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# Mathematics for 3D Game Programming and Computer Graphics, Third...

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Mathematics for 3D Game Programming and Computer Graphics, Third Edition

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Mathematics for

3D Game Programming

and Computer Graphics

Third Edition

Eric Lengyel

Course Technology PTR

A part of Cengage Learning

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Mathematics for 3D Game Programming

and Computer Graphics, Third Edition

By Eric Lengyel

Publisher and General Manager,

Course Technology PTR:

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Sarah Panella

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© 2012 Course Technology, a part of Cengage Learning.

ALL RIGHTS RESERVED. No part of this work covered by the copyright

herein may be reproduced, transmitted, stored, or used in any form or by

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All images © Cengage Learning unless otherwise noted.

Library of Congress Control Number: 2011924487

ISBN-13: 978-1-4354-5886-4

ISBN-10: 1-4354-5886-9

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Printed in China

1 2 3 4 5 6 7 13 12 11

eISBN-10: 1-4354-5887-7

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iii

Contents

Preface xiii

What’s New in the Third Edition xiii

Contents Overview xiv

Website and Code Listings xvii

Notational Conventions xvii

Chapter 1 The Rendering Pipeline 1

1.1 Graphics Processors 1

1.2 Vertex Transformation 4

1.3 Rasterization and Fragment Operations 6

Chapter 2 Vectors 11

2.1 Vector Properties 11

2.2 The Dot Product 15

2.3 The Cross Product 19

2.4 Vector Spaces 26

Chapter 2 Summary 29

Exercises for Chapter 2 30

Chapter 3 Matrices 31

3.1 Matrix Properties 31

3.2 Linear Systems 34

3.3 Matrix Inverses 40

3.4 Determinants 47

3.5 Eigenvalues and Eigenvectors 54

3.6 Diagonalization 58

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iv Contents

Chapter 3 Summary 62

Exercises for Chapter 3 64

Chapter 4 Transforms 67

4.1 Linear Transformations 67

4.1.1 Orthogonal Matrices 68

4.1.2 Handedness 70

4.2 Scaling Transforms 70

4.3 Rotation Transforms 71

4.3.1 Rotation About an Arbitrary Axis 74

4.4 Homogeneous Coordinates 75

4.4.1 Four-Dimensional Transforms 76

4.4.2 Points and Directions 77

4.4.3 Geometrical Interpretation of the w Coordinate 78

4.5 Transforming Normal Vectors 78

4.6 Quaternions 80

4.6.1 Quaternion Mathematics 80

4.6.2 Rotations with Quaternions 82

4.6.3 Spherical Linear Interpolation 86

Chapter 4 Summary 89

Exercises for Chapter 4 91

Chapter 5 Geometry for 3D Engines 93

5.1 Lines in 3D Space 93

5.1.1 Distance Between a Point and a Line 93

5.1.2 Distance Between Two Lines 94

5.2 Planes in 3D Space 97

5.2.1 Intersection of a Line and a Plane 98

5.2.2 Intersection of Three Planes 99

5.2.3 Transforming Planes 101

5.3 The View Frustum 102

5.3.1 Field of View 103

5.3.2 Frustum Planes 106

5.4 Perspective-Correct Interpolation 107

5.4.1 Depth Interpolation 108

5.4.2 Vertex Attribute Interpolation 110

5.5 Projections 111

5.5.1 Perspective Projections 112

5.5.2 Orthographic Projections 116

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