Mathematics for Computer Graphics Applications

✍ Scribed by Michael Mortenson

• Publisher: Industrial Press, Inc.
• Year: 1999
• Tongue: English
• Leaves: 369
• Edition: 2
• Category: Library

✦ Synopsis

Includes new chapters on symmetry, limit and continuity, constructive solid geometry, and the Bezier curve. Provides many new figures and exercises. Contains an annotated suggested reading list with exercises and answers in each chapter. Appeals to both academics and professionals. Offers a new solutions manual for instructors.

✦ Table of Contents

Cover......Page 1
Title page......Page 2
Copyright Page......Page 3
Dedication......Page 4
Preface......Page 5
TABLE OF CONTENTS......Page 10
1.1 Introduction......Page 15
1.2 Hypernumbers......Page 16
1.3 Geometric Interpretation......Page 18
1.4 Vector Properties......Page 22
1.6 Vector Addition......Page 24
1.7 Scalar and Vector Products......Page 25
1.8 Elements of Vector Geometry......Page 28
1.9 Linear Vector Spaces......Page 33
1.10 Linear Independence and Dependence......Page 34
1.11 Basis Vectors and Coordinate Systems......Page 35
1.12 A Short History......Page 36
2.1 Definition of a Matrix......Page 42
2.2 Special Matrices......Page 43
2.4 Matrix Arithmetic......Page 45
2.5 Partitioning a Matrix......Page 48
2.6 Determinants......Page 50
2.7 Matrix Inversion......Page 51
2.9 Eigenvalues and Eigenvectors......Page 53
2.10 Similarity Transformation......Page 55
2.12 Diagonalization of a Matrix......Page 56
3.1 The Geometries of Transformations......Page 61
3.2 Linear Transformations......Page 66
3.3 Translation......Page 68
3.4 Rotations in the Plane......Page 70
3.5 Rotations in Space......Page 74
3.6 Reflection......Page 78
3.7 Homogeneous Coordinates......Page 83
4.1 Introduction......Page 87
4.2 Groups......Page 91
4.3 Symmetry Groups......Page 92
4.4 Ornamental Groups......Page 94
4.5 The Crystallographic Restriction......Page 97
4.6 Plane Lattices......Page 98
4.7 Tilings......Page 99
4.8 Polyhedral Symmetry......Page 100
5.1 The Greek Method of Exhaustion......Page 104
5.2 Sequences and Series......Page 107
5.3 Functions......Page 112
5.4 Limit of a Function......Page 117
5.5 Limit Theorems......Page 121
5.6 Limit and the Definite Integral......Page 125
5.7 Tangent to a Curve......Page 128
5.8 Rate of Change......Page 129
5.9 Intervals......Page 133
5.10 Continuity......Page 135
5.11 Continuous Functions......Page 136
6.1 Topological Equivalence......Page 143
6.2 Topology of a Closed Path......Page 146
6.3 Piecewise Flat Surfaces......Page 151
6.4 Closed Curved Surfaces......Page 155
6.5 Euler Operators......Page 156
6.6 Orientation......Page 158
6.7 Curvature......Page 161
6.8 Intersections......Page 163
7.1 Definition......Page 166
7.2 Set Theory......Page 167
7.3 Functions Revisited......Page 168
7.4 Halfspacesin the Plane......Page 170
7.5 Halfspaces in Three Dimensions......Page 173
7.6 Combining Halfspaces......Page 174
8.1 Definition......Page 179
8.2 Arrays of Points......Page 181
8.3 Absolute and Relative Points......Page 182
8.4 Displaying Points......Page 183
8.5 Translating and Rotating Points......Page 184
9.1 Lines in the Plane......Page 188
9.2 Lines in Space......Page 191
9.3 Computing Points on a Line......Page 194
9.4 Point and Line Relationships......Page 195
9.5 Intersection of Lines......Page 197
9.6 Translating and Rotating Lines......Page 198
10.1 Algebraic Definition......Page 201
10.2 Normal Form......Page 203
10.4 Vector Equation of a Plane......Page 204
10.5 Point and Plane Relationships......Page 205
10.6 Plane Intersections......Page 206
11.1 Definitions......Page 209
11.2 Properties......Page 211
11.3 The Convex Hull of a Polygon......Page 214
11.5 Symmetry of Polygons Revisited......Page 215
11.6 Containment......Page 216
12.1 Definitions......Page 219
12.2 The Regular Polyhedra......Page 221
12.3 Semiregular Polyhedra......Page 223
12.4 Dual Polyhedra......Page 225
12.5 Star Polyhedra......Page 226
12.6 Nets......Page 227
12.8 Euler's Formula for Simple Polyhedra......Page 228
12.9 Euler's Formula for Nonsimple Polyhedra......Page 231
12.10 Five and Only Five Regular Polyhedra: The Proof......Page 232
12.11 The Connectivity Matrix......Page 233
12.12 Halfspace Representation of Polyhedra......Page 235
12.14 Maps of Polyhedra......Page 237
13.1 Set Theory Revisited......Page 240
13.2 Boolean Algebra......Page 241
13.3 Halfspaces Revisited......Page 243
13.4 Binary Trees......Page 244
13.5 Solids......Page 247
13.6 Boolean Operators......Page 249
13.7 Boolean Models......Page 254
14.1 Parametric Equations of a Curve......Page 258
14.2 Plane Curves......Page 259
14.3 Space Curves......Page 263
14.4 The Tangent Vector......Page 266
14.5 Blending Functions......Page 270
14.6 Approximating a Conic Curve......Page 272
14.7 Reparameterization......Page 273
14.8 Continuity and Composite Curves......Page 274
15.1 A Geometric Construction......Page 278
15.2 An Algebraic Definition......Page 281
15.3 Control Points......Page 284
15.4 Degree Elevation......Page 286
15.5 Truncation......Page 287
15.6 Composite Bezier Curves......Page 289
16.1 Planes......Page 291
16.2 Cylindrical Surfaces......Page 292
16.3 The Bicubic Surface......Page 293
16.4 The Bezier Surface......Page 297
16.5 Surface Normal......Page 298
17.1 Coordinate Systems......Page 300
17.2 Window and Viewport......Page 307
17.3 Line Clipping......Page 310
17.5 Displaying Geometric Elements......Page 313
17.6 Visibility......Page 322
18. Display and Scene Transformations......Page 327
18.1 Orthographic Projection......Page 328
18.2 Perspective Projection......Page 332
18.3 Orbit......Page 335
18.4 Pan......Page 337
18.5 Aim......Page 338
Bibliography......Page 340
Answers to Selected Exercises......Page 343
Index......Page 363