3rd Edition
Essential Mathematics for Games and Interactive Applications
Expert Guidance on the Math Needed for 3D Game Programming
Developed from the authors’ popular Game Developers Conference (GDC) tutorial, Essential Mathematics for Games and Interactive Applications, Third Edition illustrates the importance of mathematics in 3D programming. It shows you how to properly animate, simulate, and render scenes and discusses the mathematics behind the processes.
New to the Third Edition
Completely revised to fix errors and make the content flow better, this third edition reflects the increased use of shader graphics pipelines, such as in DirectX 11, OpenGL ES (GLES), and the OpenGL Core Profile. It also updates the material on real-time graphics with coverage of more realistic materials and lighting.
The Foundation for Successful 3D Programming
The book covers the low-level mathematical and geometric representations and algorithms that are the core of any game engine. It also explores all the stages of the rendering pipeline. The authors explain how to represent, transform, view, and animate geometry. They then focus on visual matters, specifically the representation, computation, and use of color. They also address randomness, intersecting geometric entities, and physical simulation.
An Introduction to Creating Real and Active Virtual Worlds
This updated book provides you with a conceptual understanding of the mathematics needed to create 3D games as well as a practical understanding of how these mathematical bases actually apply to games and graphics. It not only includes the theoretical mathematical background but also incorporates many examples of how the concepts are used to affect how a game looks and plays.
Web Resource
A supplementary website contains a collection of source code, supporting libraries, and interactive demonstrations that illustrate the concepts and enable you to experiment with animation and simulation applications. The site also includes slides and notes from the authors’ GDC tutorials.
Representing Real Numbers
Preliminary Concepts
Floating-Point Numbers
IEEE 754 Floating-Point Standard
Real-World Floating Point
Code
Vectors and Points
Vectors
Points
Lines
Planes
Polygons and Triangles
Linear Transformations and Matrices
Linear Transformations
Matrices
Systems of Linear Equations
Matrix Inverse
Determinant
Eigenvalues and Eigenvectors
Affine Transformations
Affine Transformations
Standard Affine Transformations
Using Affine Transformations
Object Hierarchies
Orientation Representation
Rotation Matrices
Euler Angles
Axis–Angle Representation
Quaternions
Interpolation
Interpolation of Position
Interpolation of Orientation
Sampling Curves
Controlling Speed along a Curve
Camera Control
Viewing and Projection
View Frame and View Transformation
Projective Transformation
Culling and Clipping
Screen Transformation
Picking
Management of Viewing Transformations
Geometry and Programmable Shading
Color Representation
Points and Vertices
Surface Representation
Rendering Pipeline
Shaders
Vertex Shaders
Fragment Shaders
Basic Coloring Methods
Texture Mapping
Texture Coordinates
The Steps of Texturing
Limitations of Static Shading
Lighting
Basics of Light Approximation
Measuring Light
Types of Light Sources
Surface Materials and Light Interaction
Lighting and Shading
Textures and Lighting
Advanced Lighting
Rasterization
Displays and Framebuffers
Conceptual Rasterization Pipeline
Determining the Fragments: Pixels Covered by a Triangle
Determining Visible Geometry
Computing Fragment Shader Inputs
Rasterizing Textures
From Fragments to Pixels
Random Numbers
Probability
Determining Randomness
Random Number Generators
Special Applications
Intersection Testing
Closest Point and Distance Tests
Object Intersection
A Simple Collision System
Rigid-Body Dynamics
Linear Dynamics
Numerical Integration
Rotational Dynamics
Collision Response
Efficiency
Biography
James M. Van Verth is a software engineer at Google, where he works on GPU support for the Skia 2D Graphics Library. He has worked for Insomniac Games, NVIDIA, and Red Storm Entertainment and, for the past 17 years, he has been a regular speaker at GDC, teaching the tutorials "Math for Game Programmers" and "Physics for Game Programmers." He received a BA in math/computer science from Dartmouth College, an MS in computer science from the State University of New York at Buffalo, and an MS in computer science from the University of North Carolina at Chapel Hill.
Lars M. Bishop is an engineer in the Handheld Developer Technologies group at NVIDIA. Prior to joining NVIDIA, he was the chief technology officer at Numerical Design Limited, leading the development of the Gamebryo3D cross-platform game engine. He received a BS in math/computer science from Brown University and an MS in computer science from the University of North Carolina at Chapel Hill.
Praise for Previous Editions:
"It’s the book with all the math you need for games."
—Neil Kirby, Researcher, Alcatel-Lucent"Even though I’ve worked with these systems for years, this book showed me new ways of looking at several topics that make them easier to remember and use. For someone new to 3D programming, it is extremely useful—it gives them a solid background in pretty much every area they need to understand."
—Peter Lipson, Toys for Bob, Inc.
