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Features

Explains how the mathematical concepts apply to major parts of 3D games
Covers the entire rendering pipeline, from representation and animation to light simulation and rasterization
Shows how to display and manipulate game objects
Presents techniques that make games look less structured and more organic
Includes a full set of references that encourage you to delve deeper into the mathematics involved in creating high-performance, high-quality 3D games
Offers source code, demos, tutorial slides, and other materials on a supplementary website
Summary

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.



TABLE OF CONTENTS

Introduction
What is Game Physics?
What is a Physics Engine?
Approaches to Physics Engines
The Mathematics of Physics Engines
The Source Code in the Book
How the Book is Structured

PART I Particle Physics
The Mathematics of Particles
Vectors
Calculus
Summary
Exercises
The Laws of Motion
The Particle
The First Two Laws
The Integrator
Summary
Exercises
The Particle Physics Engine
Ballistics
Fireworks
Summary
Projects

PART II Mass Aggregate Physics
Adding General Forces
D'Alembert's Principle
Force Generators
Built-In Gravity and Damping
Summary
Exercises
Springs and Spring-Like Things
Hook's Law
Spring-Like Force Generators
Stiff Springs
Summary
Exercises
Hard Constraints
Simple Collision Resolution
Collision Processing
The Contact Resolver Algorithm
Collision-Like Things
Summary
Exercises
The Mass Aggregate Physics Engine
Overview of the Engine
Using the Physics Engine
Summary
Projects

PART III Rigid-Body Physics
The Mathematics of Rotations
Rotating Objects in 2D
Orientation in 3D
Angular Velocity and Acceleration
Implementing the Mathematics
Summary
Exercises
Laws of Motion for Rigid Bodies
The Rigid Body
Newton-2 for Rotation
D'Alembert for Rotation
The Rigid-Body Integration
Summary
Exercises
The Rigid-Body Physics Engine
Overview of the Engine
Using the Physics Engine
Summary
Projects

PART IV Collision Detection
Collision Detection
The Collision Detection Pipeline
Broad-Phase Collision Detection
Bounding Volume Hierarchies
Spatial Partitioning
Summary
Exercises
Generating Contacts
Collision Geometry
Contact Generation
Simple Collision Algorithms
Separating Axis Tests
Coherence
Summary
Exercises

PART V Contact Physics
Collision Resolution
Impulse and Impulsive Torque
Collision Impulses
Resolving Interpenetration
The Collision Resolution Process
Summary
Exercises
Resting Contacts and Friction
Resting Forces
Microcollisions
Types of Friction
Implementing Friction
Friction and Sequential Contact Resolution
Summary
Exercises
Stability and Optimization
Stability
Optimizations
Summary
Putting It All Together
Overview of the Engine
Using the Physics Engine
Limitations of the Engine
Summary
Projects

PART VI Further Topics in Physics
Physics in Two Dimensions
2D or 3D?
Vector Mathematics
Particle and Mass Aggregate Physics
The Mathematics of Rotation
Rigid-Body Dynamics
Collision Detection
Collision Response
Summary
Projects
Other Programming Languages
ActionScript 3
C
Java
C#
Other Scripting Languages
Other Types of Physics
Simultaneous Contact Resolution
Reduced Coordinate Approaches
Summary
Appendix A Useful Inertia Tensors
Discrete Masses
Continuous Masses
Common Shapes
Cuboid
Sphere
Cylinder
Cone
Hemisphere
Moments of Inertia in 2D
Common 2D Shapes
Appendix B Useful Friction Coefficients

Appendix C Mathematics Summary
Vectors
Quaternions
Matrices
Integration
Physics
Other Forumlas