3DMathPrimerfor
GraphicsandGame
Development
FletcherDunn
andIanParberry
WordwarePublishing,Inc.
LibraryofCongressCataloging-in-PublicationData
Dunn,Fletcher.
3Dmathprimerforgraphicsandgamedevelopment/byFletcherDunnandIanParberry.
p.cm.
ISBN1-55622-911-9
1putergraphics.2putergames--Programming.3puterscience--Mathematics.
I.Parberry,Ian.II.Title.
T385.D8752002
006.6--dc212002004615
CIP
2002,WordwarePublishing,Inc.
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ISBN1-55622-911-9
10987654321
0205
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Contents
Acknowledgments...........xi
Chapter1
Introduction...........1
1.1Whatis3DMath........1
1.2WhyYouShouldReadThisBook1
1.3WhatYouShouldKnowBeforeReadingThisBook..3
1.4Overview.3
Chapter2
TheCartesianCoordinateSystem5
2.11DMathematics.........6
2.22DCartesianMathematics....9
2.2.1AnExample:TheHypotheticalCityofCartesia..9
2.2.2Arbitrary2DCoordinateSpaces..........10
2.2.3SpecifyingLocationsin2DUsingCartesianCoordinates........13
2.3From2Dto3D.........14
2.3.1ExtraDimension,ExtraAxis...........15
2.3.2SpecifyingLocationsin3D15
2.3.3Left-handedvs.Right-handedCoordinateSpaces16
2.3.4SomeImportantConventionsUsedinThisBook.19
2.4Exercises.20
Chapter3
MultipleCoordinateSpaces..23
3.1WhyMultipleCoordinateSpaces...........24
3.2SomeUsefulCoordinateSpaces25
3.2.1WorldSpace........25
3.2.2ObjectSpace........26
3.2.3CameraSpace.......27
3.2.4InertialSpace........28
3.3NestedCoordinateSpaces....30
3.4SpecifyingCoordinateSpaces.31
3.5CoordinateSpaceTransformations...........31
3.6Exercises.34
Chapter4
Vectors.35
4.1Vector—AMathematicalDefinition.........36
4.1.1Vectorsvs.Scalars.....36
4.1.2VectorDimension.....36
4.1.3Notation..........36
4.2Vector—AGeometricDefinition...........37
iii
4.2.1WhatDoesaVectorLookLike..........37
4.2.2Positionvs.Displacement.38
4.2.3SpecifyingVectors.....38
4.2.4VectorsasaSequenceofDisplacements.....39
4.3Vectorsvs.Points........40
4.3.1RelativePosition......41
4.3.2TheRelationshipBetweenPointsandVectors..41
4.4Exercises.42
Chapter5
OperationsonVectors.....45
5.1LinearAlgebravs.WhatWeNeed...........46
5.2TypefaceConventions......46
5.3TheZeroVector.........47
5.4NegatingaVector........48
5.4.1OfficialLinearAlgebraRules...........48
5.4.2GeometricInterpretation..48
5.5VectorMagnitude(Length)...49
5.5.1OfficialLinearAlgebraRules...........49
5.5.2GeometricInterpretation..50
5.6VectorMultiplicationbyaScalar51
5.6.1OfficialLinearAlgebraRules...........51
5.6.2GeometricInterpretation..52
5.7NormalizedVectors.......53
5.7.1OfficialLinearAlgebraRules...........53
5.7.2GeometricInterpretation..53
5.8VectorAdditionandSubtraction54
5.8.1OfficialLinearAlgebraRules...........54
5.8.2GeometricInterpretation..55
5.8.3VectorfromOnePointtoAnother.........57
5.9TheDistanceFormula......57
5.10VectorDotProduct.......58
5.10.1OfficialLinearAlgebraRules..........58
5.10.2GeometricInterpretation.59
5.10.3ProjectingOneVectorontoAnother.......61
5.11VectorCrossProduct......62
5.11.1OfficialLinearAlgebraRules..........62
5.11.2GeometricInterpretation.62
5.12LinearAlgebraIdentities...65
5.13Exercises67
Chapter6
ASimple3DVectorClass....69
6.1ClassInterface..........69
6.2ClassVector3Definition....70
6.3DesignDecisions........73
6.3.1Floatsvs.Doubles.....73
6.3.2OperatorOverloading...73
iv
Contents。