Table Of ContentThe Geometry Toolbox
for Graphics and Modeling
Editorial, Sales, and Customer Service Office
A K Peters, Ltd.
63 South Avenue
Natick. MA 01760
http:llwww.akpeters.com
Copyright C 1998 by A K Peters, Ltd.
All rights reserved. No part of the material protected by this copyright notice may be
reproduced or utilized In any form, electronic or mechanical, Including photocopying,
recording, or by any Information storage and retrieval system, without written per·
miSsion from the copyright owner.
Ubnlly of Congrna cataloglng-tn-Publlc:Mion Data
Farin, Gerald E.
The geometry toolbox for graphics and modeling I Gerald Farin,
Dianne Hansford.
p. em.
Includes bibliographical references and index.
ISBN 1·56881·074-1
1. Geometry-Study and teaching. 2. Computer graphics.
I. Hansford, Dianne. II. Title
QA462.F26 1997
516.3'5-dc21 97-41259
CIP
Printed In the United States of America
~~ 0099~ 10987654321
To our advisors
R.E. Barnhill and W. Boehm
Contents
Preface xiii
Descartes' Discovery 1 Chapter 1
1.1 Local and Global Coordinates: 2D . . . . . . . • . 2
1.2 Going from Global to Local . • . . . . • . . . . • . • . • 6
1.3 Local and Global Coordinates: 3D . . . . . . . . . . . . . 7
1.4 Stepping Outside the Box . . . . . . . . . . . . . . . . . 9
1.5 Creating Coordinates . • . . . . . . . . . . . . . . . . . . 9
1.6 Exercises . . . . . . . . . . . . . . . . • . . . . . 11
Here and There: Points and Vectors in 20 13 Chapter 2
2.1 Points and Vectors . . . . . . . . . . . . . . . . . 14
2.2 What's the Difference? . . . . . . . . . . . . . . . 15
23 Vector Fields . • . . . . . . . . . . . . . . . . . . . . . . 17
24 Combining Points . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Length of a Vector . . . . . . . . . . . . . . . . . . . • . 20
2.6 Independence . . . . . . . . . . . . . . . . . . . . . . . . 23
2.7 Dot Product . . . . . . . . . . . . . • . . . . . . . . . . . 24
2.8 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . 29
Lining Up - 20 lines 31 Chapter3
3.1 Defining a Line . . . . . . . . . . . . . . . . . . . 32
3.2 Parametric Equation of a Line . . . • . . . . . . . 33
3.3 Implicit Equation of a Line . . . . . . . . . . . . . . . . . 35
3.4 Explicit Equation of a Line. . . . . . . . . . . . . . . . . 37
3.5 Converting between Parametric and Implicit Equations . . 38
3.6 Distance of a Point to a Line . . . . . . . . . . . . . . . . 40
3.7 The Foot of a Point . . . . . . . . . • . . . . . . . . . . . 44
3.8 A Meeting Place - Computing Intersections . . . . . . . 45
3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . SO
vii
viii
Chapter 4 linear Maps in 20 53
4.1 Skew Target Boxes . . . . . . . . . . . . . . . . . . . . . 53
4.2 The Matrix Form . . . . . . • . . . . . . . . . . . . . . . 55
4.3 More about Matrices . . . . . . . . . . . . . . . . . . . • 56
4.4 Scalings • . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.5 Reflections • . . . . . . . . . . . . . . . . . • . . • . . . 61
4.6 Rotations . . . • . . . . . . . . . . . . . • . . . . . . . . 63
4.7 Shears . . . . . . . . . . . . . . . . . . . . • . . . . . . . 64
4.8 Projections • . . . . . . . . . . . . . . . . . . . . . . . . 66
4.9 Areas and Unear Maps: Determinants . . . . . . . . . . . 68
4.10 Composing Linear Maps . . . . . . . . . . . . . . . . . . 71
4.11 More on Matrix Multiplication . . . . . . . . . . . . . . . 74
4.12 Working with Matrices . . . . . . . . . . . . . . . . . • . 76
4.13 Exercises . . . • . . . . . . . . . . . . . . • . . . . . . . 77
Chapter 5 2 x 2 linear Systems 79
5.1 Coordinate Transformations . . . . . . . . . . . . . . • • 79
5.2 The Matrix Form . . . . . . . . . . . . . • • . . • . . • . 81
5.3 A Direct Approach: Cramer's Rule . . . . . . . . • . . . 82
5.4 Gauss Elimination . . . . . • . . • . • . . . . • . • . . . 82
S.S Undoing Maps: Inverse Matrices . . • . . . . . . • . . • 85
5.6 Unsolvable Systems . . . . . . • . . . . . . • . . . . . . . 90
5.7 Underdetennined Systems . . . . . . • . • . . . . . . . . 91
5.8 Homogeneous Systems • . . . . . . . . . . . . . . . . . . 91
5.9 Numerical Strategies: Pivoting . • . . . . . . . • . . . . . 92
5.10 Defining a Map . . . . . . . . . . . . . . . . . • . . . . . 94
5.11 Exercises . . . . . . . • . . . . . . . . . • . . . . . . . . 94
Chapter 6 Moving Things Around: Affine Maps 97
6.1 Affine and Linear Maps . . . . . . . . . • . . • . . . . . 97
6.2 Translations . . . . . . . . . . . • • • . . . . . . . . . • . 99
6.3 More General Affine Maps . . . . . . . • • . . . . . . . . 100
6.4 Mapping Triangles to Triangles . . • . . . . . . . . . . . 101
6.5 Composing Affine Maps . . . . • . . . . . . • . . . . . . 103
6.6 Exercises . . . . . . . . . . . . • . . . . • . . . • . . . . 107
Chapter 7 Eigen Things 109
7. I Fixed Directions . . . . . • • . . . . . . • . . . . . . • . 110
7.2 Eigenvalues . . . . . . . . . . . . . • . . . • . • . . . • . 111
7.3 Eigenvectors . . • . . . . . . . . . . . . . • . . . . . . . . 112
7.4 Special Cases . . . . . . . . . . . • . • . . . . . • . . . • 114
7.5 The Geometry of Symmetric Matrices . . . . . . . . . . . 116
ix
7.6 Repeating Maps. . . . . . . . . . . . . . . . . . . . . . . 118
7.7 The Condition of a Map . . . . . . . • . . . . . . . . . . 120
7.8 Higher Dimensional Eigen Things . . . . . . . . . . . . . 121
7. 9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Breaking It Up: Triangles 125 Chapter 8
8.1 Barycentric Coordinates . . . . . . . . . . . . . . . . . . 126
8.2 Affine Invariance . . . . . . . . . . . . . . . . . . . . . . 128
8.3 Some Special Points . . . . . . . . . . . . . • . . . . . . 128
8.4 2D Triangulations . . . . . . . . . . . . . . . . . . . . . . 130
8.5 A Data Structure . . . . . . . . . . . • . . . . . . . . . . 131
8.6 Point Location . . . . . . . . . . . . . . . . . . . . . . . 132
8.7 3D Triangulations . . . . . . . . . . . . . . . . ..... . 133
8.8 Exercises . . . . . . . . . . . . . . . . . . • . . . . . . . 135
Conics 137 Chapter 9
9.1 The General Conic . . . .. .. . .. . . . .. .. . . . . 138
9.2 Analyzing Conics . . . . . . . . . . . . . • . . . . . . . . 141
9.3 The Position of a Conic . . . . . . • . . . . . . . . . . . 142
9.4 Exercises 144
30 Geometry 147 Chapter 10
10.1 From 2D to 3D . . . . . . . . . . . . . . . . . . . . . . . 147
10.2 Cross Product . . . . . . . . . . . . . . . . . . . . . . . . 149
10.3 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
10.4 Planes . . • . . . . . . . . . . . . . . . . . . . . . . . . • 154
I 0.5 Scalar Triple Product . . . . . . . . . • . . . . . . . . . . 15 9
10.6 Exercises 160
Interactions in 30 161 Chapter 11
11.1 Distance of Point and Plane 162
11.2 The Distance between Two Lines . . . . • . . . . . . . . 163
11.3 Lines and Planes: Intersections . . . . . . . . . . . . . . . 164
11.4 Intersecting a Triangle and a Line . . . . . . . . . . . . . 166
11.5 Lines and Planes: Reflections . . . . . . . . . . . . . . . 167
11.6 Intersecting Three Planes . . . . . • . . . . . . . . . . . . 167
11.7 Intersecting Two Planes. . . . . . . . . . . . . . . . . . . 169
11.8 Exercises . . . . . . . . . . . . . . . . . . • . . . • . . . 169
