chosen weights, we change not only the shape of the rational Bézier curve, we can even change the type of conic. Thus, we can parametrize ellipses, parabolas, and hyperbolas in this way. -0.8

B2

B2

-0.6 -0.5 -0.4

B1

ϕ C ϕ

B0

-0.2

0 0.3

1

3 10

B1

B0

FIGURE 8.18. Left: A circle as a rational Bézier curve and the geometric meaning of the weight w1 . Right: Some rational quadratic Bézier curves with common control polygon and different weights of B1 .

With the help of the unit circle centered at (0, 0)T , we demonstrate how to find weights and control points in order to make the rational Bézier

403

8.2 Conics are rational quadratic Bézier curves

curve a circle. We are interested in a segment of the circle. The segment shall have the central angle 0 < 2ϕ < π. Thus, we can fix the control points B0 and B2 and the point B1 lies on the tangents at both endpoints of the curve segment. We obtain b0 = (cos ϕ, − sin ϕ)T ,

b1 = (1/ cos ϕ, 0)T ,

b2 = (cos ϕ, sin ϕ)T .

Note that the point B1 is inverse to the point [C, B1 ] ∩ [B0 , B2 ] with respect to the circle. We choose w0 = w2 = 1 and insert the latter control points into (8.26). With the unknown weight w1 , we find the parametrization

(1 − 2t + 2t2 ) cos ϕ + 2w1 (1 − t)t cos−1 ϕ (2t − 1) sin ϕ c(t)= ( , ) 2 2 (1 − t) + 2w1 (1 − t)t + t (1 − t)2 + 2w1 (1 − t)t + t2 which obviously differs from that given in (2.13) from Section 2.1 (and also from (6.6) in Example 6.3.1, Section 6.3) since it parametrizes a certain part of the curve over the unit interval. The parametrization fulfills the equation x2 +y 2 = 1 of the unit circle if, and only if, w1 = ± cos ϕ. Therefore, the weight of the control point B1 has to be ± cos ϕ if the radii at the two endpoints B0 and B2 enclose an angle of 2ϕ. Figure 8.18 (left, enclosed in the large ellipse) shows how the control points of a rational quadratic Bézier curve have to be chosen such that it becomes a circle. B2 B1

B0 B2 B1

B0 FIGURE 8.19. Control points for pieces of conics: ellipse (left), hyperbola (right).

●

Exercise 8.2.1 An equilateral hyperbola and an ellipse. Compute a rational parametrization of the part of the equilateral hyperbola starting at b0 = (1, 1)T ending at b2 = (e, 0)T for arbitrary e ∈ R. What happens if e → ∞? Where is the control point b1 and what are the weights (see Figure 8.19, right)?

●

Exercise 8.2.2 Symmetric arc on an ellipse. Give a rational parametrization of the symmetric arc of an ellipse with b0 = (a cos ϕ, −b sin ϕ)T and b2 = (a cos ϕ, b sin ϕ)T with 0 < ϕ < π/2 and a, b ∈ R {0, }. Show that b1 = ( cosa ϕ , 0)T and w1 = ± cos ϕ (cf. Figure 8.19, left).

404 ●

Chapter 8: Affine Geometry

Exercise 8.2.3 G3 splines from conics. In [51], splines consisting of conics that are glued together with G3 continuity are studied. Mainly geometric ideas solve the problem of constructing such splines. Collineations and Desargues involutions as explained in Sec. 7.4 turn out to be the right tool.

405

8.3 Conics and number theory

8.3 Conics and number theory Diophantine quadratic equations

The solutions (x, y) of a quadratic equation in two variables x and y

⎛1⎞ a00 + 2a01 x + 2a02 y + a11 x + 2a12 xy + a22 y = (1 x y)A ⎜ x ⎟ = 0 (8.27) ⎝y⎠ 2

2

with coefficients a00 , . . . , a22 ∈ F are the points of a conic in the affine plane A2 (F). Special interest was given to the case where F is replaced with the ring Z of integers. This was studied by Carl Friedrich Gauß (1777–1855) and many others before. The case a01 = a02 = a12 = 0 and a11 = a22 = −a00 = 1 leads to the unit circle with four integer solutions (±1, 0) and (0, ±1). We have derived a rational parametrization of the unit circle in Section 2.1 (and also in Section 6.3, see page 243): (

1 − t2 2t , ). 1 + t2 1 + t2

We recall that the stereographic projection was used in order to elaborate this. Now, we replace the affine parameter t with homogeneous parameters (u, v) by t = uv and then we switch to the u homogeneous representation (u2 − v 2 , 2uv, u2 + v 2 ) with

(u, v) ∈ F2 {(0, 0)}.

(8.28)

Thus, we have parametrized the Pythagorean triplets (x, y, z) satisfying x2 + y 2 = z 2 over any field F with charF ≠ 2 and even over Z.

If the integer square z 2 equals the sum x2 + y 2 of two integer squares, then the circle with radius z carries at least twelve points with integer coordinates: (0, ±z), (±z, 0), (±x, ±y), and (±y, ±x). Note that in any case x ≠ y, for 2x2 cannot be an integer square. The decomposition of integer squares into sums of integer squares is not unique (Figure 8.20). ●

Exercise 8.3.1 Rational parametrization of the pseudo-Euclidean unit circle.

Apply the stereographic projection to an equilateral hyperbola x2 − y 2 = 1, in order to obtain a rational parametrization which (after homogenizing) yields (u2 + v2 , 2uv, u2 − v2 )

with

(u, v) ∈ F2 {(0, 0)}.

This is a description of the rational points on the pseudo-Euclidean unit circle.

(8.29)

406

) 39 2, ) (5 , 33 6 (5 ) , 25 (60 6) (63, 1 (65, 0)

(36 (40 , 77) , 75 ) (5 1, 68 )

(16, 6 3) (25 , 60 ) (3 (3 3, 56 ) 9, 52 )

(0, 65)

(0, 85)

(13, 84)

Chapter 8: Affine Geometry

) 51 8, (6 ) , 40 (75 , 36) (77 (84, 13)

(85, 0)

FIGURE 8.20. The decomposition of integer squares need not be unique: The circle with radius 65 carries 36 lattice points since 652 = 162 + 632 = 252 + 602 = 332 + 562 = 392 + 522 (left). The same works with 852 = 132 + 822 = 362 + 772 = 402 + 752 + 512 + 682 (right).

Equation (8.28) gives a rational representation of the solutions of the quadratic diophantine equation (8.27) in two variables in a normal form (diagonalized). However, these solutions are in general not integer solutions.

For any n ∈ N at least one ellipse can be given such that it contains exactly n lattice points (points with integer coordinates). However, there are only finitely many lattice point on each such ellipse. For the hyperbola it is feasible that there are infinitely many integer points on it. This is also the case for the parabola: If a conic c has the normal form x2 − 2ay = 0 with a ∈ F {0} and charF ≠ 2, then it is easy to find all points with coordinates in F2 on c.

Problems of this kind are, indeed, number theoretic in nature. Results including bounds on the number of integer lattice points on arcs contained in ellipses can be found in [31]. The case of hyperbolas is discussed in [32]. In Table 8.1, we have collected the smallest ellipses passing through n = 3, 4, . . . , 20 integer lattice points. Here, an ellipse is called small if the semimajor axis is as small as possible. The attached figure illustrates some of the ellipses from the left-hand side of the Table 8.1. To be more precise: The figure shows some translata of some of the ellipses from the table in order to make it easier to illustrate the curves.

407

8.3 Conics and number theory n 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

equation of the ellipse x2+xy+y 2−x−y=0 x2+y 2−x−y=0 2x2−xy+2y 2−x+y−3=0 x2−xy+y 2−1=0 2x2−xy+2y 2−5x−4y−6=0 x2+y 2−x−y−2=0 x2−xy+3y 2−7x−6y=0 x2−xy+4y 2−5x−5y−6=0 3x2−3xy+4y2−21x−21y−10=0 x2+y 2−5x−5y=0 2x2−xy+3y 2−31x−26y−25=0 x2−xy+4y 2−12x−9y−13=0 2x2−xy+5y 2−39x−36y−41=0 x2+y 2−7x−7y−8=0 2x2−xy+3y 2−51x−51y−54=0 x2−xy+y 2−7x−7y=0 2x2−xy+3y 2−61x−61y−6=0 2x2−xy+2y 2−27x−27y−29=0

TABLE 8.1. Smallest ellipses passing through n = 3, . . . , 20 integer lattice points. A bold-faced number indicates that this particular ellipse is a circle.

For the general form of quadratic diophantine equation as given in (8.27) solutions can easily be parametrized if at least one integer solution (ξ, η) is known. The pencil of lines with vertex (ξ, η) is now parametrized by p = (ξ + t, η + kt) with the parameter t on the lines and the parameter k in the pencil. Both parameters can be replaced by homogeneous ones if necessary. Inserting p in (8.27) yields a linear equation in t because t splits off from the quadratic polynomial for p is a solution of (8.27). From the remaining linear equation we find t as rational expression in aij and k: kT Ax t = −2 T x Ax where x = (1, ξ, η) and k = (0, 1, k). Pell equation

The diophantine quadratic equation x2 − dy 2 = ±1

(8.30)

with d ∈ N {0} is usually called Pell equation after the English mathematician John Pell (1611–1685). The solutions (±1, 0) are called trivial and are obviously independent of d. The case d = 1 has no integer solution

408

Chapter 8: Affine Geometry

tC B

N

A c=n

tA

tB C

FIGURE 8.21. Just a rough idea what a conic in P2 (Z2 ) may look like: N is c’s nucleus where the three tangents of c meet each other.

(besides the trivial ones) since the difference of two integer squares cannot be 1. This special type of diophantine equation can be interpreted as the equation of a hyperbola. The task is to find integer points on the hyperbola for any d ∈ N. Once a solution (ξ0 , η0 ) is found, many others can be generated by the linear recurrence (

ξk+1 ξ dη0 ξ )=( 0 )( k ). ηk+1 η0 ξ0 ηk

However, the linear recurrence does not necessarily produce all solutions. Solving Archimedes’s cattle problem requires to solve a Pell equation (cf. [88]). Conics over finite fields

In a projective plane over a finite field F there are only finitely many points. Let N = #F be the number of elements in F. Then, any line has N + 1 points and any conic contains that many points (see Example 6.4.9 in Section 6.4). The number of points in P(F3 ) equals N 2 + N + 1. The very special case N = 2, i.e., the projective minimal plane, shows conics with a completely different behavior than anywhere else: Assume that a conic c in the projective plane is given by the homogeneous equation c ∶ x20 + x21 + x22 = 0.

409

8.3 Conics and number theory

Clearly, there are only three points on c because it contains as much points as a line does. These points are A = (0 ∶ 1 ∶ 1),

B = (1 ∶ 0 ∶ 1),

C = (1 ∶ 1 ∶ 0).

These three points lie on one line n ∶ x0 + x1 + x2 = 0 which never happens in any other projective plane! The tangents at A, B, and C are the lines tA ∶ x1 + x2 = 0,

tB ∶= x0 + x2 = 0,

t C ∶ x0 + x1 = 0

which are concurrent in one point, namely N = (1 ∶ 1 ∶ 1). The point N is called the nucleus of c, and it is the pole of n with regard to c. Note that the equations of c and n describe the same set of points. Therefore, c = n. What about an affine classification? Assume that n is the line at infinity. Then, the affine part of c is empty. Only the center of c is present. It is the point N since it is the pole of n with regard to c. Figure 8.21 shall illustrate the incidence relations of c’s tangents.

Note that polarities in P2 (Z2 ) are nullpolarities at the same time, i.e., each polar line contains the corresponding pole. For more details on conics in finite geometry we refer to [69]. ●

Exercise 8.3.2 Study the normal forms of conics in P2 (Z3 ) and give an affine classification of the conics in this plane.

The One-Seventh-conic

The rational number 1/7 can be written in decimal expansion as 1 = 0, 142857 . . . 7 Now, we take overlapping pairs of subsequent digits of the expansion and interpret them as affine coordinates of points in a plane. Thus, we have the six points (1, 4), (4, 2), (2, 8), (8, 5), (5, 7), (7, 1).

Surprisingly, these six points lie on an ellipse with the equation s ∶ 19x2 + 36xy + 41y 2 − 333x − 531y + 1638 = 0. The ellipse s is called the one-seventh-conic (see Figure 8.22).

410

Chapter 8: Affine Geometry y

y

x

x

FIGURE 8.22. The one-seventh-ellipses carry six points with integer coordinates derived from the decimal expansion of 71 .

●

Exercise 8.3.3 More one-seventh-conics

From the decimal expansion of (1, 4), (14, 42), (142, 428), (1428, 4285), ⋮

(4, 2), (42, 28), (428, 285), (4285, 2857),

1 7

we can create more sixtuples of overlapping pairs of integers: (2, 8), (28, 85), (285, 857), (2857, 8571),

(8, 5), (85, 57), (857, 571), (8571, 5714),

(5, 7), (57, 71), (571, 714), (5714, 7142),

(7, 1), (71, 14), (714, 142), (7142, 1428),

Show that these sixtuples of points are located on an ellipse if the number of digits in each coordinate is either 3k + 1 or 3k + 2 (with k ∈ N). The ellipses have the centers 9 (1, 1) , 2

99 (1, 1) , 2

9999 (1, 1) , 2

99999 (1, 1) , 2

....

If the number of digits in each place of the coordinate vectors equals 3k (with k ∈ N {0}), then the one-seventh-conics degenerate into pairs of parallel lines with the direction (2, −1). There also exists a family of one-thirteenth-conics: Write down the decimal expansion of 1/13 and define points in the same way as above. Show that these points define either ellipses or pairs of parallel lines. The centers of the ellipses defined from 1/13 are identical with the ones given above. The parallels have the direction (1, −3).

9 Special problems

A one-sheeted hyperboloid is a ruled quadric and carries two one-parameter families of lines. A top view shows the pattern of rulings as a part of a Poncelet grid. According to theoretical kinematics, the framework of crossing rods is flexible if all crossings of generators are realized as hinges.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2024 G. Glaeser et al., The Universe of Conics, https://doi.org/10.1007/978-3-662-70306-9_9

411

412

Chapter 9: Special problems

9.1 Conics in triangle geometry There are many ways to define conics related to a triangle ∆ with vertices A, B, C. We are going to pick out very few examples. All triangles have a circumcircle, i.e., the uniquely defined conic passing through the triangles vertices and the absolute points1 of Euclidean geometry. In Section 7.3, we have seen that each triangle in the Euclidean plane defines a pencil of equilateral hyperbolas including the Kiepert hyperbola, cf. Example 7.3.4. The base points of this pencil are the three vertices of the triangle together with the orthocenter. By the same token, the set of the vertices together with the orthocenter H is called an orthocentric quadrangle and has some symmetry: The orthocenter of any triangle built from three of these points is the fourth point.

C

C

k

k n

X22 k

A

X115 X22

B

X10 10

X44

X4 A

B k

FIGURE 9.1. Left: The Kiepert hyperbola k is the locus of perspectors of isosceles triangles attached to ∆’s sides and contains the centroid2 X2 and the orthocenter X4 . The three unlabeled points are the first Fermat point X13 , the outer Vecten point X485 , and the first Napoleon point X17 . Right: The Kiepert center X115 is located on the nine-point circle n. The Kiepert hyperbola also contains the Spieker center X10 . 1

The absolute points of Euclidean geometry are explained in Section 6.4. Every Euclidean circle passes through this pair of complex conjugate points on the line at infinity. Conversely, any non-degenerate conic that passes through the absolute points of Euclidean geometry is a Euclidean circle.

9.1 Conics in triangle geometry

413

Figure 9.1 shows an elementary approach to Kiepert’s hyperbola k: We attach three similar isosceles triangles to the sides of ∆ either to the outside or to the inside. The additional points of the attached triangles form triangles which are perspective to the base triangle ∆. This means that the three connecting lines with the opposite vertices have a point in common, the perspector. It turns out that all perspectors lie on k independent on whether the triangles are erected inside or outside ∆. Each triangle has four tritangent circles, the incircle and the three excircles (see Figure 9.2, left). These are the conics tangent to three lines passing through the absolute points. From Example 7.2.2, we learn that, generally speaking, there are four solutions to this problem. An exhaustive list of circles and conics related to triangles is available at [81] or [155]. In triangle geometry, special circles and conics are of interest: those that carry a huge bunch of centers or so-called central conics, i.e., conics whose equations in terms of homogeneous trilinear or homogeneous barycentric coordinates show a cyclic symmetry, cf. [80]. In the following, we list and illustrate some of the central conics, not just circles, that frequently appear in triangle geometry.

Homogeneous trilinear coordinates, barycentric coordinates In Section 5.3, we defined homogeneous coordinates in a projective plane. Here, we need a special kind of homogeneous coordinates. The plane we are dealing with is the Euclidean plane E2 . Whenever necessary, we use the projective and the complex extension. The vertices A, B, C shall serve as the base points which then have homogeneous coordinates A = (1 ∶ 0 ∶ 0), B = (0 ∶ 1 ∶ 0), C = (0 ∶ 0 ∶ 1). Any point in the plane of the triangle ∆ = ABC (except the ones on [A, B] ∪ [B, C] ∪ [C, A]) can be used for a unit point. The choice of the unit point specializes the coordinates. Two choices are frequently used: The choice of the triangle’s incenter X1 as the unit point leads to the (homogeneous) trilinear coordinates, sometimes called (homogeneous) trilinear distances. The choice of the centroid X2 for the unit point yields the (homogeneous) barycentric coordinates. 2

When we deal with triangle geometry, we shall follow the usual notation: X1 is the incenter of the triangle ∆, X2 is the centroid or barycenter, X3 is the circumcenter, X4 is the orthocenter. For a complete list of named triangle centers we refer to [81].

414

Chapter 9: Special problems e2

e2 e1 e1

u

n

C i

A

B

X5

X1

i e3

X11 e3

FIGURE 9.2. The prominent representatives of circles related to a triangle: Left: The incircle i, the three excircles e1 , e2 , e3 , and the circumcircle u. Right: The nine-point circle n touches the four tritangent circles of a triangle. The point of contact of n and i is the so called Feuerbach point X11 . Further, n passes through the midpoints of ∆’s edges and the midpoints of certain segments of the altitudes, and the feet of ∆’s altitudes.

The term homogeneous should be used with care when dealing with trilinear distances. The geometric meaning of the coordinates based on (A, B, C, X1 ) is the following: The incenter X1 has coordinates (1 ∶ 1 ∶ 1) for it is the chosen unit point. The incenter has equal distances to the three side lines of ∆, it is the radius ̺ of the incircle, thus (̺, ̺, ̺) is also a representation of X1 , expressing that X1 is actually at distance ̺ to any side of ∆. The vector (̺, ̺, ̺) is, therefore, called the actual trilinear coordinates of X1 . These coordinates are no longer homogeneous. Any point P in the plane of ∆ can be described by trilinear coordinates which are simply the oriented distances of P to the side lines [B, C], [C, A], [A, B] of ∆ in that particular ordering. These coordinates show signs in the different areas defined by ∆’s sides. Figure 9.3 shows the distribution of signs of the trilinear coordinates.

We denote the side lengths of ∆ by c = AB, a = BC, b = CA. Let further F denote the area of ∆. Then, homogeneous trilinear coordinates (p0 ∶ p1 ∶ p2 ) can be transformed into actual trilinear coordinates (d0 ∶ d1 ∶ d2 )

415

9.1 Conics in triangle geometry (− − +)

C (+ − +)

(+ − −)

A

(+ + +) (+ + −)

(− + +)

B

(− + −)

FIGURE 9.3. The distribution of signs of trilinear coordinates.

via

(d0 , d1 , d2 ) =

2F (p0 , p1 , p2 ) ap0 + bp1 + cp2 which can easily be seen by computing the area F of ∆ as the sum of the areas of the three subtriangles ABP , BCP , CAP . The scaling of coordinates fails if ap0 + bp1 + cp2 = 0

which characterizes points on the line L∞ at infinity. Thus, L∞ has (homogeneous) trilinear coordinates (a ∶ b ∶ c). A point P is called a triangle center or simply center of ∆ if its trilinear coordinates are cyclic symmetric functions in the side lengths of ∆. The first coordinate is usually called a center function. For example X1 = (1, 1, 1) (incenter) is a center. The centroid X2 = (bc ∶ ca ∶ ab) is also a center, since a → b → c → a changes bc to ca, ca to ab, and ab to bc. A central line is a line whose homogeneous trilinear coordinates are center functions.

Central circles

In terms of trilinear coordinates (x0 ∶ x1 ∶ x2 ), conics are given by homogeneous quadratic equations ⎛ q00 q01 q02 ⎞ ⎛ x0 ⎞ (x0 , x1 , x2 ) ⎜ q01 q11 q12 ⎟ ⎜ x1 ⎟ = 0 ⎝ q02 q12 q22 ⎠ ⎝ x2 ⎠

(9.1)

as is the case in terms of any kind of homogeneous coordinates (cf. Section 6.3).

416

Chapter 9: Special problems

In triangle geometry, especially to those conics is payed attention to, whose equation can be written as a cyclic sum. The terms in the sum are cyclic symmetric functions of the side lengths a, b, c of ∆. In [80], it is shown that all circles have a trilinear equation of the form (λ0 x0 + λ1 x1 + λ2 x2 )(ax0 + bx1 + cx2 ) + κ(ax1 x2 + bx2 x0 + cx0 x1 ) = 0 (9.2)

with appropriate λ0 , λ1 , λ2 , κ ∈ R. This is an affine parametrization of a pencil of conics (see Section 7.3) with the affine parameter κ ∈ R. The first term of (9.2) is a singular conic consisting of the common radical line of all circles in the pencil and the line at infinity. The representation of the pencil can be homogenized by replacing κ with κ1 ∶ κ0 . Then, κ0 = 0, or equivalently κ = ∞ yields the interpretation of the second term in (9.2): it is the circumcircle of ∆ with the equation ax1 x2 + bx2 x0 + cx0 x1 = 0. ●

Exercise 9.1.1 Center of a circle. Show that the center of a circle given by (9.2) has the following trilinear coordinates (−λ0 + λ1 cos C + λ2 cos B − κ cos A ∶ λ0 cos C − λ1 + λ2 cos A − κ cos B ∶ λ0 cos B + λ1 cos A − λ2 − κ cos C) (b2

− a2 )/(2bc)

where cos A = + is the cosine of ∆’s interior angle at A.3 The values cos B and cos C are the cosines of the interior angles at B and C. Make use of the fact that the center of a conic is the conic’s pole of the ideal line (cf. page 280). c2

●

Exercise 9.1.2 Center of an arbitrary conic. Show that the centers of the circumconics given by (9.4) have the trilinear coordinates (q01 (−aq01 + bq02 + cq12 ) ∶ q02 (aq01 − bq02 + cq12 ) ∶ q12 (aq01 + bq02 − cq12 )).

(9.3)

Simson line A well-known theorem which is erroneously ascribed to the English mathematican Robert Simson (1687–1768) is actually due to the Scottish mathematician William Wallace (1768–1843). It characterizes the points on the circumcircle: 3

This notation is standard in triangle geometry and shall not lead to any confusion.

417

9.1 Conics in triangle geometry

Theorem 9.1.1 The pedal points of the normals from a point P to the side lines of a triangle are collinear if, and only if, P is a point on the circumcircle u. The locus of collinearity is called the Simson line. A proof of this result can be found in the textbook [38]. Figure 9.4 illustrates the contents of Theorem 9.1.1 and some more results that relate to Theorem 9.1.1. s

C

P u

C u

s B

A

A

B

P FIGURE 9.4. Some results related to Theorem 9.1.1: Left: the Simson line. Middle: If the point P traverses u, then the Simson line envelopes a Steiner hypocycloid. Right: The remaining intersections of the normals through P and the circumcircle u give rise to three lines parallel to s.

Simson’s theorem appears in connection with parabolas: Theorem 9.1.2 Let p be a parabola with focus F and assume that u, v, w are three (mutually distinct and finite) tangents of p. The circumcircle of the triangle built by u, v, w passes through F . Proof: In Figure 2.13, we can see that the normals of a parabola’s tangent from the focus F meet the tangents at the tangent line of the vertex. Thus, the pedal points of the normals to the sides of the tangent triangle are collinear and F must be a point on the circumcircle of the tangent triangle according to Theorem 9.1.1.

◾

Figure 9.5 illustrates the contents of Theorem 9.1.2.

Central conics For conics that pass through the vertices, i.e., conics circumscribed to ∆, (9.1) is characterized by q00 = q11 = q22 = 0.

418

Chapter 9: Special problems

c t2

P2

p t3 t1

P1 F P3

FIGURE 9.5. The circumcircle of any tangent triangle of a parabola p passes through p’s focus (cf. theorem 9.1.2).

Therefore, the equations of the circumconics can be written in the form q01 =0 cyclic x2 ∑

(9.4)

where the cyclic sum means that all terms are obtained from the given one by a cyclic shift in all indices in {1, 2, 3} and components. For example q01 q01 q12 q20 = + + . x2 x0 x1 cyclic x2 ∑

Obviously, it is sufficient to prescribe one coefficient (center function) in order to determine the conic. In Section 7.5, we have learned that the isogonal conjugation and the isotomic conjugation are quadratic Cremona transformations. Comparing the normal form (7.21) of a Cremona transformation with three base points and (9.4), we can see that the circumconics are the isogonal transforms of certain central lines.

419

9.1 Conics in triangle geometry

● Exercise 9.1.3 Show that (9.1) is the equation of a parabola circumscribed to ∆ if, and only if, the coefficients qij satisfy 2 2 2 2abq01 q02 − a2 q01 + 2bcq02 q12 − b2 q02 + 2caq01 q12 − c2 q12 =0

together with the regularity condition det(qij ) ≠ 0. Steiner ellipses

Among the conics circumscribed to ∆ there is a special well-defined ellipse s called Steiner circumellipse. The tangents of s at ∆’s vertices are parallel to the opposite side lines. It is clear that these three line elements fit to a conic, since the tangents form the anticomplementary triangle of ∆ which is perspective to ∆ with the centroid S for the perspector. The perspector S also serves as the Brianchon point in a special version of Theorem 6.2.3 of these three line elements (see Figure 6.7).

B

A S

C

s

FIGURE 9.6. The common centroid S of ∆ = ABC and its anticomplementary triangle is the Brianchon point of ∆’s Steiner ellipse s.

We shall compare the area A∆ of an equilateral triangle ∆ with the area Au of its circumcircle u (Figure 9.6): Provided that the circumradius of √ 3 2 2π 3 3 2 ∆ equals R, we find A∆ = 2 R sin 3 = 4 R and Au = R2 π. Note that for an equilateral triangle the circumcircle is its Steiner circumellipse. Surprisingly, the ratio of these two areas is constant: Theorem 9.1.3 The area As of the Steiner circumellipse s of a triangle ∆ and the area A∆ of ∆ are related via √ A∆ ∶ As = 3 3 ∶ 4π (9.5)

420

Chapter 9: Special problems

independently of the choice of the triangle. Proof: There exist affine transformations α which map the given triangle ∆ = ABC onto an equilateral triangle A1 B1 C1 and, hence, the Steiner circumellipse s of ∆ onto the circumcircle s1 of A1 B1 C1 . Choose, e.g., A1 = A, B1 = B. Then, the affine transformations multiply all (signed) areas with a constant factor f = det A if A ∈ R2×2 represents the associated linear mapping in a Cartesian or an affine frame. Thus, the ratio A∆ ∶ As is affinely invariant.

◾

u B1

s B2

S1

A1 =A2

u

s

C2

C1

t

S2

S1

S2

z

z Z

L

T

R

Z

FIGURE 9.7. Left: The circumcircle u of an equilateral triangle can be mapped via an area preserving affine transformation (affine shearing, see page 196) to the Steiner circumellipse s of an arbitrary triangle. Right: The construction of the principal axis of the Steiner circumellipse s uses the Thales circle t centered at T on the axis of the affine mapping (shearing) and the bisector of the two centers S1 and S2 .

Figure 9.7 shows an elementary way to find the principal axis of the Steiner ellipse s being the affine image of the circumcircle u. In fact, this construction can be used for arbitrary affine mappings applied to a circle, provided that there is an axis, i.e., the affine mapping is a perspective affine mapping. Note that α is center preserving, for it preserves affine ratios. The construction in Figure 9.7 (right-hand side) uses the fact that conjugate diameters of a circle are orthogonal. Furthermore, the axes of the image s of u are conjugate and orthogonal. Thus, the fixed points L and R on the axis z (of α) have to lie on the Thales circle t through the centers of u and s. The Steiner circumellipse also appears in combination with three concurrent osculating circles of an ellipse (see page 105).

421

9.1 Conics in triangle geometry

The following theorem is usually ascribed to Morris Marden (US mathematician, 1905–1991)4 , though it was first published and proved by Jörg Siebeck (German mathematician) in [127]. It relates the Steiner inellipse of a triangle to the roots of a complex polynomial: Theorem 9.1.4 The zeros ϕ1 and ϕ2 of the derivative p′ (z) of the complex cubic polynomial p(z) = (z − α)(z − β)(z − γ) are the focal points of the Steiner inellipse of the triangle with vertices α, β, γ, provided that the complex numbers are identified with points in the Euclidean plane. A CC m B

F1

X2

F2

C FIGURE 9.8. The Steiner inellipse m of the triangle ABC with focal points F1 and F2 is centered at X2 . Proof: We assume that the Steiner inellipse of ∆ is given by the equation m∶

x2 y 2 + = 1. a2 b2

The Steiner inellipse of a triangle ∆ touches the sides at the midpoints. Therefore, we assume that one point C1 of contact has the complex coordinate γ1 = a

1 − t2 2t + ib . 1 + t2 1 + t2

Then, ∆’s vertex C opposite to C1 is given by γ = −2γ1 , because the center of the inellipse m is the centroid X2 of ∆ and divides the Cevian at the ratio 1 ∶ 2 (cf. Figure 9.8). The tangents from C to m meet the tangent [A, B] at the triangle vertices A and B with coordinate vectors √ √ 1 2 α = 1+t 3(1 − t2 )) , 2 (a(1 − t − 2 3t) + ib(2t + √ √ 1 2 β = 1+t 3(1 − t2 )) . 2 (a(1 − t + 2 3t) + ib(2t − 4

Marden published this result in 1945, never claiming that it was his result, see [91].

422

Chapter 9: Special problems

Now, we build

p(z) = (z − α)(z − β)(z − γ).

Differentiating once w.r.t. z, we find p′ (z) = 3(z 2 − a2 + b2 ) whose zeros in C are ϕ1 =

√ a2 − b2

and

√ ϕ2 = − a2 − b2

◾

which are the complex coordinates of the focal points of m.

Remark 9.1.1 Another proof of Theorem 9.1.4 can be found in [79]. Marden’s theorem can be generalized: Assume that α, β, γ are roots of a polynomial p(z) and have multiplicities r, s, t, i.e., they are the roots of p(z) = (z − α)r (z − β)s (z − γ)t . The two non-trivial zeros of p′ (z) are the focal points of the inellipse of ∆ that touches the sides at points dividing the edges of ∆ at ratios r ∶ s, s ∶ t, and t ∶ r (cf. [90]).

If Vi (with i ∈ {1, . . . , n}) are the vertices of an n-gon (not necessarily regular) that has an inscribed ellipse i, then there is a complex polynomial p(z) of degree n that vanishes at all Vi (or at the respective complex numbers). The focal points of i are among the roots of p′ (z) (see [110]). A characterizing property of the focal points of the Steiner circumellipse is given in [1].

Carnot’s theorem for conics

C c

C BC

n

c AC A

BA

CA CB B

AB A

CA

CB

B

A

B

FIGURE 9.9. Carnot’s theorem for conics: notations as used in Theorem 9.1.5 (left), the conic through the contact points of ∆’s tritangentent circles (middle). The nine point circle n passes through the midpoints of ∆’s edges and the pedal points.

A generic conic c intersects the three side lines [A, B], [B, C], [C, A] of a triangle ∆ = ABC in six points AB , AC ∈ [B, C], . . . (see Figure 9.9). Provided that c does not contain a triangle vertex, these six points form six affine ratios with ∆’s vertices. According to the French mathematician

9.1 Conics in triangle geometry

423

Lazard Nicolas Marguerite Carnot,5 the following theorem holds (cf. [64, 74]): Theorem 9.1.5 Assume that a conic c does not contain a vertex of a triangle ∆ = ABC, let CA , CB , . . . be the common points of c and [A, B], . . . , and let further ar(A, B, CA ), ar(A, B, CB ), . . . . Then, ●

ar(A, B, CA ) ⋅ ar(A, B, CB ) ⋅ ar(B, C, AB ) ⋅ ⋅ar(B, C, AC ) ⋅ ar(C, A, BC ) ⋅ ar(C, A, BA ) = 1

(9.6)

Exercise 9.1.4 Proof of Theorem 9.1.5. Elaborate a proof of Theorem 9.1.5. Hint: Since Carnot’s theorem involves affine ratios, it is favorable to attach an affine frame to ∆.

The equation (9.6) given in Theorem 9.1.5 can also be formulated in terms of lengths of segments. Then, Carnot’s equation (9.6) reads ACA ACB BAB BAC CBA CBC ⋅ ⋅ ⋅ ⋅ ⋅ = 1. BCA BCB CAB CAC ABA ABC In the middle of Figure 9.9, we can see the conic c passing through those points of contact of the tritangent circle and the sides of ∆ which lie in the interior of each side segment. In this particular case, the verification of Carnot’s theorem is rather simple since: It is elementary to verify that ACB = ABA = 12 (−a + b + c), BCB = BAB = 12 (a − b + c), CAC = CBC = 1 2 (a + b − c). The ‘inner’ contact points CA , AC , BA of the excircles are the reflections of the contact points of the incircle in the sides’ midpoints. ACA B Thus, we have, e.g., BC = BC ACB , and similarily for the others. Figure 9.9 A (right) shows the nine point circle as a very special case. Kiepert parabola

We have found the Kiepert hyperbola as the locus of perspectors of ∆ and the “new” points of triangles attached to ∆’s sides. What about the perspectrices? Surprisingly, the perspectrices turn out to be the set of tangents of a conic p, namely the Kiepert parabola (see Figure 9.10). Like in the case with the Kiepert hyperbola, we attach isosceles and similar triangles to ∆’s sides. The outermost points of the attached triangles 5

Lazare Nicolas Marquerite Carnot 1753–1823 was a French officer, mathematician, and politician.

replacemen

424

Chapter 9: Special problems

pe r sp rix ect

p e

C

C X99 u

B′

′

A C A

′

B

A

s

X3 X2 X4

X110

B

p

FIGURE 9.10. Left: The Kiepert parabola p is the envelope of all perspectrices x of ∆ and the triangles ∆x built by the “new” points A′ , B ′ , C ′ of the attached triangles. Right: The Euler line e of ∆ is the directrix of the Kiepert parabola p. The focal point of the Kiepert parabola is the triangle center X110 .

again form triangles which are perspective to ∆. Let us perform the projective closure of the Euclidean plane. This allows us to study extremal attached triangles. If the base angles of the isosceles triangles attached to ∆ reach 90○ , then the triangle of outermost points degenerates into the line at infinity. Thus, the line at infinity is the perspectrix in this case. Therefore, one of the tangents of the envelope is the line at infinity. The Brianchon point of p equals the Steiner point X99 of ∆ which is located on the Steiner circumellipse s. Nine-Point Conics and the Nine-Point Circle

The famous nine-point circle n (Figure 9.2) is centered at X5 (the ninepoint center) and is half the size of the circumcircle. The circle n is the circumcircle of the medial triangle, i.e., the triangle of the midpoints of ∆’s sides. It contains six more points: The three feet of the altitudes of ∆ and the three midpoints of the altitudes’ segments between the orthocenter and the vertices. However, the nine-point circle is a special appearance of a more general conic which is associated with any complete quadrangle, i.e., the six lines joining the four vertices of a quadrangle.

425

9.1 Conics in triangle geometry C

C

MCD n MAC

MCD

MBC

MAC Q

Q D

MAD A

R MBC

X4

R

n MBD

MAD

MBD MAB

P

B

A

P

MAB

B

FIGURE 9.11. Left: The nine-point conic n is a circumconic of the diagonal triangle of a complete quadrangle and carries the midpoints of all six sides. Right: The nine-point circle of a triangle ABC is the nine-point conic of an orthocentric quadrangle which means of the quadrangle ABCX4 with X4 being the orthocenter of ABC.

Assume that Q = ABCD is a quadrangle. Each side has a midpoint, for example, MAB is the midpoint of the edge AB. Thus, there are six midpoints of edges. Further, there are three diagonal points P = [A, B] ∩ [C, D], Q = [A, C] ∩ [B, D], and R = [A, D] ∩ [B, C]. Now, we have constructed a total of nine points out of the four vertices of Q and the following holds: Corollary 9.1.1 There exists a unique conic passing through the six midpoints of the edges and the three diagonal points of a quadrangle. The left-hand side of Figure 9.11 shows the nine-point conic of a nonconvex quadrangle. It doesn’t matter if the quadrangle is convex or not. The nine-point conics always exists, but may sometimes degenerate as is the case with a rectangle. In Section 7.5, we use a quadratic birational transformation to construct the nine-point conic as the image of a line (see page 366). ●

Exercise 9.1.5 Proof of Corollary 9.1.1. Proof the existence of the nine-point conic by assuming that A = (1 ∶ 0 ∶ 0), B = (0 ∶ 1 ∶ 0), C = (0 ∶ 0 ∶ 1), and D = (1 ∶ 1 ∶ 1). Assume further that (a ∶ b ∶ c) are the homogeneous coordinates of the line at infinity L∞ which helps to define midpoints of the edges as fourth harmonic points of the ideal points of the edges w.r.t. the edges.

426

Chapter 9: Special problems

Show that the diagonal points are P = (1 ∶ 1 ∶ 0), Q = (1 ∶ 0 ∶ 1), R = (0 ∶ 1 ∶ 1) and the midpoints are MAB = (b ∶ a ∶ 0), MAC = (c ∶ 0 ∶ a), MAD = (2a + b + c ∶ a ∶ a),

MBC = (0 ∶ c ∶ b), MBD = (b, a + 2b + c, b), MCD = (c ∶ c ∶ a + b + 2c).

The coefficient matrix of the equation of the nine-point conic equals ⎛ 2a ⎜ a+b ⎝ a+c

a+b 2b b+c

a+c ⎞ b+c ⎟ 2c ⎠

and the center equals X1125 = (bc(2a + b + c) ∶ ca(a + 2b + c) ∶ ab(a + b + 2c)). If D = X4 (the orthocenter), then n is the nine-point circle.

●

Exercise 9.1.6 Nine, ten, or eleven points on a single conic. The conic mentioned in Corollary 9.1.1 may sometimes cary even eleven points: The quadrangle Q = ABCD that delivers the diagonal points and midpoints of sides’ segments serves as the base of a pencil of conics of the first kind that induces a Desargues involution δ on a generic line l (not incident with any diagonal point or vertex of Q). If they do exist, the fixed points of δ belong to the same conic which can then be termed an eleven-point conic. The eleven-point conic (cf. Figure 9.12) is associated with the given quadrangle Q and the line l. Naturally, the midpoints of the segments are the harmonic conjugates of the ideal points of the six lines of the complete quadrangle on Q. In terms of Projective Geometry we may formulate it in the following way: The six harmonic conjugates of the intersection of the sides of a complete quadrilateral with an arbitrary line u, the three diagonal points of the associated quadrangle Q, and the fixed points of the Desargues involution δ induced by Q on a further line l ≠ u (in admissible position) lie on a single conic n. Prove this result either by synthetic reasoning or by assuming A = (1 ∶ 0 ∶ 0), B = (0 ∶ 1 ∶ 0), C = (0 ∶ 0 ∶ 1), D = (1 ∶ 1 ∶ 1), l = (l0 ∶ l1 ∶ l2 ), and u = (u0 ∶ u1 ∶ u2 ). Verify that the coeffcient matrix of n equals 2u0 −u0 − u1 −u0 − u2 ⎞ ⎛ ⎜ −u0 − u1 2u1 −u1 − u2 ⎟ . ⎝ −u0 − u2 −u1 − u2 ⎠ 2u2

The existence of these particular eleven points on a single conic is guaranteed if, and only if, the underlying field is algebraically closed. However, based on a quadrilateral Q = ABCD and with the choice of an arbitrary point Z ≠ A, B, C, D, P, Q, R, we can construct a ten-point conic (illustrated in Figure 9.13): Choose a regular conic c from the pencil of the first kind with base points A, B, C, D. Project the six poles of the lines of the complete quadrilateral onto these lines from Z and show that these six projections together with the diagonal points lie on a conic t that also passes through Z. The conic t is called the ten-point conic associated with Q and c with regard to Z. In [104], it is further shown that each conic in the pencil through Q produces its own ten-point conic and the family of ten-point conics asssociated with the pencil through Q is a again a pencil of the first kind. Further, if c is a Euclidean circle and if Z is c’s center, then the inverse of the ten point conic t in c (with center Z) is a stropohoid.

427

9.1 Conics in triangle geometry C

P

n

HAC Q A A A A A A A n F1 l

F2

QAC R

FIGURE 9.12. The eleven-point conic n contains the diagonal points P , Q, R of the quadrilateral Q together with the harmonic conjugates Hij of the intersections Qij with the line l and the fixed points F1 and F2 of the Desargues involution δ on l.

Two famous quadratic Cremona transformations Isogonal conjugation - a Cremona transform

In triangle geometry, the isogonal conjugation plays an important role. Besides the fact that many triangle centers are related via this particular mapping, it relates lines and circumconics of the base triangle ∆ = ABC. The isogonal conjugation κ is defined for all points in the Euclidean plane off the lines [A, B], [B, C], [C, A], i.e., the side lines of ∆. The mapping κ is defined as follows (see Figure 9.14): Any point X ∈/ [A, B] ∪ [B, C] ∪ [C, A] can be joined with ∆’s vertices. Then, we have α = 0. Then,

c∶

x2 y2 +ε 2 =1 2 a b

is the equation of a conic. The curve c is an ellipse/hyperbola if ε = +1/ε = −1. Show that p ∶ ε2 (x2 + y 2 )2 − 2ε2 (ux + vy)(x2 + y 2 ) − ε2 (a2 − u2 )x2 ) − (ε2 v2 − b2 )y 2 −2ε2 uvxy − 2ε2 a2 ux − 2b2 vy + a2 ε2 u2 + b2 v2

is the equation of the pedal curve of c w.r.t. F = (u, v). For the special choice F = (0, 0), the pedal curve p is Booth’s lemniscate if ǫ = +1 and it is Bernoulli’s lemniscate if ǫ = −1 (see Figure 9.31).

452 ◾

Chapter 9: Special problems

Example 9.4.1 Generalized conchoids. The concept of the conchoid constructions can be formulated in various other geometries, provided that points (or more general, geometric objects) can be joined by conics and a cross ratios can be defined (cf. [103]). This allows the construction of generalized conchoids in geometries that can be modeled within quadrics, such as the geometries of circle in the (Euclidean or pseudo-Euclidean) plane, in line and sphere geometry, or in the geometry of Euclidean motions. In this way, the conchoid construction can be applied to ruled and channel surfaces.

9.5 Poncelet porisms

453

9.5 Poncelet porisms

FIGURE 9.32. A Poncelet polygon defines a quadrangular Poncelet grid. There are two families of conics supplying the vertices of the grid.

A porism is a theorem about a closure property of a geometric figure or construction. It is a theorem of the form: If some problem has a solution for a specific choice of initial elements, then it can be solved infinitely many times, i.e., for any admissible choice of initial elements. At first we describe the Braikenridge-MacLaurin problem of closed polygons which are inscribed into one polygon and circumscribed to another

454

Chapter 9: Special problems

polygon at the same time. Then, we pay attention to the porism of Poncelet which deals with polygons which are inscribed into one conic and circumscribed to another conic. The conics appearing in the Poncelet porism can be seen as a generalization of the polygons which occur in the Braikenridge-MacLaurin problem. In this section, conics are alway assumed to be regular unless otherwise stated.

Closure problems in polygons C3

C4 C3

I2 I2

I3 I1

I3

I4

I1 C1 2′ 1′ 2

C2

1

1′ 3′2′

C1

32 1

C2

FIGURE 9.33. Closed polygons inscribed into and circumscribed to triangles or quadrangles always correspond to the solutions of quadratic equations.

For given n ≥ 3 we want to construct an n-gon that is circumscribed to a given n-gon and inscribed into a given n-lateral in P2 (R). Literally, we shall try out the problem on a triangle I1 I2 I3 and a trilateral consisting of the sides of the triangle C1 C2 C3 . We start with a point labeled by 1 on the side [C1 , C2 ]. We join 1 with I1 and extend the segment to its intersection with the side [C2 , C3 ]. Then, we extend the line from this point through I2 and intersect with the line [C3 , C1 ], and finally extend the line from this point through I3 to intersect [C1 , C2 ] again. If we have the good fortune to discover that the new point 1′ coincides with 1, then we have found a solution. Figure 9.33, left, illustrates our attempts. However, this event will not be the case in general, and so we try again a second and third time. The sequences of points 1, 2, 3, . . . and 1′ , 2′ , 3′ , . . . are related via the projectivity =1 [C2 , C3 ] ∧ =2 [C3 , C1 ] ∧ =3 [C1 , C2 ]. α∶ [C1 , C2 ] ∧ I

I

I

9.5 Poncelet porisms

455

Now, the question arises: Is there a fixed point, i.e., a point 1 which coincides with the corresponding point 1′ = α(1)? We learned in Section 5.2 that α ∶ xR → x′ R is represented by a linear map x′ = Ax with a 2 × 2 matrix A when on the line [C1 , C2 ] homogeneous coordinates are used. xR is fixed if, and only if, there is an eigenvalue λ ∈ R with Ax = λx. The eigenvalues of A are the zeros of the quadratic characteristic polynomial of A, and each eigenvalue λ gives at least one fixed point as nontrivial solution of the homogeneous system of linear equations (A − λI2 )x = 0. Hence, either two fixed points (two distinct real, or one real of multiplicity 2, or a conjugate complex pair) are to be expected, or, when A happens to be a multiple of the unit matrix I2 , then α is the identity, and the triangle closes for each choice. The latter occurs when the point triples {C1 , I1 , I2 }, {C2 , I2 , I3 } and {C3 , I3 , I1 } are collinear (Figure 9.34, left). Since the solutions of a quadratic equation can be found by straightedge and compass, the Braikenridge-MacLaurin problem can also be solved graphically by means of classical drawing tools. ●

Exercise 9.5.1 The Braikenridge-MacLaurin problem asks for triangles inscribed into the triangle C1 C2 C3 and circumscribed to I1 I2 I3 . Why is there an infinity of such triangles if, and only if, the point triples {C1 , I1 , I2 }, {C2 , I2 , I3 } and {C3 , I3 , I1 } are collinear (note Figure 9.34, left)?

The number n of vertices (or equivalently, the number of edges) of the involved polygons C1 C2 . . . Cn and I1 I2 . . . In (see the case n = 4 in Figure 9.33) plays no role at this problem; the presented procedure works in the same way for n > 3. There is always a quadratic equation to be solved, i.e., there are either two real solutions, or one real solution of multiplicity two, or two conjugate complex solutions, or even infinitely many solutions. However, the Braikenridge-MacLaurin problem is no example of a porism. There is no big difference between this problem and the modification in which the polygon C1 C2 . . . Cn is replaced by a conic c. Under the assumption that none of the vertices I1 , I2 , . . . In is a point of c, we can again start with points 1, 2, 3 ∈ c like before. Thus, we obtain on c a projectivity α which is the product of involutions with the centers I1 , I2 , and so on. Again, the fixed points of α are the starting points for closing n-gons inscribed into c and circumscribed to the n-gon I1 I2 . . . In . In the case n = 3 the projectivity α is the identity if, and only if, I1 I2 I3 is auto-polar w.r.t. c. But this implies that two vertices of this auto-polar

456

Chapter 9: Special problems I3 C3 1 =1′

I1 I2

I3

I2 I1

C1

C2

1 =1′

c

FIGURE 9.34. These are particular cases where each triangle closes which is inscribed into the triangle C1 C2 C3 (left) or to the conic c (right) and which has sides passing in turn through I1 , I2 , and I3 .

triangle are outside of c, i.e., in the area of points where two real tangent lines of c are meeting (note Figure 9.34, right).

Special porisms What about the following? We replace the outer and inner polygon by conics c and i and ask for n-gons which are inscribed into c and circumP2

c

M

I i

c

P2 P1

P1 M

I i

P3

P3

FIGURE 9.35. Left: triangle with incircle i and circumcircle c. Right: poristic family of triangles.

457

9.5 Poncelet porisms

scribed to i. A famous and well-known example comes from the elementary geometry of the triangle. Any triangle T in the Euclidean plane with vertices P1 , P2 , and P3 has a circumcircle c and an incircle i. The sides of T touch the incircle i while the vertices are located on c (Figure 9.35). Surprisingly, it turns out that the incircle i and the circumcircle c of T are incircle and circumcircle for more than just one triangle. Indeed, there are infinitely many triangles with vertices on c whose sides touch i. To put it in another way: Start with two circles, say c and i. Pick a point P0 on c and draw one of the tangents t0 to i. Then, t0 and c share two points P0 and P1 . Now, apply the same procedure to P1 and let t1 be the tangent to i that is different from t0 . In the case of the aforementioned triangle we find P3 = P0 , i.e., the polygon P0 P1 P2 is closed. Furthermore, if we start with a pair of circles that is known to be the incircle and circumcircle of a triangle T , then the polygon closes independently of the choice of P0 . There is a one-parameter family of triangles sharing the incircle and the circumcircle.

c

P2

i

i P3

P1 c

FIGURE 9.36. Triangles inscribed into a conic and circumscribed to another conic. Left: Both conics are ellipses. Right: Both conics are hyperbolas.

In the next sections we will deal with the generalization for two conics c and i (Figure 9.36). The theorem of Colin MacLaurin (1698–1746) states (note Corollary 9.5.5 on page 466) If there exists one triangle inscribed into a conic c and circumscribed to another conic i, then there are infinitely many triangles inscribed into and circumscribed to these conics c and i.

458

Chapter 9: Special problems

This result is remarkable for the following reason: We learned already in Section 6.4 that all pairs consisting of a conic and an inscribed triangle are projectively equivalent. However, for any triangle P1 P2 P3 inscribed into c there is still a two-parameter family of inscribed conics i. Nevertheless, each of them constitutes a poristic family together with c. It will turn out that by Theorem 9.5.2 for any given conic c there is a four-parameter family of conics i admitting a poristic family of triangles. E3 t

e c T1

P2 E

M

c

P1

I

P2 M

i

i P3

E2

T3 I

P1

P3 T2

E1

FIGURE 9.37. Further porisms. Left: While a triangle P1 P2 P3 is traversing its poristic family, the triangle’s excenters E1 , E2 , E3 trace a circle e. Thus, there are three nested porisms. Right: The vertices T1 , T2 , T3 of the tangent triangles of all triangles in a poristic family trace an ellipse t.

There are some more porisms involving one-parameter families of triangles (cf. [102, 108]). Figure 9.37 shows a few examples: If a triangle moves through its poristic family, then the vertices of its excentral triangle, i.e., the triangle built by the centers of the excircles, move on a circle e. The radius of e is twice the circumradius of the triangles in the poristic family. Since e’s center E is the reflection of the incenter I in the circumcenter M , e can be obtained from the circumcircle c by applying a central dilation with center I and scale factor 2. On the right-hand side of Figure 9.37, we see a kind of porism as described in Theorem 9.5: The vertices of the so-called tangent triangle move on an ellipse while the base triangle is traversing its poristic family. All poristic loci are trace three times since each point of the moving triangle plays the role of any other point at a

459

9.5 Poncelet porisms

certain instant. In the case of more general porisms (cf. [28]) with poristic triangle families interscribed in between triples of circles in a hyperbolic pencil, the poristic orbits of triangle centers are curves of three times the degree of these orbits as in the simple case (cf. [50]).

Inpolar conics There are many porisms connected with pairs of conics. This is the reason why in the coming sections we study relations between two conics which are projectively invariant. Most important in this respect is the (generalized) characteristic polynomial of two 3 × 3 matrices6 C, D FC,D (σ, τ ) ∶= det(σC + τ D) = k0 σ 3 + k1 σ 2 τ + k2 στ 2 + k3 τ 3 .

(9.25)

The substitutions τ = 0 or σ = 0 reveal k0 = det C and k3 = det D. But there are also formulas for the remaining coefficients. For regular C and D we can rewrite FC,D (σ, τ ) as det(σC + τ D) = det [C(σI3 + τ C−1 D)] = det [(σI3 + τ DC−1 )C] = = det [D(σD−1 C + τ I3 )] = det [(σCD−1 + τ I3 )D] .

Hence, the generalized characteristic polynomial is related to the (ordinary) characteristic polynomial of CD−1 and its inverse DC−1 by FC,D (1, −X) = det D ⋅ det(CD−1 − XI3 ), FC,D (−X, 1) = det C ⋅ det(DC−1 − XI3 ).

(9.26)

The coefficient of X 2 in the (ordinary) characteristic polynomial det(M − XI3 ) equals the trace tr M. This implies for the coefficients in the generalized characteristic polynomial FC,D (σ, τ ) introduced in (9.25) k0 = det C, k3 = det D,

k1 = det C ⋅ tr(C−1 D) = det C ⋅ tr(DC−1 ), k2 = det D ⋅ tr(D−1 C) = det D ⋅ tr(CD−1 ).

(9.27)

Let C, D be symmetric coefficient matrices of two conics in P2 (R). Changes of the underlying coordinate frame leave the ratio of coefficients k0 ∶ k1 ∶ k2 ∶ k3 in FC,D (σ, τ ) invariant since for C′ = TT C T and D′ = TT D T we get FC′ ,D′ (σ, τ )= det(σD′+τ C′)= (det T)2 det(σD+τ C)= (det T)2 FC,D (σ, τ ). 6

The characteristic polynomial is of course defined more generally for n × n matrices, and it is easy to rephrase the following particular results for general quadratic matrices.

460

Chapter 9: Special problems

Nevertheless, the ratios of any two coefficients are still no invariants of the pair of conics since the replacement of C by a scalar multiple λC yields FλC,D (σ, τ ) = (λ3 k0 )σ 3 + (λ2 k1 )σ 2 τ + (λk2 )στ 2 + k3 τ 3 .

(9.28)

The first binary relation on the set of conics is as follows:

Definition 9.5.1 In P2 (R), a conic p is called inpolar to the conic c if there exists a triangle ABC auto-polar w.r.t. p and inscribed into c (Figure 9.38).7 Since the conic p itself does not play a role here but only the polarity with respect to p, we want to extend this definition to the case that p has no real points. In any case, the conic c must have real points. G

A

h

g B

c

p H K C FIGURE 9.38. The conic p is inpolar to c

Below, we present a standard result of Projective Geometry. Lemma 9.5.1 Let a conic p be inpolar to a conic c. Then, there is a one-parameter family of triangles ABC which are auto-polar with respect to p and inscribed into c. Note that this example shows again a porism: If there is one auto-polar triangle inscribed into c, then there exist infinitely many. 7

Compare with [7, p. 33]. In [125, p. 148], c is called harmonically circumscribed to p. Sometimes in the literature the pair (p, c) is called apolar, but this notation does not reveal clearly that this relation is unsymmetric. In [125, p. 184], the term ‘apolar’ stands for a generalization to sets of four or five lines.

461

9.5 Poncelet porisms

Proof: Let ABC and GHK be two triangles which both are auto-polar w.r.t. p. Then, due to a result of von Staudt, the six vertices are located either on two lines or on a conic. This holds in each Pappian projective plane. A proof can, e.g., be found in [125, p. 147] or in [37, p. 87].

Let the defining triangle ABC be given. Then, specify another point G ∈ c such that its p-polar line g intersects c at a point H ≠ G (Figure 9.38). Continuity arguments guarantee the existence of G sufficiently close to A, B, or C. The line g and the polar h of H meet at a point K which completes a second auto-polar triangle GHK. There must be a conic on A, B, C, G, H, K. Since this conic is uniquely defined by the first five points, it coincides with c.

◾

The intersection points H, K of g with c are conjugate w.r.t. p, i.e., they are harmonic w.r.t. the intersection points of g with p. Hence, the polars g of points G ∈ c intersect the two conics p and c in harmonic quadruples of points. At most two of them can be conjugate complex since two pairs of conjugate complex points can never be harmonic. The desired harmonic position implies that on g the two involutions of conjugate points w.r.t. p and to c commute (note Example 5.4.4). This is the reason for a slight generalization of inpolarity which is needed in the following lemma. Definition 9.5.1’ The conic p in P2 (R) is called inpolar to c if there is a point G ∈ c, G ∈/ p, such that on the p-polar g of G the induced involutions of conjugate points w.r.t. p and c commute. In this case, the point G together with the two real or complex conjugate points of intersection between g and c forms a triangle which is auto-polar w.r.t. p. Then, Lemma 9.5.1 again implies that this holds for each choice of G ∈ (c p). In fact, Figure 9.41 on page 469 shows an example where p is inpolar to c though for all G ∈ c the p-polar g has no real intersection with c. Lemma 9.5.2 Let the conics p and c be given by the equations p ∶ xT P x = 0,

c∶ xT C x = 0

with symmetric matrices P and C. We suppose that c contains real points. Then, p is inpolar to c if, and only if, in the characteristic polynomial FP,C (σ, τ ) = det(σP + τ C) = j0 σ 3 + j1 σ 2 τ + j2 στ 2 + j3 τ 3

the coefficient j1 = det P ⋅ tr(CP−1 ) is zero.

462

Chapter 9: Special problems

Proof: We learned that the ratio j0 ∶ j1 ∶ j2 ∶ j3 of coefficients in the characteristic polynomial FP,C (σ, τ ) does not depend on the choice of the coordinate frame. For the conics p inpolar to c, we use a coordinate frame defined by the fundamental triangle ABC. Then, P is a diagonal matrix while C has vanishing diagonal entries, hence

and

⎛ σp00 σP + τ C = ⎜ τ c01 ⎝ τ c02

τ c01 σp11 τ c12

τ c02 ⎞ τ c12 ⎟ σp22 ⎠

FP,C (σ, τ ) = det(σP + τ C) = p00 p11 p22 σ3 + τ 2 (eσ + f τ ) with certain coefficients e, f . Obviously, the coefficient j1 of σ2 τ is zero, and by (9.27) j1 equals the product det P ⋅ tr(CP−1 ).

In order to prove the converse, we specify a coordinate frame which diagonalizes P and where the fundamental point A = (1 ∶ 0 ∶ 0) is located on c. This implies p00 ≠ 0 and c00 = 0.8 Suppose that in the polynomial ⎛ σp00 FP,C (σ, τ ) = det(σP + τ C) = det ⎜ τ c01 ⎝ τ c02

τ c01 σp11 + τ c11 τ c12

τ c02 ⎞ ⎟ τ c12 σp22 + τ c22 ⎠

the coefficient of σ2 τ is zero, i.e., j1 = p00 (p11 c22 +p22 c11 ) = 0. On the line a∶ x0 = 0 which is ppolar to A, the conics p and c induce (regular or singular) involutions (0 ∶ x1 ∶ x2 ) ↦ (0 ∶ x′1 ∶ x′2 ) of conjugate points, satisfying respectively p11 x1 x′1 + p22 x2 x′2 = 0 or c11 x1 x′1 + c12 (x1 x′2 + x2 x′1 ) + c22 x2 x′2 = 0.

According to Example 5.4.4, j1 = 0, i.e., p11 c22 + p22 c11 = 0, is equivalent to the condition that these two involutions commute. This proves (in the complex extension) the existence of a triangle ABC inscribed into c and auto-polar w.r.t. p.

◾

When we dualize p and c and reverse their order, i.e., when we replace the ordered pair (P, C) by (C−1 , P−1 ), then the characterization of ‘inpolarity’ given in Lemma 9.5.2 remains valid since det(σC−1 + τ P−1 ) = det [C−1 (σP + τ C)P−1 ] .

This means that there are infinitely many triangles auto-polar w.r.t. c and circumscribed to p, provided that also p has real points (and tangents). The same can be concluded if there is a polarity which exchanges p and c. Thus, we obtain9 Corollary 9.5.3 Let p and c be conics with real points. Then, p is inpolar to c if, and only if, c is “outpolar” to p, i.e., there are triangles auto-polar w.r.t. c and circumscribed to p. 8 9

Only for p = c this choice would be impossible, but under P = C we get j1 = 3 det C ≠ 0. Alternative proofs can be found in [139, p. 213], [7, p. 33–34], or [15, p. 85–86]. The three-dimensional versions of Lemmas 9.5.1 and 9.5.1 can be found in [139, p. 213], the n-dimensional versions in [124, p. 862, footnote 287].

463

9.5 Poncelet porisms

For given p, there is a two-parameter family C of conics c such that p is inpolar to c. The set C is linear, i.e., with c1 , c2 ∈ C all conics of the pencil spanned by c1 and c2 are in C since tr [P−1 (λ1 C1 + λ2 C2 )] = λ1 tr(P−1 C1 ) + λ2 tr(P−1 C2 ).

Conversely, for a given conic c the family P of conics p inpolar to c is two-parametric, and the duals of p constitute a linear family.

Triangles inscribed into and circumscribed to conics Let p be inpolar to c. The polarity w.r.t. p transforms points of c into tangents of a second conic i. The auto-polar triangles inscribed into c are at the same time circumscribed to i. So, there are infinitely many triangles sharing this property.

c

P

i

p C

G

O

FIGURE 9.39. There is an infinite set of triangles inscribed into the circle c and circumscribed to the hyperbola i. The triangles share the orthocenter O and the centroid G. Two real triangles are degenerate; the others are auto-polar w.r.t. the circle p. The centers O of p and C of c are foci of i.

◾

Example 9.5.1 Triangles sharing the circumcircle, the orthocenter and the centroid.

Figure 9.39 shows an example where p and c are circles (compare with Figure 10.20). Then, all triangles inscribed into c and auto-polar w.r.t. p share the center C of c as circumcenter and the center O of p as orthocenter. Due to the properties of the Euler line, they share also the centroid G. Moreover, the points C and O are focal points of i (compare with Figure 7.17). Proof: Each tangent t of i carries a side of a triangle which is auto-polar w.r.t. p. Therefore, the intersection points of t with c must be conjugate w.r.t. p. Let a point P be common to c

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Chapter 9: Special problems

and p (Figure 9.39). If any tangent t of the polar transform i of c passes through P , then its second intersection with c is either a point ≠ P on the tangent to p at P , or it coincides with P , which means that t is tangent to c. In other words, the tangents from P to i are identical with the tangents to p and c at P . This is also true for the absolute circle points, which in our particular case are common to c and p. Therefore, the isotropic tangents of i must pass either through the center O of p or through the center C of c. Consequently, the polar of i w.r.t. c is again a circle c′ (Figure 7.17). The circle c′ passes through the common points of c and p since they are contact points of common tangents between i and c. The c-polars of the triangles inscribed in c have their vertices on c′ and their sides tangent to c. Thus, we arrive at the well-known Poncelet porism concerning triangles that share the circumcircle and the incircle (note Euler’s condition in (9.38)).

We are going to show that for each pair of conics (c, i) as used above there exists a polarity with c ↦ i. Lemma 9.5.4 Let c and i be two different conics with real points such that there exists a triangle inscribed into c and circumscribed to i. Then, the tangent lines of i intersect c in points which are conjugate with respect to a regular or singular conic p. When C, P, and D−1 are symmetric coefficient matrices of the conics c and p and of the dual of i, resp., then up to a factor ̺ ∈ R {0} ̺P = −tr(CD−1 ) C + 2 CD−1 C.

Proof: It is easy to show that the stated representation of P is invariant against changes of the coordinate frame. We can proceed like in the proof for the influence of coordinate transformations on the characteristic polynomial on page 459. Hence, we can specify ⎛ 0 c ∶ 2(x0 x2 − x21 ) = xT C x with C = ⎜ 0 ⎝ 1

0 −2 0

1 ⎞ 0 ⎟. 0 ⎠

Any two points of c can be given in parameter form as pR = (u2 ∶ uv ∶ v2 ) and p′ R = (u′ 2 ∶ u′ v′ ∶ v′ 2 ).

The connecting line pR ∨ p′ R is represented by the vector

vv′ ⎛ ⎞ u = p × p′ = (uv′ − vu′ ) ⎜ −uv′ − vu′ ⎟ , ⎝ ⎠ uu′

which we divide by (uv′ − vu′ ). This factor does not vanish under the assumption pR ≠ p′ R. Let D−1 = (d̂ij ) with d̂ji = d̂ij be the coefficient matrix of the dual ̂ i of the conic i. Then, our chord uR of c is tangent to i if, and only if, uT D−1 u = 0 (note the points P and P ′ in Figure 9.40). This is equivalent to d̂00 v2 v′ 2 − 2d̂01 (uvv′ 2 + v2 u′ v′ ) + 2d̂02 uvu′ v′ + d̂11 (u2 v′ 2 + 2uvu′ v′ + v2 u′ 2 ) −2d̂12 (u2 u′ v′ + uvu′ 2 ) + d̂22 u2 u′ 2 = 0.

465

9.5 Poncelet porisms We can re-substitute p and p′ and obtain the vanishing symmetric bilinear form ⎛ d̂22 pT P p′ = 0 with P = ⎜ ⎜ −2d̂12 ⎝ d̂11

−2d̂12 2(d̂02 + d̂11 ) −2d̂01

d̂11 −2d̂01 d̂00

⎞ ⎟. ⎟ ⎠

This means that the points pR, p′ R ∈ c are conjugate w.r.t. a regular or singular polarity with the symmetric matrix P. By straightforward computation, we obtain due to the symmetry of D−1 ⎛ d̂20 CD−1 = ⎜ ⎜ −2d̂10 ⎝ d̂00

d̂21 −2d̂11 d̂01

d̂22 −2d̂12 d̂02

⎞ ⎟, ⎟ ⎠

⎛ d̂22 CD−1 C = ⎜ ⎜ −2d̂12 ⎝ d̂02

−2d̂21 4d̂11 −2d̂01

d̂20 −2d̂10 d̂00

⎞ ⎟, ⎟ ⎠

and we verify (d̂11 − d̂02 )C + CD−1 C = 12 P. Of course, the coefficient matrix of any polarity is unique only up to a non-vanishing factor ̺.

◾

The matrix P in Lemma 9.5.4 is a linear combination of C and CD−1 C. Therefore, the conic p belongs to the pencil spanned by c and by the polar of i w.r.t. c.10 If any point A has the same polar line a w.r.t. c and i, then a is also the polar line of A w.r.t. to p or A is a singular point of p. This can be proved as follows: Ca = λDa = u for a ≠ 0 implies D−1 Ca = λa, hence

◾

Pa = [−tr(CD−1 ) C + 2 CD−1 C] a = [−tr(CD−1 ) + 2λ] u ∈ uR.

Example 9.5.2 Rectangles inscribed into the director circle. Figure 9.40 illustrates an example where the matrix P has rank 2 : Here, c is the director circle of the ellipse i, i.e., 2 2 ⎛ −a − b 0 0 ⎞ 0 1 0 ⎟, C=⎜ ⎝ 0 0 1 ⎠

⎛ −1 0 0 ⎞ D−1 = ⎜ 0 a2 0 ⎟ , ⎝ 0 0 b2 ⎠

0 0 ⎞ ⎛ 0 P = ⎜ 0 −2b2 0 ⎟ . ⎝ 0 0 −2a2 ⎠

The tangents of i intersect c at points pR, p′ R which are placed on conjugate diameters of i. This means exactly that for each rectangle which is inscribed into the director circle c and circumscribed to the ellipse i, adjacent vertices are conjugate w.r.t. the singular polarity in p. More about pairs (c, i) of conics with in- and circumscribed quadrangles can be found in [85].

Theorem 9.5.1 Let c and i be two different conics with real points, and let the tangents of i intersect c at points conjugate w.r.t. p. Then, the following three statements are equivalent: (i) The conic p is inpolar to c in the generalized sense (Definition 9.5.1’, page 461). 10

This means that, e.g., referring to Figure 9.39 the sides of the c inscribed triangles have their c-poles on a circle which belongs together with c and p to a pencil (note Example 9.5.1).

466

Chapter 9: Special problems c P P′

i

FIGURE 9.40. There is an infinite set of rectangles circumscribed to the ellipse i and inscribed into its director circle c. Their diagonals are conjugate w.r.t. i.

(ii) There exist infinitely many triangles (including such with a pair of complex conjugate vertices) inscribed into c and circumscribed to i. (iii) The conics c and i are polar w.r.t. p. Proof: (i) ⇔ (ii): By Lemma 9.5.1, there is an infinite set of triangles ABC inscribed into c and auto-polar w.r.t. p. The vertices B and C are the only points of c which are conjugate to A. Since the tangents of i which pass through A must intersect c in points conjugate to A, the sides AB and AC are tangent to i. The same holds for the vertex B. Hence, each triangle ABC is circumscribed to i. When the two tangents from A to i are complex conjugate, then also B and C are complex conjugate, but their connecting line [B, C] is real. Conversely, according to the definition of p each triangle inscribed into c and circumscribed to i has pairwise conjugate vertices w.r.t. p. Therefore, it is auto-polar.

(ii) ⇔ (iii): The polarity w.r.t. p maps the vertices A, B, C ∈ c of the auto-polar triangles to the respectively opposite sides. Hence, the envelope i is polar to c.

Conversely, for each point A ∈ c the p-polar a is a tangent of i. Let a intersect c in two real or complex conjugate points B and C. Then, by the definition of p, the lines [A, B] and [A, C] are tangents of i, too. Hence, ABC is circumscribed to i and inscribed into c.

◾

Now, the announced theorem of Colin MacLaurin (see page 457) is just a corollary of Theorem 9.5.1. Corollary 9.5.5 If for two conics c, i there exists one triangle inscribed into c and circumscribed to i, then there are infinitely many. Proof: Suppose, ABC is such a triangle. Then, the vertices A, B, C are pairwise conjugate w.r.t. the uniquely defined polarity p (Lemma 9.5.4). Hence, ABC is auto-polar w.r.t. p, and thus, p inpolar to c. The rest follows from Theorem 9.5.1.

◾

467

9.5 Poncelet porisms

◾

Example 9.5.3 Triangles sharing the circumcircle and the centroid. For the conics displayed in Figures 9.39 and 10.20 (page 544), we use a coordinate frame with the center M of the circle c as origin and the centroid G with Cartesian coordinates (g, 0). Then, the orthocenter gets the coordinates (3g, 0). In Figure 9.39, we have 3g > r, in Figure 10.20, we have 3g < r. The equations of the involved conics are c ∶ x2 + y 2 = r 2 ,

i ∶ x2 − 3gx +

p ∶ x2 − 6gx + y 2 = −

9g 2 + r 2 . 2

r2

r2 r 2 − 9g 2 y2 = , 2 − 9g 4

In terms of homogeneous coordinates the symmetric coefficient matrices of c, i, and p are 2 ⎛ −r C=⎜ 0 ⎝ 0

⎛ P=⎜ ⎜ ⎝

0 1 0 2

9g +r 2

2

0 ⎞ 0 ⎟, 1 ⎠

−3g 0

−3g 1 0

⎛ ⎜ D=⎜ ⎜ ⎝ 0 0 1

⎞ ⎟, ⎟ ⎠

9g 2 −r 2 4 − 3g 2

− 3g 2

0

1

0

0

0

r2 r 2 −9g 2

PC−1 P =

⎞ ⎟ ⎟, ⎟ ⎠

r 2 − 9g 2 D. r2

The characteristic polynomials read FC,D (σ, τ ) = −r 2 σ3 +

9(3g 2 − r 2 )2 2 3r 2 (r 2 − 3g 2 ) r4 σ τ+ στ 2 + τ 3, 4 2(9g 2 − r 2 ) 4(9g 2 − r 2 )

r 2 − 9g 2 3 9g 2 − 3r 2 2 σ + στ − r 2 τ 3 . 2 2 We note that in the last polynomial the coefficients j1 of σ2 τ is zero (cf. Lemma 9.5.2). In FC,D (σ, τ ) the coefficients k0 , . . . , k3 satisfy k22 = 4k1 k3 , and this is in accordance with the Theorem 9.5.2 below. FP,C (σ, τ ) = det(σP + τ C) =

We conclude this section with the analytic characterization of two conics c and i with the property stated in Theorem 9.5.1,(ii). As a preparation, we pick out the statement (i) ⇒ (iii) from the aforementioned theorem and present additionally an analytical proof: Proof: By virtue of Lemma 9.5.4, there is a polarity in a regular or singular conic p with the coefficient matrix P = −tr(CD−1 ) C + 2 CD−1 C. This implies and

PC−1 = −tr(CD−1 ) I3 + 2 CD−1

(9.29)

tr(PC−1 ) = −3 tr(CD−1 ) + 2 tr(CD−1 ) = −tr(CD−1 ).

(9.30)

Let p be inpolar to c. Then, due to Lemma 9.5.2, the coefficient j1 in FP,C (σ, τ ) = det(σP+τ C) vanishes. By (9.26), we obtain FP,C (1, −X) = det C det(PC−1 − XI3 ) = j0 X 0 + j2 X 2 − j3 X 3 .

Now, we use the theorem of Cayley-Hamilton. When in the characteristic polynomial det(PC−1 − XI3 ), the indeterminate X is replaced by the matrix PC−1 , then we obtain the zero matrix O3 . This yields j0 I3 + (PC−1 )2 [j2 I3 + j3 PC−1 ] = O3

(9.31)

468 with

Chapter 9: Special problems j0 = det P,

j2 = det C tr(PC−1 ),

j3 = det C.

By (9.29) and (9.30), we obtain from (9.31)

det P I3 −(PC−1 )2 [det C tr(CD−1 )I3 −tr(CD−1 ) det CI3 −2CD−1 ]= O3 , det P I3 − 2(det C) PC−1 PC−1 CD−1 = O3 ,

hence, (det P)DC−1 = 2(det C)PC−1 PC−1 or

D = 2 det C(det P)−1 PC−1 P,

(9.32)

◾

and this shows that i is p-polar to c.

Some of these formulas will also be needed in the proof of Theorem 9.5.2 Let c and i be two conics with symmetric coefficient matrices C, D, respectively. Then, there exist infinitely many triangles (with possibly two complex conjugate vertices) inscribed into c and circumscribed to i if, and only if, in the corresponding characteristic polynomial FC,D (σ, τ ) shown in (9.25) the coefficients k1 , k2 , k3 satisfy 4k1 k3 −k22 = 0. Proof: We recall that by (9.27)

k1 k3 = det C det D tr(DC−1 ),

k2 = det D tr(CD−1 ).

Note that the condition 4k1 k3 − k22 = 0 does not change when C or D is replaced by a scalar multiple. Suppose there are infinitely many triangles inscribed into c and circumscribed to i or — equivalently — p is inpolar c. Then, Lemma 9.5.2 implies tr(CP−1 ) = 0, and with (9.32) follows 8(det C)2 2 det C det D = and DC−1 = (PC−1 )2 . (9.33) det P det P For any 3 × 3 matrix M = (mjk ), we have [ tr(M)]2 − tr(M2 ) =

∑ 0≤j 3. In view of this, note the Figures 9.40, 9.42, and 9.32 for n = 4, n = 7, and n = 36, respectively. Suppose P1 P2 . . . Pn is an n-gon inscribed into c with pairwise different vertices and with sides [P1 , P2 ], [P2 , P3 ], . . . tangent to i. Then, by virtue of Lemma 9.5.4, P1 and P3 are conjugate to P2 w.r.t. p, i.e., the diagonals [P1 , P3 ], [P2 , P4 ], . . . , [Pn−2 , Pn ] are tangent to the polar i′ of c w.r.t. p. Hence, for n = 2m the polygon P1 P2 . . . P2m closes, i.e., the last side [P2m , P1 ] is tangent to i if, and only if, the sides of the closed polygon P1 P3 . . . P2m−1 are tangent to i′ . By iteration we would be able to find characterizations for pairs of conics (c, i) with the property that there are closing n-gons inscribed into c and circumscribed to i. However, in the next section we follow the traces of Poncelet and Cayley on a direct approach to the remarkable Theorem 9.5.4. ●

Exercise 9.5.2 Formulate the dual of Lemma 9.5.4 on page 464.

●

Exercise 9.5.3 Prove the following converse of Lemma 9.5.4: For any two conics c and p there is a regular or singular conic i contacting all lines l which intersect c in points conjugate w.r.t. p. The conic i is called von Staudt’s conic of the pair ̂ of the tangent equation of i can be (c, p) (see [15, p. 91]). The symmetric coefficient matrix D expressed in terms of the symmetric coefficient matrices C and P by

●

̂ = −tr(PC−1 )C−1 + C−1 PC−1 or ̺D ̂ = −tr(CP−1 )P−1 + P−1 CP−1 for any ̺, σ ∈ R. σD

Exercise 9.5.4 Suppose there is a quadrangle P1 . . . P4 inscribed into c and circumscribed to i. Then, the coefficients in the characteristic polynomial FC,D (σ, τ ) of (9.25) satisfy 8k0 k32 − 4k1 k2 k3 + k23 = 0. Prove this statement by the following steps: 1. The matrix P of the polarity stated in Lemma 9.5.4 has to be of rank 1. 2. From det P = 0 one can conclude that

1 tr(CD−1 ) 2

is an eigenvalue of CD−1 .

9.5 Poncelet porisms

471

3. Express this eigenvalue in terms of the coefficients k1 , . . . , k3 and plug this into the characteristic polynomial.

Poncelet porisms Jean-Victor Poncelet (1788–1867) studied “porisms” and proved that MacLaurin’s theorem (see page 457) holds for general n-gons. This particular example of a porism involving two conics is called Poncelet porism.

FIGURE 9.42. Poristic figures may be regular, or unsymmetric, or symmetric. The inscribed polygons can be regular, or convex, or star-shaped.

Theorem 9.5.3 Let c and i be two conics in a projective plane. If there exists a closed polygon P0 . . . Pn−1 such that the vertices Pi are on c and the side lines [Pk , Pk+1 ] are tangent to i for all k ∈ {0, 1, . . . , n−1} with indices taken modulo n, then there are infinitely many such closed polygons. Theorem 9.5.3 gives a necessary and sufficient condition for the existence of infinitely many inscribed and circumscribed closed polygons. But it says nothing about how to specify two conics c and i such that they admit such polygons. Figure 9.42 shows some porisms of Poncelet type. The theorem as well as its first proof is due to Poncelet [112, p. 361]. He was a French mathematician and engineer who joined the French army and participated in Napoleon’s invasion of Russia from 1812 to 1814. However, he became a prisoner of war and, while imprisoned at Saratov, he found the time to

472

Chapter 9: Special problems

write his famous work on the figures in geometry, a work that could be considered one of the fundamentals of Projective Geometry, containing also Theorem 9.5.3 and its proof. It should be noted that none of the techniques that are used nowadays for the proof were developed at that time. The proof of Theorem 9.5.3 involves complex functions, differential geometry, and elliptic curves. Because of its relations to many subdisciplines of mathematics, this theorem is often called a pearl of mathematics. Below, we only give a sketch of the proof, since it is far beyond the scope of our book. Details can be found in [19, 20, 48, 42, 43]. By the same token, a proof of Poncelet’s theorem which relies only on Pascal’s theorem was presented 2015 in [62]. Proof: (Sketch) Let P and t be a point and a line in a projective plane P2 . We describe P as well as t by homogeneous coordinates, i.e., P = (p0 ∶ p1 ∶ p2 ) and t = (t0 ∶ t1 ∶ t2 ). Since P is a point of the conic c, we can represent P by quadratic functions pi = pi (u) in the parameter u for i ∈ {0, 1, 2}. The same holds true for the tangents t of c; this one-parameter family of tangents allows an analogous parametrization tj = tj (v) with v ∈ R for j ∈ {0, 1, 2}. The incidence of P with t can be written in terms of coordinates as p0 t0 + p1 t1 + p2 t2 = 0.

(9.34)

This is a biquadratic equation in u and v. It can be interpreted as the equation of a curve m in the (u, v)-plane. The points of the curve correspond to incident pairs (P, t) where P is on the conic c and t is tangent to i. The algebraic curve m of degree 4 is elliptic if the conics c and i are in general position, i.e., if they span a pencil of conics of the first kind. In this case, the conics c and i have four mutually different points and four mutually different tangents in common if we extend into the complex (u, v)-plane. The curve m is called the Poncelet correspondence. There acts an involution σ on m mapping the pair (P, t) to the pair (P ′ , t) where P and P ′ are the two points of intersection of t with c; points of m corresponding under σ share the v-coordinate. Obviously, by virtue of Lemma 9.5.4, the involution σ is the restriction of the conjugacy w.r.t. p onto the conic c. A further involution τ sends the pair (P, t) to the pair (P, t′ ) where t and t′ are the tangents of i which pass through P . The composition µ ∶= τ ○ σ acts like a translation on m; it is called Poncelet map.

Let P1 . . . Pn be a polygon inscribed into c and circumscribed to i with t1 = [P1 , P2 ], t2 = [P2 , P3 ], . . . . Then, we have µ∶ (P1 , t1 ) ↦ (P2 , t2 ), (P2 , t2 ) ↦ (P3 , t3 ), . . . .

The polygon closes if µn but not µ keeps (P1 , t1 ) fixed.

It can be proved that µn equals the identity on m as soon as it has at least five fixed points. However, four fixed points (P, t) of µ and µn are already known from the beginning: t is one of the four common tangents of c and i, and P is its point of contact with c. Hence, the existence of one closing in- and circumscribed n-gon, i.e., of one additional fixed point, implies that all these n-gons must close. The proof, sketched so far, was given for two conics in general position. It works in a similar way if the conics c and i are in a special position, i.e., if they span a pencil of conics of the second, third, fourth, or fifth kind.

◾

473

9.5 Poncelet porisms

There is another surprising result which gives a criterion on two conics for the existence of in- and circumscribed n-gons, n ≥ 3. It includes Theorem 9.5.2 and the statement presented in Exercise 9.5.4. Let C and D be the symmetric coefficient matrices of the conics c and i in any coordinate frame. Now, we recall from (9.25) on page 459 the characteristic polynomial FC,D (σ, τ ) = det(σC + τ D)

and expand the following function in a power series: √ √ FC,D (t, 1) = det(t C + D) = a0 + a1 t + a2 t2 + a3 t3 + ⋯ .

(9.35)

The following theorem dates back to Arthur Cayley.

Theorem 9.5.4 There exists an n-sided polygon inscribed into c and circumscribed to i if, and only if, the coefficients of the power series given in (9.35) satisfy ⎛ a2 . . . am+1 ⎞ ⋮ ⎟ = 0, if n = 2m + 1, m ≥ 1, det ⎜ ⋮ ⎝ am+1 . . . a2m ⎠ ⎛ a3 . . . am+1 ⎞ ⋮ ⎟ = 0, det ⎜ ⋮ ⎝ am+1 . . . a2m+1 ⎠

(9.36)

if n = 2m, m ≥ 2.

The proof of this result uses the same ideas as the proof of Theorem 9.5.3 (see, e.g., [48]). The determinant det(t C + D) is a cubic polynomial in t. It has no multiple roots if c and i span a pencil of conics of the first kind. Otherwise there would be less than three singular conics in the pencil (cf. Section 7.3). Another proof of this theorem can be found in [60]. It is based on the existence of points of finite order on elliptic curves. Cayley’s Theorem 9.5.4 is more than just a projectively invariant criterion on the equations of two conics. As a byproduct, even metric properties of poristic figures result from the formula presented in (9.36), for example for n-gons which have a circumcircle c and an incircle i. Such polygons are called bicentric (see, e.g., [118]). Theorem 9.5.4 delivers conditions on the radii and the distance of the centers of c and i to make them the circumcircle and the incircle of a

474

Chapter 9: Special problems

bicentric n-gon. Let c and i be two circles with equations in terms of Cartesian coordinates that read c∶ x2 + y 2 = R2

and i∶ (x − d)2 + y 2 = r 2 ,

(9.37)

where R, r, d are real or complex numbers and, without loss of generality, R > r (in the real case). According to Theorem 9.5.4, we have to expand the function √ −(t + 1) (t2 R2 + t(R2 + r 2 − d2 ) + r 2 ) in power series as given in (9.35). We obtain ri +

i i (R2 + 2r 2 − d2 )t + 3 (R2 − 2Rr − d2 )(R2 + 2Rr − d2 )t2 + . . . . 2r 8r

Now, we use the condition from Theorem 9.5.4: When we look for triangles circumscribed to i and inscribed into c, the determinant given in (9.36) is that of the 1 × 1 -matrix (a2 ) (compare Theorem 9.5.2). Therefore, the radii R and r and the distance d of the incenter and circumcenter of a triangle satisfy R2 − 2Rr = d2 . (9.38)

The second factor of the coefficient a2 of t2 only differs by the sign of r. The formula given in (9.38) was at least known to Leonhard Euler (1707–1783) and can also be found by elementary methods. Many other formulas relating the radii of the incircle and the circumcircle of a bicentric n-gon can be derived from Cayley’s formula. In the past, a huge amount of such formulas for arbitrary n was elaborated. For example, Carl Gustav Jacob Jacobi (1804–1851) reports 1828 in [77] about explicit formulas up to the case n = 8. Many of them were found by the Russian Privi Councillor Nicolaus Fuss (1755–1826), secretary of L. Euler. ◾

Example 9.5.5 Poncelet porisms in the most general form.

The condition (9.38) relating the radii and the central distance of two circles in order to allow for a poristic family of triangles and all the algebraic closure conditions mentioned cover cases of very special porisms. Only circles are the involved conics and there are only two of them. The most general form of a porism in the sense of Poncelet deals indeed with polygons interscribed in between n + 1 conics of one pencil (cf. [28]): Let c and di with i = 1, . . . , n be n + 1 distinct conics which belong to the same linear pencil. If an n-gone can be constructed with all vertices on c such that each side is tangent to some conic di , then there exists a porism (i.e., a one-parameter family) of n-gons interscribed in between c and di .

475

9.5 Poncelet porisms

It is clear that a condition comparable to (9.38) cannot be given for all cases of general forms of porisms. However, in the very special case of porisms interscribed in between n > 2 circles of a pencil such closure conditions can be given, see [49].

◾

Example 9.5.6 How to complete three points to a convex bicentric quadrangle (provided by Gerhard Pillwein). A bicentric quadrangle is a quadrangle that has both a circumcircle and an incircle, i.e., it is cyclic and tangential at the same time. In order to construct such a quadrangle, one usually starts with the incircle and chooses two perpendicular chords of it. The tangents at the end points are the sides of a quadrangle that has a circumcircle. By the way, the non convex bicentric quadrangles are antiparallelograms.

A d

a

D

B c

b

C FIGURE 9.43. A convex bicentric quadrangle and infinitely more . . . The following task differs a little bit: Let ABCD be a bicentric quadrilateral with the side lengths a, b, c, and d (Fig. 9.43). If three vertices A, B, and D are given arbitrarily (not collinear), where is the residual vertex C located? Vertex C must clearly lie on the (blue) circumcircle of triangle ABD (Figure 9.44). However, a + c = b + d must be fulfilled – this is the necessary and sufficient condition for convex quadrilaterals with an incircle. Since c − b = b − a, the difference c − b is constant, and therefore, C must lie on a (green) hyperbola h with foci at B and D. This branch of the hyperbola obviously also passes through A. The remaining intersection point with the circle represents the solution for the sought-after vertex C. The polarities w.r.t. the two circles reveal that the diagonals of ABCD and the (orthogonal) connections of opposite contact points with the incircle are concurrent. According to Poncelet [112, p. 365] this property holds for all bicentric n-gons with even n and their projective generalizations. More properties of quadrangles with circum- and inconic can be found in [85]. Even though we have to intersect two conics (the circumcircle and the hyperbola), the fourth vertex C can be determined elementary as follows: Let H, the midpoint of B and D, be the origin of a Cartesian coordinate system with the bisector of BD as y-axis. With the labeling of Fig. 9.44, the following relation emerges after the below given calculation: HF ∶ HG = HP ∶ HQ. Therefore, C can be constructed or calculated with a simple transfer of an affine ratio.

In order to show that HF ∶ HG = HP ∶ HQ holds, we need the following elementary calculation. Within the chosen coordinate frame, the hyperbola has the equation b2 x2 − a2 y 2 = a2 b2 , and the circle centered at M = (0, −v) is given by x2 + (y + v)2 = R2 . The semiaxes a, b, the linear

476

Chapter 9: Special problems

y h P A

F x

H

B N

C

D

M

G Q

FIGURE 9.44. The intersection of a hyperbola with foci at B and D and the circumcircle (according to Gerhard Pillwein). eccentricity e of the hyperbola, and the radius R of the circle obey the relations a2 + b2 = e2 ,

e2 + v2 = R2

⇒ R2 − v2 − a2 = b2 .

Calculation of the y-coordinates of the intersection points of the hyperbola and the circle yield b2 (y + v)2 + a2 y 2 (b2 + a2 )y 2 + 2b2 v y + b2 (v2 + a2 − R2 ) e2 y 2 + 2b2 v y − b4

= = =

b2 (R2 − a2 ), 0, 0,

and finally, y1,2 =

√ √ 1 1 b2 (−2b2 v ± 4b4 v2 + 4e2 b4 ) = 2 (−2b2 v ± 2b2 v2 + e2 ) = 2 (−v ± R) 2 2e 2e e

Thus, HF ∶ HG = y1 ∶ ∣y2 ∣ = (R − v) ∶ (R + v) = HP ∶ HQ.

Once the vertex C has been determined and the incircle has been constructed, according to Poncelet’s closure theorem, infinitely many such quadrilaterals have been found. Note that the intersection points of the hyperbola’s second branch with the circle lead to a different bicentric quadrilateral, which is just a reflection of the first in the y-axis of the coordinate system. The relation between the circumradius R, the radius r of the incircle, and the central distance d = M N is given by (R2 − d2 )2 r2 = 2(R2 + d2 ) and was found by Nicolaus Fuss in 1792. It can be elaborated with the Cayley criterion given in Thm. 9.5.4 However, it can also be found in an elementary way by utilizing Poncelet’s closure theorem: Let R, r, and d be the radii and the central distance of a generic bicentric quadrilateral. Each new vertex A on the circumcircle results in a further quadrilateral with

477

9.5 Poncelet porisms

D

r

A

r N d

x

M R

C

B

FIGURE 9.45. The symmetric solution leads to a relation between R, r, and d. the same circumcircle and incircle. If we choose A on [M, N ], the resulting quadrilateral is symmetric (see Figure 9.45). Similar triangles lead to R−d R+d = ⇐⇒ (R − d) ⋅ x = (R + d) ⋅ r. r x Then, we find which gives and finally

(R − d)2 ⋅ ((R + d)2 − r 2 ) = (R + d)2 ⋅ r 2 , (R − d)2 ⋅ (R + d)2 = r 2 ⋅ ((R + d)2 + (R − d)2 ), (R2 − d2 )2 = r 2 ⋅ (2R2 + 2d2 ).

So, if one specifies R and d, the latter formula delivers r, such that there exists at least one quadrangle inscribed into one circle and circumscribed to the other. Then, according to Poncelet’s closure theorem, there are finitely many interscribed (and thus, bicentric) quadrangles. The generalization to two fitting ellipses could be envisioned as follows: Arbitrarily specify the outer ellipse and an inner ellipse that reasonably fits by eye, then apply a perspective collineation that maps two conjugate complex intersection points of the two ellipses to the absolute circle points (the center being the Laguerre point), then adjust the radius of the inner circle according to the above formula, and finally apply the inverse collineation. This way, the estimated inner ellipse is gently adjusted. The polarity in a circle centered at the Laguerre point sends the two circles with a bicentric quadrangle to two confocal conics with a billiard as studied in Section 9.6.

◾

Example 9.5.7 Spatial ball-bearings.

We end this section with a particular porism for two circles c1 , c2 in 3-space: If there exists one closed n-gon in form of a zig-zag between c1 and c2 such that all sides are of equal length l, then there are infinitely many. The proof (see [72, 152]) is again based on Poncelet’s closure theorem since the orthogonal projections of the sides into the plane of one circle ci envelope a conic [151].

478

Chapter 9: Special problems

There is an application in mechanical engineering: Suppose the vertices of the n-gon are movable along the respective circles c1 , c2 and at the same time the centers of balls with radius l/2. Then, we obtain the spatial version of a ball-bearing [11], because each ball remains in permanent contact with the two adjacent balls which are centered on the other circle.

479

9.6 Billiards in ellipses

9.6 Billiards in ellipses Referring to Definition 8.1.2, we are going to focus on billiards in ellipses. This is a topic which has attracted the attention of many mathematicians in the past, but also in the last decades. We can confirm this by emphasizing two books: In 2005 S. Tabachnikov covered in [141] a wide variety of themes around billiards and stimulated in the sequel a lot of researchers to work on this topic. In 2011 V. Dragović and M. Radnović addressed in their book [42] billiards in conics and quadrics within the framework of dynamical systems. From Theorem 2.2.4 follows that if a ray is reflected in the ellipse e at any point P , then the incoming and the outgoing ray contact the same confocal conic c, which can be an ellipse or hyperbola (Figure 9.46), or both are focal lines. Hence, billiards in e constitute polygons P1 P2 P3 . . . inscribed in e and circumscribed to the so-called caustic c. Consequently, they are prominent examples of Poncelet porisms. Conversely, polygons inscribed in one conic and with sides tangent to another conic are often called projective billiards. From now on we change the notation: instead of c and i as symbols for the circumscribed and inscribed conic we use e and c, and we restrict the circumconics to ellipses.

e

Q2

c

Q1

P

FIGURE 9.46. If any ray is reflected in the ellipse e, then the incoming and the outgoing ray are tangent to the caustic c, which is a confocal ellipse or hyperbola.

480

Chapter 9: Special problems

When a billiard closes after n reflections in e, then we call the billiard periodic or n-periodic. If the pair (e, c) admits one n-periodic billiard, then due to Poncelet’s Theorem 9.5.3 all billiards inscribed to e and with the caustic c are n-periodic. A continuous move of an initial vertex P1 along e induces a continuous variation of the billiard which is called billiard motion, though it neither perserves angle measures or side lengths of the polygon P1 P2 . . . nor is a projective motion preserving the circumscribed ellipse e. It was C.G.J. Jacobi who proved in 1828 that there is a canonical parameter u on the ellipse e such that the simultaneous transition of each vertex to the next one is just a shift of u (see [77]). Hence, the billiard motion is induced by a continuous translation of u ∈ R (note Theorem 9.6.9). Computer experiments carried out during the last couple of years by Dan Reznik stimulated a new interest on this well studied topic, where algebraic and analytic methods are meeting. It was the beginning of a vivid research on invariants of billiard motions (note [115] and the references there) and their counterparts obtained by various geometric transformations.

Some basic formulas for elliptic billiards In this section we confine us to the cases where the caustic c is an ellipse. We briefly speak of elliptic billiards. Later, in Theorem 9.6.11, a particular three-dimensional transition will be presented which transforms elliptic billiards into those with a hyperbolic caustic while all side lengths are preserved. We recall from Definition 8.1.1, that a range of confocal central conics given as x2 y2 + = 1, k ∈ R {−a2 , −b2 }, a 2 + k b2 + k sends through each point P outside the common axes of symmetry two orthogonally intersecting conics, one ellipse and one hyperbola (Figure 2.21). The parameters (k, l) of these two conics define the elliptic coordinates of P with −a2 < l < −b2 < k .

We denote the semiaxes of the ellipse c as (ac , bc ) and assign to c the first elliptic coordinate k = 0. Then, for points P on the confocal ellipse e with semiaxes (ae , be ) and k = ke > 0, i.e., exterior to c, we have the standard

481

9.6 Billiards in ellipses

parametrization P = (x, y) = (ae cos t, be sin t), 0 ≤ t < 2π, with a2e = a2c + ke , b2e = b2c + ke .

(9.39)

For the elliptic coordinates (ke , l) of P ∈ e follows from (8.10) that ke + l = a2e cos2 t + b2e sin2 t − a2c − b2c .

After introducing the respective tangent vectors of e and c, namely te (t) ∶= (−ae sin t, be cos t), where ∥te ∥2 = ∥tc ∥2 + ke , tc (t) ∶= (−ac sin t, bc cos t),

(9.40)

we obtain11

l = l(t) = −(a2c sin2 t + b2c cos2 t) = −∥tc (t)∥2 = −∥te (t)∥2 + ke .

(9.41)

Note that points on e and c with the same parameter t share the second elliptic coordinate l and belong to the same confocal hyperbola (Lemma 8.1.3). Conversely, points of e or c on this hyperbola have a parameter out of {t, −t, π + t, π − t}. Normal vectors of e and c can be defined respectively as gradients ne (t) ∶= (

cos t sin t cos t sin t ∥tc (t)∥ , ), nc (t) ∶= ( , ), ∥nc (t)∥ = . (9.42) ae be ac bc ac b c

We complete with two useful relations between the parameter t and the second elliptic coordinate l(t). From (9.41) follows l=−

a2c tan2 t + b2c b2c + l(t) 2 , hence tan t = − . 1 + tan2 t a2c + l(t)

(9.43)

If ah and bh are the semiaxes of the hyperbola corresponding to the parameter t, i.e., a2h = a2c + l and b2h = −(b2c + l), then follows with the notation d2 ∶= a2c − b2c for the squared linear eccentricity √ −(b2c + l)(a2c + l) ah bh tan t sin t cos t = = = 2 . (9.44) 1 + tan2 t a2c − b2c d Next we compute the angle θ between two consecutive sides of any billiard (Figure 9.47). 11

The norm ∥te ∥ equals half length of e’s diameter which is parallel to te .

482

Chapter 9: Special problems R1

tP1 q1

p2 P2

θ1 /2

r P1 R5 e

p1 c

Q1

F1 Q2

Q5

q5 P5

F2 O Q4

P3

R4

Q3 P4

FIGURE 9.47. Periodic billiard P1 P2 . . . P5 inscribed in the ellipse e with the caustic c.

Lemma 9.6.1 Let P = (ae cos t, be sin t) with the elliptic coordinates (ke , l) be a point on the ellipse e with ke > 0 and c being the confocal ellipse with k = 0. Then, the angle θ(t)/2 between the tangent tP to e at P and any tangent from P to c satisfies sin2

θ ke ke = = , 2 2 ∥te (t)∥ ke − l

2ke l+ ke cos θ = 1 − = , 2 ∥te (t)∥ l− ke

tan2

θ ke =− , 2 l

√ 2 −ke l sin θ = ± . ke − l

(9.45)

(9.46)

Proof: The tangent tP to e at P = (ae cos t, be sin t) in the direction of te has the slope f ∶= tan α1 =

−be cos t . ae sin t

If s1 and s2 denote the slopes of the tangents from P to c, then they satisfy y − be sin t = si (x − ae cos t),

i = 1, 2.

As tangents of c, their homogeneous line coordinates

(u0 ∶ u1 ∶ u2 ) = ((be sin t − si ae cos t) ∶ si ∶ −1)

must satisfy the tangential equation −u20 + a2c u21 + b2c u22 = 0 of c. This results in a quadratic equation for the unknown s, namely (a2e sin2 t − ke )s2 + 2ae be s sin t cos t + (b2e cos2 t − ke ) = 0.

483

9.6 Billiards in ellipses We conclude s1 + s2 =

−2ae be sin t cos t b2 cos2 t − ke and s1 s2 = e2 . 2 2 ae sin t − ke ae sin2 t − ke

The slopes f = tan α1 of tP and tan α2 = s1 or s2 of the tangents to c imply for the enclosed signed angle θ(t)/2 (for brevity, we often suppress the parameter t) tan

θ s1 − f f − s2 = tan(α1 − α2 ) = = , 2 1 + s1 f 1 + s2 f

hence

θ (s1 − f )(f − s2 ) f (s1 + s2 ) − s1 s2 − f 2 = = . 2 (1 + s1 f )(1 + s2 f ) f (s1 + s2 ) + 1 + f 2 s1 s2 After some computations, we obtain tan2

tan2 therefore,

θ ke ke ke = = = , 2 a2e sin2 t + b2e cos2 t − ke ∥te ∥2 − ke ∥tc ∥2

cot2

θ ∥te ∥2 θ 1 = − 1 and sin2 = 2 ke 2 1 + cot2

where ke = a2e − a2c = b2e − b2c , and finally cos θ = 1 − 2 sin2 which confirms the claim.

θ 2

=

ke , ∥te ∥2

θ 2ke ∥tc ∥2 − ke =1− = . 2 2 ∥tc ∥ + ke ∥tc ∥2 + ke

●

Exercise 9.6.1 Some formulas in terms of elliptic coordinates.

●

Exercise 9.6.2 A constant product of distances.

◾

Let Q be a point of the ellipse c with semiaxes (ac , bc ) and center O. If tQ is the tangent to c at Q and κQ the corresponding curvature, while NQ denotes the intersection between the normal line to c at Q and the principal axis, then, in terms of the tangent vector tc of P from (9.40), we have ac bc bc ac bc O tQ = , ONQ = ∥tc ∥, κQ = . ∥tc ∥ ac ∥tc ∥3 Prove these formulas (among them (3.17)). Note that for the second elliptic coordinate l of Q holds l = −∥tc ∥2 by (9.41), provided that the first elliptic coordinate of c vanishes. Prove the following statement: Referring to Figure 9.47 and the notation of the previous exercise, let R denote the pole of tQ w.r.t. the ellipse e. Then, the product of the signed distances O tQ ⋅ OR = −ke is constant and equals the negative first elliptic coordinate of e w.r.t. c . The lines tQ and [Q, R] are orthogonal. Hint: Note the transformation of conjugate normals (page 367).

●

Exercise 9.6.3 3-periodic billiards in ellipses.

Suppose, that e is the ellipse with semiaxes (ae , be ), and c is the caustic for triangular elliptic √ √ billiards inscribed in e (Figure 9.48). Let (ac , bc ) = ( a2e − k, b2e − k) be the semiaxes of c. Then, after some computation, we obtain as necessary and sufficient conditions bc (a2e k + a2c b2e ) = be (a2e k − a2c b2e ),

484

Chapter 9: Special problems

and consequently, k=− where d2 = a2e − b2e .

√ a2e b2e 2 (ae + b2e − 2 d4 + a2e b2e ) 4 d

Hint: Inspect a particular position of the billiard or follow the Theorems 9.5.2 or 9.5.4.

P1 R3 Q1 P2

c

Q3

X8

Q2 P3 e

FIGURE 9.48. Triangular billiards in the ellipse e with the caustic c. The Cevians [Pi , Qi+1 ] meet at the Nagel point X8 of the triangle P1 P2 P3 (note Remark 9.6.2 on page 494).

From now on we assume for elliptic billiards P1 P2 P3 . . . in the ellipse e a counter-clockwise order and signed exterior angles θ1 , θ2 , θ3 , . . . (see Figure 9.47). The billiard’s sides P1 P2 , P2 P3 , . . . contact the caustic c at the points Q1 , Q2 , . . ., respectively. Based on the standard parametrizations (ae cos t, be sin t) of e and (ac cos t′ , bc sin t′ ) of c, we denote the respective parameters of P1 , Q1 , P2 , Q2 , P3 , . . . with t1 , t′1 , t2 , t′2 , t3 , . . . in strictly increasing order. The following two lemmas deal with sides of billiards in the ellipse e. Lemma 9.6.2 The connecting line [P1 , P2 ] of the vertices with respective parameters t1 , t2 on e contacts the caustic c if, and only if, a2c b2c t1 + t2 t1 − t2 2 t1 + t2 cos + sin2 = cos2 . 2 2 ae 2 be 2 2

485

9.6 Billiards in ellipses

This is equivalent to sin2

2 t1 − t2 ke t +t = ∥ te ( 1 2 )∥ . 2 2 ae be

Proof: The line connecting the points (ae cos ti , be sin ti ) for i = 1, 2 has the homogeneous line coordinates (u0 ∶ u1 ∶ u2 ) = (ae be (cos t1 sin t2 − sin t1 cos t2 ) ∶ be (sin t1 − sin t2 ) ∶ ae (cos t2 − cos t1 )) .

It contacts the caustic c if −u20 + a2c u21 + b2c u22 = 0, i.e., a2c b2e sin2

t1 − t2 t1 + t2 t1 − t2 t1 + t2 t2 − t1 t2 − t1 cos2 + b2c a2e sin2 sin2 = a2e b2e sin2 cos2 . 2 2 2 2 2 2

Under the condition sin[(t1 − t2 )/2] ≠ 0, we obtain the first claimed equation. The second follows after the substitutions a2c = a2e − ke and b2c = b2e − ke from 1−

ke t1 + t2 t1 + t2 t2 − t1 (b2e cos2 + a2e sin2 ) = cos2 a2e b2e 2 2 2

◾

by (9.42).

Lemma 9.6.3 Referring to the notation in Lemma 9.6.2, if the side Pi Pi+1 contacts the caustic c at Qi with parameter t′i , then sin t′i =

t +t t +t bc sin i 2i+1 ac cos i 2i+1 bc ae ti + ti+1 ′ , cos t = , tan t′i = tan . i t −t t −t i i+1 i i+1 be cos 2 ae cos 2 a c be 2

Proof: The tangent to c at Q1 has the line coordinates (u0 ∶ u1 ∶ u2 ) = (−ac bc ∶ bc cos t′1 ∶ ac sin t′1 )

which must be proportional to those in the proof of Lemma 9.6.2.

◾

Remark 9.6.1 1. The half-angle substitution ti for i = 1, 2 2 allows us to express the equation of Lemma 9.6.2 in projective coordinates on the ellipse e. We obtain a symmetric biquadratic condition τi ∶= tan

b2e ke τ12 τ22 − b2c a2e (τ12 + τ22 ) + 2(a2e ke + a2c b2e )τ1 τ2 + b2e ke = 0,

which defines a 2-2-correspondence on e between the endpoints P1 , P2 of a c contacting chord of e. Iteration yields a 2-2-correspondence between the initial point P1 and the endpoint Pn+1 of a billiard after n reflections in e. This corresponds to the elliptic curve as mentioned in the sketch of the proof for Poncelet’s Theorem 9.5.3. 2. With the aid of Jacobi’s arguments in [77, page 381], the third formula in Lemma 9.6.3 already paves the way to a representation of the billiard’s vertices in terms of Jacobian elliptic functions.

486

Chapter 9: Special problems

Some motion invariants along elliptic billiards As a direct consequence of the formulas so far, we present some motion invariants of billiards which can be periodic or aperiodic. Given any billiard P1 P2 . . . in the ellipse e, let pi = (xi , yi ) denote the position vector of Pi for i = 1, 2, . . ., while u1 , u2 , . . . denote the unit vectors of the oriented sides P1 P2 , P2 P3 , . . . (Figure 9.49). By (9.42), the vector ne∣i ∶= (xi /a2e , yi /b2e ) is orthogonal to e at Pi . According to [3, Proposition 2.1], the scalar product Je ∶= −⟨ui , ne∣i ⟩

(9.47)

is invariant along the billiard in e and called Joachimsthal integral, named after Ferdinand Joachimsthal (1818–1861) (note also [141, p. 54]). ne∣2 θi /2 θi /2

P3

Q2

ne∣1

P2 Q1

u2

∥ui ∥ = 1

u1

P1 c

e

O

FIGURE 9.49. The Joachimsthal integral Je ∶= −⟨ui , ne∣i ⟩ is constant along e.

The invariance of the Joachimsthal integral, which also holds in higher dimensions for billiards in quadrics, is the key result for the integrability of billiards, i.e., in the planar case for the existence of a caustic [3, p. 3]. In our approach, the invariance of Je follows from Lemma 9.6.1. Lemma 9.6.4 The Joachimsthal integral Je ∶= −⟨ui , ne∣i ⟩ equals √ ke Je = ae be with ke as elliptic coordinate of e w.r.t. c, i.e., ke = a2e − a2c = b2e − b2c . Proof: From (9.45) follows for the points (ae cos t, be sin t) of e Je = −⟨u, ne ⟩ = − cos(

π θ θ θ ∥te ∥ + ) ∥ne ∥ = sin ∥ne ∥ = sin , 2 2 2 2 ae be

487

9.6 Billiards in ellipses hence by (9.45), (9.41), (9.40), and (9.42) Je2 = sin2 This confirms the claim.

θ ke ke ∥ne ∥2 = ∥ne ∥2 = 2 2 . 2 ∥te ∥2 ae be

◾

The associated Poncelet grid

Each billiard is connected with a Poncelet grid as displayed in Figure 9.32. The Poncelet grid arises in a natural way. The sides of a billiard create a pattern of quadrilaterals. Surprisingly, each vertex of the quadrilateral grid lies on two conics, and the grid built by the sides can be replaced by the grid formed by conics out of a range. Below, we prove that under a billiard motion one family of conics in the Poncelet grid remains fixed. Definition 9.6.1 For the points of intersection between extended sides of a billiard . . . P0 P1 P2 . . . we use the notation (j)

Si

∶= {

[Pi−k−1 , Pi−k ] ∩ [Pi+k , Pi+k+1 ] for j = 2k, [Pi−k , Pi−k+1 ] ∩ [Pi+k , Pi+k+1 ] for j = 2k − 1,

where i = . . . , 0, 1, 2, . . . and j = 1, 2, . . ..

(j)

Note that there are j sides between those which intersect at Si , and ‘in the middle’ of these j sides, there is for even j the vertex Pi , and otherwise, the point of contact Qi (Figure 9.50). At the same token, the (j) point Si is the c-pole of the diagonal [Qi−k−1 , Qi+k ] or [Qi−k , Qi+k ] of the polygon . . . Q1 Q2 Q3 . . . of contact points (note [137]). Theorem 9.6.1 Let . . . P0 P1 P2 . . . be an elliptic billiard in the ellipse e with sides Pi Pi+1 contacting the caustic c at the respective points Qi for (j) all i ∈ Z. Then, the vertices Si of the associated Poncelet grid lie on the following conics. (1)

(3)

(i) The points Si , Si , . . . are located on the confocal hyperbola through (2) (4) Qi , while the points Si , Si , . . . are located on the confocal hyperbola through Pi . (j) (j)

(j)

(ii) For each j ∈ {1, 2, . . .}, the points . . . Si Si+(j+1) Si+2(j+1) . . . are vertices of another billiard with the caustic c inscribed in a confocal ellipse e(j) , provided that e(j) is regular. Otherwise e(j) coincides with an axis of

488

Chapter 9: Special problems

symmetry or with the line at infinity. The locus e(j) is independent of the position of the initial vertex P0 ∈ e. (1)

S3

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (2)⎪ S5 ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

P5

S2

Q3 Q1

Q5 (1)

S5

θ2 P2

Q2

Q4

(2)

S2

(1)

P3

P4

(1) S4

θ2 +θ3

θ3

Q6

P6

Q7

P1

(1)

S1

c e P7

e(2)

e(1) (1)

S7

(1)

S6

(2)

S1

FIGURE 9.50. n-periodic elliptic billiard (n = 7) with extended sides and conics of the associated Poncelet grid.

Proof: 1. The side lines [P0 , P1 ] (P0 = P7 in Figure 9.50) and [P2 , P3 ] meet at S1 , while

[P1 , P2 ] contacts c at Q1 . From Theorem 7.3.3 follows that the points Q1 and the same confocal hyperbola.

(1)

(1) S1

belong to (1)

Now, we go one step away from Q1 : The tangents from P0 and P3 to c intersect at S1

and = [P−1 , P0 ] ∩ [P3 , P4 ]. The confocal conic through and must again be the hyperbola through Q1 . This follows by continuity after choosing Q1 on one axis of symmetry. Iteration confirms the first claim. (3) S1

(1) S1

(3) S1

(2)

The tangents to c from P0 and P2 form a quadrilateral with P1 and S1 as opposite vertices. Therefore, by virtue of Theorem 7.3.3 exists a confocal conic passing through both points. This conic must be a hyperbola, as can be concluded by continuity: If P1 is specified at a (2) vertex of e, then due to symmetry the points P1 and S1 are located on an axis of symmetry. (2)

(4)

The tangents to c from P−1 and P3 form a quadrilateral with S1 and S1 as opposite vertices. Theorem 7.3.3 and continuity guarantee that this is again the confocal hyperbola (6) (2) (4) through P1 . Iteration shows the same for S1 etc. However, the points P1 , S1 , S1 , . . . need not belong to the same branch of the hyperbola. (2)

2. The tangents through P2 and S2 (1) S2

(1)

(note Figure 9.50) form a quadrilateral with S1

and

as opposite vertices. This time, continuity shows that the two points belong to the same (2)

confocal ellipse e(1) . The same holds for the tangents through P3 and S3 starting with the points P0 and P3 , we find the ellipse e

(2)

through

(2) S1

etc. Similarly, (2)

and S2 , and so on.

In order to prove that these ellipses e(1) , e(2) , . . . are independent of the choice of the initial point P1 ∈ e, we follow an argument from [3, proof of Corollary 2.2]: The claim holds for all

489

9.6 Billiards in ellipses

confocal ellipses e where billiards with the same caustic c are aperiodic and traverse e infinitely often. Since these ellipses form a dense set, the claim holds also for those with periodic billiards. The invariance of the ellipses e(1) , e(2) , . . . has already been mentioned in [3, Theorem 7]. An alternative proof consists in demonstrating that for all j the semiaxes ae∣j and be∣j of e(j) do not depend on the parameter t. However, these expressions are getting complicated with increasing j. In [136, Corollary 4.5], formulas for j = 1 and 2 are presented.

◾

It needs to be noted that there exists a Poncelet grid for all projective billiards. For example, Figure 9.47 shows the so-called outer polygon R1 R2 . . . R5 inscribed in r and circumscribed to e. This is no more a regular billiard since e and the polar transform r of c w.r.t. e are not confocal. On the other hand, the polygon Q1 Q2 . . . Q5 is inscribed in c and circumscribed to the polar transform q of e w.r.t. c. In [135], it is proved that if at the corresponding Poncelet grids the conics r (j) and c(j) are the analogues of e(j) as defined in Theorem 9.6.1, then they are in a similar mutual relation like r, c, and e: There is a projective billiard inscribed in e(j) with the caustic c(j) , while r (j) is the corresponding outer polygon. The conjugate billiard

Let us return to the original elliptic billiard. Given the two confocal ellipses c and e, there exists the affine transformation α∶ (x, y) ↦ (

ae b x, e y) ac bc

with c → e .

(9.48)

Corresponding points belong to the same confocal hyperbola (Lemma 8.1.3). The affinity α sends the contact point Qi ∈ c of the side Pi Pi+1 to ′ Pi′ , a point Pi′ ∈ e, while α−1 maps Pi to the contact point Q′i−1 of Pi−1 i.e., α∶ Qi ↦ Pi′ , Q′i−1 ↦ Pi (see Figure 9.51). This results from the symmetry between ti and t′i in the equation bc ae cos ti cos t′i + ac be sin ti sin t′i = ac bc (9.49)

which expresses that Pi ∈ e with parameter ti lies on the tangent to c at Qi with parameter t′i (note also Figure 8.11). Referring to Figure 9.51, α ′ sends the tangent [Pi−1 , Pi′ ] to c at Q′i−1 to the tangent [Ri−1 , Ri ] to e at Pi . Hence, by α the polygon Q1 Q2 . . . is mapped to P1′ P2′ . . ., and furthermore, to that of the poles R1 R2 . . . of the billiard’s sides P1 P2 , P2 P3 , . . . .

490

Chapter 9: Special problems R1

r

P1 P1′

c q

Q1 P2

e

Q′5

Q5

Q′1

Q′4 O

Q2

P3

R5 P5′

Q3

Q4 e′ P4

P5

P4′ R4

FIGURE 9.51. The periodic billiard P1 P2 . . . P5 in e with the caustic c along with the conjugate billiard P1′ P2′ . . . P5′ and the outer (projective) billiard R1 R2 . . . R5 inscribed in the dual transform r of c w.r.t. e.

Definition 9.6.2 Referring to Figure 9.51, the billiard . . . P0′ P1′ P2′ . . . is called conjugate to the elliptic billiard . . . P0 P1 P2 . . . in the ellipse e with the caustic c, when the axial scaling α ∶ c → e defined in (9.48) maps the contact point Qi of the side Pi Pi+1 to the vertex Pi′ for all i. Lemma 9.6.5 For each elliptic billiard . . . P0 P1 P2 . . . in the ellipse e with the caustic c there exists a unique conjugate billiard . . . P0′ P1′ P2′ . . ., and the relation between the two billiards in e is symmetric. Moreover, Pi Qi = Pi′ Q′i−1

and

′ Q′ . Pi Qi−1 = Pi−1 i−1

(9.50)

Proof: From the symmetry in (9.49) follows for α ∶ c → e that the preimage of Pi is the contact ′ point Q′i−1 of Pi−1 Pi′ (compare also with Figure 8.11). The congruences stated in (9.50) follow from Ivory’s Theorem for the two diagonals in the curvilinear quadrangle Pi Pi′ Qi Q′i−1 . In view of the sequence of parameters t1 , t′1 , t2 , t′2 , t3 , . . . of the vertices P1 , P1′ , P2 , P2′ , P3 , . . . on e, the switch between the original billiard and its conjugate corresponds to the interchange of ti with t′i for i = 1, 2, . . . .

◾

If . . . P0 P1 P2 . . . and . . . P0′ P1′ P2′ . . . is a pair of conjugate elliptic billiards ′ in the ellipse e, then the points of intersection [Pi , Pi+1 ] ∩ [Pi′ , Pi−1 ] are ′ located on a confocal ellipse e (Figure 9.51). This follows from Theorem 7.3.3, case (iv), applied to the tangents to c from Pi′ and in Qi as well as from Pi and at Q′i−1 .

491

9.6 Billiards in ellipses (1)

S1

e(1)

(1)

S2

2

P2 7 4 5 4

3 2

1

P2′ 1 2

v =0

e c

Q′1

P1′ Q1

P1

Q2 u =4

Q′0 7 2

3

5 2

2

3 2

1

1 2

Q0 0

P0′

(1)

S0

FIGURE 9.52. Canonical coordinates u (blue) for the confocal hyperbolas and v (red) for the confocal ellipses exterior to the caustic c such that u ± v = const. characterizes the tangents to the caustic (note Lemma 9.6.8).

From the viewpoint of the billiard’s motion, point Pi′ ‘bisects’ the way from Pi to Pi+1 for the following reason. While the billiard motion moves Pi along e to Pi′ , it takes Qi to Q′i and consequently Pi′ to Pi+1 (Figure 9.51). This illustrates that there is a canonical parameter u on e, where the so-called billiard map Pi to Pi+1 acts like a shift on u. In fact, we can iterate this way of bisecting after replacing e by e′ and using the affine transformation c → e′ . This provides an intuitive approach to the canonical parametrization of a billiard and its Poncelet grid (note Figure 9.52 and Lemma 9.6.8). Similar results can be found in [89, Sect. 4]. Explicit formulas for the parameter transformation from t to a canonical parameter u are presented in the subsection beginning on page 501.12

Some motion invariants of periodic elliptic billiards Definition 9.6.3 The sum of the oriented exterior angles θi of a periodic billiard in an ellipse e is an integer multiple of 2π, namely 2τ π. We call the factor τ ∈ N the turning number. It counts the loops of the billiard around the center O of e, anti-clockwise or clockwise. 12

Of course, canonical parameters with the property that the shift with ∆u effects Pi → Pi′ for all i, while u ↦ u + 2∆u sends Pi to Pi+1 , are unique only up to multiplicative constants.

492

Chapter 9: Special problems

If the periodic billiard P1 P2 . . . Pn has the turning number τ = 1 (Fi(1) (1) (1) gure 9.54), then the billiard S1 S3 S5 . . . in e(1) has τ = 2, that of (2) (2) S1 S4 . . . in e(2) the turning number τ = 3, and so on. In cases with g = gcd(n, τ ) > 1 the corresponding billiard splits into g ng -sided billiards, each with turning number τ /g (note [121, Theorem 1.1]). Symmetry properties of periodic elliptic billiards

The theorem below is a consequence of Theorem 9.6.1. Theorem 9.6.2 Let P1 P2 . . . Pn be an n-sided periodic elliptic billiard in the ellipse e with the caustic c and with the turning number τ ≠ 0. (i) For even n and odd τ , the billiard is centrally symmetric. (ii) For odd n = 2m + 1 and odd τ , the billiard is centrally symmetric to ′ the conjugate billiard, where Pi corresponds to Pi+m .13 (iii) If n is odd and τ is even, then the conjugate billiard coincides with ′ the original one, and Pi = Pi+m . Proof: By virtue of Theorem 9.6.1, the lines [Pi−j−1 , Pi−j , ] and [Pi+j , Pi+j+1 ] for j = 1, 2, . . . meet at the point Si (j) on the confocal hyperbola through Pi .

(i) This means for even n = 2m, odd τ and j = m − 1, that also the opposite vertex Si = Pi−m = Pi+m belongs to this hyperbola. If Pi is specified at a vertex on the minor axis of the ellipse e, then Pi+m is the opposite vertex. Continuity implies that the two points belong to different branches of the hyperbola and are symmetric w.r.t. the center O of e. (m)

(ii), (iii): If n is odd, say n = 2m + 1, then for j = m − 1 the sides [Pi−m+1 , Pi−m ] and ′ [Pi+m−1 , Pi+m ] intersect at a point on the hyperbola through Qi+m and Pi+m . For odd τ ′ (Figures 9.51 and 9.54), the same continuity argument as before proves that Pi and Pi+m are opposite w.r.t. O. If τ is even (Figure 9.53), then a specification of Pi ∈ e on an axis of symmetry shows the ′ coincidence with Pi+m ∈ e, and this must be preserved, when Pi varies continuously on e. In the case of even τ and n the billiard splits.

◾

Invariants involving distances and angles of the moving billiards

In view of invariant quantities of periodic billiards, we begin with a result that has first been proved for a much more general setting in [141, p. 103]. Theorem 9.6.3 The length L of a periodic elliptic billiard in the ellipse e is independent of the position of the initial vertex P1 ∈ e. 13

All subscripts in this section are understood modulo n.

493

9.6 Billiards in ellipses P2

P6

e(2)

(2) S4

P5

(2)

S7

Q5 Q1

(2)

S3

Q4 P3

P1 (2)

S6

Q7

c

e

P4 P7 FIGURE 9.53. Periodic billiard with n = 7 and τ = 2. The power of Pi w.r.t. the incircle that touches c at Qi−4 is motion invariant [133]. Proof: We refer to Theorem 2.2.5 treating Graves’s construction of e with a string wrapped around the caustic c (see Figure 2.29). As noted in Remark 2.2.5, the quantity D ∶= Qi−1 Pi + Pi Qi − Q̂ i−1 Qi

is constant, when Q̂ i−1 Qi stands for the length of the shorter arc between Qi−1 and Qi along c. This yields for an n-sided billiard with turning number τ the total length L = n ⋅ D + τ ⋅ Pc ,

(9.51)

◾

where Pc denotes the perimeter of the caustic c.

The following theorem on the invariant k118 in [115] deals with the lengths pi and qi of the segments Qi−1 Pi and Pi Qi (Figure 9.47). Theorem 9.6.4 In each n-sided periodic billiard, opposite segments are congruent, i.e., if n = 2m, then pi+m = pi and qi+m = qi , and if n = 2m + 1, then pi+m = qi−1 and qi+m = pi . Thus, for odd n, we have n

n

i=1

i=1

∑ q i = ∑ pi =

L . 2

Proof: By Theorem 9.6.2, for even n = 2m the central symmetry implies for opposite segments pi = pi+m and qi = qi+m .

494

Chapter 9: Special problems (1)

S2

q2 P3

e

c

Q2

(1)

e(2)

P1 (1) S9

Q9

Q4 P5

(2)

S1

e

Q1

Q3

P4

(1)

S1

P2

P9

P6

(2)

Q8

Q5

(1)

S8

Q7

Q6

S9

P8

P7

p7

(1)

S7

(2)

S8

FIGURE 9.54. Periodic billiard P1 P2 . . . P9 with τ = 1. Note that by virtue of

Ivory’s theorem q2 = P2 Q2 = Q6 P7 = p7 and S2 P3 = P7 S7 . The associated (2) (2) (2) billiard in e(2) with caustic c splits into three triangles Si Si+3 Si+6 . (1)

(1)

′ ′ If n = 2m + 1, then the distance qi = Pi Qi shows up as qi+m = Pi+m Q′i+m at the conjugate billiard and, by virtue of Ivory’s Theorem, this equals pi+m+1 = Pi+m+1 Qi+m (Figure 9.54). Similarly follows pi = qi+m .

◾

Remark 9.6.2 In the particular case n = 3, the two segments adjacent to any side are congruent. Therefore, the lines [Pi , Qi+1 ] are concurrent, also due to the Pappus’s theorem, and meet at the Nagel point X8 of the triangle P1 P2 P3 (see Figure 9.48). This has already been proved in [114] and goes hand in hand with the circles centered at Ri and tangent to c at Qi , which in this particular case are excircles of the triangle P1 P2 P3 .

Below, we recall for even n a few invariants related to the segments pi and qi . The last one is listed as k117 in [115, Table 2]. Proofs based on Jacobian elliptic functions can be found in [136]. Theorem 9.6.5 Referring to the previous notation, let P1 P2 . . . Pn be a periodic elliptic billiard with even n in the ellipse e that has the elliptic coordinate ke w.r.t. the caustic c. Then, we have for the distances pi = Qi−1 Pi , qi = Pi Qi , and for the side lengths si = Pi Pi+1 = qi + pi+1 and at

495

9.6 Billiards in ellipses ′ =p the conjugate billiard s′i = Pi′ Pi+1 i+1 + qi+1 , we have

for n = 4m ∶ pi ⋅ pi+m = qi ⋅ qi+m = ke ,

si+m qi+m pi+m+1 = = , si pi+1 qi si+m qi+m pi+m+1 for n = 4m+2 ∶ pi ⋅ qi+m = qi ⋅ pi+m+1 = ke , ′ = = . si−1 qi pi

Thus, we obtain motion invariant products p1 p2 . . . pn = q1 q2 . . . qn = ke , n/4 and for n ≡ 0 (mod 4) already p1 p2 . . . pn/2 = q1 q2 . . . qn/2 = ke . n/2

The following theorem dealing with the invariant k101 from [115, Table 2] has first been proved in [3, p. 4]. An alternative proof can be found in [10, Cor. 3.2]. We present another proof.

Theorem 9.6.6 For the exterior angles θ1 , . . . , θn of the n-periodic elliptic billiard in the ellipse e , the sum of cosines is independent of the initial vertex, namely √ n ke L, ∑ cos θi = n − Je L = n − a e be i=1 where L denotes the common perimeter of these billiards in e and Je the Joachimsthal integral. θi /2

θi /2

Pi Fi

Fi−1 Qi−1

Qi

Pi+1

Pi−1 c

e

O FIGURE 9.55. Billiard . . . Pi−1 Pi . . . with pedal points Fi−1 , Fi w.r.t. O. Proof: The pedal points Fi and Fi−1 on the sides Pi Pi+1 and Pi−1 Pi w.r.t. the center O (Figure 9.55) have the position vectors fi,i−1 = pi +

λi,i−1 ∥te ∥

(cos

θi θi ⊥ t ± sin t ) , where 2 2

0 = ⟨fi,i−1 , (cos

θi θi ⊥ t ± sin t )⟩. 2 2

Here, λi and λi+1 denote the signed distances from the vertex Pi (with position vector pi ) in one case towards Pi+1 , in the other opposite to Pi−1 , while t is a unit tangent vector at Pi to e. From ⟨pi , t⟩ = (−a2e + b2e ) cos t sin t and ⟨pi , t⊥ ⟩ = −ae be

496 follows

Chapter 9: Special problems λi,i−1 =

1 θi θi ((−a2e + b2e ) cos t sin t cos ± ae be sin ) . ∥te ∥ 2 2

By virtue of (9.45) and after reversing the orientation for λi−1 , this implies, λi − λi−1 = Pi Fi + Pi Fi−1 =

2ae be √ ke . ∥te ∥2

Since the sum over all signed lengths between Pi and the adjacent pedal points gives the total perimeter L of the billiard, we obtain by (9.46) n ae be n 2ke ae be n L = ∑ ( Pi Fi + Pi Fi−1 ) = √ = √ ∑ ∑ (1 − cos θi ), 2 ∥t ∥ k ke i=1 e e i=1 i=1

hence

n

1 L = √ 2 2ae be ke i=1 ∥te ∥ ∑

as stated.

and also

n

∑ cos θi = n −

i=1

√

ke L, ae be

(9.52)

(9.53)

◾

Remark 9.6.3 Note that the analogue result in [3] relates to the interior angles of the billiard. As already mentioned in [3, Theorem 7], the constant sum of cosines holds also for the ‘extended’ billiards in e(j) , where the exterior angles are θi +θi+1 +. . . +θi+j (note Figure 9.50).

The first equation in (9.53) together with the formulas in in Exercise 9.6.1 yields two well-known invariants: The first refers to the distances of O to the tangents to e at the vertices Pi and dates back to [10, Cor. 3.2, third equation]. The second is the invariant k119 in [115, Table 2]; it refers to the curvature κi of e at Pi and was first proved by P. Roitmann. Corollary 9.6.6 Referring to the previous notation, the squared distances from the center O to the tangents tPi to the ellipse e at the vertices Pi of the n-periodic elliptic billiard with perimeter L have the constant sum n a e be 2 ∑ O tPi = √ L. 2 ke i=1

The curvatures κi of e at the vertices Pi give rise to the invariant sum n

∑ κi i=1

2/3

L = √ (ae be )−1/3 . 2 ke

Finally, we just report about a result in [136] which again holds only for even n.

497

9.6 Billiards in ellipses

Theorem 9.6.7 The exterior angles θi of an n-periodic elliptic billiard in an ellipse satisfy for even n n

i ∑ (−1) sin θi = 0

i=1

and for n ≡ 0 (mod 4)

n/2

i ∑ (−1) sin θi = 0 .

i=1

For n ≡ 2 (mod 4) this result is trivial due to the central symmetry of periodic billiard (Theorem 9.6.2).

Velocity analysis of elliptic billiards

Let the first vertex of an elliptic billiard P1 P2 . . . move smoothly along the circumscribed ellipse e. This induces a continuous variation of all other vertices along e and also of the Poncelet grid, i.e., of intersection (j) points Si along e(j) (Theorem 9.6.1). This is what we called the billiard motion. A graphical velocity analysis as the starting point

According to Graves’s construction (Theorem 2.2.5), we take the billiard as a string that is wrapped around the caustic c without sliding on c while the vertices P1 , P2 , . . . move along e and keep the string taut. Thus, we can extend the graphical velocity analysis as presented in Figure 2.29 from a single vertex to the billiard. Let us focus on the vertex P2 (see Figure 9.56). The extended line [Q1 , P2 ] of the side Q1 P2 rolls at Q1 on c (= fixed polode) while point P2 moves along this line (= moving polode) with the velocity vector vt1 . The instantaneous rotation about Q1 with the angular velocity ω1 assigns to P2 a velocity vector vn1 orthogonal to Q1 P2 in order to keep the vector of absolute velocity v2 = vt1 + vn1 of P2 tangent to the ellipse e. The tangential component vt1 originates from the sliding of P2 relative to the string. Similarly, we have a second decomposition v2 = vt2 + vn2 , since at the same time the line [Q2 , P2 ] rotates about Q2 with the angular velocity ω2 , while P2 moves relative to this line. Due to the constant length of the string, the tangential components in these two decompositions must be of equal lengths ∥vt2 ∥ = ∥vt1 ∥. Since the tangent tP to e at P2 bisects the exterior angle of Q1 P2 Q2 , the second decomposition is symmetric w.r.t. tP to the first one. For the distances p2 ∶= P2 Q1 and q2 ∶= P2 Q2 follows ω1 q2 p2 ω1 = ∥vn1 ∥ = ∥vn2 ∥ = q2 ω2 , i.e., = , (9.54) ω 2 p2

498

Chapter 9: Special problems vt2 vn∣2 R2

v2

p3 P3 v3

θ2/2

P2

q2

θ2/2

c

R1 p2

e

Q1

ω1

Q2

θ1/2

v1 q1 P1

ω5

ω2 Q5

Q3 ω4

ω3

R3

Q4

θ1/2

R5 v5

P5

P4 v4 FIGURE 9.56. Velocities of the vertices P1 , P2 , . . . , P5 of a periodic elliptic billiard in the ellipse e with the caustic c.

and similarly for all other vertices. If the billiard is n-periodic, then the product of all ratios qi /pi for i = 1, . . . , n yields q1 q2 qn ωn ω1 ωn−1 ⋅ ⋅⋯⋅ = ⋅ ⋅⋯⋅ = 1. p1 p2 pn ω 1 ω 2 ωn This results in the equation q1 q2 . . . qn = p1 p2 . . . pn , which is listed as k116 n/2 in [115, Table 2]. According to Theorem 9.6.5, both products equal ke . Figure 9.56 shows a graphical velocity analysis for the billiard motion of a 5-sided periodic billiard. We begin this analysis by choosing an arbitrary length of the arrow representing the velocity vector v2 of P2 . This defines the two components vt2 and vn2 , the angular velocity ω2 of the side P2 P3 and furtheron the absolute velocity v3 of P3 . This can be continued. From now on, we denote the norms ∥vt1 ∥ = ∥vt2 ∥ and ∥vn1 ∥ = ∥vn2 ∥ of the respective components of the velocity vector vi of Pi with vt∣i and vn∣i , and we define vi ∶= ∥vi ∥. A new invariant related to velocities

In terms of the exterior angles θ1 , θ2 , . . . of the billiard holds by (9.54) θ 2 q 2 ω 2 p2 ω 1 θ2 vt∣2 sin = = and cos = , where v2 ∶= ∥v2 ∥. (9.55) 2 v2 v2 2 v2

499

9.6 Billiards in ellipses

Let Ri denote the pole of the line [Pi , Pi+1 ] w.r.t. e (Figure 9.56). The side P1 P2 must be orthogonal to [Q1 , R1 ] (note Exercise 9.6.2). This implies R1 Q1 = q1 tan

θ1 θ2 = p2 tan . 2 2

(9.56)

From (9.56) and (9.55) follows q1 tan

θ1 q1 ω1 θ2 p2 ω 1 = q1 = p2 tan = p2 2 vt∣1 2 vt∣2

and

vt∣2 vt∣1

=

p22 tan2 = q12 tan2

θ1 2 θ2 2

.

This implies by virtue of (9.45) and (9.41) for the products vt∣1 tan2 for i = 1, 2, . . . .

vt∣i θ1 θ2 ke = vt∣2 tan2 = . . . = vt∣i = −k e 2 2 ∥tc∣i ∥2 l(ti )

(9.57)

Lemma 9.6.7 At each instant of a motion of an elliptic billiard P1 P2 . . . in the ellipse e the tangential velocities vt∣i of the vertices Pi are proportional to the respective second elliptic coordinates l(ti ) of Pi .

We denote the invariant quantity in (9.57) temporarily with C. Instead of a free choice of the velocity v2 of the initial vertex P2 , it means no restriction of generality to set C = ke . Then, we obtain for the vertex Pi ∈ e with the elliptic coordinates (ke , l) by (9.45) vt∣i = −l,

vi =

vt∣i

cos θ2i

=

√ l(l − ke ),

vn∣i = vi sin

√ θi = −ke l. 2

(9.58)

This assignment can immediately be extended to all points of e as √ √ v(t) = −l(t) te (t) = ∥tc (t)∥ te (t) = a2c sin2 t + b2c cos2 t te (t). (9.59)

More than that, we can even extend this to all confocal ellipses of the caustic c. The assignment of a velocity vector √ v(t) = −l(t) t(t) to each point P = (a cos t, b sin t) (9.60)

with t(t) = (−a sin t, b cos t), provided that d2 = a2 − b2 = a2c − b2c and a2 − a2c ≥ 0, defines an instantaneous motion of the plane. Below, we prove

500

Chapter 9: Special problems

vn vt tQ Q

e c

FIGURE 9.57. The infinitesimal motion of the elliptic billiard assigns to each point of the Poncelet grid a velocity vector such that on each tangent tQ to the caustic c the points remain aligned. All points on a confocal hyperbola share the tangential velocity vt .

that this instantaneous motion is compatible with the associated Poncelet grid. In particular, this means that the velocities in (9.58) are also valid (j) for the induced motion of the grid points Si along e(j) and for the contact points Qi along the caustic c. This is illustrated in Figure 9.57. It shows a portion of the Poncelet grid, and for a couple of points, the velocity vectors, each represented by a scaled arrow. As indicated for an arbitrary point Q ∈ c, all points on the tangent tQ to c have velocity vectors v where the respective normal component vn = ∥vn ∥ is proportional to the distance to Q, which expresses the instantaneous rotation about Q. On the other hand, due to (9.58) all points of a confocal hyperbola share the tangential component vt = ∥vt ∥. The billiard motion as the action of a Lie group

The vector field (9.60) induces a one-parameter Lie group Γ of transformations that preserve the caustic and all confocal ellipses. There exists a canonical parameter u of Γ, i.e., for the transformations γ(u) ∈ Γ holds γ(u2 ) ○ γ(u1 ) = γ(u1 + u2 ). As an example, if γ(∆u) brings P1 to P1′ (note Definition 9.6.2) and P1′ to P2 , then (j)

γ(2∆u)∶ Pi ↦ Pi+1 , Qi ↦ Qi+1 , Si

(j)

↦ Si+1 for all i ∈ {1, 2, . . .}.

This means that u provides in particular canonical coordinates on e and on all other confocal ellipses.

501

9.6 Billiards in ellipses

Let a dot indicate the differentiation w.r.t. u. Then, follows from (9.59) v(t) = ∥tc (t)∥ te (t) = t˙ te (t) and consequently √ √ dt t˙ = = ∥tc (t)∥ = −l(t) = a2c sin2 t + b2c cos2 t . (9.61) du

Theorem 9.6.8 Let the elliptic billiard P1 P2 . . . together with the associated Poncelet grid be moving such that the field of velocity vectors (9.60) remains time independent. Then, the motion is the action of a one-parameter Lie group Γ. Each transformation γ(u) ∈ Γ preserves all confocal ellipses and permutes the confocal hyperbolas as well as the tangents to the caustic c. Proof: The first derivative t˙ in (9.61) does not depend on the choice of the confocal ellipse. Therefore, γ(u) permutes the confocal hyperbolas. On the other hand, the representation v = ∥tc ∥ t shows that all confocal ellipses are kept fixed. It remains to verify that the position of any point P on the tangent tQ to c at Q (see Figure 9.57) is preserved under the instantaneous motion: The point P = (a cos t, b sin t) lies on the tangent tQ to c at Q = (ac cos t′ , bc sin t′ ) if, and only if, by (9.49) bc a cos t′ cos t + ac b sin t′ sin t = ac bc .

This is preserved under the instantaneous motion if the differentiation by u based on (9.61) yields an identity, namely ∥tc (t′ )∥ (−bc a sin t′ cos t + ac b cos t′ sin t) = −∥tc (t)∥ (−bc a cos t′ sin t + ac b sin t′ cos t) .

(9.62)

In order to confirm this, we square both sides and substitute from the squared equation (9.49) the mixed term 2ac bc ab sin t′ cos t′ sin t cos t . After some computations, this yields for both sides d2 (sin2 t − sin2 t′ ) (a2c b2 sin2 t′ sin2 t + b2c a2 cos2 t′ cos2 t − a2c b2c ) . The velocity analysis in (9.58) for the particular ellipse e confirms, that also the signs of both sides in (9.62) are equal.

◾

Remark 9.6.4 The equation (9.61) implies in combination with the well-known formula κc (t) = ac bc /∥tc (t)∥3 for the curvature of the caustic c at the point Q (Exercise 9.6.1) that the arc length sc of c satisfies 2/3

dsc ac bc = ∥tc (t)∥ t˙ = ∥tc (t)∥2 = ( ) . du κc (t) This is Glutsyuk’s formula for the canonical parameter u, which is named Poritzky string length in [58, (1.1)]. From (9.41) follows in terms of the elliptic coordinates (k, l) the formula s˙ c = −l. s˙ c =

Canonical parametrization with Jacobian elliptic functions

In order to express the action of the transformation γ(u) ∈ Γ on any point (a cos t, b sin t), we integrate (9.61) √ √ dt t˙ = = a2c sin2 t + (a2c − d2 ) cos2 t = ac 1 − m2 cos2 t du

502

Chapter 9: Special problems

with m ∶= d/ac < 1 as the numerical eccentricity of the caustic c. After the substitution π ϕ ∶= t − , 2

the initial condition ϕ = 0 for u = 0 yields the solution u ̃(ϕ) ∶= ac u(ϕ) = F (ϕ, m) = ∫

ϕ 0

dϕ √ 1 − m2 sin2 ϕ

(9.63)

with F (ϕ, m) as the elliptic integral of the first kind with the modulus m and the quarter period K ∶= u ̃(

π/2 dϕ π π ) = ac u ( ) = ∫ √ . 2 2 0 1 − m2 sin2 ϕ

(9.64)

For the sake of simplicity, we use ̃ u(ϕ) = ac u(ϕ) as the new canonical coordinate. The inverse function of u ̃ = F (ϕ, m), namely the Jacobian amplitude ϕ = am(̃ u), leads to the Jacobian elliptic functions, the elliptic sine sn ̃ u = sin(am(̃ u)) = sin ϕ = − cos t with sn(−̃ u) = − sn u ̃ , the elliptic cosine

cn u ̃ = cos(am(̃ u)) = cos ϕ = sin t

with cn(−̃ u) = cn ̃ u and sn2 u ̃ + cn2 u ̃ = 1, and the delta amplitude √ dñ u = 1 − m2 sn2 u ̃

with dn(−̃ u) = dn u ̃ as the third elliptic base function [73]. This yields the canonical parametrization of the ellipse e with semiaxes (ae , be ) as (−ae sn u ̃, be cn u ̃) for 0 ≤ u ̃ < 4K = 4̃ u( π2 ).

Theorem 9.6.9 Let c√be the ellipse with the semiaxes (ac , bc ) and the linear eccentricity d = a2c − b2c . (i) For each confocal ellipse e with semiaxes (ae , be ), the inscribed billiards with the caustic c can be canonically parametrized using the Jacobian elliptic functions to the modulus m = d/ac in the form e(̃ u) = (−ae sn u ̃, be cn u ̃) .

(9.65)

503

9.6 Billiards in ellipses

If bc = be cn(∆̃ u), then the vertex Pi of the billiard in e belongs to the parameter u ̃ = (̃ u1 + 2i∆̃ u) for i ∈ Z and any initial u ̃1 .

(ii) Conversely, in order to obtain an ellipse e for which the billiards with the caustic c are n-periodic with turning number τ and gcd(n, τ ) = 1, we must define ae =

ac dn(∆̃ u) bc 2τ K and be = for ∆̃ u= cn(∆̃ u) cn(∆̃ u) n

(9.66)

with K from (9.64).

(iii) An elliptic billiard in e with the caustic c is periodic if, and only if, u is rational. for ∆̃ u defined by cn(∆̃ u) = bc /be the quotient K/∆̃ y

P1′

e P2 (2∆̃ u)

c

Q1 (∆̃ u)

P1 (0)

FIGURE 9.58. Dependence between the minor semiaxes be , bc and the interval ∆̃ u. Proof: If the billiard’s first vertex P1 ∈ e with u ̃ = 0 is chosen on the positive y-axis (Figure 9.58), then the first contact point Q1 has the canonical coordinate ∆̃ u, and the tangent to c at Q1 passes through P1 = (0, be ). Hence, the points P1 and Q1 are conjugate w.r.t. c, which means by (9.49) that the product of the respective y-coordinates be and bc cn(∆̃ u) equals b2c . Moreover, from dn2 (∆̃ u) = 1 − m2 sn2 (∆̃ u) and sn2 (∆̃ u) + cn2 (∆̃ u) = 1 follows dn(∆̃ u) = ae cn(∆̃ u)/ac .

◾

Remark 9.6.5 It needs to be noted that C.G.J. Jacobi presented an analogue condition for the periodicity already in 1828, when he treated the projectively equivalent bicentric case with e and c as nested circles (see [77, p. 388]). It paved directly the way to Jacobi’s short proof of Poncelet’s theorem (Theorem 9.5.3).

As stated in Theorem 9.6.1, the ellipses e(j) , j = 1, 2, . . ., of the Pon(j) celet grid pass through the points Si , i = 1, 2, . . .. According to De(j) finition 9.6.1, Si is the intersection of the extended sides [Pr−1 , Pr ] ∩ (j) [Pr+j , Pr+j+1 ] with an appropriate r. The consecutive vertex Si+j of the billiard inscribed in e(j) with the caustic c is the intersection between

504

Chapter 9: Special problems

[Pr+j , Pr+j+1 ] and [Pr+2j , Pr+2j+1 ]. Consequently, if u ̃i is the canonical pa(j) rameter of Si , then u ̃i + 2j ∆̃ u is the canonical parameter of its follower (j) Si+j . According to Theorem 9.6.9,(ii) the ellipse e(j) has the semiaxes ae∣j−1 =

ac dn(j ∆̃ u) , cn(j ∆̃ u)

be∣j−1 =

bc cn(j ∆̃ u)

(9.67)

(compare with the rational expressions for j = 1, 2 in [136, Corollary 4.5]). Q1 (̃ u1 , 0)

c h

P (̃ u, ̃ v) e

Q0 (̃ u0 , 0)

FIGURE 9.59. The half-lines along the tangents to c satisfy ̃ u±̃ v = const.

The formulas for ae∣j−1 and be∣j−1 in (9.67) reveal that the quantity ̃ v ∶= j ∆̃ u for j = 1, 2, . . . serves as a kind of canonical coordinate for the confocal ellipses e, e(1) , e(2) , . . . of the associated Poncelet grid: By definition, 2̃ v equals the shift along e(j) between two consecutive vertices of the inscribed billiard with the caustic c, and ̃ v vanishes for points of c. Now, we extend this definition of a second coordinate ̃ v to all confocal ellipses e in the exterior of c. We speak of a canonical parametrization, when we assign to the point P = (a cos t, b sin t) the canonical coordinates (̃ u, ̃ v ) with P = (−asn u ̃, bcn u ̃) = (−ac

sn u ̃ dñ v cn u ̃ , bc ). cñ v cñ v

(9.68)

We already know that u ̃ remains constant on the branch h of the confocal hyperbola through P . If P ∈ e lies on the tangent to c at Q0 , in particular on the half-line which points from Q0 = (̃ u0 , 0) into positive direction (Figure 9.59), then ̃ v =u ̃−u ̃0 equals half of the shift corresponding to e. This means that all points P on this half-line satisfy the equation u ̃−̃ v = u ̃0 = const. Similarly, for P on the negative half-line through Q1 = (̃ u1 , 0) holds ̃ v=̃ u1 − ̃ u, which is equivalent to u ̃+̃ v=u ̃1 = const.

505

9.6 Billiards in ellipses

Theorem 9.6.10 Referring to the notation in Theorem 9.6.9, the canonical parametrization, i.e., the injective mapping v sn u ̃ dñ cn u ̃ , bc ) cñ v cñ v for U ∶= {̃ u∣0≤u ̃ < 4K}, V ∶= {̃ v∣0≤̃ u < K}

C ∶ U × V → R2 ,

(̃ u, ̃ v ) ↦ (−ac

parametrizes the exterior of the caustic c with semiaxes (ac , bc ) in such a way, that the lines u ̃ = const. are branches of confocal hyperbolas. The lines ̃ v = const. are confocal ellipses. Points on tangents to c satisfy the equations u ̃ −̃ v = const. or u ̃ +̃ v = const. depending on whether they belong to the positive or negative half-line terminated by the respective contact point Q with the caustic. ̃ v

K

(3)

(2)

e(2)

S2 (1) S1

P1 Q9

e(3) = e(4)

S2 (2)

S1

0

∞

(3)

S1

Q1

(1) S2

P2

e(1) e

P3

Q2

c

4K

u ̃

FIGURE 9.60. The injective mapping C sends the square grid of points Qi , Pi (j) and Si , i = 1, . . . , 9 , j = 1, . . . , 3 , to the vertices and the diagonals to the confocal conics of the Poncelet grid depicted in Figure 9.54 on page 494.

Now, it is easy to show that this is compatible with the choice of discrete coordinates (u, v) as depicted in Figure 9.52.

Lemma 9.6.8 Referring to the previous notation, let the points with canonical coordinates P = (̃ u1 , ̃ v1 ) and Q′ = (̃ u2 , ̃ v2 ) be opposite vertices of ′ ′ an Ivory quadrangle P P Q Q with tangents of c as diagonals such that their contact points with c are outside the segments P Q′ and P ′ Q (Figure 8.12). Then, the crossing point of the diagonals has the canonical v1 +v2 2 coordinates ( u1 +u 2 , 2 ). The proof is left to the readers as Exercise 9.6.4.

506

Chapter 9: Special problems

By virtue of Theorem 9.6.10, the canonical parametrization C sends a zig-zag polygon to a billiard (Figure 9.60). Even more, C maps a square grid to a Poncelet grid.14 The Lie group Γ mentioned in Theorem 9.6.8 is the C-transform of the group of translations along the u ̃-axis. Due to Lemma 9.6.8, C sends squares to Ivory quadrangles with diagonals tangent to the caustic, and inscribed rectangles with sides parallel to the square’s diagonals to those periodic billiards in Ivory quadrangles, which were studied in Theorem 8.1.5 (Figures 8.13 and 8.14). ●

Exercise 9.6.4 Proof of Lemma 9.6.8.

Prove the statement of Lemma 9.6.8, which confirms the discrete canonical parametrization as presented in Figure 9.52. Hint: Note Theorem 9.6.10.

●

Exercise 9.6.5 Relation between canonical and elliptic coordinates.

Prove the following formulas that express the elliptic coordinates (k, l) of any point P in terms of its canonical coordinates (̃ u, ̃ v ): k = k(̃ v) =

a2c sn2 ̃ v (1 − m2 ), cn2 ̃ v

l = l(̃ u) = −a2c dn2 u ̃.

Hint: For P = (a cos t, b sin t) holds k = a2 − a2c . Note (9.41) and (9.68).

Billiards in ellipses with a hyperbola as caustic As illustrated in Figure 9.61, billiards in an ellipse e with a hyperbola as caustic c are zig-zags between an upper and lower subarc of e. Hyperbolic billiards differ from elliptic billiards in many respects. For example, the billiard motion moves the vertices to and from on each subarc. (j) At the associated Poncelet grid with vertices Si according to Definition 9.6.1, the occurrence of ellipses and hyperbolas is more complex than at elliptic billiards (Figure 9.61): (i) Let Ti be the contact point between the side line [Pi , Pi+1 ] and the (1) (3) caustic c. Then, the points Si , Si , . . . are located on the confocal ellipse (2) (4) through Ti . On the other hand, Si , Si , . . . are located on the confocal hyperbola through Pi . 14

This mapping was already studied in [44, Fig. 13]. A similar mapping and its generalization to 3-space was treated in [16] from the viewpoint of discrete confocal coordinate systems.

507

9.6 Billiards in ellipses (2)

(2)

e

e

(5)

(1)

e

P5

e(2)

S1

(3)

(5)

S5

S5

P7

S12 (3)

(3)

S7

T7

(2)

S10

S4 (1)

P9 P3

T1 (1)

c

S11

P11

T6

P1

e

T12

P6

P8

P10 P2

P4

T5

P12 T11

(1)

S7

(5)

S7

(2)

S5

(1)

S1

(2)

S3

FIGURE 9.61. Periodic billiard P1 P2 . . . P12 in the ellipse e with the hyperbola c as caustic, together with the hyperbolas e(1) , e(3) and the ellipse e(2) . The associated billiard in e(2) with the caustic c splits into four quadrangles. (j)

(ii) For odd j, the points Si , i = . . . , 1, 2, . . ., belong to a confocal hy(j) perbola e(j) or to the secondary axis of e, or they are at infinity. If Si is finite, then the tangent to e(j) bisects the angle between the sides lines (j) (j) through Si . For even j, the Si are vertices of a billiard inscribed in an ellipse e(j) (note e(2) in Figure 9.61) or at infinity. All ellipses e(j) are motion invariant. Now, we focus on n-periodic hyperbolic billiards P1 P2 . . . Pn . Since they oscillate between the upper and lower subarc of the circumscribed ellipse, n must be even. If P 1 P 2 . . . P n denotes the billiard’s image under the reflection in the principal axis of e, then the sequence of parameters t1 , t2 , t3 , t4 , . . . , tn of the vertices P1 , P 2 , P3 , P 4 , . . . , P n is cyclic. Also for hyperbolic billiards a turning number τ can be defined. It counts how often the points P1 , P 2 , P3 , . . . , P n run to and from along one subarc of e. According to [133, Definition 3.13], there exist conjugate billiards even for hyperbolic billiards.

508

Chapter 9: Special problems

For n ≡ 0 (mod 4), the hyperbolic billiards are symmetric with respect to the secondary axis of e and c (Figure 9.61). For n ≡ 2 (mod 4) and odd turning number τ , the billiards are centrally symmetric (Figure 9.63). For even τ , each hyperbolic billiard is symmetric w.r.t. the principal axis of e and c. e

c

FIGURE 9.62. Twofold covered n-periodic hyperbolic billiards in an ellipse e with n = 6, 8, 10 (turning numbers τ = 1 and 2) and n = 12.

When the initial point P1 of a billiard is chosen at any point of intersection between e and the hyperbola c, then the billiard is twofold covered (Figure 9.62). The first side P1 P2 is of course tangent to c at P1 and meets e orthogonally.15 A transition from elliptic to hyperbolic billiards

There is a surprising relation between elliptic and hyperbolic billiards: For each billiard of one type, there exists a counterpart of the other type 15

Twofold covered poses occur at projective billiards also when the tangent to e at P1 ∉ c contacts the caustic c, since P1 must be a reversal point.

509

9.6 Billiards in ellipses e(5)

P7 P9

e P5 P

e(3)

11

P3

P13

e(1) e(4)

P8 P4

c P1

P P6 10 P12

P2 P14

FIGURE 9.63. Twofold covered pose of a periodic hyperbolic billiard with n = 14 and turning number τ = 3.

with the same side lengths (Figure 9.66). An intuitive explanation for this equivalence is based on an overconstrained mechanism in the Euclidean 3-space E3 , the flexible Henrici hyperboloid, named after Olaus Henrici (1840–1918) [9]. This is a framework consisting of generators of a one-sheeted hyperboloid with spherical joints at each crossing point between intersecting generators (Figure 9.64). Here, we only report about properties of this flexible structure. For proofs, the interested reader is referred to [109, pages 87–90]. We assume that the axes of symmetry of the variable hyperboloid remain fixed. Then, during the flexion, the hyperboloid traverses a confocal family16 terminated by two flat poses, where the generators are tangents either of an ellipse or a hyperbola. These two conics constitute a pair of focal conics as treated in Section 4.2. We specify a coordinate frame such that the ellipse c′ with semiaxes (a′c , b′c ) has a standard position in the xy-plane, while its focal hyperbola c′′ with semiaxes (a′′c , b′′c ) lies in the xz-plane. During the flexion of the hyperboloid, each point of the hyperboloid remains on an ellipsoid out of the confocal family. Now, we specify the flexible structure at its flat pose in the xy-plane by choosing the rods as extended sides of a billiard in an ellipse e′ which is confocal with the caustic c′ . Then, during the flexion all vertices of the billiard remain on the 16

Quadrics are called confocal if they share the axes of symmetry and their intersections with each common plane of symmetry are confocal conics.

510

Chapter 9: Special problems

FIGURE 9.64. Henrici’s flexible hyperboloid realizes continuous transitions between isometric elliptic and hyperbolic billiards (© Collection of Kinematic Models, Vienna University of Technology, http://www.geometrie.tuwien.ac.at /kinmodelle/).

same triaxial ellipsoid E passing through e′ . Beginning as points Pi′ ∈ e′ , i = 1, 2, . . . (Figure 9.66, bottom), every second vertex goes upwards in direction of the positive z-axis, the other downwards. At the same time, the contact points Q′i with the caustic remain in the xy-plane. At each spatial pose, the vertices Pi are located on a line of curvature e ∈ E, namely the intersection of E with the confocal hyperboloid through the rods (Figure 9.65). In fact, the rods are extended sides of a so-called focal billiard in E (note [134, Definition 2]). At the second flat pose, the vertices arrive at points Pi′′ of the upper or lower subarc of the intersection e′′ between E and the xz-plane. The Pi′′ are vertices of a hyperbolic billiard since in this pose the rods are tangents of the hyperbola c′′ which is confocal with e′′ (Figure 9.66, top). If (a′e , b′e ) and (a′′e , b′′e ) are the respective semiaxes of the ellipses e′ and e′′ and if d′ and d′′ denote their linear eccentricities, then holds a′′c = d′ , a′c = d′′ ,

b′′c = b′c , b′c = b′′c ,

d′′ = a′c , d′ = a′′c ,

a′′e = a′e , a′e = a′′e ,

b′′e 2 = a′e2 − a′c2 , b′e2 = a′′e 2 − a′′c 2 .

(9.69)

511

9.6 Billiards in ellipses P3

P7

e

′ P12

e′

P11

P3′ c′

P8′

P12

P13

P2 P1

e P8

P9

P6 P5

P1′ e′ y

′ P14

x

P4

E

P5′ P14

P10

′ FIGURE 9.65. Transition from the flat periodic elliptic billiard P1′ P2′ . . . P14 ′ ′ (green) incribed in e with the caustic c and turning number τ = 3 to the isometric spatial focal billiard P1 P2 . . . P14 (red) in the ellipsoid E with vertices on the line of curvature e.

During the flexion not only the vertices Pi′ of the elliptic billiard remain on an ellipsoid of the confocal family, but also half of the other intersections (j)′ between extended sides, namely the points Si with even j, which are (j)′ located on confocal ellipses e of the Poncelet grid. If j is odd, then one (j)′ of the half-lines through the initial Si in the xy-plane goes upwards, the other downwards, so that their spatial poses do not intersect. Definition 9.6.4 Two polygons P1 P2 . . . and P1∗ P2∗ . . . in the Euclidean ∗ 3-space are called isometric if corresponding sides Pi Pi+1 and Pi∗ Pi+1 have equal lengths for all i = 1, 2, . . . During the flexion of the Henrici hyperboloid, all distances along the rods remain constant. Hence, all focal billiards and the terminating planar billiards are mutually isometric.

512

Chapter 9: Special problems P1′′

P3′′

′′ P11

′′ P13

P5′′

P7′′

P9′′ (4)′′ S5

Q′′1

T1′′

e′′

(4)′′

S6

P2′′ ′′ P12

P6′′

(2)′′

′′ P10

P8′′

′′ P14

P4′′

c′′

S3

P1′

P6′

′ P10

′ P11

P5′

c′

Q′1 ′ T1′ P2 (2)′′

S3

P7′

′ P14

P9′

(4)′

S5

(4)′ S6

′ P12

e

P3′ P8′

′

P4′

′ P13

FIGURE 9.66. A hyperbolic billiard (top) and an isometric elliptic billiard (bottom). Both are 14-periodic billiards with the turning number τ = 3.

Theorem 9.6.11 (i) For each billiard P1′ P2′ . . . inscribed in an ellipse e′ with an ellipse c′ as caustic there exists an isometric billiard P1′′ P2′′ . . . in an ellipse e′′ with a hyperbola c′′ as caustic. The latter billiard is unique only up to a reflection in the principal axis of e′′ . The semiaxes of the involved conics obey (9.69) (Figure 9.66). (ii) Conversely, for each hyperbolic billiard there exists an isometric elliptic billiard, provided that in the case of an n-periodic billiard with n ≡ 2 (mod 4) the elliptic billiard is traversed twice. ′ ′ ]→ (iii) For each side Pi′ Pi+1 of the elliptic billiard, the isometry [Pi′ , Pi+1 (j)′ ′′ ′′ [Pi , Pi+1 ] between the extended sides maps the incident points Sk with

9.6 Billiards in ellipses

513

j ≡ 0 (mod 2) of the associated Poncelet grid to the corresponding point (j)′′ Sk of the isometric hyperbolic billiard. ′ (iv) For all i, the isometric image of the contact point Q′i of the side Pi′ Pi+1 ′ ′′ ′′ ′′ with the ellipse c is the point of intersection Qi of the side Pi Pi+1 with the principal axis. On the other hand, the isometry sends the point of in′ tersection Ti′ between the side line [Pi′ , Pi+1 ] and the principal axis to the ′′ ′′ ′′ contact point Ti of the side Pi Pi+1 with the hyperbola c′′ (Figure 9.66). Remark 9.6.6 In the physical models of Henrici’s hyperboloid as depicted in [143, pp. 1318], [109, p. 88], [9, Fig. 17], or in Figure 9.64, the generators are terminated by two symmetric cross sections. For demonstrating the statements of Theorem 9.6.11 in the case of periodic billiards, the endpoints of the rods should be located on a confocal triaxial ellipsoid. Then, the endpoints remain on this ellipsoid during the flexion, and in the flat poses the rods are limited by ellipses confocal with e′ or e′′ respectively. Moreover, for each even j = 2, 4, . . . all (j)′ crossing points Si ∈ e(j)′ remain on confocal ellipsoids.

Similar to the planar version in Lemma 8.1.3, between any two poses of the flexible hyperboloid in space, there exists an affine transformation which preserves the axes of symmetry and all distances along the generators. This includes also the limiting cases where one pose is flat. Thus, we can express the mapping from the elliptic to the isometric hyperbolic billiard in coordinates. Theorem 9.6.12 Referring to Theorem 9.6.11, an isometric mapping from the elliptic billiard in the xy-plane to the hyperbolic billiard in the xz-plane sends for i = 1, 2, . . . the vertex Pi′ = (a′e cos ti , b′e sin ti ) ∈ e′ to the vertex √ ′′ ′′ ′ ⎛ b −l′ (ti ) ⎞ ′′ a d cos t i e Pi′′ = e ′ , (−1)i ∈e ac a′c ⎝ ⎠

with l′ (ti ) = −a′c2 sin2 ti − b′c2 cos2 ti as the second elliptic coordinate of Pi′ . Another possibility arises after reflecting the image in the plane z = 0. For a detailed proof see [134]. With regard to motion invariants of periodic hyperbolic billiards in ellipses, one can apply Theorem 9.6.11 in order to transfer related properties of elliptic billiards, e.g., the constant perimeter. But also other invariants like the constant sum of the cosines of the exterior angles are valid for hyperbolic billiard as well as a canonical parametrization [134].

10 Other geometries

M 3 O

−2 m o

At the ‘spherical Wankel engine’ the common path t of the three vertices of the rotor is a trochoid. It can be generated when the circle m which is attached to the rotor, is rolling along the fixed circle o. The angular velocities of the rotor’s rotations about M and additionally about the fixed center O build the constant ratio −3 ∶ 2.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2024 G. Glaeser et al., The Universe of Conics, https://doi.org/10.1007/978-3-662-70306-9_10

515

516

Chapter 10: Other geometries

10.1 Spherical conics

FIGURE 10.1. A spherical conic and its quadratic projection cone. The foci of the projected ellipse are not the images of the spherical foci.

This chapter deals with a generalization of conics on the sphere. We call curves on a sphere “spherical conics” when they fulfill the classical condition that we know from ordinary conics: When the sum or the difference of the distances from two fixed points on the sphere — measured on shortest paths on the sphere — is constant, we speak of spherical ellipses and hyperbolas. It turns out that any spherical ellipse is at the same time a spherical hyperbola, and vice versa. The borderline case of a spherical parabola,

10.1 Spherical conics

517

where the spherical distance from a point is equal to the distance from a great circle, can also be interpreted as such a spherical conic. Spherical conics are the intersections of a sphere with quadratic cones with the apex in the center of the sphere (Figure 10.1). Thus, they are algebraic curves of degree four on the sphere. Their images under orthogonal projections into their planes of symmetry are portions of planar conics. Many theorems about ordinary conics can be transferred to the sphere, e.g., the property that the tangent and the normal of a spherical conic are the bisectors of the focal rays through the spherical foci. Proclus’s (de la Hire’s) construction of a conic and Ivory’s theorem about confocal conics work on the sphere as well. Theorems of planar Projective Geometry, e.g., Pascal’s and Brianchon’s theorem can be transferred without difficulties onto the sphere.

Geometry on the sphere In the introduction we, reported that there are many analogies in the theory of conics between the spherical and the planar case. In order to be able to prove these analogies, we first have to provide the reader with some particularities of the geometry on the sphere. The reader may skip this section and refer to it later on when necessary. It is important that on the sphere we measure only angles: Since we have no straight lines on the sphere, the role of straight lines is taken over by great circles. • Each point P on the sphere has an antipodal point or antipode P ∗ , and each great circle passing through P contains also P ∗ .

• For any two different points P and Q ≠ P ∗ there exists a unique connecting great circle denoted by [P, Q]. In the sequel, when we speak of the segment, the arc, or the side P Q for any two non-antipodal points P, Q, we always mean on the great circle [P, Q] the shorter of the two arcs terminated by P and Q. ̂ • The (spherical) distance P Q is defined as length of the side P Q, mea2 ̂ sured on the unit sphere S with center O. Thus, we have 0 ≤ P Q ≤ π, ∗ when we include also the limiting cases Q = P and Q = P . The dî stance P Q equals the central angle 0.

1 − p2 , 1 + p2

Thus, we know that the intersection of our quadratic projection cone C leads to a spherical conic. Any oblique section of C is again a conic, so that the choice of the carrier plane of the ellipse involved no loss of generality, and Theorem 10.1.7 is proven in a second way. Now, it is obvious that the projections of the foci of the ellipse are √not the foci of the spherical conic. Otherwise we would obtain e = arctan p2 − q 2 .

The principal views of a spherical conic

We have seen that the orthogonal projection of a spherical conic onto the symmetry plane σ1 through its foci is a planar conic. The symmetry plane σ2 of the foci S1 and S2 is another symmetry plane. The plane σ3 perpendicular to both σ1 and σ2 that passes through O completes the set of symmetry planes of the spherical conic. We call the orthogonal projections onto these planes the principal views of the conic, and show: Theorem 10.1.8 The principal views of a spherical conic are located on ordinary (planar) conics.

533

10.1 Spherical conics Proof: The symmetry planes have the following equations σ1 ∶ z = 0,

σ2 ∶ y = 0,

σ3 ∶ x = 0.

Equations of the principal views of the spherical conic are obtained by eliminating z, y, and x from the equation (10.6) of the sphere S2 , from eq. (10.5) of the elliptic cylinder L (or likewise (10.7) of the quadratic cone C). The shape of the top view, i.e., the curve in σ1 , is given by Theorem 10.1.6, and its equation is given by (10.5). Elimination of y and x from (10.5) and (10.6) results in (1 −

cot2 e sin2 a ) x2 + z 2 = 1 − , cot2 a sin2 e

(1 −

tan2 e 2 cos2 a ) y + z2 = 1 − . tan2 a cos2 e

(10.9)

These equations can be interpreted as equations of quadratic cylinders through the spherical conic or of conics in the planes σ2 and σ3 . Because of e < a, in the latter case we obtain a hyperbola in y = 0 and an ellipse in x = 0. Of course, the principal views of the spherical conic are the restrictions to the closed unit disks. Of course, the views of the spherical conic must lie inside the closed unit disk.

◾

FIGURE 10.13. The principal projections of a spherical conic c are placed on planar conics. Hence, the curve c is the intersection of three quadratic cylinders. On the left a ∶ b = 3 ∶ 1 and a almost π/2, on the right a ∶ b = 5 ∶ 4.

We can summarize the results (Figure 10.13): Theorem 10.1.9 There are three quadratic cylinders and one quadratic cone (concentric with the sphere) passing through any spherical conic. Just to be complete, we do the same with the parametrization of our spherical conic given in (10.8):

534

Chapter 10: Other geometries

1. The top view is given by 1 p x = , y = cos u with r = r r

√

1 + p2 cos2 u + q 2 sin2 u.

Elimination of the parameter u leads to the equation (1 + q 2 )x2 +

p2 − q 2 2 y = 1. p2

2. The front view is described by y=

p q cos u, z = sin u with r = r r

√ 1 + p2 cos2 u + q 2 sin2 u.

Elimination of the parameter u leads to the equation 1 + p2 2 1 + q 2 2 y + z = 1. p2 q2 3. Finally, the right-side view is given by √ 1 q x = , z = sin u with r = 1 + p2 cos2 u + q 2 sin2 u, r r and the elimination of u leads to (1 + p2 )x2 + (1 −

p2 2 ) z = 1. q2

Figure 10.12 shows the front and right-side views of a spherical conic.

Ivory’s theorem on the sphere According to Theorem 10.1.7, each spherical conic is located on a quadratic cone C with the signature (+ − −). By (10.7), we have C ∶ x2 (

cos2 e sin2 e 2 − 1) − y (1 − ) − z 2 = 0, cos2 a sin2 a

0 w2 , we can solve this for the quantities e, a, and b associated with the conic (see Figure 10.4, right): sin2 e =

v 2 − w2 , u2 + v 2

sin2 a =

v2 , u2 + v 2

sin2 b =

w2 . w 2 + u2

(10.11)

Therefore, all conics located on quadratic cones with equations x2 y2 z2 − − = 0, u2 + k v 2 − k w 2 − k

k ∈ R {−u2 , v 2 , w2 }

(10.12)

have focal points in the set {S1 , S2 , S1∗ , S2∗ }. Therefore, we speak of a family of confocal spherical conics.

Under our assumption that w2 < v 2 , there are two types of conics to be distinguished in this family: • For −u2 < k < w2 , the focal points are (S1 , S2 ) or (S1∗ , S2∗ ). In view of Figure 10.5, left, these are ‘spherical ellipses’. • For w2 < k < v 2 , we obtain branches of ‘spherical hyperbolas’ with respect to the focal points (S1 , S2 ) or (S1∗ , S2∗ ).

For k < −u2 or k > v 2 , the cones have the signature (+ + +), they don’t contain any real point except 0. The limit for k → −u2 is the great circle in x = 0 (as a, b → π2 ). The great circle in y = 0 occurs as limit for k → v 2 (as a → 0 ). The limits for k → w2 (as b → 0 ) from k < w2 are the segments S1 S2 or S1∗ S2∗ . As limits from k > w2 , we obtain the complementary segments S1∗ S2 or S1 S2∗ on the great circle in z = 0. The family of confocal conics together with these limiting curves forms an orthogonal net on the sphere (see Figure 10.14). Through each point P ∈ S2 pass two mutually orthogonal net curves, and if P lies outside of any symmetry plane then there is precisely one such conic of each type. This follows from Theorem 10.1.5 since the tangent line of an ellipse

536

Chapter 10: Other geometries

bisects the exterior angle between the focal rays, i.e., the segments P S1 and P S2 . Below, we prove the spherical analogue of Ivory’s theorem 2.2.3. Theorem 10.1.10 Ivory’s theorem on S2 . The orthogonal net of confocal conics on the sphere has the property that in each curvilinear quadrangle the diagonals are of equal length.

P′ c′ P

Q′ c Q S2

′Q = P ̂ FIGURE 10.14. Ivory’s theorem on the sphere states that P̂ Q′ .

Proof: For the sake of simplicity, we identify points P with their position vectors p. Therefore, we briefly speak of points p, q ∈ S2 . Let l be a linear map l ∶ R3 → R3 with p ↦ l(p) = Ap,

⎛κ 0 0⎞ A = ⎜ 0 λ 0 ⎟, ⎝0 0 µ⎠

where κ, λ, µ ∈ R and (κ ∶ λ ∶ µ) ≠ (1 ∶ 1 ∶ 1). We look for points p = (x, y, z)T ∈ S2 whose image points are again on S2 . This holds if, and only if, x2 + y 2 + z 2 = κ2 x2 + λ2 y 2 + µ2 z 2 = 1, or equivalently

∥p∥ = 1 and (κ2 − 1)x2 + (λ2 − 1)y 2 + (µ2 − 1)z 2 = 0.

A comparison with the equation of the cone C in (10.10) shows that if

κ2 − 1 > 0 > λ2 − 1 > µ2 − 1, i.e., κ2 > 1 > λ2 > µ2 ≥ 0

537

10.1 Spherical conics these points p lie on the pair of antipodal conics given by x2 y 2 z2 1 1 − − = 0 with 2 = κ2 − 1, = 1 − λ2 , u2 v 2 w 2 u v2 after adjusting u, v, and w by an appropriate factor. C∶

1 = 1 − µ2 , w2

As a consequence, when µ2 > 0, the image points l(p) = (x′ , y ′ , z ′ )T = (κx, λy, µz)T are located on the pair of antipodal conics of C′ ∶

x′ 2 y′ 2 z′ 2 x′ 2 y′ 2 z′ 2 − − = − − = 0. κ2 u2 λ2 v2 µ2 w 2 u2 + 1 v2 − 1 w 2 − 1

We learn from (10.12) that the conics on C and C ′ are confocal (with k = 1) and of the same type (Figure 10.14), since w 2 > 1 > −u2 . Our linear mapping l ∶ p ↦ Ap is self-adjoint, since due to the symmetry of the matrix A, the adjoint mapping l⋆ ∶ q ↦ AT q equals l. Now, we use a well-known rule from Linear Algebra. It is a consequence of the following sequence of equations with switches between the scalar product ⟨⋅, ⋅⟩ and its matrix representation: ⟨l(p), q⟩ = l(p)T q = (Ap)T q = pT AT q = pT l⋆ (q) = ⟨p, l⋆ (q)⟩.

′ q = pq ̂ ̂′ . It only remains to confirm This expresses directly the statement of Ivory’s theorem p that corresponding points p ∈ C and p′ = l(p) ∈ C ′ are always located on the same conic from the other type of the confocal family, in our case on conics (10.12) with k between v2 and w 2 :

Suppose that simultaneously x2 y 2 z2 − − =0 u2 v 2 w 2 and k ≠ 0. Then, also

and

x2 y2 z2 − − = 0, u2 + k v 2 − k w 2 − k

x′ 2 y′ 2 z′ 2 κ2 x2 λ2 y 2 µ2 z 2 − − = − − = 0, u2 + k v 2 − k w 2 − k u2 + k v 2 − k w 2 − k because

κ2 u2 + 1 1 1 k−1 1 = 2 2 = + , + k u (u + k) k u2 k u2 + k which finally yields an affine combination, i.e., a linear combination with coefficients summing up to 1, u2

x′ 2 y′ 2 z′ 2 1 x2 y 2 z2 k−1 x2 y2 z2 − − = ( − − )+ ( 2 − − ). u2 +k v2 −k w 2 −k k u2 v2 w 2 k u +k v2 −k w 2 −k When the second conic through p degenerates (i.e., when k → v2 or k → w 2 ), then it is still true that l(p) lies on the same degenerate conic, since l maps each plane of symmetry onto itself.

◾

Remark 10.1.1 Ivory’s statement holds also for degenerate conics in the family of confocal conics since the linear map l in the proof above is singular if µ = 0. Then, one conic on C ′ is the segment S1 S2 between the focal points of C. Since the endpoints S1 and S2 are the l-images of the vertices A, B ∈ C, we obtain ′A + P ′ B = AB ̂ ̂ = 2a. P̂ S1 + P̂ S2 = P̂ A′ + P̂ B ′ = P̂

In this way, Ivory’s theorem can be used to prove directly that the intersection curves between a quadratic cone and the unit sphere satisfy the properties of a spherical ellipse.

538

Chapter 10: Other geometries

c0 c1

FIGURE 10.15. An incircular net of great circles tangent to the spherical conic c0 together with the related spherical Poncelet grid.

After inverting the coefficient matrices of the confocal conics in (10.12), we obtain their tangential equations (see Section 7.1) (u2 + k)x2 − (v 2 − k)y 2 − (w2 − k)z 2 = (u2 x2 − v 2 y 2 − w2 z 2 ) + k(x2 + y 2 + z 2 ) = 0.

We notice: The duals of confocal spherical conics belong to a range which is spanned by one of the conics and the conic defined by the isotropic cone satisfying the equation x2 + y 2 + z 2 = 0 .

The previously excluded parameter values k = {−u2 , v 2 , w2 } define the singular conics in this pencil, among them the pencils of great circles through the focal points. Hence, it is no surprise that the spherical analogue of Theorem 7.3.3 is also true. Figure 10.15 shows the spherical counterpart of Figure 7.44. It is based on a periodic billiard inscribed in c1 and with the caustic c0 . The nine extended sides of the billiard form a grid of great circles, where any two pairs of adjacent great circles form an incircular

539

10.1 Spherical conics

spherical quadrangle as depicted in the figure. By the way, incircular quadrangles arise also when from the great circles in cyclic order those with one side between are coupled or those with two sides between.

Two particular spherical conics A bundle in the projective three-space, i.e., the set of lines and planes through a fixed point O, is a projective plane if lines and planes are seen as ‘points’ and ‘lines’ (cf. Section 5.1). In this sense quadratic cones are the ‘conics’ of the bundle. Hence, all projective properties of conics occur again at quadratic cones and at their spherical visualization in the form of antipodal pairs of spherical conics. For example, the spherical version of Theorem 7.3.3 is also true as well as all spherical analogues of theorems concerning porisms for planar conics (note the Figures 10.15 and 10.16). However, the metric properties of spherical conics often differ from those of their planar counterparts. We have already seen this in the previous sections. A deeper reason for this difference lies in the fact that the isotropic lines (see Section 6.4, especially Example 6.4.8 on page 266) in the complex extension of the Euclidean plane form a singular dual conic while the quadratic cone of isotropic lines in the bundle is regular. In Cartesian coordinates, the isotropic cone with the apex at the origin O obeys the equation x2 + y 2 + z 2 = 0, as already mentioned. The corresponding symmetric coefficient matrix is the unit matrix I3 . Two vectors v1 = (x1 , y1 , z1 )T and v2 = (x2 , y2 , z2 )T are conjugate with respect to this cone if, and only if, 0 = x1 x2 + y1 y2 + z1 z2 = ⟨v1 , v2 ⟩,

i.e., the vectors are orthogonal. The polarity in the isotropic cone is also called absolute polarity of the bundle. We set up the Cartesian equation of any quadratic cone with apex O as C ∶ c00 x2 + c11 y 2 + c22 z 2 = 0 with c00 > 0 > c11 ≥ c22 .

(10.13)

The intersections of this quadratic cone with the planes z = 0 and y = 0 show that the semiaxes a and b of the corresponding spherical conic in the halfspace x > 0 obey tan2 a = −

c00 c00 c00 c00 , sin2 a = , tan2 b = − , sin2 b = . (10.14) c11 c00 − c11 c22 c00 − c22

540

Chapter 10: Other geometries

P1

e

c

P7

P2 P6

P3

P5

P4

FIGURE 10.16. A 7-periodic billiard in the spherical conic e with the caustic c together with a second pose obtained by the spherical billiard motion and two trajectories (yellow) of the intersections between the sides’ extensions.

This follows also upon comparison with (10.7). Among the quadratic cones we pick out the following two with special metrical properties: • equilateral cones have a vanishing trace, i.e., c00 + c11 + c22 = 0; • normal cones are characterized by c00 + c22 = c11 . We use these names also for the corresponding spherical conics.

Equilateral spherical conics

As a consequence of (10.14), equilateral spherical conics are characterized by sin2 a sin2 b = . (10.15) 3 sin2 a − 1

541

10.1 Spherical conics

The condition b < a for proper conics implies the limits √ √ 2 π 2 π arcsin < a < , arcsin >b< . 3 2 3 4 In one limiting √ case, our spherical conic becomes a circle with radius a = b = arcsin 2/3. At the other limit, with a = π/2 and b = π/4, i.e., c11 = 0, the quadratic cone splits into two perpendicular planes obeying x2 − z 2 = 0. K

H

M

c G y

z

FIGURE 10.17. The regular right triangle GHK can move around in the equilateral spherical conic c while all three vertices G, H, and K run along c.

Theorem 10.1.11 Within any equilateral spherical conic c we can inscribe an infinite number of octants, i.e., of regular triangles all of whose sides are of length π/2 and whose angles are all π/2. Proof: For the matrix C = diag(c00 , c11 , c22 ) of c we obtain

0 0 ⎛ σ + τ c00 ⎞ det(σI3 + τ C) = det ⎜ 0 σ + τ c11 0 ⎟. ⎝ 0 0 σ + τ c22 ⎠

The coefficient of σ2 τ vanishes. Hence, by Lemma 9.5.2 (page 461), the isotropic cone is inpolar to c. In the bundle of lines and planes through O, each triple of lines which is selfpolar with respect to the isotropic cone consists of three pairwise orthogonal lines. Therefore, by Lemma 9.5.1 (page 460), an infinite set of octants can be inscribed into the equilateral conic (Figure 10.17).

◾

542

Chapter 10: Other geometries

The octant performs a periodic continuous spherical motion while the three vertices GHK simultaneously traverse the same equilateral conic c (Figure 10.18). During one full turn, the moving triangle GHK returns twice to its initial position, however rotated under 120○ and 240○ .2 The sides of the moving octant envelope again a spherical conic i. It is located on the cone which corresponds to the cone of c in the absolute polarity (compare also Lemma 9.5.4).

K

m G y

M i

c

H

z FIGURE 10.18. Orthogonal projection into the yz-plane of the motion of the octant GHK turning around in the equilateral spherical conic c, together with the path m of the triangle’s center M and the envelope i of the three sides.

Thus, we obtain again a spherical Poncelet porism, namely a 3-periodic projective billiard, where the triangles are even mutually congruent. This means that in this particular case the billiard motion is indeed an overconstrained spherical motion where three points trace the same curve. This is very special. There exists no counterpart in the plane. In the limiting case a = b, the octant’s motion is a pure rotation about its axis of 2

The generic point paths are of spherical degree 24, i.e., they are projected from the origin by cones of degree 24. For details see [130].

543

10.1 Spherical conics

threefold symmetry. At the other limit, with the splitting cone, the continuous motion consists of rotations which keep one of the three vertices fixed. There are poses which allow a bifurcation between these rotations. Another spherical motion with the three vertices of a regular triangle running along the same curve t occurs in the spherical version of a Wankel engine (Figure 10.19): The curve t is a spherical trochoid generated as a point path when the rotor rotates about its center M with angular velocity −2 while additionally it rotates about the fixed center O with angular velocity 3. During this spherical motion the circle m rolls along the fixed circle o. In this case, the rotor needs not be a spherical octant. However, the common point path t is not a spherical conic but a curve of spherical degree 6.

c

M

3

-2

O

m

o

FIGURE 10.19. At the ‘spherical Wankel engine’ the common path t of the three vertices of the rotor is a spherical trochoid. It can be generated when the circle m which is attached to the rotor, is rolling along the fixed circle o. The angular velocities of the rotor’s rotations about M and additionally about the fixed center O have the constant ratio −3 ∶ 2.

Let us return to the equilateral conic c: Suppose, we project the moving octant from the origin into a plane ε which intersects the cone of c

544

Chapter 10: Other geometries

along a circle c0 (Figure 4.23). Then, we obtain a one-parameter family of triangles with a common circumcircle c0 and a common orthocenter O, which is the pedal point of ε w.r.t. the center of the sphere (Figure 10.20)3 . The triangles in ε, which envelope again a conic i0 , share also the centroid G. Conversely, these triangles could serve for an elementary approach to Theorem 10.1.11.

c0

i0 C

G

O

FIGURE 10.20. The triangles share the circumcircle c0 , the centroid G, and the orthocenter O. They are circumscribed to the conic i0 with O and the center C of c0 as focal points. Normal spherical conics

By (10.14), normal spherical conics are characterized by sin a = tan b

or

sin2 b =

sin2 a . sin2 a + 1

(10.16)

On the other hand, the condition c00 + c22 = c11 is equivalent to the fact that the corresponding cone C from (10.13) intersects the plane of symmetry y = 0 in lines which are perpendicular to planes √ √ x c00 − c11 ± z c11 − c22 = 0, 3

Figure 9.39 on page 463 shows a version with the orthocenter O outside of c. In this case, the envelope of the triangles is a hyperbola. In Exercise 9.5.1 it is proved that O and C are the foci of i0 .

545

10.1 Spherical conics

which cut C along circles. We conclude this section with spherical analogues of well-known theorems from planar geometry concerning the Thales circle and the angle of circumference (compare with Figure 6.9 on page 238). Theorem 10.1.12 Let A, B be two fixed points on the sphere S2 with ̂ = 2λ < π/2. The set of points P ∈ S2 with 1 an ellipse, for ε = 1 a parabola and for ε < 1 a nonequilateral hyperbola which intersects the axis of symmetry through F in real points. (Note that in the Euclidean case we had ε < 1 for ellipses and ε > 1 for hyperbolas.) After applying pseudo-Euclidean motions, all obtained ellipses and hyperbolas have a center and a spacelike and a timelike axis of m-symmetry. The obtained parabolas have either a spacelike or a timelike axis of m-symmetry. The lightlike lines through F intersect the directrix l at two points which satisfy the Apollonian condition, since their m-distances to F and l vanish. These lightlike lines cannot meet the conic c at another point since such a point must again have a zero-distance to l. Therefore, a point F is called a focal point or focus of a conic c in M2 if, and only if, the two lightlike lines through F are tangent to c (Figure 10.24).

553

10.2 Conics in non-Euclidean planes y

y

ch

T

Q

P l

tP

T F l

F x

tP

Q ce

x

tQ

P FIGURE 10.24. The ellipse ce (left) satisfies the Apollonian definition in M2 with ε = 1.80 , the hyperbola ch (right) with ε = 0.80 . The tangent tP at P intersects the directrix l at T such that [P, F ] and [T, F ] are m-orthogonal. Equally marked angles are e-congruent.

Lemma 10.2.1 In the pseudo-Euclidean plane M2 , the locus c of points satisfying the Apollonian definition dm (P, F ) = ε ⋅ dm (P, l) with a nonlightlike directrix l, F /∈ l and ε > 0 is a conic with an axis of symmetry. We obtain for ε > 1 an ellipse, for ε = 1 a parabola and for ε < 1 a hyperbola with non-lightlike asymptotes. The point F is a focus of c; the directrix l is polar to F with respect to c. Due to properties of the polarity with respect to c (Section 7.1), the tangent tP to c at P can be constructed by virtue of the pseudo-Euclidean version of Theorem 2.1.4 via the pole T of the line [P, F ] (Figure 10.24).

According to Lemma 10.2.1, all conics in M2 which satisfy the Apollonian definition have an axis of symmetry. However conversely, not all conics c in M2 admit an axial symmetry. For ellipses and hyperbolas, the two axes of m-symmetry must be conjugate with respect to c and harmonic with respect to the lightlike directions. Such a pair always exists for ellipses; for hyperbolas it exists only, when the asymptotes do not separate the lightlike directions (note Figure 6.30). This gives rise to a classification of conics in M2 as given below:

554

Chapter 10: Other geometries

Up to m-motions and a commutation of the coordinate axes, there are six types of conics to be distinguished: 1. Circles: Their standard equation is x2 − y 2 = σ ≠ 0. Given two points A, B with dm (A, B) ≠ 0, e.g., A = (a, 0) and B = (−a, 0), the locus of points P = (x, y), for which the lines [P, A] and [P, B] are m-orthogonal, is the circle x2 − a2 − y 2 = 0 with the diameter AB. This means, Thales’s theorem is also valid in M2 . y F3

l1

X

3′′

3

c

l2

l3 1′ F1

2′

c′ 1

X

′

2

c′′

F2

x

l4 4

Y

Y ′′ F4 4′′ FIGURE 10.25. The ellipse c is of type 2 in M2 . Points X ∈ c satisfy either dm (X, F1 ) + dm (X, F2 ) = dm (1, 2) or ∣dm (X, F1 ) − dm (X, F2 )∣ = dm (1, 2), as well as dm (X, F3 ) + dm (X, F4 ) = dm (3, 4) or ∣dm (X, F3 ) − dm (X, F4 )∣ = dm (3, 4).

2. Ellipses or hyperbolas with two axes of symmetry and standard equation x2 y 2 + = 1, where στ (σ + τ ) ≠ 0. σ τ

Under σ +√τ > 0 these conics have four real focal points (±e, 0) and (0, ±e) with e ∶= σ + τ , like the ellipse c depicted in Figure 10.25. With respect to the two foci F1 and F2 on the x-axis and the associated directrices l1

555

10.2 Conics in non-Euclidean planes

and l2 , points X ∈ c satisfy

dm (X, F1 ) = ε ⋅ dm (X, l1 ) and dm (X, F2 ) = ε ⋅ dm (X, l2 ).

Consequently, points X ∈ c between l1 and l2 have a constant sum of distances dm (X, F1 ) + dm (X, F2 ) = ε ⋅ dm (l1 , l2 ) = dm (1, 2), while points outside the parallel strip bounded by l1 and l2 satisfy ∣dm (X, F1 ) − dm (X, F2 )∣ = ε ⋅ dm (l1 , l2 ) = dm (1, 2). Points on l1 or l2 satisfy both condititions. 3. Hyperbolas with a spacelike and a timelike asympote: These conics (see Figure 10.26) have a center, but no axis of symmetry. Their standard equation is σ(y 2 − x2 ) + (1 − σ 2 )xy = τ with τ ≠ 0.

Here, σ with −1 < σ < 1 is the e-slope of the spacelike asymptote. The pairwise complex conjugate m-focal points are located on the e-isotropic lines y = ±ix . y

1′

c′

1 c

X′

X

2 Y′

Y 4′

2′

3′ =4

x

3

FIGURE 10.26. Confocal hyperbolas without axis of symmetry (type 3) in M2 .

4. Hyperbolas with one lightlike asymptote: We specify the non-lightlike asymptote as x-axis and obtain the standard equation xy − y 2 = σ with σ ≠ 0.

556

Chapter 10: Other geometries

√ √ In the case σ > 0 they have two real focal points F1 = (± 2σ, ± 2σ).

5. Parabolas with an axis of symmetry: We choose the axis of symmetry as x-axis and get the standard form y22 − 4σx = 0, where σ > 0.

with the focal point (−σ, 0).

6. Parabolas with lightlike diameters: Their standard equation is (x + y)2 − 2σ(x − y) = 0 with σ ≠ 0.

They have no focal point (Figure 10.27).

y

c′

c

A′ B

B′

X ′ =Y

A X

x

Y′

FIGURE 10.27. Confocal parabolas c, c′ with lightlike diameters in M2 and dm (A′ , B) = dm (A, B ′ ), dm (X ′ , Y ) = dm (X, Y ′ ) = 0.

A comparison with Lemma 10.2.1 reveals that in M2 only the conics of type 5 and those of type 2 with real foci satisfy the Apollonian definition. In the case of hyperbolas of type 2 (Figure 10.24, right) only foci on the principal axis are admitted, i.e., on the axis which intersects the hyperbola in real points. Otherwise, one of the distances dm (P, F ) and dm (P, l) would be spacelike, the other timelike. Confocal conics in M2

Two conics are called confocal in M2 if, and only if, their tangential equations span a linear system which contains the two pencils of lightlike

10.2 Conics in non-Euclidean planes

557

lines as a singular curve. Therefore, confocal conics share the lightlike tangent lines and the foci. Also in the pseudo-Euclidean plane, a family of confocal conics forms an orthogonal net. We prove this below with the aid of Desargues’s involution theorem. Let P be a point of any conic c included in a confocal range. When P is not located on one of the common lightlike tangents of the family, then the tangent to c at P is one of the two fixed lines of the induced Desargues involution at P (Section 7.4). The fixed lines separate the lightlike lines through P harmonically. Therefore, there exists a second conic of the range which intersects c m-orthogonally at the point P .

F4

F2

F1

F3

FIGURE 10.28. Ranges of confocal conics of type 2 in M2 with real foci (left) or imaginary foci (right).

Figure 10.28 shows two ranges of confocal conics of type 2, on the left ellipses and hyperbolas sharing four real focal points, on the right hyperbolas with complex conjugate foci. When on the left-hand side one semiaxis of an included ellipse tends to zero, then the limit of the ellipse is the segment F1 F2 and F3 F4 . The analogous limits of the included hyperbolas consist of pairs of disjoint aligned half-lines, each terminated by a focus. In the standard model of M2 , which is used in all figures, the pseudo-Euclidean foci belong to the e-director circle or e-orthoptic circle (Theorem 2.2.6) of all ellipses and hyperbolas included in the confocal family.

558

Chapter 10: Other geometries

An important property of confocal conics in the Euclidean plane E2 is known as Ivory’s theorem (Theorem 2.2.3, Figure 2.23): In each curvilinear quadrangle formed by two pairs of confocal conics the two diagonals have equal lengths. Another formulation of this theorem uses the fact that for any two confocal conics c, c′ of the same type an affine transformation α ∶ c → c′ can be defined such that curves of the confocal family which intersect c and c′ orthogonally pass through corresponding points X ∈ c and α(X) ∈ c′ (note Theorem 8.1). Then, Ivory’s theorem in E2 states X α(Y ) = α(X) Y for all X, Y ∈ c.

This statement holds even for singular α, when c′ = α(c) degenerates into a point-set on an axis of symmetry (Figure 2.24). y F4 =4′′′

X ′′′

X

4

Y

4′

′ X′ Y

F1 ′ 1′′ 1

5

5′ =6

X ′′

1

6′ 2′ 7′

2 7

c c′

F2 = 2′′ x

8′

3′ 8 3 F3 =3′′′

FIGURE 10.29. Ivory’s theorem in M2 , applied to confocal ellipses of type 2.

Analogous results are valid for confocal spherical conics, as stated in Theorem 10.1.10 (see Figure 10.14). Moreover it has been proved in [131] that even in the pseudo-Euclidean plane M2 Ivory’s theorem is valid for all types of confocal sets. According to [138] confocal conics c and c′ are of

559

10.2 Conics in non-Euclidean planes

the same type if within the confocal range there is a continuous transition from c to c′ without passing a singular curve (note Figure 10.28, right).

In terms of an appropriate affine transformation α ∶ c → c′ with X ↦ X ′ and Y ↦ Y ′ , the pseudo-Euclidean version of Ivory’s theorem claims dm (X, Y ′ ) = dm (X ′ , Y ) for all X, Y ∈ c.

Figure 10.29 shows Ivory’s theorem in several cases. The distances dm (X, Y ′ ) = dm (X ′ , Y ) and dm (7, 8′ ) = dm (7′ , 8) are measured on timelike lines, while 5′ = 6 implies that dm (5, 6′ ) = dm (5′ , 6) = 0. Hence, [5, 6′ ] must be lightlike.

In Figure 10.26, two confocal hyperbolas of type 3 are depicted. The distance dm (X, Y ′ ) = dm (X ′ , Y ) is timelike and dm (1, 2′ ) = dm (1′ , 2) is spacelike. Another example with confocal parabolas of type 6 is shown in Figure 10.27. Figure 10.25 presents singular affine transformations with c′ as x-axis and c′′ as y-axis. The equations dm (X, 1′ )+dm (X, 2′ ) = dm (X ′ , 1)+dm (X ′ , 2) = dm (1, 2) and dm (Y, 3′′ ) − dm (Y, 4′′ ) = dm (Y ′′ , 3) − dm (Y ′′ , 4) = dm (3, 4) confirm the constant sum or difference of the pseudo-Euclidean focal distances. Also in Figure 10.29 singular affine transformations X ↦ X ′′ and X ↦ X ′′′ between the ellipse c and its axes c′′ and c′′′ are indicated. Remark 10.2.2 According to Theorem 10.2.1 in each Ivory quadrangle of M2 the diagonal lines are tangent to the same curve of the confocal range.

●

Exercise 10.2.1 Invariants of pseudo-Euclidean motions.

(i) Verify with the aid of (10.19) that direct pseudo-Euclidean motions keep the directions of lightlike lines fixed, while in the indirect case these directions are exchanged. (ii) Confirm that under the map (10.22) all points on the axis y = tanh other points remain on lines m-orthogonal to this axis.

τ 2

x are fixed, while

●

Exercise 10.2.2 Pseudo-Euclidean orthocenter. Prove that in M2 the three altitudes of a triangle P1 P2 P3 are concurrent. Hint: Focus on the pairs of lines consisting of one side of the triangle P1 P2 P3 and its corresponding m-altitude, i.e., the pseudo-Euclidean normal through the opposite vertex. Two pairs span a pencil of conics with the base points P1 , P2 , P3 and the intersection Om of two pseudo-Euclidean altitudes. Use Desargues’s involution theorem.

●

Exercise 10.2.3 Euclidean and pseudo-Euclidean circles.

In the standard-model of M2 the m-circles are e-equilateral hyperbolas with lightlike asymptotes. Prove the pseudo-Euclidean counterpart to Example 7.3.4 (Figure 7.34, right): The m-orthocenter Om of any triangle P1 P2 P3 lies on the e-circumcircle of this triangle.

560

Chapter 10: Other geometries

Hint: Look for the remaining reducible curves in the pencil spanned by the e-circumcircle ce and one pair of lines consisting of one side of the triangle and its m-altitude.

Om

ce

P2

ce

P2 P3

m

P3

Ce M

cm

P4

Q Cm

Oe

P

Cm P1

cm P1

FIGURE 10.30. Intersecting a pseudo-Euclidean circle cm with a Euclidean circle ce (Exercises 10.2.4 and 10.2.5).

●

Exercise 10.2.4 Intersection of a circle with an equilateral hyperbola.

Prove the following statement: If an e-circle ce and an e-equilateral hyperbola cm share three real points P1 , P2 , and P3 , then the fourth common point P4 is opposite to the e-orthocenter Oe of P1 P2 P3 w.r.t. the m-circumcenter Cm of P1 P2 P3 . Similarly, P4 is opposite to the morthocenter Om w.r.t. to the e-circumcenter Ce (Figure 10.30, left, note [119] and the references there). Hint: The conics through P1 , P2 , P3 , and the e-orthocenter Oe are e-equilateral hyperbolas and belong to a pencil. Their centers lie on a conic m (Theorem 7.5.4) which contains the fixed points of the Desargues’s involution on the line at infinity and passes through the pedal points of the e-altitudes. Consequently, m is the Feuerbach circle of P1 P2 P3 (as a particular case of the nine-point conic), and its center is the midpoint M between Oe and the center Ce of the e-circumcircle (Figure 10.30, left). The Feuerbach circle contains also the midpoints of the segments Pi M , i = 1, 2, 3. The dilatation with center Oe and ratio 1 ∶ 2 sends the Feuerbach circle to the e-circumcircle ce of P1 P2 P3 and the center Cm ∈ m of the hyperbola cm to the opposite point of Oe on cm . This is the remaining point of intersection. Similarly follows the second statement after exchanging the attributes e- and m-.

●

Exercise 10.2.5 An equilateral triangle among the points of intersection If P, Q are opposite points of an e-equilateral hyperbola cm , then the e-circle ce which is centered at Q and passes through P intersects cm in the vertices of an equilateral triangle P1 P2 P3 (Figure 10.30, right, note [119] and the references there).

Hint: Oe = Ce . There is no pseudo-Euclidean counterpart since there does not exist an equilateral triangle in M2 .

10.2 Conics in non-Euclidean planes

561

Conics in the hyperbolic plane The axioms of the hyperbolic plane H2 and the Euclidean plane E2 differ only in the parallel postulate. In E2 , there is a unique parallel h to a given line g through any point P . In H2 , the opposite is true: Through each point P ∈/ g, there pass (at least) two lines which do not meet the line g.

We refer to the projective or Cayley-Klein model of H2 : Given a conic ω as the so-called absolute conic in a projective plane P2 , points of H2 are the points in the interior of ω, and the lines of H2 are secants of ω. Hyperbolic motions are projective transformations in P2 which map the absolute conic ω onto itself — however restricted to H2 . Hyperbolic reflections are the restrictions of harmonic homologies of ω. For further details see, e.g., [35, 82, 84, 92]. A standard model of H2 is embedded into the projective extension of the Euclidean plane E2 . We use homogeneous Cartesian coordinates (x0 ∶ x1 ∶ x2 ) = (1 ∶ x ∶ y) and specify the absolute conic ω as the unit circle. Then, the polar form with respect to ω can be written as ⟨p, q⟩h = p0 q0 − p1 q1 − p2 q2 .

Points in H2 have coordinate vectors p where ⟨p, q⟩h > 0. The hyperbolic distance dh (P, Q) can be computed by virtue of RRR RRR ⟨p, q⟩h RRR cosh dh (P, Q) = RRRR √ RRR ⟨p, p⟩ ⟨q, q⟩ RRRR h hR R

and satisfies the axioms of a metric space.

There is another expression for dh (P, Q) which is valid in all projective models of hyperbolic geometry and based on the natural logarithm of a cross ratio: If the line [P, Q] intersects the absolute conic ω at the points U1 and U2 , then

(note Figure 10.31).

dh (P, Q) =

1 ∣ ln [cr(P QU1 U2 )] ∣ 2

Conics c in H2 are projective conics which contain interior points of H2 . One can classify all conics c in the hyperbolic plane H2 by classifying in P2 (R) the pencil of conics spanned by ω and c and, in addition, by

562

Chapter 10: Other geometries

considering the different possibilities concerning the reality of the base points.

A point M ∈ H2 is a center of c if the polars of M with respect to c and ω coincide. A line in H2 is an axis of c if its poles with respect to ω and c are the same. According to results in Section 7.1, conics c in H2 need not have a center or axis. Figure 10.32 shows a conic c without center and axis; c osculates the absolute conic ω. A point F ∈ H2 is a focal point of the conic c, if F is the point of intersection of two common (complex conjugate) tangents of c and ω. Two conics c, c′ in H2 are called confocal if, and only if, their tangential equations span a linear system which contains the absolute conic. In this case, common tangents of c and ω are also tangent to c′ . Therefore, confocal conics share the focal points, but this is not always sufficient. Also in the hyperbolic plane, a family of confocal conics forms an orthogonal net. Similarily to the pseudo-Euclidean plane, this can be concluded from Desargues’s involution theorem. In [138], it is proved that Ivory’s theorem is valid in the hyperbolic plane H2 . This is illustrated in Figure 10.31 and means that cr(XY ′ U1 U2 ) = cr(X ′ Y U1′ U2′ ),

when the ellipse ω serves as the absolute conic. By the way, in [8, p. 118] there is an explicit formula for this cross ratio. However, Figure 10.31 has an Euclidean background. In the specified position, all depicted conics are confocal in the Euclidean sense, too. This means according to Definition 7.1.5 that they share the four isotropic tangents passing through the focal points F1 and F2 . Consequently, the conics are also confocal in the hyperbolic sense provided that the absolute conic ω is included in the family. If the confocal conic c0 which contacts the diagonal [X, Y ′ ] is chosen as the absolute conic of H2 , then the cross ratio cr(XY ′ U1 U2 ) equals 1, and the same must hold for cr(X ′ Y U1′ U2′ ). This confirms again the statement of Theorem 8.1.4. Similar reasoning for ranges of conics which are confocal in the sense of M2 or H2 gives rise to the following generalisation. Theorem 10.2.1 In the Minkowski plane M2 and in the hyperbolic plane H2 , each Ivory quadrangle has diagonal lines that are tangent to the same curve c0 of the confocal range.

563

10.2 Conics in non-Euclidean planes U1 P

c

Q c U1′

P′

′

Q′

c0 F1

ω

U2′

O

F2

U2

FIGURE 10.31. Ivory’s theorem in the hyperbolic plane H2 with the absolute conic ω. The diagonals in the Ivory quadrangle satisfy dh (P, Q′ ) = dh (P ′ , Q).

When a conic c in H2 has an axis s of symmetry with two real focal points, then similar to previous cases (see, e.g., Figure 10.25), one can use Ivory’s theorem for proving that the sum or difference of distances to the focal points on the axis s is constant in the hyperbolic plane. Just as in the spherical case, the Apollonian definition of conics does not work in H2 . In general, the geometric locus defined by dh (P, F ) = ε ⋅ dh (P, l) is not even algebraic. Another version of Ivory’s theorem in H2 is depicted in Figure 10.32. The involved confocal conics belong to a range of fourth kind; they osculate the absolute conic and share one more absolute tangent. Also Theorem 7.3.3 is valid in H2 due to its projective character. However, in the case of confocal conics c0 and c1 , as depicted in Figure 7.42, the absolute points (E2 , F2 ) of E2 have to be replaced by the absolute conic ω. Consequently, in the proof of Theorem 7.3.3 (note Figure 7.43) ω belongs to the range Rc , and the range Rt spanned by ω and the pencil T consists of conics which contact ω at points with tangents passing through T . This characterizes d2 ∈ Rt as a hyperbolic circle. It depends on whether T is

564

Chapter 10: Other geometries

c

ω

c′

Y X Y′

X′

FIGURE 10.32. Ivory’s theorem in the hyperbolic plane H2 in the case of confocal conics c and c′ which osculate the absolute conic ω.

an interior point of ω or on ω or in the exterior, that the circle d2 has a hyperbolic center, or is a horo- or hypercircle. In any case, this confirms the validity of Theorem 7.3.2 in H2 , too.

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Index

A absolute circle point, 266, 334, 464, 477 conic, 561, 563 point, 266, 334 points, 119, 266, 267, 288, 289, 303, 304, 308, 322, 327, 342, 347, 348, 367, 372, 376, 412, 413, 436, 447, 448, 477 polarity, 285, 539 acceleration vector, 25 actual trilinear coordinates, 414 adjoint mapping, 276 affine combination, 213, 378, 401 coordinates, 205, 206, 423 evolute, 111–113 geometry, 378 mapping, 378, 420 perspective, 420 normal, 109–111 parameter, 241 part, 241 plane, 378 ratio, 205, 212, 378, 420, 475 transformation, 111, 378, 420 algebraic curve, 3, 101, 126, 240, 334, 356, 357, 362, 429, 472, 517 algorithm of de Casteljau, 400 angle, 441, 519 bisector, 219 of circumference, 60, 237, 352, 547 Antarctic Circle, 6 anti-focus, 290 anti-holomorphic function, 375 anti-inversion, 222, 224, 279 anti-polarity, 279 anti-projectivity, 375 antiparallelogram, 49, 63, 66 antipodal points, 192, 517 antipode, 517 apex of quadr. cone, 163, 529 Apollonian circles, 352 definition, 13, 17, 23, 552 hyperbola, 370, 443 Apollonius of Perga, 2, 9, 13, 352 arc, 517 arc length, 24, 27, 113 of an ellipse, 85 Archimedes, 408

Arctic Circle, 6 area of a spherical triangle, 519 areal velocity, 74, 79 Aronhold, Siegfried H., 66 associated net, 357 associative, 192 astroid, 95 asymptote, 92, 132, 244, 284, 387 asymptotic direction, 132 line, 132 tangent, 131 auto-polar triangle, 280, 460 automorphic collineation of a conic, 261, 280 of the Veronese variety, 293 automorphism of a conic, 261, 263 of the Veronese variety, 293 auxiliary vertex, 521, 546 axiom of Desargues, 194 of Fano, 193 of Pappus, 194 axioms of a collineation, 246 of a projective plane, 189 axis, 6, 13, 38, 562 major, 521 minor, 521 of a collineation, 247 of a conic, 13, 38, 285 of a coordinate system, 205 of a hyperbola, 387 of a parabola, 250, 285 of a pencil of circles, 349 of a perspectivity, 200 of a projectivity, 203, 261 of a spherical conic, 521 of an ellipse, 383, 384 of curvature, 135 of the Earth, 6 semimajor, 17, 92, 521 semiminor, 17, 92, 521

B Böhm, Wolfgang, 333 ball-bearing, 477 barycenter, 413 barycentric coordinates, 205, 413 base point, 309, 310, 317, 320 of a conic’s generation, 230 of a pencil, 309, 310, 320 of birat. map., 356, 364

belt drive, 50 Bernoulli’s lemniscate, 119 Bernstein polynomial, 401 Bézier curve, 97, 400 Bézier, Pierre Étienne, 401 bicentric, 473–475, 503 bicircular curve, 372 bilinearform, 360 billiard, 395 elliptic, 480 hyperbolic, 506 in an ellipse, 479 in an Ivory quadrangle, 395, 506 motion, 480, 491, 497, 501, 506 projective, 479, 489 table, 395 twofold covered, 508 bipolar coordinates, 355 birational mapping, 221, 281, 356 transformation, 221, 281, 356 bisector, 36, 67, 68, 349 spherical, 519, 547 Booth’s lemniscate, 119 Boy’s surface, 187, 229 Boy, Werner, 187, 229 Brahe, Tycho, 70 Braikenridge, William, 455 Brianchon point, 235 Brianchon, Charles J., 197, 234 bundle model of a proj. plane, 190

C canonical coordinate, 500 parameter, 480, 502, 504 parametrization, 491, 502, 504 Cardan circles, 62 Cardano, Gerolamo, 62 cardioid, 446 Carnot, Lazard N.M., 422 carrier of a bundle, 190 of a pencil of lines, 188, 295 Cartesian coordinates, 205, 475 Cassini’s curves, 119 Cassini, Giovanni Domenico, 119 cattle problem, 408 caustic, 479 Cayley line, 244

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2024 G. Glaeser et al., The Universe of Conics, https://doi.org/10.1007/978-3-662-70306-9

573

574 Cayley’s curve, 334 Cayley’s theorem, 476 Cayley, Arthur, 245, 334, 473, 476 Cayley-Klein model, 561 center, 284, 562 conic, 15 equation, 19 function, 413, 415 of a circle, 280, 416 of a collineation, 247 of a conic, 280, 409, 416 of a hyperbola, 250, 378 of a polarity, 279 of a projection, 165, 297 of a projectivity, 203 of a reflection, 224 of a triangle, 415 of an ellipse, 250, 378, 521 of an inversion, 223, 370 of an involution, 264, 265, 268 of curvature, 25, 87, 91, 95, 98, 99, 102, 112, 393 of a hyperbola, 97, 102 of a parabola, 98, 102 of an ellipse, 96, 97, 102 of perspectivity, 200 ofcurvature, 134 central circle, 415 conic, 15, 388, 413, 417 line, 415, 418 perspectivity, 454 projection, 165, 296 similarity, 196 centroid, 323, 413, 419, 431, 467, 544 Ceres, vi, 185 Cesàro, Ernesto, 114 Cesàro curve, 113, 114 channel surface, 157, 158 characteristic cross ratio, 225, 248 of a field, 192, 193 polynomial, 459 generalized, 459 Chasles’s parabola, 46, 334, 369 Chasles, Michel, 45, 333, 369 circle, 57, 66, 132, 237, 238, 242– 244, 250, 255, 266, 267, 315, 327, 342, 347, 372, 379, 380, 402, 416, 446 Antarctic, 6 Arctic, 6 Euclidean, 267 hyperosculating, 89, 101, 134 nine point, 422, 423 of Apollonius, 49 osculating, 102, 104, 109, 111, 112, 393 point, 266 pseudo-Euclidean, 551, 559

Index tritangent, 304, 412, 414, 422, 423 circular cubic curve, 450 section, 165 circumcenter, 413 circumcircle, 101, 246, 412, 416, 430, 457, 467, 475, 544 of an ellipse, 20 of a triangle, 457 circumconic, 418 circumference of a circle, 85 of an ellipse, 85 cissoid, 451 closed polygon, 453 closest point, 442, 443 clothoid, 115 collinear, 189 collineation, 246, 255, 259, 328 harmonic, 250 perspective, 246, 247, 258, 263 projective, 246, 261 collision detection, 442 common polar triangle, 301 commutative, 192 commuting involutions, 226 complete quadrangle, 424 complex plane, 374 concentric circles, 327, 347, 355 conchoid, 452 generalized, 452 of a circle, 447 concurrent, 189 concyclic points, 220, 254 cone, 135 oblique circular, 165 of degree 4, 546 of revolution, 176, 184 quadratic, 163, 528, 529, 534 configuration, 195–197, 235 confocal central conics, 388 conics, 38, 332, 367, 388, 537, 562 hyperbolic plane, 562 pseudo-Eucl., 557 same type, 390, 392, 559 parabolas, 390 quadrics, 509 range, 394 spherical conics, 535 conformal, 374, 375 congruent, 329 pencils, 237 conic, 13, 230, 291, 312 absolute, 561 compass, 56 degenerate, 178, 316, 330, 398 empty, 312 hyperosculating, 101, 111

irreducible, 306 on five points, 254 on six points, 254 osculating, 101, 102 projective, 230 reducible, 306 regular, 293, 306, 398 singular, 292, 306, 316, 330, 398, 464, 470, 473 spherical, 521, 532, 534, 542 conjugate diameters, 284, 379, 382, 385 hyperbolas, 132, 387 lines, 278 normals, 89, 368 pair, 280 pencils, 352 points, 278 surface tangents, 132 conjugation, 313 contact condition, 254 fourth-order, 111 of conic and line, 254 order, 94, 373 third-order, 101, 102 transformations, 373 contour, 170 of a 1-sheeted hyperboloid, 181 of a cone, 140, 145, 176, 177, 184 of a cylinder, 173, 177 of a hyperb. paraboloid, 182 of a parabolic cylinder, 178 of a quadric, 179 of a sphere, 149, 170, 171 of a torus, 125 of an ell. paraboloid, 180 of an ellipsoid, 180 control point, 400 polygon, 400 convex, 378 convolution surface, 432 coordinates actual trilinear, 414 affine, 205, 206, 423 barycentric, 205, 413 bipolar, 355 Cartesian, 205, 206, 467 homogeneous, 205–207, 413 inhomogeneous, 205, 241 trilinear, 413, 414 Cornu spiral, 115 Cornu, Marie Alfred, 115 coupler, 63 Cremona transformation, 356, 427 Cremona, Antonio Luigi Gaudenzio Giuseppe, 356 cross ratio, 212, 213, 215, 216, 267, 368, 452, 454 characteric, 225

575

Index characteristic, 250 cube duplication, 2, 306 cubic curve, 46, 450 form, 320 polynomial, 108 twisted, 291 curvature, 25, 135 diagram, 113, 114 function, 113, 127 Gaussian, 130, 135 geodesic, 135 Mean, 130 principal, 130 radius, 29, 88, 91, 92, 114, 126, 127 vector, 27 curve of degree two, 20 algebraic, 3, 94, 101, 126, 240, 334, 356, 357, 362, 429, 472, 517 bicircular, 372 cubic, 46, 450 geodesic, 520 irreducible, 20, 306 isoptic, 353 quartic, 436 reducible, 20, 306 regular, 306 singular, 306 cusp, 95 of the first kind, 127, 362, 446 of the second kind, 127 cycle, 357 cyclic sum, 418 symmetry, 413 cylinder of revolution, 172, 173

D Dandelin, Germinal Pierre, 9, 140 de la Hire, Philippe, 20, 379, 517 Delian problem, 2, 306 deltoid, 431 Desargues configuration, 195, 248 figure, 195–197, 247 involution, 336, 342, 427, 549, 562 Desargues, Gérard, 336 Desarguesian plane, 195 Descartes, René, 206 development, 135 diagonal, 217 line, 367 of a quadrilateral, 193 point, 189, 217, 371, 424, 426, 427 triangle, 317 diagonal point, 292 diameter of a circle, 280, 355 of a conic, 280

diophantine equations, 405 director circle, 48, 466 directrix, 13, 110, 369 of a parabola, 110, 346 pseudo-Euclidean, 552 discretization, 101 distance function, 378 distorsion ratio, 382 distributive, 192 double contact, 183, 325, 343 line, 297, 320 point isolated, 362, 446 ordinary, 446 with real tangents, 362 doubly conjugate lines, 366 touching conics, 310 dual conic, 234, 238, 346 curve, 287 duality, 239, 330 dualization, 313 Dürer, Albrecht, 9 Dupin cyclide, 38, 157–159, 161 parabolic, 38 indicatrix, 128, 130 Dupin, Charles, 130

E eccentricity linear, 16 numerical, 86 eigenvector, 223 eight loop, 49 elation, 248, 259, 343 eleven-point conic, 426, 427 ellipse, 2, 13, 57, 59, 65, 67, 131, 172, 250, 254, 255, 259, 284, 325, 378– 380, 383, 403, 434, 477, 520, 528 compass, 67 ellipsoid, 180, 182 of revolution, 180 elliptic billiard, 480 coordinates, 389 cosine, 502 curve, 472 disks, 50 gears, 65 integral, 86 involution, 265, 350 motion, 56, 57, 59 orbits, 74 paraboloid, 180 pencil of circles, 347, 349 point, 131, 132 polarity, 278, 299, 303 sine, 502 wheels, 63

empty conic, 299, 312 set, 312, 520 entangled pairs, 267 envelope, 97, 125, 126, 158, 424, 542 epicycloid, 115, 116 equation of a circle, 18 of a conic, 18, 20, 240 of a hyperbola, 14 of a line, 210 of a parabola, 18 of a spherical conic, 526 of an ellipse, 14 of the dual conic, 287 equi-anharmonic, 219 equiform geometry, 348, 378 motion, 348 plane, 378 transformation, 378 equilateral cone, 540 hyperbola, 114, 237, 244, 323, 324, 403, 405, 412, 443, 551, 559 spherical conic, 542 equioptic curve, 441 Erlangen program, 378 Euclidean circle, 267 geometry, 327 plane, 12, 348, 378 rotation, 266, 387 transformation, 348 Euler line, 424 Euler’s formula, 130 evolute, 111 affine, 112, 113 of a conic, 368, 393 of a hyperbola, 97, 99 of a parabola, 97–99 of an ellipse, 93–95, 97, 121 excentral triangle, 459 exceptional point, 429 point of a birat. map., 356 set, 356 excircle, 414, 423 exponential curve, 124 pencil, 313 exterior contact of circles, 37 of a conic, 43 point, 182, 268 product, 211

F Fano plane, 189, 193, 283

576 Fano’s axiom, 217 Fano, Gino, 189 Fermat point, 119, 412, 413 Feuerbach circle, 413, 560 point, 413, 414 Feynman, Richard Phillips, 80 fibration, 309 field, 192 algebraically closed, 308 finite, 408 finite field, 408 projective plane, 268, 408 five-fold intersection, 111 fixed line, 247 point, 204, 223, 247, 262, 268, 283, 344, 431, 455 of a coll., 247, 270 of a proj., 204, 223 of quadr. transf., 365 flatpoint, 132 Fluchtgerade, 255 focal billiard, 510, 511 conics, 509 involution, 367 point, 13, 117, 184, 288, 302, 367, 384, 421, 450, 520, 562 hyperbolic plane, 562 of a hyperbola, 290, 387 of a parabola, 290, 438 of an ellipse, 65, 290 pseudo-Euclidean, 552 ray, 536 focus, 13, 184 pseudo-Euclidean, 552 form, trivariate, 292 four-bar linkage, 63 four-conics-theorem, 318 fourth common point, 102 harmonic, 217 fourth-order contact, 111 Frégier conic, 127, 269, 270 Frenet frame, 25, 109 Fresnel integral, 115 Fresnel, Augustin-Jean, 115 front view, 138, 534 function quadratic, 324 rational, 170, 322, 324, 329, 355 fundamental form first, 130 second, 130 triangle, 209 Fundamental Theorem, 201 Fuss, Nicolaus, 476

Index

G gardener’s construction, 7, 66, 520 Gauß, Carl Friedrich, vi, 405 Gaussian curvature, 130 generalized conchoids, 452 geodesic curvature, 135 curve, 520 geometric continuity, 94 Gergonne point, 235, 236 Gergonne, Joseph Diaz, 192, 236 Ghys’s theorem, 108 Ghys, Étienne, 107, 108 Graves, Charles, 46, 154 gravitational constant, 70 gravity, 65 great circle, 192, 517, 525 group, 192 of automorphisms, 261

H Hamilton, William Rowan, 70, 76 harmonic collineation, 263 conjugate, 217, 220, 285 conjugates, 427 homology, 263, 274, 276, 561 quadruple, 217, 225, 285 Hein, Piet, 122 Henrici’s hyperboloid, 509, 510, 513 Henrici, Olaus, 509 Hesse, Ludwig Otto, 334 Hessenberg, Gerhard, 198 Hessian curve, 334 hexagrammum mysticum, 244 Hilbert, David, 187, 229 Hirst’s inversion, 375 Hirst, Thomas Archer, 375 hodograph, 76 Hoecken’s mechanism, 57 homofocal conics, 38 parabolas, 304 homogeneous Cartesian coordinates, 209 coordinates, 104, 205, 206 equation, 164 mass density, 71 parameters, 210 polynomial, 163, 164 trilinear coordinates, 413 distances, 413 homology, 248 harmonic, 250, 264, 280, 561 involutive, 264 horn cyclide, 38 hyperbola, 2, 13, 131, 250, 254, 255, 258, 260, 284, 325, 378, 379, 384, 387, 520, 522

equilateral, 238, 244, 322– 325, 327, 403, 412, 551 hyperbolic billiard, 506 cylinder, 175 involution, 351, 355 motion, 561 paraboloid, 180 pencil of circles, 347, 350 plane, 561 point, 131, 132 polarity, 278, 280, 303 hyperboloid, one-sheeted, 180 hypercycloid, 115, 116 hyperosculating, 258 circle, 89, 101, 134 of a hyperbola, 89, 93 of a parabola, 93 conic, 101, 111, 184, 185 hyperbola, 183 parabola, 108–110, 183, 321 hyperosculation, 89, 101, 182, 248, 257–259, 321, 331, 332, 359 hyperplane, 292 hypocycloid, 115, 116, 431

I ideal line, 190, 196, 211, 248, 267, 279, 280, 378, 412, 424, 436, 448 point, 190, 196, 207, 209, 280, 378 identity mapping, 204, 222 image plane of a projection, 165 incenter, 413 incidence, 210 geometry, 188 incident, 189 incircle, 236, 457, 475 of an ellipse, 20 of a triangle, 457 incircular, 54 spherical quadrangle, 539 inhomogeneous coordinates, 205, 378 inpolar, 460, 461 instantaneous pole, 60, 63, 66 interior contact circles, 37 of a conic, 43, 268 point, 182, 264, 268, 281– 283, 344 inverse element, 192 points, 370, 372 inversion, 222, 223, 281, 370, 372, 374, 426 in a circle, 372 in a conic, 370 in a line, 375 involution, 223, 225, 226, 267, 283, 336, 344, 472 center of, 265

577

Index elliptic, 223, 265, 267, 268, 350 hyperbolic, 223, 267, 268, 351, 355 of conj. lines, 282 of conj. points, 282, 283 of right angles, 265, 279, 285, 288 on a conic, 268 involutive, 223, 248, 283, 363, 430 collineation, 248 motion, 551 projectivity, 338, 343, 344 isogonal conjugation, 427, 430 isolated double point, 446 isometry, 135 isoptic curve, 50, 353, 434, 545 of a hyperbola, 439 of a parabola, 437 of an ellipse, 434 of a segment, 353 isotomic conjugation, 431 isotropic cone, 538, 539 line, 288, 303, 353, 450, 539 tangent, 303, 304 Istanbul, 7 Ivory quadrangle, 40, 391, 394, 395, 505 Ivory’s theorem, 39, 151, 152, 392, 534, 556, 558, 562 hyperbolic plane, 563 pseudo-Eucl. plane, 556

J Jacobi, Carl Gustav Jacob, 480 Jacobian elliptic function, 502 Joachimsthal, Ferdinand, 486 joint, 58

K Kennedy, Alexander B.W., 66 Kepler’s equation, 79, 80 First Law, 72 Second Law, 74 Third Law, 76 Kepler, Johannes, vi, 10, 70 Kiepert center, 413 hyperbola, 323, 412, 413, 423 parabola, 423, 424 Kirkman point, 244 Kirkman, Thomas Penyngton, 245 Klein, Felix, 10, 378 Konika, 2

L Laguerre point, 265, 266, 279, 342, 350, 477 Laguerre, Edmond, 266 Lamé, Gabriel, 121 Lamé curves, 121

latitude, 527 lattice point, 406 level set, 119, 132, 441 Lie group, 500 lightlike, 549 line, 188, 189 at infinity, 190, 211, 248, 250, 255, 267, 279, 303, 412, 424, 430, 431, 436, 448 element, 109, 230, 239, 320, 331, 353 projective, 206, 291 repeated, 178, 277, 293, 312, 319, 327 self-conjugate, 274 line-conic-intersection, 236 line-saving, 258 line-to-point transformation, 376 linear eccentricity, 16, 521, 528 image, 171 normal, 432 pencil, 313 perspective, 165 Liouville, Joseph, 91 LN surfaces, 432 local expansion, 133 logarithmic spiral, 124 longitude, 527 Lorentz transformation, 551

M MacLaurin’s trisectrix, 451 MacLaurin, Colin, 454 major axis, 15, 521 Mannheim curve, 113 diagram, 114 Mannheim, Amédée, 114 mapping affine, 174, 379, 420 projective, 104, 199, 222, 225, 246, 259 self-adjoint, 537 via equal coordinates, 243 Marden’s theorem, 421 Marden, Morris, 421 mean curvature, 130 mechanism, 56 medial triangle, 424 Menaichmos, 3, 9 Meusnier sphere, 134 Meusnier’s theorem, 133 midpoint of a segment, 218 minimal projective plane, 189 Minkowski, Hermann, 549 Minkowski geometry 236, 327, 549 minor axis, 15, 521 model of a projective plane, 189 Möbius group, 375 Möbius, August Ferdinand, 205 monofocal conics, 144 motion

hyperbolic, 561 pseudo-Euclidean, 550 multifocal curve, 116 ellipse, 116, 117

N n-body problem, 71 n-ellipse, 117 n-gon, 422, 454, 471, 473 Napoleon point, 412, 413 natural parameter, 27 needle cyclide, 38 negative pedal curve, 376 plane, 167 net of conics, 333 Newton’s Law of Gravitation, 70 Second Law of Motion, 71 Newton, Isaac, vi, 10, 70 nine-point center, 424 circle, 412–414, 422–424 conic, 366, 412, 414, 424, 426 Noether, Max, 356 non-orientable, 297 normal acceleration, 25 affine, 109–111 cone, 540 curvature, 128 of a hyperbola, 443 of a parabola, 444 of an ellipse, 442 nucleus, 409 null circle, 350 nullpolarity, 278, 409 numerical eccentricity, 13, 17, 86, 502

O offset, 124 generalized, 127, 270 of a conic, 124 of a parabola, 126 of an ellipse, 125, 126 one-parameter family of conics, 306 of lines, 90 one-seventh-conic, 409 one-sheeted hyperboloid, 179, 180 one-sided, 297 one-thirteenth-conic, 410 opposite points, 192 optical property, 30 of a spherical conic, 525 order of a projective plane, 268 ordinary double point, 446 orientation reversing, 374, 375 oriented distances, 414

578 origin, 205, 378 of a coordinate system, 205 orthocenter, 323, 346, 413, 424, 544 orthocircle, 351 orthogonal, 279, 382 conjugate lines, 367 net, 39, 351, 536 projection, 173, 528 orthogonality, 280, 378 orthonormal pseudo-Euclidean, 549 orthoptic circle, 436 curve, 48, 369, 434 orthotomic circle, 33 curve, 34 osculating, 258 circle, 25, 87, 89, 93, 94, 101, 102, 104, 109, 111, 112, 258, 259, 373, 374, 393 of a curve, 101 of a hyperbola, 97, 102, 107 of a parabola, 93, 98, 102, 103 of an ellipse, 89, 91, 96, 97, 102, 105, 133 conic, 101, 102 parabola, 102, 108 paraboloid, 132 plane, 27, 134 osculation, 94, 248, 257, 321, 331, 332, 359 outer point, 264, 268, 281–283, 344

P pair of conj. diameters, 280, 383 conj. hyperbolas, 131, 387 conj. semidiameters, 387 lines, 277, 293, 312, 319 parallel lines, 131 Pappian projective plane, 197, 198 Pappus configuration, 197 figure, 197 parabola, 2, 13, 68, 107, 108, 183, 250, 254, 255, 259, 260, 285, 314, 324, 325, 339, 346, 378, 390, 437, 451, 520 Chasles’s, 334 compass, 68 hyperosculating, 108–110, 321 osculating, 102, 108 semicubical, 97 parabolic cylinder, 133, 176 Dupin cyclide, 158 pencil of circles, 347 point, 131, 132 projectivity, 223 paraboloid, 133

Index elliptic, 133, 180 hyperbolic, 133, 180 paracycloid, 115, 116 parallel, 378 curve, 125 lines, 190, 378 projection, 380 parameter, 13, 17 homogeneous, 241 parametrization, 401 of a spherical conic, 529 canonical, 504 linear, 432 of a conic, 91, 240, 241, 295 of a hyperbola, 91 of a parabola, 91 of an ellipse, 22, 91, 94 polynomial, 97 rational, 22, 85, 170, 242, 254, 297, 400, 401, 403, 405, 450 trigonometric, 22, 91, 297 Pascal line, 244 Pascal’s limaçon, 446, 449 Pascal, Etienne, 449 Pascal, Blaise, 10, 245 pedal curve, 34, 376, 446 negative, 376 of a circle, 446 of a parabola, 449 point, 376, 442 transformation, 376 Pell equation, 407 Pell, John, 407 pencil exponential, 313 linear, 313 of circles, 316, 323, 347 conjugate, 352 elliptic, 347, 349 hyperbolic, 347, 350 parabolic, 347, 353 of conics, 306, 308, 312, 313, 357, 472 of the fifth kind, 321, 342, 343, 359, 364 of the first kind, 309, 317, 348, 364 of the fourth kind, 321, 342, 359, 364 of the second kind, 310, 320, 348, 353, 364 of the third kind, 310, 320, 348, 355, 364 of hyperosc. conics, 321, 359 of lines, 188, 295, 406 of osc. conics, 321, 359 of planes, 238 of polarities, 302 of doubly touching con., 320 permutation, 216 Perseus, 119 perspective, 165

affine mapping, 174 affinity, 378, 379 collineation, 248, 250, 257, 263, 328, 342, 378 perspectivity, 199, 200, 260, 263 perspector, 195–197, 199, 412, 413 of a conic, 236 perspectrix, 195–197, 199, 200, 424 photographic mapping, 167 Pillwein, Gerhard, 66, 475 pinch point, 297 Plücker line, 244 Plücker, Julius, 245 plane affine, 378 equiform, 378 Euclidean, 12, 378 osculating, 134 projective, 188 pseudo-Euclidean, 549 point, 188 absolute, 266, 289, 304, 322, 327, 348, 367, 372, 376, 412, 436, 447, 448, 477 at infinity, 190, 207 exterior, 182 interior, 182 of inflection, 101 of intersection, 189 of maximal curvature, 88 of minimal curvature, 88 proper, 378 self-conjugate, 274 singular, 119, 465 point model, 291 polar equation, 23 form, 281 line, 274, 285 w.r.t. a triangle, 221 system, 274, 302 of a conic, 274 triangle, 280, 281, 299, 317 polarity, 185, 274, 277, 296 degenerate, 276 elliptic, 278, 303 hyperbolic, 280, 303 regular, 465 singular, 276, 282, 465 pole, 274, 285 instantaneous, 63, 66 pole curve, 62, 64, 66 fixed, 62, 66 moving, 62, 66 polygon closed, 454 polynomial complex, 421 cubic, 108, 421 function, 107 homogeneous, 163, 164

579

Index parametrization, 97, 401 Poncelet correspondence, 472 grid, 333, 335, 487, 489, 506 map, 472 porism, 453, 471, 542 Poncelet, Jean-Victor, 471 porism, 453, 456, 472, 542 Poritzky string length, 501 power, 373 w.r.t. a circle, 50, 493 w.r.t. a sphere, 140 principal axis, 164, 383, 397, 399, 420 circle, 20 curvature, 130 distance, 165 line, 174 normal vector, 25 plane, 174 point, 165 tangent, 130 vertex, 258 view, 532 principle of duality, 192, 280, 330, 366 Proclus, 20, 517 Proclus’s construction, 105 product rule, 214, 216 projection central, 165, 296 orthogonal, 441 parallel, 171 stereographic, 243, 401, 405 projective conic, 230 coordinate, 214 coordinate system, 214 extension, 205 frame, 214 generation of a conic, 230 inversion, 370 line, 241 plane, 188 finite, 268, 408 minimal, 193, 408 reflection, 248 scale, 208 projectivity, 199, 200, 223, 224, 260, 262, 263, 455, 547 between conics, 260 elliptic, 224 hyperbolic, 104, 222–224 in a pencil, 204 involutive, 225, 226, 264, 265, 343, 344 on a conic, 261, 263, 264 on a line, 204, 261 parabolic, 223, 226, 264 pseudo-Euclidean circumcircle, 559 geometry, 236, 327

orthocenter, 559 plane, 549 unit circle, 405 Pythagoras’s theorem, 378 Pythagorean triplet, 405

Q quadrangle, 189, 217, 475 bicentric, 475 complete, 424 Ivory, 391, 394, 395, 505 quadratic birat. mapping, 221, 356 quadratic Bézier curve, 400 cone, 163, 528, 534 cylinder, 528 function, 324, 325 mean, 86 surface, 238 transformation, 221, 356, 432 quadric, 452 quadrilateral, 54, 193, 292, 331, 426, 427 quartic curve, 372, 436 ruled surface, 59, 238

R radical axis, 51, 308, 349 center, 52 radius of curvature, 25, 79, 109 Ramanujan, Srinivasa Iyengar, 86 range confocal, 394 of confocal conics, 332 of conics, 330, 366 of the fifth kind, 331 of the first kind, 330, 332, 366 of the fourth kind, 331, 332 of the second kind, 331, 332 of the third kind, 331 of points, 188 rational rational Bézier curve, 402 cubic curve, 450 function, 170, 322, 324, 329, 355 normal curve, 291 parametrization, 22, 85, 170, 242, 243, 254, 297, 400, 401, 403, 405 real projective plane, 190 representative, 279 reducible algebraic curve, 357 curve of degree two, 155, 306, 337 surface of degree two, 163 reduction theorem, 201 reflection, 222, 224, 250, 368

hyperbolic, 561 in a circle, 223, 281, 372 in a line, 248, 250 in a parabola, 33 in a point, 224, 250 in the unit circle, 281 projective, 248 pseudo-Euclidean, 551 regular conic, 293 curve point, 89 polarity, 465 surface point, 128 reparametrization, 432 repeated line, 277, 293, 297, 312, 319, 320, 327 representative, 210 of a line, 207 of a point, 206 Reuleaux triangle, 63 Reznik, Dan, 480 rhomb, 67, 68 right angle, 206, 244, 265, 266, 269, 279, 285, 288, 349, 350 cissoid, 451 right-side view, 184, 534 ring cyclide, 38 Roman surface, 297 rotation Euclidean, 21, 266, 387 pseudo-Euclidean, 551 Rytz von Brugg, David, 383 Rytz’s construction, 383

S Salmon point, 244 Salmon, George, 245 scalar product, 211 pseudo-Euclidean, 549 segment, 517 self-adjoint mapping, 277, 537 self-conjugate, 274 line, 278 point, 278, 299 self-dual, 195, 197, 332 self-orthogonal line, 265 self-polar triangle, 280 self-similar, 353 semicubical parabola, 97 semimajor axis, 17, 521 semiminor axis, 17, 521 shearing, 196, 197, 420 side view, 134, 138 Siebeck, Jörg, 421 signature, 529, 534, 535 similar, 196 similarity, 196 Simson line, 416 Simson, Robert, 416 singular

580 conic, 292, 308, 312, 320, 464, 470, 473 pencil, 330 point, 119, 465 polarity, 282, 465 six-fold intersection, 111 spacelike, 549 spatial interpretation, 172, 342, 345 sphere, 192, 517, 529, 546 Meusnier, 134 model of a proj. plane, 192 spherical bisector, 519, 547 conic, 521, 532, 534 conics, 516 confocal, 535 degree, 542 distance, 518 ellipse, 517, 521, 535 focus, 521 hyperbola, 535 law of cosines, 519 law of sines, 519 Pythagoras theorem, 519 tangent, 525 trigonometry, 519 Wankel engine, 515, 543 Spieker center, 412 spine curve, 158 spiric curve, 119, 441 standard equation ellipse, 19 hyperbola, 19 parabola, 18 Steiner circle, 263, 265 circumellipse, 105, 419, 420, 422, 424 conic, 263, 265, 341, 345 cycloid, 417 ellipse, 431 inellipse, 421 point, 244, 424 surface, 297 Steiner’s generation, 199, 299 hypocycloid, 431 Roman surface, 296 Steiner, Jakob, 230, 245, 263, 296, 297 stereographic proj., 243, 401, 405 straight line, 3, 210 straightedge and compass, 455 strophoid, 46, 334, 426 subnormal, 93 superellipse, 122 support function, 90 surface of degree two, 163 symmetric arc, 403 tensor, 293 symmetry, cyclic, 413

Index synthetic geometry, 188 proof, 198

T tangent, 295 asymptotic, 131 circle, 422 isotropic, 303, 304 line, 24, 28 of a conic, 230, 234, 254, 274, 332 of the Veronese, 295 triangle, 459 tangent plane, 135, 296 tangential acceleration, 25 Taylor approximation, 107, 108 ten-point conic, 426 tensor, 293 symmetric, 293 Thales circle, 244, 266, 351, 353, 420, 545, 546 Thales of Miletus, 243 theorem of Graves, 46, 154 Aronhold & Kennedy, 66 Brianchon, 197, 234, 517 Cayley, 473 Chasles, 263 Desargues, 194, 336 Frégier, 60, 269 Ghys, 107, 108 Graves, 65 Ivory, 39, 534, 536 Pappus, 194, 197, 234 Pascal, 517 Poncelet, 471 Pythagoras, 378 Thales, 60, 243, 265, 269, 349, 554 the angle of circumference, 60, 352, 430, 545, 547 third-order contact, 102 three-conics-theorem, 318 timelike, 549 top view, 138, 534, 542 Topkapı, 7 Torricelli point, 119 torus, 125 trace, 226 trammel construction, 57, 380 transformation line-to-point, 376 Lorentz, 551 of conj. normals, 89, 367 of doubly conj. points, 363 translation, 196, 222, 223, 328, 329, 378 triangle, 236, 384, 457, 519 auto-polar, 280 center, 413, 415 fundamental, 209 geometry, 412 inequality, 518

polarity, 220 self-polar, 280 trigon. param., 22, 297 trilinear coordinates, 413, 414 distances, 413 triple point, 297, 448 tritangent circle, 423 trivariate form, 292 trochoid, 515, 543 turning number, 507 two-body problem, 71 two-param. fam. of circles, 352

U umbilical point, 132 unfolding, 135 uniform scaling, 348 unit circle, 243, 280, 561 Euclidean, 280, 281, 405, 561 pseudo-Euclidean, 551 length, 205 point, 205, 209, 378, 413 points, 205 sphere, 518 tangent vector, 24

V vanishing line, 247, 255, 259, 342 plane, 165 Vecten point, 412, 413 vector space model, 188, 192 velocity, 24 areal, 74, 79 diagram, 76 vector, 24 Veronese manifold, 293 map, 292 variety, 291 Veronese, Giuseppe, 291 Verschwindungsgerade, 255 vertex, 14, 89, 101, 321, 374, 521 auxiliary, 87, 88, 546 equation, 18 of a curve, 373 of a hyperbola, 92, 285 of a parabola, 92, 93, 285 of a pencil of lines, 188 of a planar curve, 87 of an ellipse, 87, 285 principal, 87, 88, 258

W Wallace, William, 416 Wankel engine, 515, 543 Wantzel, Pierre Laurent, 307 Weingarten map, 130 Weingarten, Julius, 130