Essential Mathematics for Games and Interactive Applications: A Programmer's GuidePublisher

705 Pages·9.033 MB·English

by  Kaufmann M.

This excellent volume is unique in that it covers not only the basic techniques of computer
graphicsandgamedevelopment,butalsoprovidesathoroughandrigorous—yetveryreadable—
treatmentoftheunderlyingmathematics.Fledglinggraphicsandgamesdeveloperswillfindit
avaluableintroduction;experienceddeveloperswillfinditaninvaluablereference.Everything
ishere,fromthedetailednumericissuesofIEEEfloatingpointnotation,tothecorrectwayto
usequaternionsandsphericallinearinterpolationtorepresentorientation,tothemathematics
ofcollisiondetectionandrigid-bodydynamics.
—DavidLuebke,UniversityofVirginia,
co-authorofLevelofDetailfor3DGraphics
Whenitcomestosoftwaredevelopmentforgamesorvirtualreality,youcannotescapethemath-
ematics.Thebestperformancecomesnotfromsuperfastprocessorsandterabytesofmemory,
butfromwell-chosenalgorithms.Withthisinmind,thetechniquesmostusefulfordeveloping
production-qualitycomputergraphicsforHollywoodblockbustersarenotthebestchoicefor
interactiveapplications.Whenrenderingtimesaremeasuredinmillisecondsratherthanhours,
youneedanentirelydifferentperspective.
Essential Mathematics for Games and Interactive Applications provides this perspective.
Whilethemathematicsarerigorousandperhapschallengingattimes,VanVerthandBishop
providethecontextforunderstandingthealgorithmsanddatastructuresneededtobringgames
andVRapplicationstolife.ThismaynotbetheonlybookyouwilleverneedforgamesandVR
softwaredevelopment,butitwillcertainlyprovideanexcellentframeworkfordevelopingrobust
andfastapplications.
—IanAshdown,President,ByHeartConsultantsLimited
WithEssentialMathematicsforGamesandInteractiveApplications,VanVerthandBishophave
providedinvaluableassistanceforprofessionalgamedeveloperslookingtoshoreupweaknesses
intheirmathematicaltraining.Evenifyouneverintendtowritearendererortuneaphysics
engine,thisbookprovidesthemathematicalandconceptualgroundingneededtounderstand
manyofthekeyconceptsinrendering,simulation,andanimation.
—DaveWeinstein,Microsoft,RedStormEntertainment
Geometry,trigonometry,linearalgebra,andcalculusareallessentialtoolsfor3Dgraphics.Math-
ematicscoursesinthesesubjectscovertoomuchground,whileatthesametimeglossingoverthe
bread-and-butteressentialsfor3Dgraphicsprogrammers.InEssentialMathematicsforGames
andInteractiveApplications,VanVerthandBishopbringjusttherightlevelofmathematicsout
ofthetrenchesofprofessionalgamedevelopment.Thisbookprovidesanaccessibleandsolid
mathematicalfoundationforinteractivegraphicsprogrammers.Ifyouareworkinginthearea
of3Dgames,thisbookisa“musthave.”
—JonathanCohen,DepartmentofComputerScience,
JohnsHopkinsUniversity,
co-authorofLevelofDetailfor3DGraphics
It’sthebookwithallthemathyouneedforgames.
—NeilKirby,BellLabs
As games become ever more sophisticated, mathematics and technical programming skills
becomeincreasinglyimportanttohaveinyourtoolbox.EssentialMathprovidesasolidfoun-
dationinmanycriticalareas.Youwillfindmanytopicscoveredindetail:fromlinearalgebra
to calculus, from physics to rasterization. Some of this will be review material, but you will
undoubtedlylearnsomethingnewand,mostimportantly,somethinguseful.
—ErinCatto,BlizzardEntertainment
Thispageintentionallyleftblank
Essential Mathematics
for Games and
Interactive Applications
A Programmer’s Guide
Second Edition
Thispageintentionallyleftblank
Essential Mathematics
for Games and
Interactive Applications
A Programmer’s Guide
Second Edition
James M. Van Verth
Lars M. Bishop
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Dedications
ToMurandFiona,forallowingmetoslaythemonsteronemoretime.—Jim
ToJen,whoconstantlyhelpsmegeteverythingdone;andtoNadiaandTasha,
whoconstantlyhelpmeavoidgettinganyofitdoneontime.—Lars
About the Authors
James M. Van Verth is an OpenGL Software Engineer at NVIDIA, where
he works on device drivers for NVIDIA GPUs. Prior to that, he was a found-
ing member of Red Storm Entertainment, where he was a lead engineer for
eightyears.Forthepastnineyearshealsohasbeenaregularspeakeratthe
GameDevelopersConference,teachingtheall-daytutorials“MathforGame
Programmers” and “Physics for Game Programmers,” on which this book
is based. His background includes a B.A. in Math/Computer Science from
Dartmouth College, an M.S. in Computer Science from the State University
ofNewYorkatBuffalo,andanM.S.inComputerSciencefromtheUniversity
ofNorthCarolinaatChapelHill.
LarsM.BishopisanengineerintheHandheldDeveloperTechnologiesgroup
at NVIDIA. Prior to joining NVIDIA, Lars was the Chief Technology Officer
at Numerical Design Limited, leading the development of the Gamebryo3D
cross-platform game engine. He received a BS in Math/Computer Science
fromBrownUniversityandanMSinComputerSciencefromtheUniversity
of North Carolina at Chapel Hill. His outside interests include photography,
drumming,andplayingbassguitar.
Contents
Preface
xix
Introduction
xxiii
Chapter 1
Real-World Computer Number Representation
1
1.1 Introduction 1
1.2 RepresentingRealNumbers 2
1.2.1 Approximations 2
1.2.2 PrecisionandError 3
1.3 Floating-PointNumbers 4
1.3.1 Review:ScientificNotation 4
1.3.2 ARestrictedScientificNotation 5
1.4 Binary“ScientificNotation” 6
1.5 IEEE754Floating-PointStandard 9
1.5.1 BasicRepresentation 9
1.5.2 RangeandPrecision 11
1.5.3 ArithmeticOperations 13
1.5.4 SpecialValues 16
1.5.5 VerySmallValues 19
1.5.6 CatastrophicCancelation 22
1.5.7 DoublePrecision 24
1.6 Real-WorldFloating-Point 25
1.6.1 InternalFPUPrecision 25
1.6.2 Performance 26
1.6.3 IEEESpecificationCompliance 29
1.6.4 GraphicsProcessingUnitsandHalf-Precision
Floating-PointFormats 31
1.7 Code 32
1.8 ChapterSummary 33
ix