Geometric Algebra for Computer Science (Revised Edition): An Object-Oriented Approach to Geometry (The Morgan Kaufmann Series in Computer Graphics)

622 Pages·2007·10.52 MB·English

by  Leo Dorst

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09 10 11 5 4 3 2
List of Figures
1.1 Exampleoftheuseofgeometricalgebra 2
1.2 CodetogenerateFigure1.1 5
1.3 Exampleoftheuseofgeometricalgebra 6
1.4 Theouterproductanditsinterpretations 11
2.1 Spanninghomogeneoussubspacesina3-Dvectorspace 25
2.2 Imaginingvectoraddition 27
2.3 Bivectorrepresentations 32
2.4 Imaginingbivectoradditionin2-Dspace 33
2.5 Bivectoradditionin3-Dspace 34
2.6 Theassociativityoftheouterproduct 35
2.7 Solvinglinearequationswithbivectors 40
2.8 Intersectinglinesintheplane 41
2.9 Codefordrawingbivectors 58
2.10 Drawingbivectorsscreenshot(Example1) 59
2.11 Theorientationoffront-andback-facingpolygons 59
2.12 Awire-frametoruswithandwithoutbackfaceculling 60
2.13 Thecodethatrendersamodelfromits2-Dvertices(Exercise2) 61
2.14 Samplingavectorfieldandsummingtrivectors 62
2.15 Codetotestforsingularity(Example3) 63
2.16 Ahelix-shapedsingularity,asdetectedbyExample3 64
3.1 Computingthescalarproductof2-blades 70
3.2 Fromscalarproducttocontraction 72
3.3 Thecontractionofavectorontoa2-blade 76
3.4 Dualityofvectorsin2-D 81
xx
LIST OF FIGURES xxi
3.5 Dualityofvectorsandbivectorsin3-D 82
3.6 Projectionontoasubspace 84
3.7 Threeusesofthecrossproduct 87
3.8 Dualityandthecrossproduct 89
3.9 Orthonormalizationcode(Example1) 93
3.10 Orthonormalization 94
3.11 Reciprocalframecode 96
3.12 Colorspaceconversioncode(Example4) 97
3.13 Colorspaceconversionscreenshot 98
4.1 Thedefiningpropertiesofalineartransformation 100
4.2 Projectionontoalineaintheb-direction 104
4.3 Arotationaroundtheoriginofunitvectorsintheplane 105
4.4 Projectionofavectorontoabivector 121
4.5 Matrixrepresentationofprojectioncode 122
4.6 Transformingnormalsvector 123
5.1 Theambiguityofthemagnitudeoftheintersectionoftwoplanes 126
5.2 Themeetoftwoorientedplanes 130
5.3 Alinemeetingaplaneintheorigin 131
5.4 Whenthejoinoftwo(near-)parallelvectorsbecomesa2-blade(Example3) 140
6.1 Non-invertibilityofthesubspaceproducts 142
6.2 Ratiosofvectors 146
6.3 Projectionandrejectionofavector 156
6.4 Reflectingavectorinaline 158
6.5 Gram-Schmidtorthogonalization 163
6.6 Gram-Schmidtorthogonalizationcode(Example2) 164
7.1 Lineandplanereflection 169
7.2 ArotationinaplaneparalleltoIistworeflectionsinvectorsinthatplane 170
7.3 Arotorinaction 171
7.4 Senseofrotation 175
7.5 Theuniquerotor-basedrotationsintherange(cid:2)=[0,4π) 176
7.6 (a)Asphericaltriangle.(b)Compositionofrotationsthroughconcatenation
ofrotorarcs 180
7.7 Areflectorinaction 189
7.8 TherotorproductinEuclideanspacesasaTaylorseries 197
xxii LIST OF FIGURES
7.9 InteractiveversionofFigure7.2 205
7.10 Rotationmatrixtorotorconversion 207
7.11 2-DJuliafractalcode 210
7.12 A2-DJuliafractal,computedusingthegeometricproductofrealvectors 211
7.13 3-DJuliafractal 212
8.1 Directionaldifferentiationofavectorinversion 227
8.2 Changesinreflectionofarotatingmirror 229
8.3 Thedirectionalderivativeofthesphericalprojection 241
10.1 Atrianglea+b+c=0inadirectedplaneI 249
10.2 Theanglebetweenavectorandabivector(seetext) 252
10.3 Asphericaltriangle 253
10.4 Interpolationofrotations 259
10.5 Interpolationofrotations(Example1) 266
10.6 Crystallography(Example2) 267
10.7 Externalcameracalibration(Example3) 268
11.1 Theextradimensionofthehomogeneousrepresentationspace 274
11.2 RepresentingoffsetsubspacesinRn+1 280
11.3 Definingoffsetsubspacesfullyinthebasespace 288
11.4 ThedualhyperplanerepresentationinR2andR1 290
11.5 TheintersectionoftwooffsetlinesLandMtoproduceapoint 293
11.6 Themeetoftwoskewlines 295
11.7 Therelativeorientationoforientedflats 296
11.8 Thecombinationsoffourpointstakeninthecrossratio 300
11.9 Thecombinationsoffourlinestakeninthecrossratio 301
11.10 Conicsinthehomogeneousmodel 308
11.11 Findingalinethroughapoint,perpendiculartoagivenline 310
11.12 Theorthogonalprojectioninthehomogeneousmodel(seetext) 315
11.13 Thebeginningofarowofequidistanttelegraphpoles 319
11.14 Example2inaction 323
11.15 Perspectiveprojection(Example4) 325
12.1 Plu¨ckercoordinatesofalinein3-D 329
12.2 Apinholecamera 337
12.3 Theepipolarconstraint 342
12.4 TheplaneofraysgeneratedbyalineobservationL 343
LIST OF FIGURES xxiii
12.5 Theprojectionoftheopticalcenterontoallraysgeneratesaneyeball 348
12.6 Reconstructionofmotioncapturedata 351
12.7 Reconstructionofmarkers 352
12.8 Crossinglinescode 354
13.1 Euclideantransformationsasmultiplereflectionsinplanes 366
13.2 Flatelementsintheconformalmodel 373
13.3 Planarreflectionintheconformalmodel 377
13.4 Chasles’screw 382
13.5 Computationofthelogarithmofarigidbodymotion 384
13.6 Rigidbodymotioninterpolation 385
13.7 Reflectioninarotatingmirror 387
13.8 TheoutputofthesolutiontoExample2 393
13.9 Example4inaction:theinterpolationofrigidbodymotions 394
14.1 Dualroundsintheconformalmodel 398
14.2 Intersectionoftwospheresofdecreasingradii 405
14.3 Visualizationofa2-DEuclideanpointontherepresentativeparaboloid 411
14.4 Therepresentationofacircleontherepresentativeparaboloid 412
14.5 Crosssectionoftheparabolaofnullvectors 413
14.6 Visualizationoftheintersectionofcirclesontherepresentativeparaboloid 414
14.7 AVoronoidiagramintheconformalmodel 416
14.8 Innerproductasdistancemeasure 418
14.9 Forwardkinematicsofarobotarm 421
14.10 Inversekinematicsofarobotarm 422
14.11 AVoronoidiagramofasetofpoints,ascomputedbyExample1 429
14.12 Euclid’selements(Example2) 432
14.13 Example3inaction 433
14.14 Fitting-a-spherecode 435
15.1 Themeetandplungeofthreespheres 439
15.2 Theplungeofdiverseelements 441
15.3 Themeetandplungeoftwospheresatdecreasingdistances 442
15.4 Visualizationofflatsasplunge 443
15.5 Orbitsofaduallineversor 444
15.6 Tangentsofelements 445
15.7 Factorizationofrounds 447
xxiv LIST OF FIGURES
15.8 Affinecombinationofconformalpoints 448
15.9 Affinecombinationofcirclesandpointpairs 449
15.10 OrthogonalprojectionsintheconformalmodelofEuclideangeometry 450
15.11 Variouskindsofvectorsintheconformalmodel 452
15.12 DefinitionofsymbolsfortheVoronoiderivations 456
15.13 Constructionofacontourcircle 462
15.14 ScreenshotofExample2 463
15.15 ScreenshotofExample3onprojectionandplunge 464
16.1 Inversioninasphere 467
16.2 Reflectioninasphere 469
16.3 Generationofasnailshell 472
16.4 Swappingscalingandtranslation 473
16.5 Computationofthelogarithmofapositivelyscaledrigidbodymotion 475
16.6 Loxodromes 478
16.7 Conformalorbits 479
16.8 Hyperbolicgeometry 481
16.9 Sphericalgeometry 482
16.10 Imagingbytheeye 484
16.11 Reflectioninapointpair 485
16.12 Dupincycloidastheinversionofatorusintoasphere 486
16.13 MetricalMysteryTour 487
16.14 Functionmatrix4×4 ToVersor() 489
16.15 Functionlog(const TRSversor &V) 493
16.16 ScreenshotofExample4 495
19.1 Functioncanonical ReorderingSign(int a, int b) 514
19.2 Functiongpop (BasisBlade a, BasisBlade b) 515
20.1 Matricesforgeometricproduct,outerproduct,andleftcontraction 525
20.2 Implementationoftheouterproductofmultivectors 527
21.1 Venndiagramsillustratingunion,intersection,andthedeltaproductof
twosets 536
21.2 Venndiagramsillustratingmeet,join,andthedeltaproductoftwoblades 537
22.1 Basictool-chainfromsourcecodetorunningapplication 544
22.2 CodegeneratedbyGaigen 2 551
22.3 Generatedmatrix-pointmultiplicationcode 553
LIST OF FIGURES xxv
23.1 Teapotpolygonalmesh 560
23.2 Screenshotoftheuserinterfaceofthemodeler 567
23.3 Rotatinganobject 570
23.4 Thespaceballinterface 571
List of Tables
2.1 Geometricalpropertiesofasubspace 43
2.2 Pascal’striangleofthenumberofbasisk-bladesinn-dimensionalspace 45
2.3 NotationalconventionsforbladesandmultivectorsforPartIofthisbook 47
2.4 C++Operatorbindings 55
5.1 Theorderoftheargumentsforameetmayaffectthesignoftheresult 134
7.1 ReflectionofanorientedsubspaceXinasubspaceA 190
8.1 Directionaldifferentiationandvectorderivatives 226
8.2 Elementaryresultsofmultivectordifferentiation 237
10.1 Thepointgroup2H4 255
11.1 Thegeometricalgebraofthehomogeneousmodelof3-DEuclideanspace 273
11.2 Thenumberofbladesrepresentingsubspacesanddirections 287
11.3 NonzerobladesinthehomogeneousmodelofEuclideangeometry 291
11.4 Specializedmultivectortypesintheh3ga 320
12.1 CommonPlu¨ckercoordinatecomputations 330
12.2 Transformationoftheflatsinthehomogeneousmodel 335
13.1 Multiplication table for the inner product of the conformal model of 3-D
EuclideangeometryE3,fortwochoicesofbasis 361
13.2 Theinterpretationofvectorsintheconformalmodel 363
13.3 Alistofthemostimportantspecializedmultivectortypesinc3ga 391
13.4 Constantsinc3ga 392
14.1 NonzerobladesintheconformalmodelofEuclideangeometry 407
16.1 Basicoperationsintheconformalmodelandtheirversors 476
16.2 Commonpropertransformationsofsomeofthestandardelementsofthe
conformalmodel 477
xxvi
LIST OF TABLES xxvii
18.1 MatrixrepresentationsofCliffordalgebrasofsignatures(p,q) 507
19.1 Thebitmaprepresentationofbasisblades 512
19.2 BitwisebooleanoperatorsusedinJavacodeexamples 513
19.3 Reversion,gradeinvolution,andCliffordConjugateforbasisblades 519
22.1 Performancebenchmarksfortheraytracer 555