Essential Mathematics for Games and Interactive Applications, Third Edition

599 Pages·2015·13.23 MB·English

by  James M. Van Verth

Computer Game Development VA N V E R T H Third Edition
B I S H O P
Essential Mathematics
THIRD EDITION
Essential Mathematics for Games
and Interactive Applications E Games Interactive
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Developed from the authors’ popular Game Developers Conference (GDC) tutorial, e
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Essential Mathematics for Games and Interactive Applications, Third Edition d n Applications
illustrates the importance of mathematics in 3D programming. It shows you how
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to properly animate, simulate, and render scenes and discusses the mathematics i
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behind the processes. n
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Completely revised to fix errors and make the content flow better, this third edition e M
reflects the increased use of shader graphics pipelines, such as in DirectX 11,
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OpenGL ES (GLES), and the OpenGL Core Profile. It also updates the material on a a James M. Van Verth
real-time graphics with coverage of more realistic materials and lighting.
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The book covers the low-level mathematical and geometric representations and t Lars M. Bishop
algorithms that are the core of any game engine. It also explores all the stages of i e
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the rendering pipeline. The authors explain how to represent, transform, view, and
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animate geometry. They then focus on visual matters, specifically the representa- a

tion, computation, and use of color. They also address randomness, intersecting
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geometric entities, and physical simulation. i
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Features p s
• Explains how the mathematical concepts apply to major parts of 3D games l
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• Covers the entire rendering pipeline, from representation and f
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animation to light simulation and rasterization
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• Shows how to display and manipulate game objects r
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• Presents techniques that make games look less structured and more organic
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• Includes source code, demos, tutorial slides, and other materials o
on a supplementary website n a
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This updated book provides you with a conceptual understanding of the mathe-
matics needed to create 3D games as well as a practical understanding of how e
these mathematical bases actually apply to games and graphics. It not only s
includes the theoretical mathematical background but also incorporates many
examples of how the concepts are used to affect how a game looks and plays.
THIRD EDITION
K23795
ISBN: 978-1-4822-5092-3
90000
9 781482 250923
Third Edition
Essential Mathematics
Games Interactive
for and
Applications
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Third Edition
Essential Mathematics
Games Interactive
for and
Applications
James M. Van Verth
Lars M. Bishop
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CRC Press
Taylor & Francis Group
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© 2016 by Taylor & Francis Group, LLC
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Version Date: 20150713
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ToDadandMom,
forallyourloveandsupport.—Jim
ToJen,
whoconstantlyhelpsmegeteverythingdone,and
toNadiaandTasha,
whoconstantlyhelpmeavoidgettinganyofitdoneontime.—Lars
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Contents
Preface xix
Authors xxiii
Introduction xxv
1. Representing Real Numbers 1
1.1 Introduction .................................................... 1
1.2 PreliminaryConcepts ............................................ 2
1.2.1 Fixed-PointNumbers .................................... 2
1.2.2 Approximations ......................................... 3
1.2.3 PrecisionandError ...................................... 3
1.3 Floating-PointNumbers.......................................... 5
1.3.1 Review:ScientificNotation ............................... 5
1.3.2 RestrictedScientificNotation ............................. 5
1.3.3 BinaryScientificNotation ................................ 6
1.4 IEEE754Floating-PointStandard ................................. 8
1.4.1 BasicRepresentation..................................... 8
1.4.2 RangeandPrecision ..................................... 10
1.4.3 ArithmeticOperations.................................... 11
1.4.3.1 AdditionandSubtraction........................ 12
1.4.3.2 Multiplication ................................. 13
1.4.3.3 RoundingModes .............................. 14
1.4.4 SpecialValues .......................................... 15
1.4.4.1 Zero ......................................... 15
1.4.4.2 Infinity....................................... 16
1.4.4.3 NonnumericValues ............................ 17
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viii Contents
1.4.5 VerySmallValues ....................................... 18
1.4.5.1 NormalizedMantissasandtheHoleatZero ........ 18
1.4.5.2 DenormalsandGradualUnderflow ............... 19
1.4.6 CatastrophicCancelation ................................. 20
1.4.7 ComparingFloating-PointNumbers ........................ 23
1.4.8 DoublePrecision ........................................ 24
1.4.9 HalfPrecision .......................................... 24
1.5 Real-WorldFloatingPoint ........................................ 26
1.5.1 InternalFPUPrecision ................................... 26
1.5.2 Performance............................................ 27
1.5.2.1 PerformanceofDenormalizedNumbers ........... 27
1.5.2.2 SoftwareFloating-PointEmulation ............... 29
1.5.3 IEEESpecificationCompliance............................ 29
1.5.4 PrecisioninGraphicsProcessingUnits...................... 30
1.6 Code .......................................................... 31
1.7 ChapterSummary ............................................... 31
2. Vectors and Points 33
2.1 Introduction .................................................... 33
2.2 Vectors ........................................................ 33
2.2.1 GeometricVectors....................................... 34
2.2.2 RealVectorSpaces ...................................... 36
2.2.2.1 Definition .................................... 36
2.2.2.2 Examples..................................... 37
2.2.2.3 Subspaces .................................... 39
2.2.3 LinearCombinationsandBasisVectors ..................... 39
2.2.4 BasicVectorClassImplementation......................... 42
2.2.5 VectorLength........................................... 44
2.2.6 DotProduct ............................................ 46
2.2.7 Gram–SchmidtOrthogonalization.......................... 50
2.2.8 CrossProduct........................................... 51
2.2.9 TripleProducts.......................................... 55
2.2.10 GrassmanAlgebra....................................... 58
2.3 Points ......................................................... 58
2.3.1 PointsasGeometry ...................................... 58
2.3.2 AffineSpaces........................................... 60
2.3.3 AffineCombinations..................................... 62
2.3.4 PointImplementation .................................... 63
2.3.5 PolarandSphericalCoordinates ........................... 64
2.4 Lines.......................................................... 67
2.4.1 Definition.............................................. 67
2.4.2 ParameterizedLines ..................................... 68
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Contents ix
2.4.3 GeneralizedLineEquation................................ 69
2.4.4 CollinearPoints......................................... 71
2.5 Planes......................................................... 71
2.5.1 ParameterizedPlanes .................................... 71
2.5.2 GeneralizedPlaneEquation ............................... 72
2.5.3 CoplanarPoints ......................................... 73
2.6 PolygonsandTriangles .......................................... 73
2.7 ChapterSummary ............................................... 76
3. LinearTransformations and Matrices 77
3.1 Introduction .................................................... 77
3.2 LinearTransformations .......................................... 78
3.2.1 Definitions ............................................. 78
3.2.2 LinearTransformationsandBasisVectors ................... 79
3.2.3 RangeandNullSpace.................................... 80
3.3 Matrices ....................................................... 82
3.3.1 IntroductiontoMatrices .................................. 82
3.3.2 SimpleOperations....................................... 84
3.3.2.1 MatrixAdditionandScalarMultiplication ......... 84
3.3.2.2 Transpose .................................... 84
3.3.3 VectorRepresentation.................................... 85
3.3.4 BlockMatrices.......................................... 86
3.3.5 MatrixProduct.......................................... 87
3.3.6 TransformingVectors .................................... 89
3.3.7 CombiningLinearTransformations......................... 91
3.3.8 IdentityMatrix.......................................... 93
3.3.9 PerformingVectorOperationswithMatrices................. 94
3.3.10 Implementation ......................................... 95
3.4 SystemsofLinearEquations ...................................... 97
3.4.1 Definition.............................................. 97
3.4.2 SolvingLinearSystems .................................. 98
3.4.3 GaussianElimination .................................... 100
3.5 MatrixInverse .................................................. 103
3.5.1 Definition.............................................. 103
3.5.2 SimpleInverses ......................................... 105
3.6 Determinant.................................................... 106
3.6.1 Definition.............................................. 106
3.6.2 ComputingtheDeterminant............................... 108
3.6.3 DeterminantsandElementaryRowOperations ............... 110
3.6.4 AdjointMatrixandInverse................................ 112
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