3D Math Primer For Graphics And Game Development

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Overview

3D Math Primer for Graphics and Game Development covers fundamental 3D math concepts that are especially useful for computer game developers and programmers. The authors discuss the mathematical theory in detail and then provide the geometric interpretation necessary to make 3D math intuitive. Working C++ classes illustrate how to put the techniques into practice, and exercises at the end of each chapter help reinforce the concepts.

This book explains basic concepts such as vectors, coordinate spaces, matrices, transformations, Euler angles, homogenous coordinates, geometric primitives, intersection tests, and triangle meshes. It discusses orientation in 3D, including thorough coverage of quaternions and a comparison of the advantages and disadvantages of different representation techniques. The text describes working C++ classes for mathematical and geometric entities and several different matrix classes, each tailored to specific geometric tasks. Also included are complete derivations for all the primitive transformation matrices.

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Product Details

  • ISBN-13: 9781556229114
  • Publisher: Jones & Bartlett Learning
  • Publication date: 6/21/2002
  • Edition description: 1E
  • Pages: 429
  • Product dimensions: 7.65 (w) x 9.29 (h) x 0.87 (d)

Table of Contents

Acknowledgments xi
Chapter 1 Introduction 1
1.1 What is 3D Math? 1
1.2 Why You Should Read This Book 1
1.3 What You Should Know Before Reading This Book 3
1.4 Overview 3
Chapter 2 The Cartesian Coordinate System 5
2.1 1D Mathematics 6
2.2 2D Cartesian Mathematics 9
2.3 From 2D to 3D 14
2.4 Exercises 20
Chapter 3 Multiple Coordinate Spaces 23
3.1 Why Multiple Coordinate Spaces? 24
3.2 Some Useful Coordinate Spaces 25
3.3 Nested Coordinate Spaces 30
3.4 Specifying Coordinate Spaces 31
3.5 Coordinate Space Transformations 31
3.6 Exercises 34
Chapter 4 Vectors 35
4.1 Vector--A Mathematical Definition 36
4.2 Vector--A Geometric Definition 37
4.3 Vectors vs. Points 40
4.4 Exercises 42
Chapter 5 Operations on Vectors 45
5.1 Linear Algebra vs. What We Need 46
5.2 Typeface Conventions 46
5.3 The Zero Vector 47
5.4 Negating a Vector 48
5.5 Vector Magnitude (Length) 49
5.6 Vector Multiplication by a Scalar 51
5.7 Normalized Vectors 53
5.8 Vector Addition and Subtraction 54
5.9 The Distance Formula 57
5.10 Vector Dot Product 58
5.11 Vector Cross Product 62
5.12 Linear Algebra Identities 65
5.13 Exercises 67
Chapter 6 A Simple 3D Vector Class 69
6.1 Class Interface 69
6.2 Class Vector3 Definition 70
6.3 Design Decisions 73
Chapter 7 Introduction to Matrices 83
7.1 Matrix--A Mathematical Definition 83
7.2 Matrix--A Geometric Interpretation 91
7.3 Exercises 98
Chapter 8 Matrices and Linear Transformations 101
8.1 Transforming an Object vs. Transforming the Coordinate Space 102
8.2 Rotation 105
8.3 Scale 112
8.4 Orthographic Projection 115
8.5 Reflection 117
8.6 Shearing 118
8.7 Combining Transformations 119
8.8 Classes of Transformations 120
8.9 Exercises 124
Chapter 9 More on Matrices 125
9.1 Determinant of a Matrix 125
9.2 Inverse of a Matrix 130
9.3 Orthogonal Matrices 132
9.4 4x4 Homogenous Matrices 135
9.5 Exercises 146
Chapter 10 Orientation and Angular Displacement in 3D 147
10.1 What is Orientation? 148
10.2 Matrix Form 149
10.3 Euler Angles 153
10.4 Quaternions 159
10.5 Comparison of Methods 179
10.6 Converting between Representations 180
10.7 Exercises 193
Chapter 11 Transformations in C++ 195
11.1 Overview 196
11.2 Class EulerAngles 198
11.3 Class Quaternion 205
11.4 Class RotationMatrix 215
11.5 Class Matrix4x3 220
Chapter 12 Geometric Primitives 239
12.1 Representation Techniques 239
12.2 Lines and Rays 241
12.3 Spheres and Circles 246
12.4 Bounding Boxes 247
12.5 Planes 252
12.6 Triangles 257
12.7 Polygons 269
12.8 Exercises 275
Chapter 13 Geometric Tests 277
13.1 Closest Point on 2D Implicit Line 277
13.2 Closest Point on Parametric Ray 278
13.3 Closest Point on Plane 279
13.4 Closest Point on Circle/Sphere 280
13.5 Closest Point in AABB 280
13.6 Intersection Tests 281
13.7 Intersection of Two Implicit Lines in 2D 282
13.8 Intersection of Two Rays in 3D 283
13.9 Intersection of Ray and Plane 284
13.10 Intersection of AABB and Plane 285
13.11 Intersection of Three Planes 286
13.12 Intersection of Ray and Circle/Sphere 286
13.13 Intersection of Two Circles/Spheres 288
13.14 Intersection of Sphere and AABB 291
13.15 Intersection of Sphere and Plane 291
13.16 Intersection of Ray and Triangle 293
13.17 Intersection of Ray and AABB 297
13.18 Intersection of Two AABBs 297
13.19 Other Tests 299
13.20 Class AABB3 300
13.21 Exercises 316
Chapter 14 Triangle Meshes 319
14.1 Representing Meshes 320
14.2 Additional Mesh Information 328
14.3 Topology and Consistency 330
14.4 Triangle Mesh Operations 331
14.5 A C++ Triangle Mesh Class 336
Chapter 15 3D Math for Graphics 345
15.1 Graphics Pipeline Overview 346
15.2 Setting the View Parameters 349
15.3 Coordinate Spaces 354
15.4 Lighting and Fog 358
15.5 Buffers 372
15.6 Texture Mapping 373
15.7 Geometry Generation/Delivery 374
15.8 Transformation and Lighting 377
15.9 Backface Culling and Clipping 380
15.10 Rasterization 383
Chapter 16 Visibility Determination 385
16.1 Bounding Volume Tests 386
16.2 Space Partitioning Techniques 390
16.3 Grid Systems 392
16.4 Quadtrees and Octrees 393
16.5 BSP Trees 398
16.6 Occlusion Culling Techniques 402
Chapter 17 Afterword 407
Appendix A Math Review 409
Summation Notation 409
Angles, Degrees, and Radians 409
Trig Functions 410
Trig Identities 413
Appendix B References 415
Index 417
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