Engineering Drawing and Descriptive Geometry [Reprint 2014 ed.] 9780674366428, 9780674365261

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ENGINEERING

DRAWING

AND

DESCRIPTIVE

GEOMETRY

By

C. J. Walsh Supervisor of Engineering Drawing Department of Engineering Sciences Harvard University

Harvard University Press, Cambridge, Massachusetts

1945

Copyright, 1943, 1944 B y C H A R L E S JAMES W A L S H

Copyright,

1945

B y t h e P R E S I D E N T A N D F E L L O W S OF H A R V A R D

COLLEGE

P H I N ' I M ) I N T H I ; W N I T I ' D STATES OF A M E R I C A

PREFACE

The subject matter in this book has been made brief and concise, so that the important elements of each phase of the subject may be covered in one college semester.

In many technical schools, engineering drawing

is divided into two distinct parts.

Part 1, is devoted to "Engineering

Drawing," which is a combination of "Mechanical Drawing" and a special kind of free-hand drawing, called "Technical Sketching," that is considered so essential to the engineer.

Part 2, Descriptive Geometry is a more

advanced subject than the former, and deals with the solutions of problems of a more complicated nature. It is not intended that this book should be used for self-instruction. Like any language (drawing can certainly be termed the language of architects and engineers) it can be learned with any degree of success only from a teacher and not from a textbook. The author of this book, has kept in mind the adoption of the text in courses of an accelerated nature, where the reading matter must be condensed so that the greater part of the time allotted for the course can be devoted to the practical application of the subject in the drafting room.

Cambridge, 1944

C. J. Walsh.

iii

Part 1

ENGINEERING

DRAWING

CHAPTER 1 - - INSTRUMENTS AND MATERIALS

Page 1

Introduction. Learning to draw. Description and use of drafting equipment. Importance of pencil drawings. Pencil. T-Square and Triangle. Scales. Dividers. Compasses. Sharpening Compass Leads. CHAPTER 2 — LETTERING ON ENGINEERING DRAWINGS

Page 9

Uses of Lettering. Type of lettering in common use. Vertical System of Lettering. Inclined System of Lettering. CHAPTER 3 - - DRAWING BOARD GEOMETRY

Page 12

Ruling straight lines. Drawing parallel lines. Angles. Drawing perpendicular lines. Drawing circles. How to Rule a Series of Parallel Lines at a Given Distance Apart. How to Rule Parallel Lines Dividing A Space Into Equal Parts. How to lay off an Angle. How to divide a circle into equal parts. How to transfer a given arc length to another line. How to rectify a given arc. How to draw a smooth curve through a series of given points. Tangencies. To draw a straight line tangent to a given circle at a point on its circumference. Circle Tangent to Another Circle. To draw a straight line tangent to a given circle from a point outside its circumference. How to Join two Straight Lines by a Tangent Arc. How to Join Two Circles by a Tangent Arc. How to Join a Straight Line and a Circle by a Tangent Arc. Curves. Ellipse. Parabola. Rectangular Hyperbola. The Family of " Cycloids." Construction of Cycloid. Construction of Epi- and Hypocycloids. Involute. Cissoid. Helix. Application of the Curves to Machine Design. CHAPTER 4 — FUNDAMENTALS OF ORTHOGRAPHIC PROJECTION-Page 28 Working drawings. Perspective. Orthographic Projection. Foreshortening. Number of orthographic views required. The principal views of Orthographic Projection. The thiee fundamental rules for identifying points. Distances and coordinates. Reading the Views. Visibility of lines. Rules of Visibility. Theorems of Orthographic Projection. Drafting procedure. Practical suggestions. CHAPTER 5 — AUXILIARY VIEWS

Page 46

Auxiliary views. Face perpendicular to Η and oblique to V. Face perpendicular to V and to H. CHAPTER 6 — SECTIONAL VIEWS

-

Symbols for section lining. Cutting plane. Full sections and half sections. Revolved sections. Broken out sections. Phantom sections. Thin sections. Exceptions. ν

Page 50

CHAPTER 7 — DIMENSIONING

Page 56

General. Dimension and Extension Lines. Dimension Figures. Other Suggestions. Finish Marks. Angles. Circles. Dimensions with Tolerances or Limits. Curves and Angles. Tapers. Standard Tapers. Special Tapers. Precision Work. Interchangeable manufacture. Allowance and interference. Classes of fits. Basic Rules for Dimensioning. CHAPTER 8 — FASTENERS

Page 67

Bolts, Nuts, and Screws. Screw-thread Terminology. To draw a screw thread. Screw thread forms. Thread Symbols, Regular. Thread Symbols, Simplified. Fits. American Standard Screw Thread Fits. American Standard Screw Threads. Pipe Threads. CHAPTER 9 — WORKING DRAWINGS Detail Drawings. Projections.

Page 79

Assembly Drawings.

Violations of

CHAPTER 10 — TECHNICAL FREEHAND SKETCHING Line Technique. objects.

Circles and Ellipses.

Page 81

Sketching from

CHAPTER 11 — INKING

Page 85

Preparation of tracing cloth. Care of the ruling pen and compass pen. Use of the ruling pen. Kinds of lines. Tangents and joints. Use of the scroll or French curve. Lettering. Dimensions. CHAPTER 12 - - PICTORIAL DRAWING

Page 8»

Axometric projection. Procedure for drawing an axometrlc view. Projection ratio and inclination. Procedure for drawing an axometrlc view. Isometric projection. Procedure for making an isometric drawing. Oblique projection. Types of oblique projection. Procedure for drawing an oblique view. Summary of Systems of Projection.

vi

Part 2

DESCRIPTIVE

GEOMETRY

CHAPTER 1 - PROJECTION OF A POINT The approach to Descriptive Geometry. Location of a Point in Space The Frame of Reference and Principal Planes.. Axes and Coordinates. The Principal Views. The Χ, Υ, Ζ laws of Orthographic Projection. Position of a Point in Space as shown by the Principal Views: Important Facts about the Location of Points. Lecture Notes.

Page 98

CHAPTER 2 - PROJECTION OF A LINE Definition. General and Special Cases of a Line. Lines in Relation to the Principal Planes. Point on Line. Piercing Points. Special cases of Perpendicular Lines. Parallel Lines. Intersecting Lines. Non-intersecting Lines. Lecture Notes. Exercises.

Page 105

CHAPTER 3 - PLANES AND THEIR TRACES The Trace of a·Plane. Relation of a Plane to the Principal Planes as Revealed by its Traces. Oblique, Perpendicular and Parallel Planes. To Assume a Line in a Given Plane. To Assume a Point In a Given Plane. Line Perpendicular to a Plane. To Pass a Plane through a Given Line so that the Plane will be Parallel to aihother Given Line. Intersection of Two . Planes. Lecture Notes. Exercises.

Page 121

CHAPTER 4 . AUXILIARY VIEV^S - REVOLUTION Projection upon an Auxiliary Plane. Revolution of a Point about a line True Length of a Line. Slope and Bearing of a Line True Angle between two Lines. Shortest Distance between Parallel Lines. Shortest distance between Skew Lines. General Case of Perpendicular Lines. Intersection of a Line and an Oblique Plane. Angle between Two Planes: True Size of a Plane Figure which is Perpendicular to Η and V. Counter Revolution of a Plane Figure. Lecture Notes. Exercises.

Page 138

CHAPTER 5 - INTERSECTIONS AND DEVELOPMENTS Surfaces Ruled, Plane, Single-Curved, Warped, Double-Curved Surfaces.

Page 170

Developments Development of a Plane Surface. Definitions. Development of a Cylinder; of a Right Cone; of an Oblique Cone; of a Transition Surface; of a Warped Cone; of a Sphere. Laps and Seams.

««

vn

Intersections (Elementary) Intersection of prisms: of two cylinders; of a cone and a cylinder; of a plane and a surface of revolution; Intersection of a cylinder and a cone; of a cylinder and a cylinder; a cone and a cone; a single and a double curved surface of revolution. Tangent planes To pass a Plane Tangent to a Cone at a Given Point on its Surface; to a Cone at a Given point Outside its Surface; to a Cone and Parallel to a Given Line; to a Cylinder through a Given Point on its Surface; to a Cylinder through a Given Point outside its Surface; to a Curved Surface of Revolution through a Given Point on its Surface. Intersections (Advanced) To determine the Line of Intersection of a Double-Curved Surface of Revolution with an Oblique Plane. Intersections of Cones and Cylinders. To draw the Curves of Intersection of two Cones. Visibility of the Curve of Intersection. Critical Points. To draw the Curve of Intersection of Two cylinders. To draw the Curve of Intersection of a Cone and Cylinder. To draw the Curve of Intersection between a Sphere and a Cone. Lecture Notes. Exercises. CHAPTER 6 - PERSPECTIVE Procedure for making a Perspective Drawing. Fundamental or Visual Ray Method. Theorems of Perspective Lines. Line or Vanishing Point Method. Perspective of Curved Lines. Lecture Notes.

Page 215

CHAPTER 7 - SHADES AND SHADOWS Directions of the Rays. Shadows cast on the Principal Planes. Lecture Notes.

Page 224

APPENDIX

I

Limiting Dimensions and Tolerances. Screw threads. Wrench-Head Bolts and Nuts. Screw thread Fits. Pipe threads. Decimal Equivalents of Fractions of an inch. Greek Alphabet.

viii

Page 23?

Part 1

ENGINEERING

DRAWING

1

CHAPTER 1 INSTRUMENTS AND MATERIALS Introduction.-

The term "mechanical drawing" is generally used

to denote drawing with mechanical aids and precision instruments as contrasted with "free-hand drawing." A more inclusive term "engineering drawing" is now becoming common, denoting both kinds of drawing -- in fact, all graphic processes which are useful to the engineer.

This includes

a kind of free-hand drawing which is indispensable to the engineering designer -- namely, preliminary sketches made to convey his ideas to the draftsman. Learning to draw.-

Many people are inclined to regard any kind

of drawing as an art, and to believe that skill can only be acquired by those who have a natural bent for it.

This view is quite erroneous so far as

"mechanical" drawing is concerned - i.e., drawing by means of mechanical aids and precision instruments.

For this, the principal requirements are

an interest in geometrical ideas, a liking for neat and precise work, and the will to organize one's own practice into a system. Description and use of drafting equipment.-

So much has been

written on instruments that it hardly seems necessary to describe them fully here, but those points will be reviewed which will help the beginner to avoid the most common faults in their use.

For more detailed informa-

tion, the reader is referred to the standard texts on this subject, and to the pamphlets published by the manufacturers of drafting equipment. Importance of pencil drawings.-

Pencil drawings are of first

importance for several r e a s o n s .

(1) They a r e the means by which the

d e s i g n e r ' s ideas f i r s t appear on paper.

(2) When ink t r a c i n g s a r e made

on transparent paper or cloth for blue-printing, their quality depends upon the accuracy of the original pencil work.

(3) T h e r e a r e many drawings

not requiring inking on account of the transient nature of the work. (4) Photographic methods of duplicating the original drawing a r e now developed by which many copies can be reproduced directly f r o m the pencil work, thus saving the expense of tracing.

For these r e a s o n s the

beginner should realize that in acquiring skill and accuracy in pencil work, he is learning the g r e a t e r part of mechanical drawing.

It is understood

that the instruction in this book will imply pencil work unless otherwise stated. Pencil - The hardness of lead is denoted by Η, 2H, etc., Η being soft, 9H very hard.

The 3H lead is for ruled lines; the 1H for lettering.

Sharpen both leads to a conical point.

(See Fig. 1).

With light p r e s s u r e and frequent polishing of the lead on the sandp a p e r , one can draw very fine lines with this type of point.

"A pencil

well sharpened is a drawing half done."

1. Expose about J?" inch of lead (L) and 8 rub it back and forth on sand paper block, holding pencil almost horizontal. 2. Turn pencil and repeat for several sides of lead. 3. Finish off to conical point by twisting pencil while rubbing. Test by looking at the point endwise in a strong light. So long as a bright spot is visible, the pencil r e q u i r e s f u r t h e r sharpening. The alphabet of lines used in making an ink drawing i s shown in Fig. 123, and s i m i l a r lines a r e used in pencil work.

3 Τ-Square and Triangle. The T-square is used for ruling horizontal lines, sliding up and down on left* edge of board. Vertical lines are ruled by means of triangles resting on upper edge of T-square. The blade should be straight. The lower edge is not used because it is not made parallel to the upper edge. (See Fig. 2).

Test the T-square for straightness of upper edge. (a) First fasten the paper to the drawing board. Use four tacks and place paper well toward upper left portion of board. (See Fig. 2).

(b) Draw a line ab (with 3H pencil along upper edge of T-square in regular position) and a line ?' - b • with T-square upside down. (See Fig. 3). Any bulge or depression is then detected by comparing these two lines.

FIG. 3

Test the triangles for accuracy of right angles. Draw and compare lines a and a (the latter with triangle reversed). See Fig. 4. If angle is exactly 90° a and a ' will be parallel.

* If left-handed, substitute "right" for "left."

FIG. 4

Scales.-

A "scale" is a precise measuring stick for setting off

given distances on the drawing. _It should not be used as a guide for ruling lines.

There are two kinds according to the subdivisions of one

inch:- Architect's, divided into 16ths, 24ths, 48ths, etc.; (Fig. 5). Engineer's, divided into lOths, 20ths, 30ths, etc. (Fig. 6). The "scale of a drawing" is a statement of the length on the drawing which represents a given distance on the object.

For example,

if a drawing is marked SCALE: A" » 1· it was constructed so that 1 inch on the paper stands for one foot on the object, thus implying a 4 ^ ^ ratio of 1 to 48.

Architects call this a "quarter scale" drawing. To

expedite the work of drawing at this scale a s e r i e s of quarter-inch interval» a r e found on the Architect's scale, one at the end being subdivided into 12 parts to denote inches. On the same stick other scales are found, such as ^ " = 1", which implies the ratio of 1 to 16. In this 4 case a common carpenter's rule could be used in an emergency since 1 inch represents one inch. 16 /TY^v^v y y ν y ν 6

V—A

\

ν ν V y?

? \

a

y

50 FIG. 5.

ARCHITECTS' SCALE On the Engineer's scale the marks 20, 30, etc., denote scales which a surveyor uses for long distances: Note:

for example 1" * 50'.

When constructing an object from a drawing, if there is any

discrepancy between the dimension stated in figures on the drawing and the corresponding scaled length on the paper, the f o r m e r should always be followed.

^

5

engineer's scale Dividers.- The dividers are used principally for subdividing a line into a given number of equal parts and for transferring a given distance from one part of the drawing to another.

Two kinds are shown in

Fig. 7 - (Right) "Hair spring" dividers for long distances, (Left) "Bow" dividers for short distances. The hair-spring type is so named because It has a spring In one leg, controlled by a screw so that one of the points can be adjusted a little as suggested by dotted lines in Fig. 7 (Right).

This device allows

very fine adjustment after the dividers have been opened to the approximate distance. The bow dividers derive their name from the fact that both legs form a continuous spring somewhat resembling a bow. likewise obtained by a screw.

The fine adjustment is

When once opened to a given distance, the

points are held at that distance by the screw. In both cases the two metal points should be sharp and not bent. Test this by seeing if the two points come absolutely together when the dividers are closed. Compasses.-

The compasses shown in Fig. 8 are for drawing

circles of medium radii, say 4 to 4 inches.

See if the joints are tight at

the handle and at the "knees " (E and F) so as to avoid any change of radius while the circle is being drawn.

Also see that the threads on

adjusting screws are in good condition, and that points C (Center) and Ρ (Pencil) come together when closed.

When drawing, pull compass around gently. Do not push or bear down. When compasses are open wide, crook the knee (E) to avoid boring hole at (C) Also crook (F)to keep point (P) level with (C).

7 For very large circles insert the "lengthening bar" and hold the compasses with both hands.

(See Fig. 9c).

For very small circles use

the "bow" compasses which are similar to the bow dividers previously described.

(See Fig. 9b).

Sharpening Compass Leads. furnished Is usually too soft.

In a new set of Instruments the lead

Substitute a piece of harder lead fastening

It with the thumb-screw so as to expose not less than 1 inch of lead. 8 Then sharpen the lead by rubbing one side of the lead on the sandpaper block, i.e., the side which is farther from the center point C. The rubbing should continue until the polished surface extends way across the lead and takes the form of a long thin ellipse.

If this surface is fre-

quently polished on the sandpaper, the point will give very sharp, light circles.

(See Fig. 9a).

The directions given here are purposely made as brief as possible *with the expectation that refinements will be demonstrated by the instructor in person.

The use of the instruments Is taken up In Chapter 3.

8

FIG. U

9 CHAPTER 2 LETTERING ON ENGINEERING DRAWINGS Uses of Lettering.-

Lettering and numerals are necessary on

engineering drawings to supplement the information given by the drawing itself.

They appear principally in the following: (a) (b)

Dimensions. stating the size of the objects represented. Part Lists or Bills of Material, are tabulations of all the parts

of the object represented.

Such a list gives the number of duplicate parts,

the material to be used, and usually includes a reference number for identification of that part in the shop and drawing office (c)

Descriptive notes, stating method of production, and other facts.

(d)

Title.

This gives various matters of record for filing purposes.

(Scale, name of draftsman, date, etc.)

For an example of lettering seethe

shop drawings shown at reduced size on opposite page. (Courtesy of General Electric Co. and General Motors) Type of lettering in common use.

The function of lettering is to give

information necessary to prevent error and misunderstanding.

The im-

portant requirements are (1) legibility, (2) ease and rapidity of execution, (3) pleasing appearance which will make it less tedious to read. These qualities are obtained in the simple Gothic letter, now almost universally used on Engineering drawings.

This style was developed by

omitting the curls and loops of the Gothic used in metal type.. The letters are drawn free-hand except in occasional formal titles of large maps and plans. The alphabet for this style is made in two forms, the vertical system and the inclined system.

While the former is often used for capital letters

in formal titles on large maps and drawings, the inclined system is generally used on engineering drawings.

(See Fig. 12a and 12b).

10 BASIC STRUCTURE OF THE VERTICAL SYSTEM OF LETTERING

® B: C D E\ -ΘΞ

J; K LM N 'QPQR

[xmummmu: ;.·'

n r a y c cJ e r g t i ιj

I

-

/

^ r v m

,) i«r Fig. 12 A. Box.

Each capital letter is designed with a definite relation to a

rectangular box or frame.

The beginner is to visualize the relationships

without drawing the boxes. Strokes.

Success requires drawing the letters with a definite order

of successive strokes, starting and stopping each stroke at definite points as indicated by the numbered arrows. Height and width of letters. of the letters.

The height is determined by the function

The width is a matter of good ratio to the height adopted.

Good appearance requires slight variations of widths.

π BASIC STRUCTURE OF THE INCLINED SYSTEM OF LETTERING

Fig. 12 B. Capital letters and numbers.

The structure differs from the vertical

system only in the fact that each box is a rhombus with sides inclined from the vertical by a small angle, specified by 2 horizontal on 5 vertical. Lower case letters. above.

These conform to the same slope as stated

In the case of V, X and W the center line of the letter has the

given slope rather than the strokes of the letter itself. Note that the stems of such lower case letters as b, d, f, etc. extend 50% above the round body of the letter. as

y, g> P> and q extend 50% below.

Similarly the tails of such letters

12 CHAPTER 3. DRAWING BOARD GEOMETRY Ruling straight lines.

(Fig. 13).

(a) Lines are to be fine and uniform, i.e., not too faint at one place and too heavy at another. (b) Left hand holds T-square firmly

supported here.

against edge of board. FIG. 13

(c) Hold pencil nearly vertical with conical point against edge of T-square. (d) Support the weight of hand on the last two fingers only, letting them slide along on the finger nails.

See Fig. 13.

(e) Whenever possible draw horizontal lines from left to right and vertical lines from bottom to top of sheet.

FIG. 14

Drawing parallel lines. To draw line Β parallel to A in Fig. 14. Fit a triangle to the given line A and bring a straight edge (T-square or other triangle) up to the triangle. Then slide the triangle (along the straight edge as a guide) to required position (shown dotted) and draw the parallel line B.

FIG. 15

Angles of 15°, 75°, etc. Both triangles combined give angles of 15°, 75°, 105°, etc.

(See Fig. 15).

Drawing perpendicular lines. To draw line C perpendicular to A in Fig. 16. C Fit the hypotenuse of a triangle against the given line A. and bring straight edge up to th

^mm

Triangle. Then turn triangle to the dotted position and draw the perpendicular line C. Drawing circles. (a) Correct method of holding compasses is suggested by Fig. 17. Show such dotted lines and sections as will aid the person who is to "read" the drawing, (c) Do not erase the original lines of the blockout.

FREE HAND SKETCHING. FIG. 120

STAGE m - DIMENSION LINES (a) Consult pages 56-66, 79 and 80 for general Ideas and conventions for dimensioning. STAGE IV - DIMENSION FIGURES AND LETTERING (a) Insert figures in dimension lines. (b) Denote by (V) any surfaces which are to be "finished" (i.e., machined after coming from the foundry or forge shop). (c) Add necessary notes and title, using letters of standard height.

η

A correctly ruled line

Too much pressure against T-square

Pen slanting toward T-squarp

*ι FIG. 121

Pen slanting away from T-square.

FIG. 122

85

CHAPTER 11 INKING Ink drawings are made either by (1) inking directly on a pencil drawing or by (2) tracing on transparent paper or glazed cloth.

Unless otherwise

specified the instructions will refer to the process of tracing on cloth. Preparation of tracing cloth.

The cloth has a dull and a glossy side.

The inking should be done on the dull side.

This surface should be prepared

for inking by rubbing it with chalk (or a special powder called " pounce.") Be sure to rub off all the powder with a cloth before applying inkCare of the ruling pen and compass pen. (a) The ruling pen has two points called ' nibs;' which should be oval shaped and not too sharp lest they cut the tracing cloth. (See Fig. 121 A). (b) Ink is filled into the space between the nibs by means of a quill which is a part of the stopper of the ink bottle.

Do not fill the

space with too much ink and be sure that no ink is left on the outside of the nibs. (c) Clean the inside of the nibs occasionally during use and at the end of the day's wo.k because the ink is corrosive on steel. (d) Keep the tracing cloth f r e e from dust and dirt while inking. Use of the ruling pen.

As in the pencil work, horizontal lines are ruled

with T-square held firmly against left edge of the drawing board; verticals with triangle resting on upper edge of the T-square.

There is one important

difference between inking and pencil work - i.e., the T-square or triangle is placed at a slight distance back from the line to be inked. may touch the T-square or triangle and make a blot.

Otherwise the nibs

(See Fig. 122).

86

Kinds of lines.

Three weights of lines are suggested which will be

designated by L (light) ,· Η (heavy) and Μ (medium). Dotted lines are used to denote portions of the object which are hidden by other portions.

The term "dotted/' though generally used, is a mis-

nomer because such lines are really a series of short dashes.

To ensure

uniform length of the dashes and spacing (also to avoid fatigue), make the dashes in groups, say four at a time, pause and draw four more, repeat until the line is complete«

With experience, increase to five or six at a time.

Outline of Paris

T h e outline should be the outstanding feature and the thickness may v a r y to suit size of drawing

Section lines

Spaced evenly to make a shaded effect

Hidden lines Broken line, made up of long and short dashes, alternately spaced

Center lines Dimension and Extension lines

LIGHT -

3?

-

Lines unbroken, except at dimensions Broken line made up of one long and two short dashes, alternately spaced

Cutting Plane line

Free hand line f o r short breaks. Break lines

Ruled line and free hand zigzag f o r long breaks

Adjacent Parts and Alternate Positions

Broken line made up of dashes

long

Indication of repeated detail

FIG. 123 Tangents and joints.

In cases where straight lines are tangent to

circles or where any two lines must be joined smoothly, special precautions must be taken to avoid a hump or other evidence of discontinuity. following suggestions are given:-

The

87 (a) Before starting the ink work, mark with pencil on the tracing cloth the exact point of tangency or junction.

(Your knowledge of drawing board

geometry should be helpful in doing this). (b) Then be sure to draw the circle or arc before the straight line, because the latter can be brought up to join the former by slightly shifting the inclination of the ruling pen as It approaches the tangent or junction mark. A

A

FIG. 124

Use of Uie scroll or French curve.

As in the case of the T-square and

triangle, the scroll should be placed at a slight distance back from the curve to be traced in order to avoid a blot. Lettering. fairly coarse pen.

This work is done free hand with a common penholder and a Many prefer a ball pointed pen.

Before doing any lettering in ink, always draw horizontal guide lines to preserve uniform alignment and height of letters. pencil directly on the tracing cloth.

These lines are drawn in

Even when such lines have been drawn

on the original sheet, they are generally not a sufficiently clear guide.

88 Dimensions. pen.

Dimension and leader lines are ruled lightly with the ruling

The arrow heads are then added free hand, using the same pen as for

lettering.

Finally the numbers are inserted free hand, ordinarily without the

aid of any pencil guide lines. Cleaning the tracing cloth.

When a tracing on cloth is completed, all

pencil marks can be removed by wiping with a cloth slightly moistened with carbon-tetrachloride.

Without guide lines Wrong

FIG. 125

89

CHAPTER 12 PICTORIAL Axometric projection.

DRAWING

An axometric view of an object is an ortho-

graphic projection of it upon a plane, the object being intentionally placed so that its principal faces or axes are oblique to that plane.

This causes the

projection to reveal more than one face of the object in a single view. (See Fig. 126B) Since the axometric view is an orthographic projection, it follows that parallel lines of the object will be represented by parallel lines in the view. In this respect, it differs from a perspective view. Procedure for drawing an axometric view. object be a cube with edges 2" long.

(First Method).

Let the

The desired axometric view is ob-

tained by setting up the cube in oblique position relative to the V plane and making the usual orthographic projection of it upon that plane. (1) Draw Top and Front Views of cube in first position with edges parallel to principal axes as shown by light lines of Fig. 127. 3h

4h

ο I ι ύζ

I?

±s FIG. 126B

FIG. 126A

5s

STANDARD VIEWS

aJ

AXOMETRIC VIEW

90 (2) Rotate the cube to a second position, the axis of rotation being perpendicular to the Η plane and passing through the corner 1.

Since each corner

moves on a circle which is parallel to H, its V projection must move in a horizontal line.

Hence, locate l v 2 V , etc., at same levels as in first position

and complete the Front View as shown by the heavy lines of Fig. 127. (3) Draw Right Side View of Fig. 127 in the usual way. (4) Rotate the cube to a third position, as shown in the Right Side View of Fig. 128.

The axis of rotation in this case is perpendicular to the S plane

and passes through the corner 1.

Since each corner moves on a circle which

is parallel to S,.lts V projection must move In a vertical line. Hence, in Fig. 128 locate l v a n d 2 V

in same vertical lines with l v a n d 2 V of Fig. 127 and join them

by a line to show, the edge x*.

Similarly,locate the V projections of the other

corners and complete the Front View of Fig. 128 . etric view. TOP VIEW

This is the required axom-

91

FIG. 128 SHOWING ROTATION FROM 2nd TO 3rd POSITION Let φ = any angle (45° in this case)

RIGHT SIDE VIEW Projection ratio and inclination.

Let χ y ζ be the true lengths of

the edges of the cube or rectangular prism, and let x' y' ζ ' be the foreshortened lengths of their projections.

Then the ratio x'/x is the "pro-

jection ratio" for the edge χ and the angle ( cc ) between x ' and a horizontal line on tHe paper, is its " inclination."

This angle should not be

confused with the true angle in space between the line and the horizontal plane). In the case of Fig. 127 and 128 the values are a s follows: for -θ- = 60o and

x'/x = 1.58 /2

.79 and CC = 51°

z ' / z = 1.88 /2

.94 and β - 22°

φ =45°

y'/y . 1.42 /2

.71 and inclination = 9CP

These values are valid only for the particular position of the object relative to the plane a s given by angle Φ and φ.

Other positions would give

an axometric view with different projection ratios and inclinations, showing some faces with less foreshortening and other faces with more. The following table gives four sets of consistent values which are multiples of 15°.

This is done so that lines parallel to x' and ζ ' can be readily

drawn with the regular draftsman's triangles resting on the T-square.

92

β

No. of set

0c deg.

deg.

x'/x

Projection z'/z

I

15

45

.92

.65

.86

η

45

15

.65

.92

.86

m

15

60

.96

.73

.73

IV

60

15

.73

.96

.73

V

30

30

.81

.81

.81

Procedure for drawing an axometric view.

Ratios y'/y

(Second Method).

Suppose it is

desired to draw an axometric view of the same cube as used for illustration of the first method.

The edges are two inches long.

(1) Choose consistent values of projection ratios and inclinations, as found by previous graphical work. based on Λ = 15° and β (2)

Let us take Set No. I from the table,

= 45°.

From point 1 in Fig. 129 draw line χ ' at angle OC =. 15° and on

it locate point 2 by making distance x' =. 2"x.92 = 1.84".

Draw z' at angle

β = 45° and locate point 4 by making ζ · = 2 "x.65 >1.30".

Then locate

5 by making y' - 2 " x.86 » 1.72 " vertical. (3)

Locate points 3, 6 and 8 by similar steps and complete the axo-

metric view bf Fig. 129.

4X x*/x = .92 FIG. 129

y/y

= .86

z ' / z a .65 AXOMETRIC For a more elaborate example, see Figs.l30AandlSOB^he latter having the same projection ratios and inclinations as in Fig. 129.

In such cases use

the same procedure, locating the points by coordinates measured along axes which are parallel to x* y 1 and z*.

93

(1) From point 1 in Fig.l30Bdraw axes X ' , Y 1 , and Z '

at given

inclinations. (2) On these locate points 1, 3, 4, 5, 6, 8 using coordinates derived from the given projection ratios.

These coordinates can be computed by slide rule

or found from a diagram of projection scales constructed as shown in Fig. 130B (3) Similarly, locate points 9, 10, 11 and 12 and complete the axometric view of the block without the circular hole. (4) Locate center (c) of the circle in the top face by similar steps. (5) Locate (p) by making cf = .92 χ (c h f h ) and fp = .65 χ ( f V 1 ) , the lengths in parenthesis being scaled off from Fig,130A.

Similarly, locate other points

and join them by a scroll to make the top ellipse (w ρ η e s). (6) Locate (t) by making pt = .86 χ .

Locate several other points

at same distance below the top ellipse and draw the lower ellipse. AXOMETRIC 4

.?"

iCVd" I

oa = ob = oc - 1" oa' = .92 (oa)

gv

τρ -•r

ob' = .65 (ob)

I

oc' = .86 (oc) FIG. 130 A

DATA

FIG. 130B

94 Isometric projection.

Again using a cube f o r illustration, an

isometric view is a special kind of axometric projection in which the three rectangular edges of the cube are foreshortened in the same ratio. For this condition the edges x ' y' z ' will be 120° apart in the isometric view, causing oC and ß to be 30° each.

The projection ratio

f o r each of these edges will be .816 as can be easily proved.

See the

light lines of Fig. 131. Procedure for making an isometric drawing.

The construction of an

isometric projection of a cube could be carried out as described for the case of the axometric -view in a previous paragraph using the equal projection ratios of .816.

In practice, however, the distances are set off using a

ratio of unity to avoid the labor of computation.

This short cut produces

a drawing as shown by the heavy lines of Fig. 131 which is similar to, but larger than, the real Isometric projection.

The larger view Is some-

times called an " Isometric drawing " to distinguish it from the real

95

In Fig. 132 is shown an isometric drawing of the same block as in Figs. 130A and 130B. The construction of it should proceed by steps similar to those of Fig.l30Bwith the use of three projection scales which are all unity. Note: An approximate method of drawing the ellipse n, e, s, w is shown in Fig. 133 where four circular arcs take the place of an elliptical curve through plotted points. Oblique Projection. An oblique projection or view of an object is a projection of it made upon a plane by means of a system of parallel rays, which are oblique to that plane, not perpendicular, as in orthographic projections.

As in the case of an axometric view, the purpose is to reveal

more than one face of the object in a single view. For example, in Fig. 134 the cube is set up so that its front face (1 2 β 5) is parallel to the plane of projection (V). This causes the projection of that f a c e d ' itself.

2'

6'

5') to have the same shape ahd size as the face

But the edge (1,4), being perp. to the plane V, has a projection

(1', 4') whose length (z'> is shorter than its true length (z), the projection ratio z'/z depending on the assumed direction of the rays in space. inclination ( ß ) also depends on the same assumption.

The

In this or any type

of oblique projection, parallel lines of the object have the same projection ratio and same inclination. Types of oblique projection.

There are two types to which names

have been applied: 'Cabinet projection" - Inclination ( 0 ) = any convenient angle, generally 45°.

Projection ratio z'/z =• .5.

"Cavalier projection" = Inclination ( @ ) = any convenient angle, generally 45°.

Projection ratio z'/z = unity.

96

Procedure for drawing an oblique view. * Suppose an oblique view i desired of a cube with edges 2" long, one of its square faces having a concentric circle of 3 / 4 " radius. (See Fig. 134) (1) Choose the "cavalier"

type with (3 = 45° and ζ'/ζ = unity.

(2) Draw the face 2 " square with circle on it true size. (3) Draw 45° lines from 1' 2 ' and 6 ' and locate 4' 3' and 7' at distance Ζ' - 2." Join by lines to complete the view.

FIG. 134

97

SUMMARY OF SYSTEMS OF PROJECTION /— WORKING DRAWINGS 3 planes at 90°. Object set up square with planes. ORTHOGRAPHIC PROTECTION — Parallel rays


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