Elementary Geometry in Hyperbolic Space [Reprint 2011 ed.] 9783110849455, 9783110117349

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de Gruyter Studies in Mathematics 11 Editors: Heinz Bauer · Jerry L. Kazdan · Eduard Zehnder

Werner Fenchel

Elementary Geometry in Hyperbolic Space

w

Walter de Gruyter G Berlin · New York 1989 DE

Author Werner Fenchel t Editorial Consultant Christian Siebeneicher Universität Bielefeld Fakultät für Mathematik D-4800 Bielefeld 1, FRG Series Editors Heinz Bauer Mathematisches Institut der Universität Bismarckstrasse 1 '/i D-8520 Erlangen, FRG

Jerry L. Kazdan Department of Mathematics University of Pennsylvania 209 South 33rd Street Philadelphia, PA 19104-6395, USA

Eduard Zehnder ETH-Zentrum/Mathematik Rämistrasse 101 CH-8092 Zürich Switzerland

1980 Mathematics Subject Classification (1985 Revision): Primary: 51-02; 51M10. Secondary: 15A24; 20H10; 51M25. Library of Congress Cataloging-in-Publication Data Fenchel, W. (Werner), 1905Elementary geometry in hyperbolic space / Werner Fenchel. p. cm. - (De Gruyter studies in mathematics: 11) Bibliography: p. Includes index. ISBN 0-89925-493-4 (U.S.): 1. Geometry, Hyperbolic. I. Title II. Series. QA685.F38 1989 516.9 - dc20

89-7650 CIP

Deutsche Bibliothek Cataloging-in-Publication Data Fenchel, Werner:

Elementary geometry in hyperbolic space / Werner Fenchel. Berlin; New York: de Gruyter, 1989 (De Gruyter studies in mathematics; 11) ISBN 3-11-011734-7 NE:GT

© Copyright 1989 by Walter de Gruyter & Co., Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form - by photoprint, microfilm, or any other means - nor transmitted nor translated into a machine language without written permission from the publisher. Printed in Germany. Cover design: Rudolf Hübler, Berlin. Typesetting and Printing: Tutte Druckerei GmbH, SalzwegPassau. Binding: Dieter Mikolai, Berlin.

Editorial

HABENT SUA FATA LIBELLI. This proverbial statement of Terentianus Maurus applies in a very particular sense to the present volume of the de Gruyter Studies in Mathematics. Werner Fenchel, Professor emeritus at the University of Copenhagen, one of the pioneers in the theory of convex bodies and in duality theory for convex functions, well-known also by his contributions to global differential geometry and other fields of mathematics, died on January 24,1988 quite unexpectedly. At that time it was clear to both the publisher and the editors that Fenchel's monograph "Elementary Geometry in Hyperbolic Space" would be accepted for the de Gruyter Studies. Unfortunately, the publisher's official letter of acceptance did not reach Werner Fenchel in time. Fenchel's monograph appears in a period of revival of interest in hyperbolic geometry. The book contains a substantial account of the parts of the theory basic to the study of Kleinian groups. But Fenchel has a lot more to say about a subject which always belonged to his favorite fields. Consequently, the publisher and editors are convinced that Werner Fenchel's monograph will interest mathematicians working in the field of hyperbolic geometry, those needing a handy reference and others who intend to study the subject. It was a happy coincidence that Dr. Christian Siebeneicher of the University of Bielefeld was in close scientific contact with Werner Fenchel during preparation of the manuscript. Thanks to Siebeneicher's effort, Fenchel's manuscript was brought into its final form for printing. Werner Fenchel was born in Berlin in 1905. In Berlin he studied mathematics and physics during the years 1923-1928. He had to leave Germany in 1933, a few months after Hitler's take-over. The publisher and editors are happy that - more than half a century later - Werner Fenchel's last mathematical work will appear in his hometown Berlin. January 1989

Heinz Bauer

Preface

There exist many excellent books on non-Euclidean geometry. To add another one is motivated by the fact that these books contain very little about the geometry in hyperbolic space which has found various applications. Most of what is known is to be found in very old, not easily accessible papers. Many of these have shortcomings. In general the proofs do not cover exceptional cases, often of special interest. Sometimes one refers to a passage to the limit, an exact proof of which may be tedious. There are also cases in which it is not even obvious what can be expected to be valid in the limit. Further, it had already been noted in old papers that satisfactory results in the geometry of lines, which actually covers the elementary geometry of the hyperbolic plane, can only be obtained if oriented lines are considered. The proofs applied yield however often only squares of the relations aimed at, and the determination of the correct signs is not convincing. In the following presentation of various aspects of the elementary geometry in hyperbolic space the axiomatic point of view has been completely neglected. Everything is based on the conformal model. The use of the projective model would, of course, have made it possible to obtain some of the results as consequences of theorems of projective geometry. Many of the relations would formally be simpler, but their geometric interpretation more involved. In any case, if only one model is to be used, the conformal one seems to be preferable. The tools which are needed are very modest: apart from the elements of algebra and analysis, only few facts from elementary Euclidean geometry. What goes beyond can be found in the first chapter. The intention to prove statements under the weakest assumptions made it frequently necessary to replace well-known simple proofs, valid only under certain restrictions, by others, or to deal with exceptional cases separately. Therefore, the reader will certainly find a considerable part of the exposition rather elaborate. If it were to serve as a text-book and not primarily as a reference, much of the content could have been stated in the form of problems. The author hopes that the references to the various sections and the index make it possible to pick out what one is interested in and to skip the rest up to a few definitions and results. Clearly most of the statements are known, in any case under more restrictive assumptions or for the plane. The notes to the various chapters contain some historical remarks and references. To trace everything back to its origin would be an impossible task. Therefore only the sources of ideas which have been used and which seem not to be well known have been quoted. There are, of course, also

VIII

Preface

some references to publications in which the subject in question is developed further. For efficient help in many respects the author offers cordial thanks to Christian Siebeneicher, Bielefeld. Sincere thanks are also due to Mrs. Obershelp for typing the manuscript. August 1986

Werner Fenchel

Contents

/. Preliminaries

l

1. Quaternions 2. The hyperbolic functions 3. Trace relations 4. The fractional linear group and the cross ratio Notes to Chapter I

l 3 8 10 16

II. The Möbius Group

17

1. Similarity transformations 2. The extended space. Orientation. Angular measure 3. Inversion 4. Circle- and sphere-preserving transformations 5. The Möbius group of the upper half-space Notes to Chapter II

17 18 20 22 24 27

///. The Basic Notions of Hyperbolic Geometry

28

1. Lines and planes. Convexity 2. Orthogonality 3. The invariant Riemannian metric 4. The hyperbolic metric 5. Transformation to the unit ball Notes to Chapter III

28 31 34 36 40 42

IV. The Isometry Group of Hyperbolic Space

44

1. Characterization of the isometry group 2. Classification of the motions 3. Reversals 4. The isometry group of a plane 5. The spherical and cylindric surfaces Notes to Chapter IV

44 45 48 54 56 60

X

Contents

V. Lines

61

1. Line matrices 2. Oriented lines 3. Double crosses 4. Transversals 5. Pencils and bundles of lines Notes to Chapter V

61 63 67 70 72 78

VI. Right-Angled Hexagons

79

1. Right-angled hexagons and pentagons 2. Trigonometric relations for right-angled hexagons 3. Trigonometric relations for polygons in a plane 4. Determination of a hexagon by three of its sides 5. The amplitudes of a right-angled hexagon 6. Transversals of a right-angled hexagon 7. The bisectors and radii of a right-angled hexagon 8. The medians of a right-angled hexagon 9. The altitudes of a right-angled hexagon Notes to Chapter VI

79 81 85 93 102 107 Ill 123 127 138

VII. Points and Planes

140

1. Point and plane matrices 2. Incidence and orthogonality 3. Distances and angles 4. Pencils of points and planes 5. Bundles of points and planes 6. Tetrahedra Notes to Chapter VII

140 144 148 155 159 164 174

VIII. Spherical Surfaces

175

1. Equations of spherical surfaces 2. An invariant of a pair of spherical surfaces 3. The power of a point with respect to a spherical surface 4. The radical plane of a pair of spherical surfaces 5. Linear families of spherical surfaces Notes to Chapter VIII

175 177 182 185 191 201

Contents

XI

IX. Area and Volume

202

1. Various coordinate systems 2. Area 3. Volume of some bodies of revolution 4. Volume of polyhedra Notes to Chapter IX

202 206 209 213 220

References

221

Index

223

The reader should take notice of the following: In Chapters I, II, III all terms denoting geometrical notions are to be understood in the Euclidean sense. In Chapter III those denoting notions of hyperbolic geometry are provided with the prefix h. In the following chapters terms denoting geometrical notions are to be understood in the sense of hyperbolic geometry. Those denoting Euclidean notions are provided with prefix e. The values of square roots of positive numbers are always assumed to be positive.

I. Preliminaries

I.I. Quaternions Let C denote the field of complex numbers. As customary, the imaginary unit is denoted by / and the complex conjugate of a e C by . Also Re α — ^ (a + a), lma>= ^i(a — a}. We consider the set C χ C of pairs a = (a, a) of complex numbers. Addition of pairs is defined in the usual way: a + b - (a, a) + (b, β) = (a + b, a + β),

so (C x C, +) is an abelian group. Multiplication is defined by ab = (a, a) (b, ) = (ab - αίβ, a + αϊ) .

The distributive law is obviously satisfied, and it is easily verified that the multiplication is assciative. Further, it is seen that (1,0) acts as unity and that the pairs (a, 0) form a field isomorphic with C. Therefore we may write a instead of (a, 0). Since(0,a) = (a,

or, with the notation (0, 1) =j, (α, α) = a + a/.

Computations with these quaternions can now be performed according to the usual rules together with the special cases j2 = — 1 , ja = j

for α e C

of the definition of the multiplication. The set C x C provided with this ring structure will be denoted by IH. Defining the conjugate of a quaternion a = a + · χ + a for a ε IH, χ ι—>· bx and χ ι—> xb for b e IH \ {0}, χ ι-> χ"1, χ ι-> χ are topological mappings of IH^ onto itself. The subspace of IH spanned by 1 and i is the complex plane C and

1.2 The hyperbolic functions

3

CDO = Cu{oo} the extended complex plane in the usual sense. The 3-dimensional subspace of IH generated by 1 , i and j, that is, J = {x + £ / £ l H | f e l R } = {xe IH|(/x = x//} and its extension

which is homeomorphic to a 3-sphere, will play an essential role in the sequel. For the sake of brevity the points of J) will be called ^-quaternions. The J-quaternions x + ξ/ witn ^ e (R form a field isomorphic with C. We shall write IR^ for the extended real line [R u {oo}.

1.2. The hyperbolic functions Consider the additive group A = C/(2niZ) = IR 0 i!R/(2wZ) of the complex numbers modulo 2π/'. Provided with the ordinaty topology it is homeomoφhic to an open cylinder and will be called the complex cylinder. It may be compactified by adjoining two points, + oo and — oo, and defining a basis for the open sets to consist of the open subsets of A and the sets ρ} u {-00}

for all ρ e IR. The extended complex cylinder A«, = A u ( + 00, -00}

is homeomorphic to a 2-sphere. To avoid confusion with the point oo of C^, the points + oo and — oo of A^ will always be written with their signs. In A all the usual rules for addition, subtraction and multiplication by integers are valid. Division by integers is multivalued. In the following only division by 2 is needed. For each ξ e A there are two elements ζ differing by πι of A such that 2ζ = ξ. If ξ is not specified, we write ^ξ for one of them, choosen arbitrarily but once for all in the investigation in question. However, if ζ e A is determined by a specified complex number, \ξ shall denote the element of A determined by % of this number (for example, ξ = πϊ, \ ξ = %πΐ). Clearly, it makes sense to talk of the conjugate ζ of an element ξ of A.

I. Preliminaries For computation with + oo and — oo the obvious conventions are adopted:

-00 + ξ = -οο -(+00) = - o o , i(+oo) = + o o , + 00 = + 0 0 ,

for £ e A \ { + oo}, -(-co) = +oo, i(—00

=—00.

The complex cylinder A may be considered as the domain of the exponential function, and if we define exp(— oo) = 0,

exp(+oo) = oo,

then

exp: A« -> C^ is a homeomorphism with inverse l o g i C » - » A,,. The hyperbolic functions sinh ξ = ^(βξ — β~ξ), cosh£ = \(el· + β~ξ), tanh ξ = (e2* - \)/(e2* + 1), coth ξ = (ε2ξ + 1)/(