## Book description

This engaging book presents the essential mathematics needed to describe, simulate, and render a 3D world. Reflecting both academic and in-the-trenches practical experience, the authors teach you how to describe objects and their positions, orientations, and trajectories in 3D using mathematics. The text provides an introduction to mathematics for

1. Preliminaries
2. Dedication
3. Epigraph
4. Acknowledgments
5. Introduction
6. Chapter 1 Cartesian Coordinate Systems
1. 1.1 1D Mathematics
2. 1.2 2D Cartesian Space
3. 1.3 3D Cartesian Space
4. 1.4 Odds and Ends
5. 1.5 Exercises
7. Chapter 2 Vectors
1. 2.1 Mathematical Definition of Vector, and Other Boring Stuff
2. 2.2 Geometric Definition of Vector
3. 2.3 Specifying Vectors with Cartesian Coordinates
4. 2.4 Vectors versus Points
5. 2.5 Negating a Vector
6. 2.6 Vector Multiplication by a Scalar
7. 2.7 Vector Addition and Subtraction
8. 2.8 Vector Magnitude (Length)
9. 2.9 Unit Vectors
10. 2.10 The Distance Formula
11. 2.11 Vector Dot Product
12. 2.12 Vector Cross Product
13. 2.13 Linear Algebra Identities
14. 2.14 Exercises
8. Chapter 3 Multiple Coordinate Spaces
1. 3.1 Why Bother with Multiple Coordinate Spaces?
2. 3.2 Some Useful Coordinate Spaces
3. 3.3 Basis Vectors and Coordinate Space Transformations
4. 3.4 Nested Coordinate Spaces
5. 3.5 In Defense of Upright Space
6. 3.6 Exercises
9. Chapter 4 Introduction to Matrices
1. 4.1 Mathematical Definition of Matrix
2. 4.2 Geometric Interpretation of Matrix
3. 4.3 The Bigger Picture of Linear Algebra
4. 4.4 Exercises
10. Chapter 5 Matrices and Linear Transformations
1. 5.1 Rotation
2. 5.2 Scale
3. 5.3 Orthographic Projection
4. 5.4 Reflection
5. 5.5 Shearing
6. 5.6 Combining Transformations
7. 5.7 Classes of Transformations
8. 5.8 Exercises
11. Chapter 6 More on Matrices
1. 6.1 Determinant of a Matrix
2. 6.2 Inverse of a Matrix
3. 6.3 Orthogonal Matrices
4. 6.4 4 × 4 Homogeneous Matrices
5. 6.5 4 × 4 Matrices and Perspective Projection
6. 6.6 Exercises
12. Chapter 7 Polar Coordinate Systems
1. 7.1 2D Polar Space
2. 7.2 Why Would Anybody Use Polar Coordinates?
3. 7.3 3D Polar Space
4. 7.4 Using Polar Coordinates to Specify Vectors
5. 7.5 Exercises
13. Chapter 8 Rotation in Three Dimensions
1. 8.1 What Exactly is “Orientation”?
2. 8.2 Matrix Form
3. 8.3 Euler Angles
4. 8.4 Axis-Angle and Exponential Map Representations
5. 8.5 Quaternions
6. 8.6 Comparison of Methods
7. 8.7 Converting between Representations
8. 8.8 Exercises
14. Chapter 9 Geometric Primitives
1. 9.1 Representation Techniques
2. 9.2 Lines and Rays
3. 9.3 Spheres and Circles
4. 9.4 Bounding Boxes
5. 9.5 Planes
6. 9.6 Triangles
7. 9.7 Polygons
8. 9.8 Exercises
15. Chapter 10 Mathematical Topics from 3D Graphics
1. 10.1 How Graphics Works
2. 10.2 Viewing in 3D
3. 10.3 Coordinate Spaces
4. 10.4 Polygon Meshes
5. 10.5 Texture Mapping
6. 10.6 The Standard Local Lighting Model
7. 10.7 Light Sources
8. 10.8 Skeletal Animation
9. 10.9 Bump Mapping
10. 10.10 The Real-Time Graphics Pipeline
11. 10.11 Some HLSL Examples
13. 10.13 Exercises
16. Chapter 11 Mechanics 1: Linear Kinematics and Calculus
1. 11.1 Overview and Other Expectation-Reducing Remarks
2. 11.2 Basic Quantities and Units
3. 11.3 Average Velocity
4. 11.4 Instantaneous Velocity and the Derivative
5. 11.5 Acceleration
6. 11.6 Motion under Constant Acceleration
7. 11.7 The Integral
8. 11.8 Uniform Circular Motion
9. 11.9 Exercises
17. Chapter 12 Mechanics 2: Linear and Rotational Dynamics
1. 12.1 Newton’s Three Laws
2. 12.2 Some Simple Force Laws
3. 12.3 Momentum
4. 12.4 Impulsive Forces and Collisions
5. 12.5 Rotational Dynamics
6. 12.6 Real-Time Rigid Body Simulators
8. 12.8 Exercises
18. Chapter 13 Curves in 3D
1. 13.1 Parametric Polynomial Curves
2. 13.2 Polynomial Interpolation
3. 13.3 Hermite Curves
4. 13.4 Bézier Curves
5. 13.5 Subdivision
6. 13.6 Splines
7. 13.7 Hermite and Bézier Splines
8. 13.8 Continuity
9. 13.9 Automatic Tangent Control
10. 13.10 Exercises
19. Chapter 14 Afterword
20. Appendix A Geometric Tests
1. A.1 Closest Point on 2D Implicit Line
2. A.2 Closest Point on a Parametric Ray
3. A.3 Closest Point on a Plane
4. A.4 Closest Point on a Circle or Sphere
5. A.5 Closest Point in an AABB
6. A.6 Intersection Tests
7. A.7 Intersection of Two Implicit Lines in 2D
8. A.8 Intersection of Two Rays in 3D
9. A.9 Intersection of a Ray and Plane
10. A.10 Intersection of an AABB and Plane
11. A.11 Intersection of Three Planes
12. A.12 Intersection of Ray and a Circle or Sphere
13. A.13 Intersection of Two Circles or Spheres
14. A.14 Intersection of a Sphere and AABB
15. A.15 Intersection of a Sphere and a Plane
16. A.16 Intersection of a Ray and a Triangle
17. A.17 Intersection of Two AABBs
18. A.18 Intersection of a Ray and an AABB
21. Appendix B Answers to the Exercises
22. Bibliography

## Product information

• Title: 3D Math Primer for Graphics and Game Development, 2nd Edition
• Author(s): Fletcher Dunn, Ian Parberry
• Release date: November 2011
• Publisher(s): A K Peters/CRC Press
• ISBN: 9781498759892