Game Physics Engine Development 1st Edition by Ian Millington ISBN 9780123694713 012369471X

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. Game 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. This book will serve as a introduction to more mathematically advanced books on game physics, such as Dave Eberly’s Game Physics.

Game Physics Engine Development 1st Edition Table of contents:

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 Weaknesses of a Physics Engine

1.3 Approaches to Physics Engines

1.3.1 Types of Objects

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

Figure 1.1 Trigonometry and coordinate geometry.

1.4.2 The Math We’ll Review

1.4.3 The Math I’ll Introduce

1.5 The Source Code in the Book

1.6 How the Book is Structured

1.6.1 Exercises and Projects

PART I Particle Physics

2 The Mathematics of Particles

2.1 Vectors

Figure 2.1 3D coordinates.

Excerpt from file include/cyclone/core.h

Excerpt from file include/cyclone/precision.h

2.1.1 The Handedness of Space

Figure 2.2 Left- and right-handed axes.

2.1.2 Vectors and Directions

Figure 2.3 A vector as a movement in space.

Excerpt from file include/cyclone/core.h

2.1.3 Scalar and Vector Multiplication

Excerpt from file include/cyclone/core.h

Figure 2.4 The geometry of scalar-vector multiplication.

Figure 2.5 The geometry of vector addition.

2.1.4 Vector Addition and Subtraction

Excerpt from file include/cyclone/core.h

Excerpt from file include/cyclone/core.h

Excerpt from file include/cyclone/core.h

2.1.5 Multiplying Vectors

2.1.6 The Component Product

Excerpt from file include/cyclone/core.h

2.1.7 The Scalar Product

Excerpt from file include/cyclone/core.h

The Trigonometry of the Scalar Product

The Geometry of the Scalar Product

Figure 2.6 Geometric interpretation of the scalar product.

2.1.8 The Vector Product

Excerpt from file include/cyclone/core.h

The Trigonometry of the Vector Product

Commutativity of the Vector Product

The Geometry of the Vector Product

Figure 2.7 Geometric interpretation of the vector product.

2.1.9 The Orthonormal Basis

2.2 Calculus

2.2.1 Differential Calculus

Velocity

Figure 2.8 Same average velocity, different instantaneous velocity.

Acceleration

Vector Differential Calculus

Velocity, Direction, and Speed

2.2.2 Integral Calculus

Vector Integral Calculus

2.3 Summary

2.4 Exercises

3 The Laws of Motion

3.1 The Particle

Excerpt from file include/cyclone/particle.h

3.2 The First Two Laws

The First Law

Excerpt from file include/cyclone/particle.h

The Second Law

3.2.1 The Force Equations

3.2.2 Adding Mass to Particles

Excerpt from file include/cyclone/particle.h

3.2.3 Momentum and Velocity

3.2.4 The Force of Gravity

The Value of g

3.3 The Integrator

3.3.1 The Update Equations

Position Update

Velocity Update

3.3.2 The Complete Integrator

Excerpt from file include/cyclone/particle.h

Excerpt from file include/cyclone/precision.h

Excerpt from file src/particle.cpp

3.4 Summary

3.5 Exercises

4 The Particle Physics Engine

4.1 Ballistics

4.1.1 Setting Projectile Properties

Figure 4.1 Screenshot of the ballistic demo.

4.1.2 Implementation

Excerpt from file src/demos/ballistic/ballistic.cpp

Excerpt from file src/demos/ballistic/ballistic.cpp

Figure 4.2 Screenshot of the bigballistic demo.

Figure 4.3 Screenshot of the fireworks demo.

4.2 Fireworks

4.2.1 The Fireworks Data

Excerpt from file src/demos/fireworks/fireworks.cpp

4.2.2 Firework Rules

Excerpt from file src/demos/fireworks/fireworks.cpp

Excerpt from file src/demos/fireworks/fireworks.cpp

4.2.3 The Implementation

Excerpt from file src/demos/fireworks/fireworks.cpp

Excerpt from file src/demos/fireworks/fireworks.cpp

Excerpt from file src/demos/fireworks/fireworks.cpp

4.3 Summary

4.4 Projects

Mini-Project 4.1

Mini-Project 4.2

Mini-Project 4.3

Mini-Project 4.4

Project 4.1

Project 4.2

PART II Mass Aggregate Physics

5 Adding General Forces

5.1 D’Alembert’s Principle

Excerpt from file include/cyclone/particle.h

Excerpt from file src/particle.cpp

Excerpt from file include/cyclone/particle.h

Excerpt from file src/particle.cpp

5.2 Force Generators

5.2.1 Interfaces and Polymorphism

5.2.2 Implementation

Excerpt from file include/cyclone/pfgen.h

Excerpt from file include/cyclone/pfgen.h

Excerpt from file src/pfgen.cpp

5.2.3 A Gravity Force Generator

Excerpt from file include/cyclone/pfgen.h

Excerpt from file src/pfgen.cpp

5.2.4 A Drag Force Generator

Excerpt from file include/cyclone/pfgen.h

Excerpt from file src/pfgen.cpp

5.3 Built-In Gravity and Damping

5.4 Summary

5.5 Exercises

Exercise 5.1

Exercise 5.2

Exercise 5.3

Exercise 5.4

6 Springs and Spring-Like Things

6.1 Hook’s Law

Spring Compression

The Limit of Elasticity

Spring-Like Things

Figure 6.1 The game’s camera attached to a spring.

6.2 Spring-Like Force Generators

6.2.1 A Basic Spring Force Generator

Excerpt from file include/cyclone/pfgen.h

Excerpt from file include/cyclone/precision.h

Excerpt from file src/pfgen.cpp

6.2.2 An Anchored Spring Generator

Excerpt from file include/cyclone/pfgen.h

Figure 6.2 A rope bridge held up by springs.

Excerpt from file src/pfgen.cpp

6.2.3 An Elastic Bungee Generator

Excerpt from file include/cyclone/pfgen.h

Excerpt from file src/pfgen.cpp

6.2.4 A Buoyancy Force Generator

Figure 6.3 A buoyant block submerged and partially submerged.

Excerpt from file include/cyclone/pfgen.h

Excerpt from file src/pfgen.cpp

6.3 Stiff Springs

6.3.1 The Stiff Springs Problem

Figure 6.4 A non-stiff spring over time.

Figure 6.5 A stiff spring over time.

6.3.2 Faking Stiff Springs

Harmonic Motion

Damped Harmonic Motion

Implementation

Excerpt from file include/cyclone/pfgen.h

Excerpt from file include/cyclone/precision.h

Excerpt from file src/pfgen.cpp

Zero Rest Lengths

Velocity Mismatches

Figure 6.6 The rest length and the equilibrium position.

Interacting with Other Forces

6.4 Summary

6.5 Exercises

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

Figure 7.1 Contact normal is different from the vector between objects in contact.

7.1.4 Impulses

7.2 Collision Processing

Excerpt from file include/cyclone/pcontacts.h

Excerpt from file include/cyclone/pcontacts.h

Excerpt from file src/pcontacts.cpp

7.2.1 Collision Detection

Figure 7.2 Interpenetrating objects.

7.2.2 Resolving Interpenetration

Excerpt from file include/cyclone/pcontacts.h

Figure 7.3 Interpenetration and reality.

Excerpt from file include/cyclone/pcontacts.h

Excerpt from file src/pcontacts.cpp

7.2.3 Resting Contacts

Figure 7.4 Vibration on resting contact.

Velocity and the Contact Normal

Excerpt from file src/pcontacts.cpp

Other Approaches to Resting Contact

7.3 The Contact Resolver Algorithm

7.3.1 Resolution Order

Figure 7.5 Resolving one contact may resolve another automatically.

Excerpt from file include/cyclone/pcontacts.h

Excerpt from file src/pcontacts.cpp

7.3.2 Time-Division Engines

7.4 Collision-Like Things

7.4.1 Cables

Excerpt from file include/cyclone/plinks.h

Excerpt from file src/plinks.cpp

7.4.2 Rods

Excerpt from file include/cyclone/plinks.h

Excerpt from file src/plinks.cpp

7.5 Summary

7.6 Exercises

Exercise 7.1

Exercise 7.2

Exercise 7.3

Exercise 7.4

Exercise 7.5

Exercise 7.6

8 The Mass Aggregate Physics Engine

8.1 Overview of the Engine

Excerpt from file include/cyclone/pworld.h

Excerpt from file include/cyclone/pworld.h

Excerpt from file include/cyclone/pcontacts.h

Excerpt from file include/cyclone/pworld.h

Excerpt from file src/pworld.cpp

8.2 Using the Physics Engine

8.2.1 Rope Bridges and Cables

Figure 8.1 Screenshot of the bridge demo.

Figure 8.2 Screenshot of the platform demo.

8.2.2 Friction

8.2.3 Blob Games

8.3 Summary

8.4 Projects

Mini-Project 8.1

Mini-Project 8.2

Mini-Project 8.3

Project 8.1

Project 8.2

PART III Rigid-Body Physics

9 The Mathematics of Rotations

9.1 Rotating Objects in 2D

9.1.1 The Mathematics of Angles

Figure 9.1 Angle that an object is facing.

9.1.2 Angular Speed

Figure 9.2 The circle of orientation vectors.

9.1.3 The Origin and the Center of Mass

The Origin of an Object

Figure 9.3 The relative position of a car component.

Rotations

Figure 9.4 The car is rotated.

The Composition of Rotations and Translations

Rigid Bodies

Center of Mass

9.2 Orientation in 3D

9.2.1 Euler Angles

Figure 9.5 Aircraft rotation axes.

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

Excerpt from file include/cyclone/core.h

Excerpt from file include/cyclone/core.h

9.4.2 Matrix Multiplication

Excerpt from file include/cyclone/core.h

Matrices as Transformations

Three-by-Four Matrices

Excerpt from file include/cyclone/core.h

Multiplying Two Matrices

Excerpt from file include/cyclone/core.h

Excerpt from file include/cyclone/core.h

9.4.3 Matrix Inverse and Transpose

Excerpt from file include/cyclone/core.h

Excerpt from file include/cyclone/core.h

Excerpt from file src/core.cpp

The Matrix Transpose

Excerpt from file include/cyclone/core.h

9.4.4 Converting a Quaternion to a Matrix

Excerpt from file include/cyclone/core.h

Excerpt from file include/cyclone/core.h

9.4.5 Transforming Vectors

Excerpt from file include/cyclone/core.h

Excerpt from file include/cyclone/core.h

9.4.6 Changing the Basis of a Matrix

Figure 9.6 A matrix basis is changed.

9.4.7 The Quaternion Class

Excerpt from file include/cyclone/core.h

9.4.8 Normalizing Quaternions

Excerpt from file include/cyclone/core.h

9.4.9 Combining Quaternions

Excerpt from file include/cyclone/core.h

9.4.10 Rotating

Excerpt from file include/cyclone/core.h

9.4.11 Updating by the Angular Velocity

Excerpt from file include/cyclone/core.h

9.5 Summary

9.6 Exercises

Exercise 9.1

Exercise 9.2

Exercise 9.3

Exercise 9.4

Exercise 9.5

Exercise 9.6

Exercise 9.7

10 Laws of Motion for Rigid Bodies

10.1 The Rigid Body

Excerpt from file include/cyclone/body.h

Excerpt from file include/cyclone/body.h

Excerpt from file src/body.cpp

10.2 Newton-2 for Rotation

10.2.1 Torque

Figure 10.1 A force generating no torque.

10.2.2 The Moment of Inertia

The Inverse Inertia Tensor

Excerpt from file include/cyclone/body.h

Excerpt from file scr/body.cpp

10.2.3 Inertia Tensor in World Coordinates

Figure 10.2 The moment of inertia is local to an object.

Excerpt from file src/body.cpp

10.3 D’Alembert for Rotation

Excerpt from file include/cyclone/body.h

Excerpt from file src/body.cpp

Excerpt from file include/cyclone/body.h

Excerpt from file src/body.cpp

Excerpt from file include/cyclone/body.h

Excerpt from file src/body.cpp

10.3.1 Force Generators

Excerpt from file include/cyclone/fgen.h

Excerpt from file src/fgen.cpp

Excerpt from file include/cyclone/fgen.h

Excerpt from file src/fgen.cpp

Torque Generators

10.4 The Rigid-Body Integration

Excerpt from file include/cyclone/body.h

Excerpt from file src/body.cpp

10.5 Summary

10.6 Exercises

Exercise 10.1

Exercise 10.2

Exercise 10.3

Exercise 10.4

Exercise 10.5

11 The Rigid-Body Physics Engine

11.1 Overview of the Engine

Excerpt from file include/cyclone/world.h

Excerpt from file include/cyclone/world.h

Excerpt from file src/world.cpp

Excerpt from file include/cyclone/world.h

Excerpt from file src/world.cpp

11.2 Using the Physics Engine

11.2.1 A Flight Simulator

The Aerodynamic Tensor

The Aerodynamic Surface

Excerpt from file include/cyclone/fgen.h

Excerpt from file src/fgen.cpp

Excerpt from file include/cyclone/fgen.h

Excerpt from file src/fgen.cpp

Excerpt from file include/cyclone/core.h

Excerpt from file src/core.cpp

Putting It Together

Figure 11.1 Screenshot of the flightsim demo.

11.2.2 A Sailing Simulator

Buoyancy

Figure 11.2 Different centers of buoyancy.

Excerpt from file include/cyclone/fgen.h

The Sail, Rudder, and Hydrofoils

Excerpt from file include/cyclone/fgen.h

Figure 11.3 Screenshot of the sailboat demo.

The Sailing Example

11.3 Summary

11.4 Projects

Mini-Project 11.1

Mini-Project 11.2

Mini-Project 11.3

Project 11.1

Project 11.2

PART IV Collision Detection

12 Collision Detection

12.1 The Collision Detection Pipeline

Figure 12.1 The collision detection pipeline.

12.2 Broad-Phase Collision Detection

Figure 12.2 A three-stage collision detection pipeline.

12.2.1 Requirements

12.3 Bounding Volume Hierarchies

Figure 12.3 A spherical bounding volume.

12.3.1 Hierarchies

Figure 12.4 A spherical bounding volume hierarchy.

Excerpt from file include/cyclone/collide_broad.h

Excerpt from file include/cyclone/collide_broad.h

Excerpt from file src/collide_broad.cpp

12.3.2 Building the Hierarchy

Figure 12.5 Bottom-up hierarchy building in action.

Figure 12.6 Top-down hierarchy building in action.

Figure 12.7 Insertion hierarchy building in action.

Excerpt from file include/cyclone/collide_broad.h

Excerpt from file include/cyclone/collide_broad.h

Figure 12.8 Working out a parent bounding sphere.

Excerpt from file include/cyclone/collide_broad.h

Figure 12.9 Removing an object from a hierarchy.

12.3.3 Subobject Hierarchies

Figure 12.10 A subobject bounding volume hierarchy.

12.4 Spatial Partitioning

12.4.1 Binary Space Partitioning

Figure 12.11 A BSP for level geometry.

12.4.2 Oct-Trees and Quad-Trees

Figure 12.12 Identifying an object’s location in a quad-tree.

12.4.3 Grids

Figure 12.13 A quad-tree forms a grid.

Figure 12.14 An object may occupy up to four same-sized grid cells.

12.4.4 Multiresolution Maps

12.5 Summary

12.6 Exercises

Exercise 12.1

Exercise 12.2

Exercise 12.3

Exercise 12.4

Exercise 12.5

Exercise 12.6

13 Generating Contacts

13.1 Collision Geometry

Figure 13.1 An object approximated by an assembly of primitives.

13.1.1 Primitive Assemblies

13.2 Contact Generation

Figure 13.2 Collision detection and contact generation.

13.2.1 Contact Types

Figure 13.3 3D cases of contact.

Figure 13.4 Ignoring a vertex–vertex contact.

13.2.2 Contact Data

Figure 13.5 The relationship among the collision point, collision normal, and penetration depth.

Excerpt from file include/cyclone/contacts.h

13.2.3 VertexFace Contacts

Figure 13.6 The vertex–face contact data.

13.2.4 EdgeEdge Contacts

13.2.5 EdgeFace Contacts

Figure 13.7 The edge–edge contact data.

Figure 13.8 The edge–face contact data.

13.2.6 FaceFace Contacts

Figure 13.9 The face–face contact data.

13.2.7 Testing Before Generating Contacts

13.3 Simple Collision Algorithms

Excerpt from file include/cyclone/collide_narrow.h

13.3.1 Colliding Two Spheres

Excerpt from file src/collide_narrow.cpp

13.3.2 Colliding a Sphere and a Plane

Excerpt from file src/collide_narrow.cpp

Excerpt from file src/collide_narrow.cpp

Figure 13.10 The difference in contact normal for a plane and a half-space.

13.3.3 Colliding a Box and a Plane

Figure 13.11 Contacts between a box and a plane.

Excerpt from file src/collide_narrow.cpp

Figure 13.12 The half-sizes of a box.

Excerpt from file src/collide_narrow.cpp

13.3.4 Colliding a Box and a Sphere

Figure 13.13 Contacts between a box and a sphere.

Excerpt from file src/collide_narrow.cpp

Excerpt from file src/collide_narrow.cpp

Excerpt from file src/collide_narrow.cpp

Excerpt from file src/collide_narrow.cpp

13.4 Separating Axis Tests

Figure 13.14 A separating axis test.

13.4.1 Generating Contact Data with SATs

Figure 13.15 A separating axis test showing maximum interpenetration.

13.4.2 Colliding Two Boxes

Contact Based on Face Axis

Contact Based on Edge–Edge

13.4.3 Colliding Convex Polyhedra

13.5 Coherence

Figure 13.16 Sequence of contacts over two frames.

13.6 Summary

13.7 Exercises

PART V Contact Physics

14 Collision Resolution

14.1 Impulse and Impulsive Torque

14.1.1 Impulsive Torque

Figure 14.1 The rotational and linear components of a collision.

14.1.2 Rotating Collisions

Figure 14.2 Three objects with different bounce characteristics.

14.1.3 Handling Rotating Collisions

14.2 Collision Impulses

14.2.1 Change to Contact Coordinates

Figure 14.3 The three sets of coordinates: world, local, and contact.

The Contact Coordinate Axes

The Basis Matrix

Excerpt from file include/cyclone/contacts.h

Excerpt from file src/contacts.cpp

Excerpt from file include/cyclone/core.h

Inverse Transformation

14.2.2 Velocity Change by Impulse

The Linear Component

The Angular Component

Putting It Together

Excerpt from file src/contacts.cpp

14.2.3 Impulse Change by Velocity

14.2.4 Calculating the Desired Velocity Change

Calculating the Closing Velocity

Calculating the Desired Velocity Change

14.2.5 Calculating the Impulse

Excerpt from file src/contacts.cpp

Excerpt from file src/contacts.cpp

14.2.6 Applying the Impulse

14.3 Resolving Interpenetration

14.3.1 Choosing a Resolution Method

Linear Projection

Figure 14.4 Linear projection causes realism problems.

Velocity-Based Resolution

Figure 14.5 Velocity-based resolution introduces apparent friction.

Nonlinear Projection

Relaxation

Figure 14.6 Nonlinear projection is more believable.

Figure 14.7 Nonlinear projection does not add friction.

14.3.2 Implementing Nonlinear Projection

Calculating the Components

Excerpt from file src/contacts.cpp

Applying the Movement

14.3.3 Avoiding Overrotation

Figure 14.8 Angular motion cannot resolve the interpenetration.

Figure 14.9 Angular resolution causes other problems.

14.4 The Collision Resolution Process

Figure 14.10 Data flow through the physics engine.

14.4.1 The Collision Resolution Pipeline

Excerpt from file include/cyclone/contacts.h

Excerpt from file src/contacts.cpp

Excerpt from file include/cyclone/contacts.h

14.4.2 Preparing Contact Data

Excerpt from file include/cyclone/contacts.h

Excerpt from file include/cyclone/contacts.h

Excerpt from file src/contacts.cpp

Excerpt from file include/cyclone/contacts.h

Excerpt from file src/contacts.cpp

Swapping Bodies

Excerpt from file include/cyclone/contacts.h

Excerpt from file src/contacts.cpp

Calculating Relative Velocity

Excerpt from file include/cyclone/contacts.h

Excerpt from file src/contacts.cpp

14.4.3 Resolving Penetration

Figure 14.11 Resolution order is significant.

Figure 14.12 Repeating the same pair of resolutions.

Iterative Algorithm Implemented

Figure 14.13 Resolving penetration can cause unexpected contact changes.

Updating Penetration Depths

Excerpt from file include/cyclone/contacts.h

Excerpt from file src/contacts.cpp

14.4.4 Resolving Velocity

Updating Velocities

14.4.5 Alternative Update Algorithms

Performance

14.5 Summary

14.6 Exercises

Exercise 14.1

Exercise 14.2

Exercise 14.3

Exercise 14.4

Exercise 14.5

15 Resting Contacts and Friction

15.1 Resting Forces

Figure 15.1 A reaction force at a resting contact.

15.1.1 Force Calculations

Figure 15.2 The long-distance dependence of reaction forces.

15.2 Microcollisions

Figure 15.3 Microcollisions replace reaction forces.

15.2.1 Removing Accelerated Velocity

Excerpt from file include/cyclone/body.h

Excerpt from file src/body.cpp

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

Dynamic Friction

Figure 15.4 A microscopic view of dynamic and static friction.

Rolling Friction

15.3.2 Isotropic and Anisotropic Friction

Figure 15.5 Anisotropic friction.

15.4 Implementing Friction

15.4.1 Friction as Impulses

15.4.2 Modifying the Velocity Resolution Algorithm

Velocity from Angular Motion

Excerpt from file include/cyclone/core.h

Excerpt from file src/contacts.cpp

Velocity from Linear Motion

15.4.3 Putting It All Together

Excerpt from file include/cyclone/contacts.h

Excerpt from file src/contacts.cpp

15.5 Friction and Sequential Contact Resolution

Figure 15.6 The problem with sequential contact resolution.

15.6 Summary

15.7 Exercises

Exercise 15.1

Exercise 15.2

Exercise 15.3

16 Stability and Optimization

16.1 Stability

16.1.1 Quaternion Drift

16.1.2 Interpenetration on Slopes

Figure 16.1 Objects drift down angled planes.

Excerpt from file src/contacts.cpp

16.1.3 Integration Stability

Newton-Euler 2

Runga-Kutta 4

16.1.4 The Benefit of Pessimistic Collision Detection

Figure 16.2 Collisions can be missed if they are not initially in contact.

16.1.5 Changing Mathematical Accuracy

Excerpt from include/cyclone/precision.h

16.2 Optimizations

16.2.1 Sleep

Adding Sleep State

Excerpt from file include/cyclone/body.h

Excerpt from file include/cyclone/body.h

Excerpt from file src/body.cpp

Putting Objects to Sleep

Waking Objects Up

Excerpt from file include/cyclone/contacts.h

Excerpt from file src/contacts.cpp

Figure 16.3 A chain of collisions is awakened.

16.2.2 Margins of Error for Penetration and Velocity

Figure 16.4 Iterative resolution makes microscopic changes.

16.2.3 Contact Grouping

Figure 16.5 Sets of independent contacts.

16.2.4 Code Optimizations

Caching Contact Data

Vectorizing Mathematics

Twizzling Rigid-Body Data

Grouping Data for Areas of the Level

16.3 Summary

17 Putting It All Together

17.1 Overview of the Engine

Figure 17.1 Data flow through the physics engine.

17.2 Using the Physics Engine

17.2.1 Ragdolls

Figure 17.2 Screenshot of the ragdoll demo.

Figure 17.3 Closeup of a ragdoll joint.

Excerpt from file include/cyclone/joints.h

Excerpt from file src/demos/ragdoll/ragdoll.cpp

More Complex Joints

17.2.2 Fracture Physics

Figure 17.4 Precreated fractures can look very strange for large objects.

Figure 17.5 Screenshot of the fracture demo.

Figure 17.6 The fractures of a concrete block.

Excerpt from file src/demos/fracture/fracture.cpp

Excerpt from file src/demos/fracture/fracture.cpp

17.2.3 Explosive Physics

Implosion

Excerpt from file include/cyclone/fgen.h

Concussion Wave

Figure 17.7 The force cross-section across a compression wave.

Excerpt from file include/cyclone/fgen.h

Convection Chimney

Excerpt from file include/cyclone/fgen.h

Figure 17.8 Screenshot of the explosion demo.

17.3 Limitations of the Engine

17.3.1 Stacks

17.3.2 Reaction Force Friction

17.3.3 Joint Assemblies

17.3.4 Stiff Springs

17.4 Summary

17.5 Projects

Mini-Project 17.1

Mini-Project 17.2

Mini-Project 17.3

Project 17.1

Project 17.2

PART VI Further Topics in Physics

18 Physics in Two Dimensions

18.1 2D or 3D?

18.2 Vector Mathematics

18.3 Particle and Mass Aggregate Physics

18.4 The Mathematics of Rotation

18.4.1 Representing Rotation

18.4.2 Matrices

18.5 Rigid-Body Dynamics

18.6 Collision Detection

Figure 18.1 The three types of contact in 2D.

18.6.1 VertexEdge Contacts

18.6.2 EdgeEdge Contacts

18.6.3 Contact Generation

18.7 Collision Response

18.8 Summary

18.9 Projects

Mini-Project 18.1

Mini-Project 18.2

Mini-Project 18.3

Project 18.1

Project 18.2

19 Other Programming Languages

19.1 ActionScript 3

19.2 C

19.3 Java

19.4 C#

19.5 Other Scripting Languages

20 Other Types of Physics

20.1 Simultaneous Contact Resolution

20.1.1 The Jacobian

20.1.2 The Linear-Complementarity Problem

How It Is Used

20.2 Reduced Coordinate Approaches

20.3 Summary

Back Matter

Appendix A Useful Inertia Tensors

A.1 Discrete Masses

A.2 Continuous Masses

A.3 Common Shapes

A.3.1 Cuboid

A.3.2 Sphere

A.3.3 Cylinder

A.3.4 Cone

A.3.5 Hemisphere

A.4 Moments of Inertia in 2D

A.4.1 Common 2D Shapes

Appendix B Useful Friction Coefficients

Appendix C Mathematics Summary

C.1 Vectors

C.2 Quaternions

C.3 Matrices

C.4 Integration

C.5 Physics

C.6 Other Formulas

Glossary

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