The Geometry Toolbox for Graphics and Modeling

305 Pages·1998·69.668 MB·\305

by  Gerald Farin (Author), Dianne Hansford (Author)

The Geometry Toolbox
for Graphics and Modeling

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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