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Computer Graphics Through OpenGL
From Theory to Experiments
Third Edition
Computer Graphics Through OpenGL
From Theory to Experiments
Third Edition

Sumanta Guha
Asian Institute of Technology, Thailand
Cover Designed by Sumanta Guha

CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742

© 2019 by Taylor & Francis Group, LLC

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Version Date: 20181022

International Standard Book Number-13: 978-1-138-61264-8 (Hardback)

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Library of Congress Cataloging-in-Publication Data

Names: Guha, Sumanta,

author.
Title: Computer graphics through openGL / Sumanta Guha.
Description: Third edition. | Boca Raton : Taylor & Francis, a CRC title,
part of the Taylor & Francis imprint, a member of the Taylor & Francis
Group, the academic division of T&F Informa, plc, 2018. | Includes
bibliographical references and index.
Identifiers: LCCN 2018027365 | ISBN 9781138612648 (hardback : acid-free paper)
Subjects: LCSH: Computer graphics. | Microcomputers--Programming. | OpenGL. |
BISAC: COMPUTERS / Computer Graphics. | MATHEMATICS / General.
Classification: LCC T385 .G85 2018 | DDC 006.6/633--dc23
LC record available at https://lccn.loc.gov/2018027365

Visit the Taylor & Francis Web site at

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To my parents Utpal Chandra and
Santa Guha and to Kamaladi
Contents

P REFACE XVII

A BOUT THE AUTHOR XXVII

I Hello World 1
1 A N I NVITATION TO C OMPUTER G RAPHICS 3
1.1 Brief History of Computer Graphics . . . . . . . . . . . . . . . . . . . 5
1.2 Overview of a Graphics System . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Input Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 Output Devices . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Quick Preview of the Adventures Ahead . . . . . . . . . . . . . . . . . 14

2 O N TO O PEN GL AND 3D C OMPUTER G RAPHICS 17

2.1 First Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Orthographic Projection, Viewing Box and World Coordinates . . . . 19
2.3 The OpenGL Window and Screen Coordinates . . . . . . . . . . . . . 24
2.4 Clipping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Color, OpenGL State Machine and Interpolation . . . . . . . . . . . . 26
2.6 OpenGL Geometric Primitives . . . . . . . . . . . . . . . . . . . . . . 29
2.7 Approximating Curved Objects . . . . . . . . . . . . . . . . . . . . . . 35
2.8 Three Dimensions, the Depth Buffer and Perspective Projection . . . . 37
2.8.1 A Vital 3D Utility: The Depth Buffer . . . . . . . . . . . . . . 38
2.8.2 A Helix and Perspective Projection . . . . . . . . . . . . . . . 40
2.9 Drawing Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.10 Approximating Curved Objects Once More . . . . . . . . . . . . . . . 46
2.11 An OpenGL Program End to End . . . . . . . . . . . . . . . . . . . . 49
2.12 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . . 51

II Tricks of the Trade 53

3 A N O PEN GL T OOLBOX 55
3.1 Vertex Arrays and Their Drawing Commands . . . . . . . . . . . . . . 56
3.2 Vertex Buffer Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Vertex Array Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 Display Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 vii
CONTENTS 3.5 Drawing Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.6 Programming the Mouse . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.7 Programming Non-ASCII Keys . . . . . . . . . . . . . . . . . . . . . . 69
3.8 Menus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.9 Line Stipples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.10 FreeGLUT Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.11 Clipping Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.12 Frustum, Differently . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.13 Viewports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.14 Multiple Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.15 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . . 78

III Movers and Shapers 79

4 T RANSFORMATION , A NIMATION AND V IEWING 81
4.1 Modeling Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.1.1 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.1.2 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.1.3 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.2 Composing Modeling Transformations . . . . . . . . . . . . . . . . . . 88
4.3 Placing Multiple Objects . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4 Modelview Matrix Stack and Isolating Transformations . . . . . . . . 101
4.5 Animation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.5.1 Animation Technicals . . . . . . . . . . . . . . . . . . . . . . . 103
4.5.2 Animation Code . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.5.3 Animation Projects . . . . . . . . . . . . . . . . . . . . . . . . 115
4.6 Viewing Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.6.1 Understanding the Viewing Transformation . . . . . . . . . . . 117
4.6.2 Simulating a Viewing Transformation with Modeling Transfor-
mations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.6.3 Orientation and Euler Angles . . . . . . . . . . . . . . . . . . . 130
4.6.4 Viewing Transformation and Collision Detection in Animation 133
4.7 More Animation Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.7.1 Animating an Articulated Figure . . . . . . . . . . . . . . . . 136
4.7.2 Simple Orthographic Shadows . . . . . . . . . . . . . . . . . . 138
4.8 Selection and Picking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.8.1 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.8.2 Picking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.9 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . . 145

5 I NSIDE A NIMATION : T HE T HEORY OF T RANSFORMATIONS 147

5.1 Geometric Transformations in 2-Space . . . . . . . . . . . . . . . . . . 148
5.1.1 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.1.2 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.1.3 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.1.4 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.2 Affine Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.2.1 Affine Transformations Defined . . . . . . . . . . . . . . . . . . 154
5.2.2 Affine Transformations and OpenGL . . . . . . . . . . . . . . 158
5.2.3 Affine Transformations and Homogeneous Coordinates . . . . 160
viii 5.3 Geometric Transformations in 2-Space Continued . . . . . . . . . . . . 162
5.3.1 Affine Geometric Transformations . . . . . . . . . . . . . . . . 162 CONTENTS
5.3.2 Euclidean and Rigid Transformations . . . . . . . . . . . . . . 166
5.4 Geometric Transformations in 3-Space . . . . . . . . . . . . . . . . . . 173
5.4.1 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.4.2 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.4.3 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.4.4 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.4.5 Affine Geometric Transformations . . . . . . . . . . . . . . . . 187
5.4.6 Accessing and Manipulating the Current Modelview Matrix . 189
5.4.7 Euclidean and Rigid Transformations . . . . . . . . . . . . . . 191
5.5 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . . 196

6 A DVANCED A NIMATION T ECHNIQUES 197

6.1 Frustum Culling by Space Partitioning . . . . . . . . . . . . . . . . . . 197
6.1.1 Space Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . 199
6.1.2 Quadtrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
6.1.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.1.4 More about Space Partitioning . . . . . . . . . . . . . . . . . . 202
6.2 Occlusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
6.2.1 Querying and Culling . . . . . . . . . . . . . . . . . . . . . . . 202
6.2.2 Conditional Rendering . . . . . . . . . . . . . . . . . . . . . . 205
6.3 Timer Queries and Performance Measurememt . . . . . . . . . . . . . 205
6.4 Animating Orientation Using Euler Angles . . . . . . . . . . . . . . . . 206
6.4.1 Euler Angles and the Orientation of a Rigid Body . . . . . . . 206
6.4.2 Animating Orientation . . . . . . . . . . . . . . . . . . . . . . 207
6.4.3 Problems with Euler Angles: Gimbal Lock and Ambiguity . . 208
6.5 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6.5.1 Quaternion Math 101 . . . . . . . . . . . . . . . . . . . . . . . 210
6.5.2 Quaternions and Orientation . . . . . . . . . . . . . . . . . . . 213
6.6 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . . 221

IV Geometry for the Home Office 223

7 C ONVEXITY AND I NTERPOLATION 225
7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7.2 Convex Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
7.3 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
7.4 Convexity and the Convex Hull . . . . . . . . . . . . . . . . . . . . . . 234
7.5 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . . 239

8 T RIANGULATION 241
8.1 Definition and Justification . . . . . . . . . . . . . . . . . . . . . . . . 241
8.2 Steiner Vertices and the Quality of a Triangulation . . . . . . . . . . . 244
8.3 Triangulation in OpenGL and the Trouble with Non-Convexity . . . . 245
8.4 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . . 248

9 O RIENTATION 249
9.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
9.2 OpenGL Procedure to Determine Front and Back Faces . . . . . . . . 250
9.3 Consistently Oriented Triangulation . . . . . . . . . . . . . . . . . . . 255
9.4 Culling Obscured Faces . . . . . . . . . . . . . . . . . . . . . . . . . . 259
9.5 Transformations and the Orientation of Geometric Primitives . . . . . 261 ix
CONTENTS 9.6 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . . 262

V Making Things Up 263

10 M ODELING IN 3D S PACE 265
10.1 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
10.1.1 Specifying Plane Curves . . . . . . . . . . . . . . . . . . . . . 266
10.1.2 Specifying Space Curves . . . . . . . . . . . . . . . . . . . . . 269
10.1.3 Drawing Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 271
10.1.4 Polynomial and Rational Parametrizations . . . . . . . . . . . 274
10.1.5 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
10.1.6 Curves More Formally . . . . . . . . . . . . . . . . . . . . . . . 277
10.2 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
10.2.1 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
10.2.2 Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
10.2.3 Planar Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 285
10.2.4 General Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 286
10.2.5 Drawing General Surfaces . . . . . . . . . . . . . . . . . . . . . 287
10.2.6 Swept Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
10.2.7 Drawing Projects . . . . . . . . . . . . . . . . . . . . . . . . . 296
10.2.8 Ruled Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
10.2.9 Quadric Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 299
10.2.10 GLU Quadric Objects . . . . . . . . . . . . . . . . . . . . . . . 301
10.2.11 Regular Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . 303
10.2.12 Surfaces More Formally . . . . . . . . . . . . . . . . . . . . . . 307
10.3 Bézier Phrase Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
10.3.1 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
10.3.2 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
10.4 Importing Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
10.5 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
10.6 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . . 324

VI Lights, Camera, Equation 327

11 C OLOR AND L IGHT 329
11.1 Vision and Color Models . . . . . . . . . . . . . . . . . . . . . . . . . . 330
11.1.1 RGB Color Model . . . . . . . . . . . . . . . . . . . . . . . . . 331
11.1.2 CMY and CMYK Color Models . . . . . . . . . . . . . . . . . 333
11.1.3 HSV (or HSB) Color Model . . . . . . . . . . . . . . . . . . . 334
11.1.4 Summary of the Models . . . . . . . . . . . . . . . . . . . . . . 334
11.2 Phong’s Lighting Model . . . . . . . . . . . . . . . . . . . . . . . . . . 335
11.2.1 Phong Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
11.2.2 Specifying Light and Material Values . . . . . . . . . . . . . . 337
11.2.3 Calculating the Reflected Light . . . . . . . . . . . . . . . . . 337
11.2.4 First Lighting Equation . . . . . . . . . . . . . . . . . . . . . . 342
11.3 OpenGL Light and Material Properties . . . . . . . . . . . . . . . . . 344
11.3.1 Light Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 345
11.3.2 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . 346
11.3.3 Experimenting with Properties . . . . . . . . . . . . . . . . . . 347
11.3.4 Color Material Mode . . . . . . . . . . . . . . . . . . . . . . . 350
11.4 OpenGL Lighting Model . . . . . . . . . . . . . . . . . . . . . . . . . . 351
x 11.5 Directional Lights, Positional Lights and Attenuation of Intensity . . . 353
11.6 Spotlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 CONTENTS
11.7 OpenGL Lighting Equation . . . . . . . . . . . . . . . . . . . . . . . . 356
11.8 OpenGL Shading Models . . . . . . . . . . . . . . . . . . . . . . . . . 357
11.9 Animating Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
11.10 Partial Derivatives, Tangent Planes and Normal Vectors 101 . . . . . 358
11.11 Computing Normals and Lighting Surfaces . . . . . . . . . . . . . . . 365
11.11.1 Polygons and Planar Surfaces . . . . . . . . . . . . . . . . . . 365
11.11.2 Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
11.11.3 General Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 368
11.11.4 FreeGLUT, Bézier and Quadric Surfaces . . . . . . . . . . . . 371
11.11.5 Transforming Normals . . . . . . . . . . . . . . . . . . . . . . 373
11.11.6 Normalizing Normals . . . . . . . . . . . . . . . . . . . . . . . 375
11.12 Phong’s Shading Model . . . . . . . . . . . . . . . . . . . . . . . . . . 375
11.13 Lighting Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
11.14 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . 378

12 T EXTURE 379
12.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
12.2 Texture Space, Coordinates and Map . . . . . . . . . . . . . . . . . . . 382
12.3 Repeating and Clamping . . . . . . . . . . . . . . . . . . . . . . . . . . 385
12.4 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
12.5 Specifying the Texture Map . . . . . . . . . . . . . . . . . . . . . . . . 393
12.5.1 Parametrized Surfaces . . . . . . . . . . . . . . . . . . . . . . . 393
12.5.2 Bézier and Quadric Surfaces . . . . . . . . . . . . . . . . . . . 394
12.5.3 Spheres and Mercator Projection . . . . . . . . . . . . . . . . . 395
12.6 Texture Matrix and Animating Textures . . . . . . . . . . . . . . . . . 397
12.7 Lighting Textures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
12.8 Multitexturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
12.9 Rendering to Texture with Framebuffer Objects . . . . . . . . . . . . . 401
12.10 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . 405

13 S PECIAL V ISUAL T ECHNIQUES 407

13.1 Blending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
13.1.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
13.1.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
13.1.3 Opaque and Translucent Objects Together . . . . . . . . . . . 412
13.1.4 Blending Textures . . . . . . . . . . . . . . . . . . . . . . . . . 414
13.1.5 Creating Reflections . . . . . . . . . . . . . . . . . . . . . . . . 415
13.1.6 Motion Blur in World Space . . . . . . . . . . . . . . . . . . . 416
13.2 Fog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
13.3 Billboarding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
13.4 Antialiasing Points and Lines, Multisampling Polygons . . . . . . . . . 419
13.4.1 Antialiasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
13.4.2 Multisampling . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
13.5 Point Sprites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
13.6 Environment Mapping: Sphere Mapping and Cube Mapping . . . . . . 422
13.6.1 Sphere Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . 423
13.6.2 Cube Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
13.7 Stencil Buffer Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 429
13.7.1 OpenGL Buffers . . . . . . . . . . . . . . . . . . . . . . . . . . 429
13.7.2 Using the Stencil Buffer . . . . . . . . . . . . . . . . . . . . . . 430
13.8 Image Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 xi
CONTENTS 13.9 Bump Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
13.10 Shadow Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
13.11 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . 442

VII Pixels, Pixels, Everywhere 443

14 R ASTER A LGORITHMS 445
14.1 Cohen-Sutherland Line Clipper . . . . . . . . . . . . . . . . . . . . . . 445
14.2 Sutherland-Hodgman Polygon Clipper . . . . . . . . . . . . . . . . . . 449
14.3 DDA and Bresenham’s Line Rasterizers . . . . . . . . . . . . . . . . . 451
14.4 Scan-Based Polygon Rasterizer . . . . . . . . . . . . . . . . . . . . . . 456
14.4.1 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
14.4.2 Optimizing Using Edge Coherence – Active Edge List . . . . . 462
14.5 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . . 466

VIII Programming Pipe Dreams 467

15 O PEN GL 4.3, S HADERS AND THE P ROGRAMMABLE P IPELINE : L IFT -
OFF 469
15.1 New Pipeline for OpenGL . . . . . . . . . . . . . . . . . . . . . . . . . 470
15.1.1 Shaders in the Rendering Pipeline . . . . . . . . . . . . . . . . 470
15.1.2 New OpenGL . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
15.2 GLSL Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
15.3 First Core GL 4.3 Program (Dissected) . . . . . . . . . . . . . . . . . 476
15.4 Animation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
15.5 Lighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
15.5.1 Per-Vertex Lighting . . . . . . . . . . . . . . . . . . . . . . . . 487
15.5.2 Per-Pixel Lighting . . . . . . . . . . . . . . . . . . . . . . . . . 490
15.6 Textures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
15.6.1 Procedural Textures . . . . . . . . . . . . . . . . . . . . . . . . 493
15.6.2 Specular Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
15.6.3 Normal Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
15.7 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . . 498

16 O PEN GL 4.3, S HADERS AND THE P ROGRAMMABLE P IPELINE : E S -

CAPE V ELOCITY 499
16.1 Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
16.1.1 VAOs and Instanced Rendering Instead of Display Lists . . . . 499
16.1.2 Clipping Planes . . . . . . . . . . . . . . . . . . . . . . . . . . 502
16.1.3 Rest of the Toolbox . . . . . . . . . . . . . . . . . . . . . . . . 503
16.2 Shader Control Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
16.2.1 Shader Subroutines . . . . . . . . . . . . . . . . . . . . . . . . 505
16.2.2 Multiple Program Objects . . . . . . . . . . . . . . . . . . . . 506
16.3 Special Visual Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 508
16.3.1 Blending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
16.3.2 Points and Sprites . . . . . . . . . . . . . . . . . . . . . . . . . 509
16.3.3 Remaining Effects . . . . . . . . . . . . . . . . . . . . . . . . . 511
16.4 More Animation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
16.4.1 Picking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
16.4.2 Transform Feedback . . . . . . . . . . . . . . . . . . . . . . . . 515
xii 16.5 Tessellation Shaders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518
16.5.1 TCS (Tessellation Control Shader) . . . . . . . . . . . . . . . . 520 CONTENTS
16.5.2 TES (Tessellation Evaluation Shader) . . . . . . . . . . . . . . 522
16.5.3 TPG (Tessellation Primitive Generator) . . . . . . . . . . . . . 524
16.6 Geometry Shaders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
16.6.1 Particle System with Transform Feedback . . . . . . . . . . . . 535
16.7 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . . 538

IX Anatomy of Curves and Surfaces 539

17 B ÉZIER 541
17.1 Bézier Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
17.1.1 Linear Bézier Curves . . . . . . . . . . . . . . . . . . . . . . . 542
17.1.2 Quadratic Bézier Curves . . . . . . . . . . . . . . . . . . . . . 543
17.1.3 Cubic Bézier Curves . . . . . . . . . . . . . . . . . . . . . . . . 545
17.1.4 General Bézier Curves . . . . . . . . . . . . . . . . . . . . . . . 547
17.2 Bézier Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
17.3 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . . 556

18 B-S PLINE 557

18.1 Problems with Bézier Primitives: Motivating B-Splines . . . . . . . . . 558
18.2 B-Spline Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
18.2.1 First-Order B-Splines . . . . . . . . . . . . . . . . . . . . . . . 562
18.2.2 Linear B-Splines . . . . . . . . . . . . . . . . . . . . . . . . . . 563
18.2.3 Quadratic B-Splines . . . . . . . . . . . . . . . . . . . . . . . . 566
18.2.4 Cubic B-Splines . . . . . . . . . . . . . . . . . . . . . . . . . . 569
18.2.5 General B-Splines and Non-uniform Knot Vectors . . . . . . . 571
18.3 B-Spline Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583
18.4 Drawing B-Spline Curves and Surfaces . . . . . . . . . . . . . . . . . . 584
18.4.1 B-Spline Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 584
18.4.2 B-Spline Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 585
18.4.3 Lighting and Texturing a B-Spline Surface . . . . . . . . . . . 586
18.4.4 Trimmed B-Spline Surface . . . . . . . . . . . . . . . . . . . . 587
18.5 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . . 589

19 H ERMITE 591
19.1 Hermite Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
19.2 Natural Cubic Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
19.3 Cardinal Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596
19.4 Hermite Surface Patches . . . . . . . . . . . . . . . . . . . . . . . . . . 597
19.5 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . . 599

X Well Projected 601

20 A PPLICATIONS OF P ROJECTIVE S PACES : P ROJECTION T RANSFOR -
MATIONS AND R ATIONAL C URVES 603
20.1 OpenGL Projection Transformations . . . . . . . . . . . . . . . . . . . 604
20.1.1 Viewing Box to Canonical Viewing Box . . . . . . . . . . . . . 606
20.1.2 Viewing Frustum to Canonical Viewing Box . . . . . . . . . . 607
20.1.3 Projection Matrix in the Pipeline . . . . . . . . . . . . . . . . 610
20.2 Rational Bézier and NURBS Curves and Surfaces . . . . . . . . . . . . 612
20.2.1 Rational Bézier Curves Basics . . . . . . . . . . . . . . . . . . 612
20.2.2 Drawing Rational Bézier Curves . . . . . . . . . . . . . . . . . 615 xiii
CONTENTS 20.2.3 Rational Bézier Curves and Conic Sections . . . . . . . . . . . 616
20.2.4 Properties of Rational Bézier Curves . . . . . . . . . . . . . . 617
20.2.5 Rational Bézier Curves and Projective Invariance . . . . . . . 618
20.2.6 Rational Bézier Curves in the Real World . . . . . . . . . . . . 622
20.2.7 Rational Bézier Surfaces . . . . . . . . . . . . . . . . . . . . . 622
20.2.8 The ‘R’ in NURBS . . . . . . . . . . . . . . . . . . . . . . . . 623
20.3 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . . 624

XI Time for a Pipe 625

21 P IPELINE O PERATION 627
21.1 Synthetic-Camera Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . 627
21.1.1 Pipeline: Preliminary Version . . . . . . . . . . . . . . . . . . 628
21.1.2 Perspective Division by Zero . . . . . . . . . . . . . . . . . . . 629
21.1.3 Rasterization with Perspectively Correct Interpolation . . . . 632
21.1.4 Revised Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . 636
21.1.5 OpenGL Fixed-function Pipeline . . . . . . . . . . . . . . . . . 637
21.1.6 1D Primitive Example . . . . . . . . . . . . . . . . . . . . . . 638
21.1.7 Exercising the Pipeline . . . . . . . . . . . . . . . . . . . . . . 641
21.2 Ray Tracing Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
21.2.1 Going Global: Shadows . . . . . . . . . . . . . . . . . . . . . . 643
21.2.2 Going Even More Global: Recursive Reflection and Transmission 645
21.2.3 Implementing Ray Tracing . . . . . . . . . . . . . . . . . . . . 648
21.3 Radiosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649
21.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649
21.3.2 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650
21.3.3 Computing Form Factors . . . . . . . . . . . . . . . . . . . . . 652
21.3.4 Solving the Radiosity Equation to Determine Patch Brightnesses 654
21.3.5 Implementing Radiosity . . . . . . . . . . . . . . . . . . . . . . 655
21.4 Summary, Notes and More Reading . . . . . . . . . . . . . . . . . . . . 656

Appendices 657
A P ROJECTIVE S PACES AND T RANSFORMATIONS 657
A.1 Motivation and Definition of the Projective Plane . . . . . . . . . . . . 658
A.2 Geometry on the Projective Plane and Point-Line Duality . . . . . . . 660
A.3 Homogeneous Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 660
A.4 Structure of the Projective Plane . . . . . . . . . . . . . . . . . . . . . 662
A.4.1 Embedding the Real Plane in the Projective Plane . . . . . . . 662
A.4.2 A Thought Experiment . . . . . . . . . . . . . . . . . . . . . . 662
A.4.3 Regular Points and Points at Infinity . . . . . . . . . . . . . . 663
A.5 Snapshot Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 665
A.6 Homogeneous Polynomial Equations . . . . . . . . . . . . . . . . . . . 669
A.6.1 More About Point-Line Duality . . . . . . . . . . . . . . . . . 670
A.6.2 Lifting an Algebraic Curve from the Real to the Projective Plane 671
A.6.3 Snapshot Transformations Algebraically . . . . . . . . . . . . . 674
A.7 The Dimension of the Projective Plane and Its Generalization to Higher
Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674
A.8 Projective Transformations Defined . . . . . . . . . . . . . . . . . . . . 675
A.9 Projective Transformations Geometrically . . . . . . . . . . . . . . . . 676
A.10 Relating Projective, Snapshot and Affine Transformations . . . . . . . 681
xiv A.10.1 Snapshot Transformations via Projective Transformations . . . 682
A.10.2 Affine Transformations via Projective Transformations . . . . 683 CONTENTS
A.11 Designer Projective Transformations . . . . . . . . . . . . . . . . . . . 685

B M ATH S ELF -T EST 691

C M ATH S ELF -T EST S OLUTIONS 697
B IBLIOGRAPHY 705
S UBJECT I NDEX 713
P ROGRAM I NDEX 730

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