3D Math Primer for Graphics and Game Development Book

	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