Mathematics for Game Programming and Computer Graphics: Explore the essential mathematics for creating, rendering, and manipulating 3D virtual environments

445 Pages·2022·34.146 MB·English

by  Penny de Byl

Mathematics for Game
Programming and
Computer Graphics
Explore the essential mathematics for creating, rendering, and
manipulating 3D virtual environments
Penny de Byl
BIRMINGHAM—MUMBAI
Mathematics for Game Programming and Computer
Graphics
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Life-long learning has always been my goal. But I couldn’t have done it alone.
I’d like to thank Adrian who has been my friend, technical reviewer, and code
tester throughout. He possesses the attention to detail I lack. And, I might add,
if the code doesn’t work – it’s his fault. Next, I’d like to thank my wonderful
daughter, Tabytha, who is in her first year of studying theoretical mathematics
and computer science at university. She’s been a great sounding board in
helping me understand mathematical notations that I still struggle with. A
mother could not be prouder than I am of her.
– Penny de Byl
Co n t r i b u t o r s
About the author
Penny de Byl is a full-stack developer with an honors degree in graphics and a PhD in artificial
intelligence for games. She has a passion for teaching, and has been teaching games development
and computer graphics for over 25 years at universities in Australia and Europe. Her best-selling
textbooks, including Holistic Game Development with Unity, are used in over 100 institutions. She
has won numerous awards for teaching, including an Australian Government Excellence in Teaching
Award and the Unity Mobile Game Curriculum Competition. Her approach to teaching computer
science and related fields is project-based, giving you hands-on workshops you can immediately get
your teeth into. The full range of her teaching interests can be found at h3dlearn.com.
I’d like to thank the wonderful publishing crew and editors at Packt Publishing,
who’ve been attentive, patient, highly professional, and available throughout the
writing process.
About the reviewers
Amit Verma is a game developer and WebGL graphics engineer from Dehradun, Uttarakhand, who
has developed web versions of the Pixlr photo editor. He has rich experience in the gaming industry,
with development experience of more than 60 games and 30 WebGL applications, using Unity3D and
WebGL. His passion and curiosity toward gaming technologies and their challenges have helped his
clients to launch games on various platforms, such as web, mobile, and consoles. He holds a BTech
degree in computer science and engineering from Uttar Pradesh Technical University, where he studied
computer science and graphics programming. Currently, he is working at Excellarate as a lead WebGL
graphics engineer for the development of metaverse content using WebGL.
I would like to thank my best friend, my wife, and the love of my life, Simran
Gogna, for supporting me in my hobbies and professional career. Her selfless
love and wise advice gave me the strength to reach my goals. I also wish to
thank my mentors, Sumeet Arora and Shiwani Arora, for providing inspiration
and direction in my early career days.
When Soumya Mondal is not saving the world, he cosplays as a student at the Technical University
of Munich and as a visual AI researcher at a start-up. He has shipped a few games, raising awareness
of social issues, and has a strong belief in the power of games as a medium. In his free time, he travels
and runs.
Table of Contents
Preface xiii
Part 1 – Essential Tools
1
Hello Graphics Window: You’re On Your Way 3
Technical requirements 4 Let’s do it… 15
Getting Started with Python, Your Turn… 19
PyCharm, and Pygame 5
Summary 19
Creating a Basic Graphics Window 8
Answers 20
Working with Window and Cartesian
Coordinates 14
2
Let’s Start Drawing 21
Technical requirements 22 Your turn… 32
Color 22 Polygons 33
Let’s do it… 23 Let’s do it… 34
Your turn… 25 Your turn… 35
Lines 25 Raster Images 35
Let’s do it… 25 Let’s do it… 36
Your turn… 26
Summary 38
Let’s do it… 27
Answers 38
Text 29
Let’s do it… 31
vi Table of Contents
3
Line Plotting Pixel by Pixel 41
Technical requirements 41 Let’s do it… 47
The Naïve Way: Drawing a line Drawing circles the Bresenham way 52
with brute force 42
Let’s do it… 53
Let’s do it… 43
Anti-aliasing 56
Enter Bresenham: The
Summary 59
improved approach 45
4
Graphics and Game Engine Components 61
Technical requirements 61 Understanding 3D Coordinate
Exploring the OpenGL Graphics Systems in OpenGL 73
Pipeline 62 Perspective Projection 74
Let’s do it… 63 Your turn… 76
Projecting into the NDC 76
Drawing Models with Meshes 65
Let’s do it… 79
Let’s do it… 67
Your turn… 82
Your turn… 69
Summary 82
Viewing the Scene with Cameras 69
Answers 83
Projecting Pixels onto the Screen 73
5
Let’s Light It Up! 85
Technical requirements 85 Let’s do it… 93
Adding Lighting Effects 85 Summary 97
Let’s do it… 87 Answers 98
Your turn… 90
Placing Textures on Meshes 90
Let’s do it… 90
Table of Contents vii
6
Updating and Drawing the Graphics Environment 99
Technical requirements 100 Measuring time 108
Introducing the main game loop 100 Let’s do it… 109
Updating and drawing objects 102 Summary 110
Let’s do it… 104 Answers 110
Your turn… 108
7
Interactions with the Keyboard and Mouse for Dynamic Graphics
Programs 113
Technical requirements 113 Your turn… 130
Working with mouse interactions 114 Let’s do it… 131
Let’s do it… 115 Adding keyboard commands 132
Your turn… 119 Let’s do it… 132
Combining 2D and 3D environments 120 Your turn… 133
Let’s do it… 120 Let’s do it… 134
Converting the mouse to projection
Summary 135
coordinates 126
Let’s do it… 128 Answers 135
Part 2 – Essential Trigonometry
8
Reviewing Our Knowledge of Triangles 141
Technical requirements 142 Calculating angles with sine, cosine,
Comparing similar triangles 142 and tangent 148
Working with right-angled triangles 145 Investigating triangles by drawing 3D objects 151
Your turn… 147 Summary 156
Answers 156
viii Table of Contents
9
Practicing Vector Essentials 159
Technical requirements 160 Exploring the relationship between
Understanding the difference between angles and vectors 172
points and vectors 160 The dot product 174
Defining vector key operations 165 The cross product 174
Let’s do it… 169 Summary 176
Working out a vector’s magnitude 171 Answers 177
Your turn… 172
10
Getting Acquainted with Lines, Rays, and Normals 179
Technical requirements 180 Your turn… 189
Defining lines, segments, and rays 180 Let’s do it… 189
Your turn… 193
Using the parametric form of lines 182
Your turn… 183 Summary 193
Let’s do it… 184 Answers 194
Calculating and displaying normals 187
11
Manipulating the Light and Texture of Triangles 197
Technical requirements 198 Let’s do it… 206
Displaying Mesh Triangle Normals 198 Let’s do it… 208
Let’s do it… 199 Exploring How Normals Affect
Lighting 214
Defining polygon sides with normals 202
Let’s do it… 215
Working with visible sides of a polygon 203
Summary 220
Culling Polygons According
to the Normals 205
Table of Contents ix
Part 3 – Essential Transformations
12
Mastering Affine Transformations 223
Technical requirements 224 Rotating points around a pivot 231
Translating points in 3D 225 Let’s do it… 233
Your turn… 226 Exploring transformation orders 236
Scaling points with x, y, and z 227 Shearing and reflections 239
Let’s do it… 228 Summary 241
Your turn… 231 Answers 241
13
Understanding the Importance of Matrices 243
Technical requirements 244 Moving from linear equations to matrix
operations 252
Defining matrices 244
Compounding affine transformations 254
Performing operations on matrices 245
Combining transformation matrices
Adding and subtracting matrices 245
for complex maneuvers 256
Multiplying by a single value 246
Dividing matrices 248 Let’s do it… 257
Creating matrix representations of Summary 263
affine transformations 252 Answers 264
14
Working with Coordinate Spaces 267
Technical requirements 268 Let’s do it… 276
Understanding OpenGL’s Matrix Stack 268 Working with the Projection Matrix 280
Working with the Model Matrix 270 Let’s do it… 281
Let’s do it… 271 Your turn… 283
Your turn… 275
Summary 284
Working with the View Matrix 276 Answers 284