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Dorst L., Fontijne D., Mann S. Ч Geometric algebra for computer science
Dorst L., Fontijne D., Mann S. Ч Geometric algebra for computer science

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Ќазвание: Geometric algebra for computer science

јвторы: Dorst L., Fontijne D., Mann S.


Within the last decade, Geometric Algebra (GA) has emerged as a powerful alternative to classical matrix algebra as a comprehensive conceptual language and computational system for computer science. This book will serve as a standard introduction and reference to the subject for students and experts alike. As a textbook, it provides a thorough grounding in the fundamentals of GA, with many illustrations, exercises and applications. Experts will delight in the refreshing perspective GA gives to every topic, large and small. -David Hestenes, Distinguished research Professor, Department of Physics, Arizona State University Geometric Algebra is becoming increasingly important in computer science. This book is a comprehensive introduction to Geometric Algebra with detailed descriptions of important applications. While requiring serious study, it has deep and powerful insights into GA's usage. It has excellent discussions of how to actually implement GA on the computer. -Dr. Alyn Rockwood, CTO, FreeDesign, Inc. Longmont, Colorado Until recently, almost all of the interactions between objects in virtual 3D worlds have been based on calculations performed using linear algebra. Linear algebra relies heavily on coordinates, however, which can make many geometric programming tasks very specific and complex-often a lot of effort is required to bring about even modest performance enhancements. Although linear algebra is an efficient way to specify low-level computations, it is not a suitable high-level language for geometric programming. Geometric Algebra for Computer Science presents a compelling alternative to the limitations of linear algebra. Geometric algebra, or GA, is a compact, time-effective, and performance-enhancing way to represent the geometry of 3D objects in computer programs. In this book you will find an introduction to GA that will give you a strong grasp of its relationship to linear algebra and its significance for your work. You will learn how to use GA to represent objects and perform geometric operations on them. And you will begin mastering proven techniques for making GA an integral part of your applications in a way that simplifies your code without slowing it down. Features Explains GA as a natural extension of linear algebra and conveys its significance for 3D programming of geometry in graphics, vision, and robotics. Systematically explores the concepts and techniques that are key to representing elementary objects and geometric operators using GA. Covers in detail the conformal model, a convenient way to implement 3D geometry using a 5D representation space. Presents effective approaches to making GA an integral part of your programming. Includes numerous drills and programming exercises helpful for both students and practitioners. Companion web site includes links to GAViewer, a program that will allow you to interact with many of the 3D figures in the book, and Gaigen 2, the platform for the instructive programming exercises that conclude each chapter. About the Authors Leo Dorst is Assistant Professor of Computer Science at the University of Amsterdam, where his research focuses on geometrical issues in robotics and computer vision. He earned M.Sc. and Ph.D. degrees from Delft University of Technology and received a NYIPLA Inventor of the Year award in 2005 for his work in robot path planning. Daniel Fontijne holds a Master's degree in artificial Intelligence and is a Ph.D. candidate in Computer Science at the University of Amsterdam. His main professional interests are computer graphics, motion capture, and computer vision. Stephen Mann is Associate Professor in the David R. Cheriton School of Computer Science at the University of Waterloo, where his research focuses on geometric modeling and computer graphics. He has a B.A. in Computer Science and Pure Mathematics from the University of California, Berkeley, and a Ph.D. in Computer Science and Engineering from the University of Washington. * The first book on Geometric Algebra for programmers in computer graphics and entertainment computing * Written by leaders in the field providing essential information on this new technique for 3D graphics * This full colour book includes a website with GAViewer, a program to experiment with GA

язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 2007

 оличество страниц: 674

ƒобавлена в каталог: 04.02.2014

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ѕредметный указатель
$<>_{k}$      47
$E_{0}$      272
$f^{*}$      111
$\ast$      67 154
$\bar{f}$      109
$\bigwedge$      50
$\bigwedge\mathds{R}^{n}$      47
$\bigwedge^{k}\mathds{R}^{n}$      47
$\bullet$      589
$\bullet_{H}$      590
$\cap, \cup$      127
$\lfloor$      73 154
$\mathds{E}^{n}$      356
$\mathds{R}^{n + 1, 1}$      586
$\mathds{R}^{n, 0}$      586
$\mathds{R}^{n}$      586
$\mathds{R}^{p, q}$      586
$\partial$      221Ч236
$\rfloor$      77 154
$\star$      289 374
$\subseteq$      43
$\times$      215
$\triangle$      538
$\widehat{}$      47
$\widetilde{}$      47
*      80
0-blades      37 38 44
2-blades      29 see
2-blades, angle between subspaces      69
2-blades, lines as      278Ч283
2-blades, pure rotors as exponentials      183Ч184
2-blades, reshapability      31Ч32 279
2-blades, squared norm of subspace      68
2-blades, squaring to scalars      184
3-blades, planes as      283Ч285
3-D cross product      86Ч89 94Ч95
3-D rotations      see also "Quaternions"
3-D rotations, multiple      179
3-D rotations, visualizing      179Ч180
3-D rotors, computing with      256Ч260
3-D rotors, determining      256Ч258
3-D rotors, logarithm of      258Ч259
3-D rotors, rotation interpolation      259Ч260
3-D space, angular relationships in      251Ч254
3-D space, defined      8
3-D space, Grassmann algebra for      48
3-D space, quadvectors in      37
3-D space, rotors in      181
3-D space, translations in      219
Addition in linear operations implementation      522
Addition of lines      282Ч283 324Ч325
Addition of points      276Ч278
Addition, bivector      32Ч33 199
Addition, products distributive over      197
Adjoint as derivative      233Ч234
Adjoint of linear transformation      108Ч109
Affine combinations in conformal model      447Ч449
Affine combinations of circles      448Ч449
Affine combinations of flats      448
Affine combinations of points      307Ч308 463Ч464
Affine combinations, defined      277
Affine combinations, example      463
Affine distance ratio      299
Affine transformations      299 306Ч308 334 see
Affine transformations, defined      306
Affine transformations, Euclidean motions as      307
Algebra of directions      247Ч270 498
Algebra specification      546Ч547
Algebra specification, constants      547
Algebra specification, dimension      546
Algebra specification, metric      546Ч547
Algebra specification, specialized types      547
Angles between subspaces      68Ч71
Angles, cosine law of      254
Angles, rotor determination from      256Ч257
Angular relationships      248Ч256
Angular relationships between three directions      252Ч253
Angular relationships in 3-D      251Ч254
Angular relationships, geometry of planar triangles      249Ч251
Angular relationships, rotation groups and      254Ч256
Angular relationships, two bivectors      251Ч252
Angular relationships, two vectors      251
Angular relationships, vector and bivector      252
Antisymmetry      29 51 143
Applications, all kinds of vectors      451Ч455
Applications, fitting a sphere to points      417Ч420
Applications, homogeneous model      327Ч354
Applications, homogeneous Plucker coordinates      328Ч336
Applications, imaging by multiple cameras      336Ч346
Applications, intersecting planar lines      41
Applications, inversion      467 468Ч469
Applications, kinematics      420Ч426
Applications, radius of curvature of planar curve      223Ч224
Applications, ray-tracing      557Ч581
Applications, reciprocal frames      89Ч90
Applications, solving linear equations      39Ч40
Applications, tilting a mirror      228Ч230
Applications, Voronoi cell analysis      455Ч460
Applications, Voronoi diagrams      415Ч417
Area elements      28 30
Area elements in 3-D      30
Area elements, oriented      251
Area elements, standard      30
Associativity of outer product      35Ч36
Associativity, geometric product of multivectors      148
Associativity, geometric product of vectors      144
Attitudes, defined      43
Attitudes, directions      248
Attitudes, homogeneous planes      27Ч28
Attitudes, homogeneous spaces      133
Attitudes, lines      26
Attitudes, points      38 275Ч276
Attitudes, sine measure      133
Attitudes, volumes      33
Automatic code generator      543
Backface culling      57Ч60 430Ч431
Barycentric coordinates      316
Base space, defined      272
Base space, elements      274
Base space, projective transformations      299
Base space, pseudoscalar      289
Basis blades      511Ч519 see
Basis blades, bitmap representation      512
Basis blades, Clifford conjugate      519
Basis blades, commutator product of      518
Basis blades, defined      512
Basis blades, geometric product      516Ч517
Basis blades, grade involution      519
Basis blades, grade-dependent signs on      518Ч519
Basis blades, list of      526Ч527
Basis blades, metric products      518
Basis blades, outer product      513Ч515
Basis blades, product evaluations      526
Basis blades, reversion      519
Basis blades, unit, representing with bitmaps      512Ч513
Basis vectors, geometric product      144Ч145
Benchmarks      554Ч556
Bilinear form      66 148 585
Bitwise boolean operators      513
Bivector addition      199
Bivector addition in 2-D space      32Ч33
Bivector addition in 3-D space      34
Bivector addition, geometrical representation      32
Bivector addition, visualizing      31Ч33
Bivector angle      174
Bivectors in angular relationships      251Ч252
Bivectors, defined      29
Bivectors, drawing      57
Bivectors, drawing code      58
Bivectors, drawing screenshot      59
Bivectors, duality of      82
Bivectors, geometric reshapability      31Ч32 39Ч40 41
Bivectors, representations      32
Bivectors, rotors as exponentials      185Ч187
Bivectors, visualizing      31Ч32
blades      see also "k-blades"
Blades in versor product      195
Blades, basis      511Ч519
Blades, defined      44 195
Blades, delta product      537 538
Blades, direction      408Ч409
Blades, even unit      195
Blades, factorization      533Ч535 554
Blades, grade-reducing product between      72
Blades, intersection      127
Blades, inverses      79Ч80 155 529Ч530 554
Blades, invertible      195
Blades, join of      536Ч540 554
Blades, ladder of      51
Blades, linear transformations      101Ч107 302
Blades, location      409
Blades, meet of      536Ч540 554
Blades, notational convention      47
Blades, null      80 587
Blades, orientation      408Ч409
Blades, rounds as      410Ч415
Blades, squared size      408
Blades, test      532Ч533
Blades, union of      127
Blades, vs. multivectors      46 198
Blades, weight      43 408Ч409
Boolean operators      511 513
Bounding sphere, computing      563Ч564
Bounding sphere, test      577
BSP tree, construction      564Ч565
BSP tree, traversing      577Ч579
C++ operator bindings      55
Cameras, calibration      260Ч263 268Ч270 351Ч354
Cameras, multiple, imaging by      336Ч346
Cameras, orbiting      572
Cameras, pinhole      337Ч339 347
Cameras, rays      575Ч576
Cameras, rotating      571Ч572
Cameras, stereo vision and      340Ч342
Cameras, translating      570Ч571
Carriers      445
Chain rule      234
Chasles' theorem      381 383
circles      see also "Rounds"
Circles, affine combination of      448Ч449
Circles, circum-      457Ч459
circles, creating      432
Circles, direct      402
Circles, dual      399
Circles, Euclidean, intersection as planes      414Ч415
Circles, intersection      414
Circles, oriented      402
Circles, reflection      468Ч469
Circles, representation      412Ч413
Circles, rotation      479
Circles, through points      2 404
Classification, conformal blades      407
Classification, homogeneous blades      291
Classification, multivector      532Ч533 554
Clifford algebra      141
Clifford algebra of spaces      507
Clifford algebra, 4-D representation space      336
Clifford algebra, defined      198
Clifford algebra, dormant results      199
Clifford algebra, geometric algebra versus      198Ч199
Clifford algebra, matrix representation      506Ч507
Clifford conjugation      53 519 604
Clifford group      196
Clifford, W.K.      6 143
Coincident lines      294
Color space conversion      95Ч98
Color space conversion, code      97
Color space conversion, screenshot      98
Commutativity, geometric product for vectors      143
Commutator product      215Ч217 see
Commutator product of basis blades      518
Commutator product, defined      215
complex numbers      177Ч178
Composition of orthogonal transformations      200
Composition of rotations      176Ч182
Composition, multiplicative      180
Compression, per-coordinate      548
Compression, per-grade      548 549
Compression, per-group      548Ч549
Concatenated transformations      191
Conformal geometries      487 498Ч499
Conformal model      9 355Ч396
Conformal model, applications      377Ч379 385Ч388
Conformal model, basic operations      476
Conformal model, blade parameters      407Ч409
Conformal model, computing geometric product in      517
Conformal model, defined      355Ч359
Conformal model, differential planar reflections      386Ч388
Conformal model, directions      376
Conformal model, equations      607
Conformal model, Euclidean transformations      364Ч370
Conformal model, flats      370Ч376
Conformal model, general planar reflection      377Ч379
Conformal model, general vectors represent      361Ч364
Conformal model, homogeneous model embedded in      374
Conformal model, hyperbolic geometry      480Ч481
Conformal model, inner product      361
Conformal model, interpolation of rigid body motions      385Ч386
Conformal model, intersecting circles      415
Conformal model, meet      438
Conformal model, metrics      390Ч492
Conformal model, nonzero blades      407
Conformal model, null vectors      359
Conformal model, orthogonal projections in      450
Conformal model, points as null vectors      359Ч361
Conformal model, pseudoscalar of      374
Conformal model, representational space and metric      356Ч359
Conformal model, rigid body motions      379Ч385
Conformal model, rounds      398Ч404
Conformal model, spherical geometry      482Ч483
Conformal operators      465Ч495
Conformal operators, conformal transformations      477Ч480
Conformal operators, scaling      469Ч475
Conformal operators, spherical inversion      465Ч467
Conformal operators, transformations of standard blades      477
Conformal operators, transversions      475Ч477
Conformal transformations      466 477Ч480 see
Conformal transformations, circular rotations      479
Conformal transformations, loxodromes      478Ч479
Conformal transformations, Moebius      479Ч480
Constructions in conformal model      437Ч464
Constructions in Euclidean geometry      437Ч464
Constructions in homogeneous model      279 285 291 305Ч306
Constructions, coordinate-free parameterized      309Ч312
Constructions, covariance of      13 367Ч370
Contact geometry      499
Containment, in subspace      76
Contractions from geometric product      598Ч599
Contractions from scalar product to      71Ч75
Contractions, containment      85
Contractions, defined      71Ч72
Contractions, explicit definition      73Ч74 591Ч593
Contractions, geometric interpretation      75Ч77
Contractions, implicit definition      71Ч73 591Ч593
Contractions, left      77 518 590
Contractions, linear transformation of      109Ч110
Contractions, nonassociativity in      78Ч79
Contractions, orthogonal transformation of      110Ч111
Contractions, right      77
Conversion, matrix to versor      491
1 2 3 4 5 6
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