2nd Edition

3D Math Primer for Graphics and Game Development

By Fletcher Dunn Copyright 2012
    846 Pages
    by A K Peters/CRC Press

    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 game designers, including the fundamentals of coordinate spaces, vectors, and matrices. It also covers orientation in three dimensions, calculus and dynamics, graphics, and parametric curves.

    Cartesian Coordinate Systems
    1D Mathematics
    2D Cartesian Space
    3D Cartesian Space
    Odds and ends

    Vector — mathematical definition and other boring stuff
    Vector — a geometric definition
    Specifying vectors using Cartesian coordinates
    Vectors vs. points
    Negating a vector
    Vector multiplication by a scalar
    Vector addition and subtraction
    Vector magnitude (length)
    Unit vectors
    The distance formula
    Vector dot product
    Vector cross product
    Linear algebra identities

    Multiple Coordinate Spaces
    Why multiple coordinate spaces?
    Some useful coordinate spaces
    Coordinate space transformations
    Nested coordinate spaces
    In defense of upright space

    Introduction to Matrices
    Matrix — a mathematical definition
    Matrix — a geometric interpretation
    The bigger picture of linear algebra

    Matrices and Linear Transformations
    Orthographic projection
    Combining transformations
    Classes of transformations

    More on Matrices
    Determinant of a matrix
    Inverse of a matrix
    Orthogonal matrices
    4 x 4 homogeneous matrices
    4 x 4 matrices and perspective projection

    Polar Coordinate Systems
    2D Polar Space
    Why would anybody use Polar coordinates?
    3D Polar Space
    Using polar coordinates to specify vectors

    Rotation in Three Dimensions
    What exactly is "orientation?"
    Matrix form
    Euler angles
    Axis-angle and exponential map representations
    Comparison of methods
    Converting between representations

    Geometric Primitives
    Representation techniques
    Lines and rays
    Spheres and circles
    Bounding boxes

    Mathematical Topics from 3D Graphics
    How graphics works
    Viewing in 3D
    Coordinate spaces
    Polygon meshes
    Texture mapping
    The standard local lighting model
    Light sources
    Skeletal animation
    Bump mapping
    The real-time graphics pipeline
    Some HLSL examples
    Further reading

    Mechanics 1: Linear Kinematics and Calculus
    Overview and other expectation-reducing remarks
    Basic quantities and units
    Average velocity
    Instantaneous velocity and the derivative
    Motion under constant acceleration
    Acceleration and the integral
    Uniform circular motion

    Mechanics 2: Linear and Rotational Dynamics
    Newton's three laws
    Some simple force laws
    Impulsive forces and collisions
    Rotational dynamics
    Real-time rigid body simulators
    Suggested reading

    Curves in 3D
    Parametric polynomial curves
    Polynomial interpolation
    Hermite curves
    Bezier curves
    Hermite and Bezier splines
    Automatic tangent control

    What next?

    Appendix A: Geometric Tests
    Appendix B: Answers to the Exercises



    Exercises appear at the end of each chapter.


    Fletcher Dunn has been programming video games professionally since 1996. He served as principle programmer at Terminal Reality in Dallas, where he was one of the architects of the Infernal engine and lead programmer on BloodRayne. He was a technical director for the Walt Disney Company at Wideload Games in Chicago, where he was the lead programmer for Disney Guilty Party, which won IGN's Family Game of the Year at E3 2010. He is currently a developer at Valve Software in Bellevue, Washington.

    Ian Parberry is a professor in the Department of Computer Science and Engineering at the University of North Texas. Dr. Parberry has more than a quarter century of experience in research and teaching and is nationally known as one of the pioneers of game programming in higher education.

    "With solid theory and references, along with practical advice borne from decades of experience, all presented in an informal and demystifying style, Dunn & Parberry provide an accessible and useful approach to the key mathematical operations needed in 3D computer graphics."
    —Eric Haines, author of Real-Time Rendering

    "The book describes the mathematics involved in game development in a very clear and easy to understand way, layered on the practical background of years of game engine programming experience."
    —Wolfgang Engel, editor of GPU Pro