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By Ian Millington, Ian Millington is a consultant specializing in research and development for the games industry. He has done a good deal of work on physics
engines and developing physics code. He was previously the CEO of a games technology company, and before that he ran one of the first
web-based games companies. Millington's background is in artificial intelligence, and he has published academically on a range of topics.
Ian Millington is author of Artificial Intelligence for Games (MK).
Description Simulating physics helps cutting-edge games distinguish themselves by making virtual objects behave as we expect them to in the real world.
Physics engines are the software programs that run these simulations. Building an engine is difficult, however. There are a large number
of new developers (and hobbyists) coming into this market who need help through this complex process. Current introductory books are
inadequate; they don't bring enough real-world programming experience to the task. There is a need for an introductory book on game physics
with solid coding guidance but which limits the math content. Ian Millington brings his extensive professional programming experience
to this problem. He has developed games since 1987, has studied AI and mathematics at the PhD level, and founded Mindlathe Ltd., a company
that designed and built commercial physics engines. Physics Engine Development carefully describes each step in the creation of a robust,
usable physics engine. It introduces the mathematical concepts in a clear and simple manner, keeping to high school level topics and
building a physics code library as it goes. Each new concept is explained in diagrams and code to make sure that even the most novice
of game programmers understands. The companion CD-ROM includes the source code for a complete physics engine of commercial quality. This
book will serve as a introduction to more mathematically advanced books on game physics, such as Dave Eberly's Game Physics.
Audience
Game developers, esp. novice developers, students of game development, and game hobbyists
Contents
Preface
About the Author
About the CD-ROM
1 Introduction
1.1 What is Game Physics
1.2
What is a Physics Engine
1.2.1 Advantages of a Physics Engine
1.2.2 Weaknessess of a Physics Engine
1.3
Approaches to Physics Engines
1.3.1 Types of Object
1.3.2 Contact Resolution
1.3.3 Impulses and Forces
1.3.4 What We're Building
1.4 The Mathematics of Physics Engines
1.4.1 The Math You Need to Know
1.4.2 The Math We'll Review
1.4.3 The Mathematics We'll Introduce
1.5 The Source Code in the Book
1.6 How the Book is Structured
I Particle Physics
2 The Mathematics of Particles
2.1 Vectors
2.1.1 The Handedness of Space
2.1.2 Vectors and Directions
2.1.3 Scalar and Vector Multiplication
2.1.4 Vector Addition and Subtraction
2.1.5 Multiplying Vectors
2.1.6 The Component Product
2.1.7 The
Scalar Product
2.1.8 The Vector Product
2.1.9 The Orthonormal Basis
2.2 Calculus
2.2.1 Differential
Calculus
2.2.2 Integral Calculus
3 The Laws of Motion
3.1 A Particle
3.2
The First Two Laws
3.2.1 The Force Equations
3.2.2 Adding Mass to Particles
3.2.3 Momentum and Velocity
3.2.4 The Force of Gravity
3.3 The Integrator
3.3.1 The Update Equations
3.3.2 The Complete Integrator
4 The Particle Physics Engine
4.1 Ballistics
4.1.1 Setting Projectile Properties
4.1.2 Implementation
4.2 Fireworks
4.2.1 The Firework Data
4.2.2 Firework Rules
4.2.3 The Implementation
II Mass Aggregate Physics
5 Adding General Forces
5.1 D'Alembert's Principle
5.2 Force
Generators
5.2.1 Interfaces and Polymorphism
5.2.2 Implementation
5.2.3 A Gravity Force Generator
5.2.4
A Drag Force Generator
5.3 Built-In Gravity and Damping
6 Springs and Spring-like Things
6.1 Hook's Law
6.2 Spring-Like Force Generators
6.2.1 A Basic Spring Generator
6.2.2 An Anchored Spring
Generator
6.2.3 An Elastic Bungee Generator
6.2.4 A Buoyancy Force Generator
6.3 Stiff Springs
6.3.1
The Problem of Stiff Springs
6.3.2 Faking Stiff Springs
7 Hard Constraints
7.1
Simple Collision Resolution
7.1.1 The Closing Velocity
7.1.2 The Coefficient of Restitution
7.1.3 The Collision
Direction and the Contact Normal
7.1.4 Impulses
7.2 Collision Processing
7.2.1 Collision Detection
7.2.2 Resolving Interpenetration
7.2.3 Resting Contacts
7.3 The Contact Resolver Algorithm
7.3.1
Resolution Order
7.3.2 Time-Division Engines
7.4 Collision-Like Things
7.4.1 Cables
7.4.2
Rods
8 The Mass Aggregate Physics Engine
8.1 Overview of the Engine
8.2 Using
the Physics Engine
8.2.1 Rope Bridges and Cables
8.2.2 Friction
III Rigid
Body Physics
9 The Mathematics of Rotations
9.1 Rotating Objects in 2D
9.1.1 The Mathematics of
Angles
9.1.2 Angular Speed
9.1.3 The Origin and the Centre of Mass
9.2 Orientation in 3D
9.2.1
Euler Angles
9.2.2 Axis-Angle
9.2.3 Rotation Matrices
9.2.4 Quaternions
9.3 Angular Velocity
and Acceleration
9.3.1 Velocity of a Point
9.3.2 Angular Acceleration
9.4 Implementing the Mathematics
9.4.1 The Matrix Classes
9.4.2 Matrix Multiplication
9.4.3 Matrix Inverse and Transpose
9.4.4 Converting
a Quaternion to a Matrix
9.4.5 Transforming Vectors
9.4.6 Changing the Basis of a Matrix
9.4.7 The Quaternion
Class
9.4.8 Normalising Quaternions
9.4.9 Combining Quaternions
9.4.10 Rotating
9.4.11 Updating by
The Angular Velocity
10 Laws of Motion for Rigid Bodies
10.1 The Rigid Body
10.2
Newton 2 for Rotation
10.2.1 Torque
10.2.2 The Moment of Inertia
10.2.3 Inertia Tensor in World-Coordinates
10.3 D'Alembert for Rotation
10.3.1 Force Generators
10.4 The Rigid Body Integration
11 The Rigid Body Physics Engine
11.1 Overview of the Engine
11.2 Using the Physics Engine
11.2.1 A
Flight Simulator
11.2.2 A Sailing Simulator
IV Collision Detection
12
Collision Detection
12.1 Collision Detection Pipeline
12.2 Coarse Collision Detection
12.3 Bounding Volumes
12.3.1 Hierarchies
12.3.2 Building the Hierarchy
12.3.3 Sub-object Hierarchies
12.4 Spatial
Data Structures
12.4.1 Binary Space Partitioning (BSP)
12.4.2 Oct-Trees and Quad-Trees
12.4.3 Grids
12.4.4
Multi-Resolution Maps
12.5 Summary
13 Generating Contacts
13.1 Collision Geometry
13.1.1 Primitive Assemblies
13.1.2 Generating Collision Geometry
13.2 Contact Generation
13.2.1
Contact Data
13.2.2 Point-Face Contacts
13.2.3 Edge-Edge Contacts
13.2.4 Edge-Face Contacts
13.2.5
Face-Face Contacts
13.2.6 Early Outs
13.3 Primitive Collision Algorithms
13.3.1 Colliding two Spheres
13.3.2 Colliding a Sphere and a Plane
13.3.3 Colliding a Box and a Plane
13.3.4 Colliding a Sphere and a Box
13.3.5 Colliding two Boxes
V Contact Physics
14 Collision Resolution
14.1 Impulses and Implusive Torques
14.1.1 Impulsive Torque
14.1.2 Rotating Collisions
14.1.3 Handling
Rotating Collisions
14.2 Collision Impluses
14.2.1 Change to Contact-coordinates
14.2.2 Velocity
change by Impulse
14.2.3 Impulse change by Velocity
14.2.4 Calculating the Desired Velocity Change
14.2.5
Calculating the Impulse
14.2.6 Applying the Impulse
14.3 Resolving Interpenetration
14.3.1 Choosing
a Resolution Method
14.3.2 Implementing Non-linear Projection
14.3.3 Avoiding Over-Rotation
14.4
The Collision Resolution Process
14.4.1 The Collision Resolution Pipeline
14.4.2 Preparing Contact Data
14.4.3
Resolving Penetration
14.4.4 Resolving Velocity
14.4.5 Alternative Update Algorithms
14.4.6 Summary
15 Resting Contacts and Friction
15.1 Resting Forces
15.1.1 Force Calculations
15.2 Micro-Collisions
15.2.1 Removing Accelerated Velocity
15.2.2 Lowering the Restitution
15.2.3 The New Velocity Calculation
15.3 Types of Friction
15.3.1 Static and Dynamic Friction
15.3.2 Isotropic and Anisotropic Friction
15.4 Implementing Friction
15.4.1 Friction as Impulses
15.4.2 Modifying the Velocity Resolution Algorithm
15.4.3
Putting it All Together
15.5 Friction and Sequential Contact Resolution
16 Stability and
Optimization
16.1 Stability
16.1.1 Quaternion Drift
16.1.2 Interpenetration on Slopes
16.1.3 Integration
Stability
16.1.4 The Benefit of Pessimistic Collision Detection
16.1.5 Changing Mathematical Accuracy
16.2 Optimizations
16.2.1 Sleep
16.2.2 Margins of Error for Penetration and Velocity
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