Table Of ContentFoundations
of
Game Engine Development
Foundations
of
Game Engine Development
2
V
OLUME
R
ENDERING
by Eric Lengyel
Terathon Software LLC
Lincoln, California
Foundations of Game Engine Development
Volume 2: Rendering
ISBN-13: 978-0-9858117-5-4
Copyright © 2019, by Eric Lengyel
All rights reserved. No part of this publication may be reproduced, stored in an
information retrieval system, transmitted, or utilized in any form, electronic or
mechanical, including photocopying, scanning, digitizing, or recording, without
the prior permission of the copyright owner.
Fourth printing
Published by Terathon Software LLC
www.terathon.com
Series website: foundationsofgameenginedev.com
About the cover: The cover image is a scene from a game entitled The 31st,
developed with the Tombstone Engine. Artwork by Javier Moya Pérez.
0
Contents
Preface ix
Chapter 5 Graphics Processing 1
5.1 Pixels 1
5.2 Color Science 4
5.2.1 �e CIE RGB Color Space 6
5.2.2 �e CIE XYZ Color Space 8
5.2.3 �e sRGB Color Space 12
5.3 Gamma Correction 18
5.4 World Structure 21
5.4.1 Coordinate Spaces 22
5.4.2 Transform Hierarchy 26
5.4.3 Vertex Transformations 28
5.5 �e Graphics Pipeline 32
5.5.1 Geometry Processing 32
5.5.2 Pixel Processing 38
5.5.3 Frame Buffer Operations 45
Exercises for Chapter 5 51
Chapter 6 Projections 53
6.1 �e View Frustum 53
6.2 Perspective-Correct Interpolation 58
6.3 Projection Matrices 64
v
vi Contents
6.3.1 Perspective Projection Matrices 66
6.3.2 Infinite Projection Matrices 69
6.3.3 Projected Depth Precision 71
6.3.4 Orthographic Projections 75
6.3.5 Frustum Plane Extraction 78
6.4 Oblique Clipping Planes 80
Exercises for Chapter 6 86
Chapter 7 Shaders 89
7.1 Rendering Fundamentals 90
7.1.1 Luminance 90
7.1.2 �e Rendering Equation 94
7.2 Diffuse Reflection 96
7.3 Specular Reflection 100
7.4 Texture Mapping 104
7.4.1 Texture Coordinates 104
7.4.2 Conventional Texture Mapping 105
7.4.3 Cube Texture Mapping 106
7.5 Tangent Space 110
7.6 Bump Mapping 116
7.6.1 Normal Map Construction 117
7.6.2 Rendering with Normal Maps 120
7.6.3 Blending Normal Maps 122
7.7 Parallax Mapping 123
7.8 Horizon Mapping 127
7.8.1 Horizon Map Construction 129
7.8.2 Rendering with Horizon Maps 133
7.8.3 Ambient Occlusion Mapping 138
Exercises for Chapter 7 139
Chapter 8 Lighting and Shadows 141
8.1 Light Sources 141
8.1.1 Point Lights 142
8.1.2 Spot Lights 148
8.1.3 Infinite Lights 152
Contents vii
8.2 Extent Optimization 153
8.2.1 Scissor Rectangle 155
8.2.2 Depth Bounds 160
8.3 Shadow Maps 170
8.3.1 2D Shadow Maps 170
8.3.2 Cube Shadow Maps 174
8.3.3 Cascaded Shadow Maps 178
8.3.4 Shadow Depth Offset 190
8.4 Stencil Shadows 194
8.4.1 Rendering Algorithm 194
8.4.2 Variant Selection 200
8.4.3 Shadow Volumes 203
8.4.4 Optimizations 209
8.5 Fog 213
8.5.1 Absorption and Scattering 213
8.5.2 Halfspace Fog 217
Exercises for Chapter 8 223
Chapter 9 Visibility and Occlusion 225
9.1 Polygon Clipping 225
9.2 Polyhedron Clipping 230
9.3 Bounding Volumes 240
9.3.1 Bounding Spheres 241
9.3.2 Bounding Boxes 245
9.4 Frustum Culling 253
9.4.1 Visibility Regions 254
9.4.2 Sphere Visibility 257
9.4.3 Box Visibility 259
9.5 Light Culling 261
9.6 Shadow Culling 264
9.7 Portal Systems 269
9.7.1 Zones and Portals 270
9.7.2 Light Regions 275
9.8 Occluders 278
viii Contents
9.9 Fog Occlusion 286
Exercises for Chapter 9 295
Chapter 10 Advanced Rendering 297
10.1 Decals 297
10.2 Billboards 300
10.2.1 Spherical Billboards 301
10.2.2 Cylindrical Billboards 305
10.2.3 Polyboards 307
10.2.4 Trimming 310
10.3 �e Structure Buffer 312
10.4 Volumetric Effects 316
10.4.1 Halos 318
10.4.2 Shafts 321
10.5 Ambient Occlusion 328
10.5.1 �e Occlusion Buffer 329
10.5.2 Depth-Aware Blurring 337
10.6 Atmospheric Shadowing 340
10.6.1 �e Atmosphere Buffer 340
10.6.2 Sample Randomization 346
10.6.3 Anisotropic Scattering 347
10.6.4 Implementation 349
10.7 Motion Blur 355
10.7.1 �e Velocity Buffer 356
10.7.2 Image Postprocessing 363
10.8 Isosurface Extraction 370
10.8.1 Marching Cubes 372
10.8.2 Preferred Polarity 375
10.8.3 Implementation 378
Exercises for Chapter 10 392
Index 393
0
Preface
�is book explores the mathematical foundations of real-time rendering methods
used in modern game engines. �is vast subject includes an enormous variety of
techniques that must balance the goals of visual quality and performance. �ese
goals are achieved through the application of solid engineering principles, and we
emphasize that fact by including a large amount of technical detail. Because de-
tails require a lot of space, we could not hope to cover every major topic in the
field of computer graphics in a single volume, so we focus primarily on the most
important fundamentals. But even those topics have become rather complex, and
this book is filled with some lengthy expositions as a result.
�e six chapters in this book are numbered five through ten, and this number-
ing is a continuation of the four chapters appearing in the first volume. In this
second volume, we assume that the reader is familiar with the subject matter cov-
ered in Chapters 1–3, which includes vectors, matrices, transforms, and basic 3D
geometry. We do not rely on any knowledge of the advanced algebra covered in
Chapter 4. �e first volume is not required reading for someone who is already
proficient in mathematics, but there are occasional references in this book to spe-
cific locations in Volume 1. �ose can be identified by a first number of 1, 2, 3, or
4 in the x.y notation used for numbering sections, equations, etc.
We begin the second volume with a general overview of graphics processing
in Chapter 5. �is includes an introduction to color science, a discussion of game
world structure, and a light review of graphics hardware capabilities. Chapter 6 is
much more mathematical, and it contains a thorough treatment of projections. In
Chapter 7, we move on to the topic of shading, where we describe the details of
basic rendering concepts as well as some advanced texture mapping techniques.
�at is followed by a rather long discussion of lighting and shadows in Chapter 8.
ix
x Preface
�e focus changes somewhat in Chapter 9 where we address performance on a
large scale by visiting the topics of visibility determination and occlusion culling.
Finally, Chapter 10 presents several advanced rendering techniques in extreme
detail. Much of this combines the author’s own original research with existing
methods to provide highly practical engineering advice.
�is book does not teach the use of any specific graphics API or shading lan-
guage, and it is not intended to be a primary source of information about how to
use a graphics processor in general. Many excellent resources on these topics are
available, and we prefer not to include redundant material here. However, it was
necessary to choose certain conventions employed by graphics APIs and shading
languages to be used throughout this book. �e choices of axes in various coordi-
nate systems generally agree with Direct3D and Vulkan. Shader code is written in
a generic hybrid of HLSL and GLSL. �e types and built-in functions found in
our shaders use the names defined by HLSL, but uniform constants and texture
functions use the GLSL syntax for its succinctness. Conversion to the reader’s
shading language of choice is expected to be a simple task.
�ere are many example code listings in this book, and most of them are
much more complicated than the listings found in Volume 1. Programs that run
on the CPU are written in C++, and programs that run on the GPU are written in
our generic shading language. None of these examples are written in “pseudo-
code”. With the exception of minor tweaks that might be necessary to compile for
a specific environment, all of the examples consist of production-ready code that
works. �e code listings can be downloaded from the website given below, and a
license for its usage is included. To save space, a lot of the code appearing in the
book is written in a highly compact form, sometimes to the detriment of readabil-
ity. �e code on the website is more loosely formatted. In the case of the march-
ing cubes algorithm discussed at the end of Chapter 10, some data tables have
been omitted from the code listings in the book, but they are included with the
code on the website.
We assume that the reader has a solid understanding of basic trigonometry
and the linear algebra topics covered by Chapters 1–3 of the first volume. We
also assume a proficiency in calculus (including Newton’s method) because its
use is necessary at many places throughout this book. On the engineering side,
we assume that the reader has a working knowledge of the C++ language, some
experience using a GPU shading language, and an understanding of how standard
floating-point numbers operate.
Important equations and key results appearing in the text are boxed with a
blue outline. This is intended both to highlight the most valuable information and
to make it easier to find when using the book as a reference.
Preface xi
Each chapter concludes with a set of exercises, and many of those exercises
ask for a short proof of some kind. The exercises are designed to provide addi-
tional educational value, and while many of them have easy solutions, others are
more involved. To ensure that getting stuck doesn’t deprive any reader of a valu-
able learning experience, the answers to all of the exercises are provided on the
website cited below.
This book is the second volume in a series that covers a wide range of topics
related to game engine development. The official website for the Foundations of
Game Engine Development series can be found at the following address:
foundationsofgameenginedev.com
This website contains information about all of the books in the series, including
announcements, errata, code listings, and answers to exercises.
Mathematical Conventions
We continue to use the mathematical conventions and much of the same notation
that was introduced in the first volume:
■ A vector v representing a direction vector is written in a bold style, and a
vector representing a point is written in a bold script style. �ese are differ-
entiated because, when extending to 4D homogeneous coordinates, a direc-
tion v acquires a w coordinate of zero, and a point acquires a w coordinate
of one.
■ The notation (p|w) represents a homogeneous point with p=(x,y,z), the
notation [n|d] represents a plane with normal direction n and distance to
origin d, and the notation {v|m} represents a line with direction v and mo-
ment m.
■ Indexes referring to rows and columns of a matrix are zero-based. For exam-
ple, the notation M refers to the upper-left entry of the matrix M.
00
■ �e notation M refers to the zero-based j-th column of the matrix M. If M
[j]
is a 4×4 transformation matrix having a fourth row of [0 0 0 1], then M ,
[0]
M , and M are direction vectors, and M is a point.
[1] [2] [3]
■ Ordinary vectors are treated as column matrices. Normals and planes, which
are antivectors, are treated as row matrices.
■ A matrix M transforms a vector v as Mv. A normal is transformed as nM−1.
■ If v is a 4D vector, then the notation v refers to the 3D vector consisting of
xyz
the x, y, and z components.
