Essential mathematics for games and interactive applications: a programmer's guide
Gespeichert in:
| Hauptverfasser: | , |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Amsterdam [u.a.]
Elsevier, Morgan Kaufmann Publ.
2008
|
| Ausgabe: | 2. ed. |
| Schlagworte: | |
| Online-Zugang: | Publisher description Table of contents Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | XXX, 670 S. Ill., graph. Darst. CD-ROM (12 cm) |
| ISBN: | 9780123742971 0123742978 9780123742988 |
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| 245 | 1 | 0 | |a Essential mathematics for games and interactive applications |b a programmer's guide |c James M. van Verth ; Lars M. Bishop |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Amsterdam [u.a.] |b Elsevier, Morgan Kaufmann Publ. |c 2008 | |
| 300 | |a XXX, 670 S. |b Ill., graph. Darst. |e CD-ROM (12 cm) | ||
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| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Computer games |x Programming | |
| 650 | 4 | |a Three-dimensional display systems |x Mathematics | |
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Datensatz im Suchindex
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| adam_text | ESSENTIAL MATHEMATICS FOR GAMES AND INTERACTIVE APPLICATIONS A
PROGRAMMER S GUIDE SECOND EDITION JAMES M. VAN VERTH LARS M. BISHOP
AMSTERDAM * BOSTON. HEIDELBERG * LONDON NEW YORK * OXFORD * PARIS. SAN
DIEGO SAN FRANCISCO * SINGAPORE * SYDNEY * TOKYO 14 ~~ ELSEVIER MORGAN
KAUHNANN PUBLISHERS IS AN IMPRINT OF ELSEVIER HORGAN KAUFMANN PUBLISHERS
CONTENTS PREFACE INTRODUCTION XIX XXIII CHAPTER 1 REAL-WORLD COMPUTER
NUMBER REPRESENTATION 1 1. 1 INTRODUCTION 1 1.2 REPRESENTING REAL
NUMBERS 2 1.2.1 APPROXIMATIONS 2 1.2.2 PRECISION AND ERROR 3 1.3
FLOATING-POINT NUMBERS 4 1.3.1 REVIEW: SCIENTIFIC NOTATION 4 1.3.2 A
RESTRICTED SCIENTIFIC NOTATION 5 1.4 BINARY SCIENTIFIC NOTATION 6 1.5
IEEE 754 FLOATING-POINT STANDARD 9 1.5.1 BASIC REPRESENTATION 9 1.5.2
RANGE AND PRECISION 11 1.5.3 ARITHMETIC OPERATIONS 13 1.5-4 SPECIAL
VALUES 16 1.5.5 VERY SMALL VALUES 19 1.5.6 CATASTROPHIC CANCELATION 22
1.5.7 DOUBLE PRECISION 24 1.6 REAL-WORLD FLOATING-POINT 25 1.6.1
INTERNAL FPU PRECISION 25 1.6.2 PERFORMANCE 26 1.6.3 IEEE SPECIFICATION
COMPLIANCE 29 1.6-4 GRAPHICS PROCESSING UNITS AND HALF-PRECISION
FLOATING-POINT FORMATS 31 1.7 CODE 32 1.8 CHAPTER SUMMARY 33 IX X
CONTENTS CHAPTER 2 VECTORS AND POINTS 2.1 INTRODUCTION 35 2.2 VECTORS 36
2.2.1 GEOMETRIE VECTORS 36 2.2.2 LINEAR COMBINATIONS 39 2.2.3 VECTOR
REPRESENTATION 40 2.2-4 BASIC VECTOR CLASS IMPLEMENTATION 42 2.2.5
VECTOR LENGTH 44 2.2.6 DOT PRODUCT 47 2.2.7 GRAM-SCHMIDT
ORTHOGONALIZATION 51 2.2.8 CROSS PRODUCT 53 2.2.9 TRIPIE PRODUCTS 56
2.2.10 REAL VECTOR SPACES 59 2.2.11 BASIS VECTORS 62 2.3 POINTS 63 2.3.1
POINTS AS GEOMETRY 64 2.3.2 AFFINE SPACES 66 2.3.3 AFFINE COMBINATIONS
68 2.3-4 POINT IMPLEMENTATION 70 2.3.5 POLAR AND SPHERICAL COORDINATES
72 2-4 LINES 75 2-4.1 DEFINITION 75 2-4.2 PARAMETERIZED LINES 76 2-4.3
GENERALIZED LINE EQUATION 77 2-4-4 COLLINEAR POINTS 79 2.5 PLANES 80
2.5.1 PARAMETERIZED PLANES 80 2.5.2 GENERALIZED PLANE EQUATION 80 2.5.3
COPLANAR POINTS 82 2.6 POLYGONS AND TRIANGLES 82 2.7 CHAPTER SUMMARY 86
CHAPTER 3 MATRICES AND LINEAR TRANSFORMATIONS 3.1 INTRODUCTION 87 3.2
MATRICES 88 3.2.1 INTRODUCTION TO MATRICES 88 3.2.2 SIMPLE OPERATIONS 90
3.2.3 VECTOR REPRESENTATION 92 3.2-4 BLOCK MATRICES 92 3.2.5 MATRIX
PRODUCT 94 3.2.6 IDENTITY MATRIX 96 35 87 CONTENTS XL 3.2.7 PERFORMING
VECTOR OPERATIONS WITH MATRICES 97 3.2.8 IMPLEMENTATION 98 3.3 LINEAR
TRANSFORMATIONS 101 3.3.1 DEFINITIONS 101 3.3.2 NULL SPACE AND RANGE 103
3.3.3 LINEAR TRANSFORMATIONS AND BASIS VECTORS 104 3.3-4 MATRICES AND
LINEAR TRANSFORMATIONS 106 3.3.5 COMBINING LINEAR TRANSFORMATIONS 108
3-4 SYSTEMS OF LINEAR EQUATIONS 110 3+1 DEFINITION 110 3-4.2 SOLVING
LINEAR SYSTEMS 112 3-4.3 GAUSSIAN ELIMINATION 113 3.5 MATRIX INVERSE 117
3.5.1 DEFINITION 117 3.5.2 SIMPLE INVERSES 120 3.6 DETERMINANT 121 3.6.1
DEFINITION 121 3.6.2 COMPUTING THE DETERMINANT 123 3.6.3 DETERMINANTS
AND ELEMENTARY ROW OPERATIONS 126 3.6-4 ADJOINT MATRIX AND INVERSE 128
3.7 EIGENVALUES AND EIGENVECTORS 129 3.8 CHAPTER SUMMARY 130 CHAPTER 4
AFFINE TRANSFORMATIONS 4.1 INTRODUCTION 133 4.2 AFFINE TRANSFORMATIONS
134 4.2.1 MATRIX DEFINITION 134 4.2.2 FORMAL DEFINITION 136 4.2.3 FORMAL
REPRESENTATION 138 4.3 STANDARD AFFINE TRANSFORMATIONS 139 4.3.1
TRANSLATION 139 4.3.2 ROTATION 141 4.3.3 SCALING 150 4.3-4 REFLECTION
151 4.3.5 SHEAR 154 4.3.6 APPLYING AN AFFINE TRANSFORMATION AROUND AN
ARBITRARY POINT 156 4.3.7 TRANSFORMING PLANE NORMALS 158 4-4 USING
AFFINE TRANSFORMATIONS 159 4+ 1 MANIPULATION OF GAME OBJECTS 159 4-4.2
MATRIX DECOMPOSITION 164 4-4.3 AVOIDING MATRIX DECOMPOSITION 166 133 XII
CONTENTS 4.5 OBJECT HIERARCHIES 169 4.6 CHAPTER SUMMARY 171 CHAPTER 5
ORIENTATION REPRESENTATION 5.1 INTRODUCTION 173 5.2 ROTATION MATRICES
174 5.3 FIXED AND EULER ANGLES 174 5.3.1 DEFINITION 174 5.3.2 FORMAT
CONVERSION 177 5.3.3 CONCATENATION 178 5.3-4 VECTOR ROTATION 178 5.3.5
OTHER ISSUES 179 5.4 AXIS-ANGLE REPRESENTATION 181 5 -4. 1 DEFINITION
181 5-4.2 FORMAT CONVERSION 182 5-4-3 CONCATCNATION 184 5-4.4 VECTOR
ROTATION 184 5.4.5 AXIS-ANGLE SUMMARY 185 5.5 QUATEMIONS 185 5.5.1
DEFINITION 185 5.5.2 QUATERNIONS AS ROTATIONS 186 5.5.3 ADDITION AND
SCALAR MULTIPLICATION 187 5.5-4 NEGATION 188 5.5.5 MAGNITUDE AND
NORMALIZATION 188 5.5.6 DOTPRODUCT 189 5.5.7 FORMAT CONVERSION 189 5.5.8
CONCATENATION 193 5.5.9 IDENTITY AND INVERSE 195 5.5.10 VECTOR ROTATION
197 5.5.11 SHORTEST PATH OF ROTATION 199 5.5.12 QUATERNIONS AND
TRANSFORMATIONS 200 5.6 CHAPTER SUMMARY 201 173 CHAPTER 6 VIEWING AND
PROJECTION 203 6.1 INTRODUCTION 203 6.2 VIEW FRAME AND VIEW
TRANSFORMATION 205 6.2.1 DEFINING A VIRTUAL CAMERA 205 6.2.2
CONSTRUCTING THE VIEW-TO-WORLD TRANSFORMATION 206 6.2.3 CONTROLLING THE
CAMERA 208 6.2-4 CONSTRUCTING THE WORLD-TO-VIEW TRANSFORMATION 211
CONTENTS XIII 6.3 PROJECTIVE TRANSFONNATION 212 6.3.1 DEFINITION 212
6.3.2 NORMALIZED DEVICE COORDINATES 216 6.3.3 VIEW FRUSTUM 216 6.3.4
HOMOGENEOUS COORDINATES 220 6.3.5 PERSPECTIVE PROJECTION 221 6.3.6
OBLIQUE PERSPECTIVE 228 6.3.7 ORTHOGRAPHIE PARALLEL PROJECTION 231 6.3.8
OBLIQUE PARALLEL PROJECTION 232 6-4 CULLING AND CLIPPING 235 6+ 1 WHY
CULL OR CLIP? 235 6+2 CULLING 238 6+3 GENERAL PLANE CLIPPING 239 6+4
HOMOGENEOUS CLIPPING 244 6.5 SCREEN TRANSFONNATION 246 6.5.1 PIXEL
ASPECT RATIO 248 6.6 PICKING 249 6.7 MANAGEMENT OF VIEWING
TRANSFONNATIONS 252 6.8 CHAPTER SUMMARY 254 CHAPTER 7 GEOMETRY AND
PROGRAMMABLE SHADING 255 7.1 INTRODUCTION 255 7.2 COLOR REPRESENTATION
257 7.2.1 RGB COLOR MODEL 257 7.2.2 COLORS AS VECTORS 257 7.2.3 COLOR
RANGE LIMITATION 258 7.2-4 OPERATIONS ON COLORS 259 7.2.5 ALPHA VALUES
260 7.2.6 COLOR STORAGE FORMATS 264 7.3 POINTS AND VERTICES 266 7.3.1
PER-VERTEX ATTRIBUTES 266 7.3.2 AN OBJECT S VERTICES 267 7-4 SURFACE
REPRESENTATION 270 7,{.1 VERTICES AND SURFACE AMBIGUITY 270 7+2
TRIANGLES 271 7+3 CONNECTING VERTICES INTO TRIANGLES 271 7-4-4 DRAWING
GEOMETRY 274 7.5 RENDERING PIPELINE 275 7.5.1 FIXED-FUNCTION VERSUS
PROGRAMMABLE PIPELINES 277 7.6 SHADERS 278 7.6.1 USING SHADERS TO MOVE
FROM VERTEX TO TRIANGLE TO FRAGMENT 278 }CI ~ORLTELLTS 7.6.2 SHADER
INPUT AND OUTPUT VALUES 279 7.6.3 SHADER OPERATIONS AND LANGUAGE
CONSTRUCTS 280 7.7 VERTEX SHADERS 280 7.7.1 VERTEX SHADER INPUTS 280
7.7.2 VERTEX SHADER OUTPUTS 281 7.7.3 BASIC VERTEX SHADERS 282 7.704
LINKING VERTEX AND FRAGMENT SHADERS 282 7.8 FRAGMENT SHADERS 283 7.8.1
FRAGMENT SHADER INPUTS 283 7.8.2 FRAGMENT SHADER OUTPUTS 284 7.8.3
COMPILING, LINKING, AND USING SHADERS 284 7.804 SETTING UNIFORM VALUES
286 7.9 BASIC COLORING METHODS 287 7.9.1 PER-OBJECTCOLORS 288 7.9.2
PER-VERTEX COLORS 288 7.9.3 PER-TRIANGLE COLORS 290 7.904 SHARP EDGES
AND VERTEX COLORS 290 7.9.5 MORE ABOUT BASIC SHADING 291 7.9.6
LIMITATIONS OF BASIC SHADING METHODS 292 7.10 TEXTURE MAPPING 292 7.10.1
INTRODUCTION 292 7.10.2 SHADING VIA IMAGE LOOKUP 293 7.10.3 TEXTURE
IMAGES 294 7.1004 TEXTURE SAMPIERS 297 7.11 TEXTURE COORDINATES 297
7.11.1 MAPPING TEXTURE COORDINATES ONTO OBJECTS 298 7.11. 2 GENERATING
TEXTURE COORDINATES 300 7.11.3 TEXTURE COORDINATE DISCONTINUITIES 301
7.11-4 MAPPING OUTSIDE THE UNIT SQUARE 302 7.11.5 TEXTURE SAMPIERS IN
SHADER CODE 309 7.12 THE STEPS OFTEXTURING 309 7.12.1 OTHER FORMS OF
TEXTURE COORDINATES 310 7.12.2 FROM TEXTURE COORDINATES TO A TEXTURE
SAMPIE COLOR 311 7.13 LIMITATIONS OF STATIC SHADING 312 7.14 CHAPTER
SUMMARY 313 CHAPTER 8 LIGHTING 8.1 INTRODUCTION 315 8.2 BASICS OF LIGHT
APPROXIMATION 316 8.2.1 MEASURINGLIGHT 317 8.2.2 LIGHT AS A RAY 318 8.3
A SIMPLE APPROXIMATION OF LIGHTING 318 315 369 CONTENTS XV 8-4 TYPES OF
LIGHT SOURCES 319 8-4.1 DIRECTIONAL LIGHTS 320 8-4.2 POINT LIGHTS 321
8-4.3 SPOTLIGHTS 327 8-4-4 OTHER TYPES UF LIGHT SOURCES 330 8.5 SURFACE
MATERIALS AND LIGHT INTERACTION 331 8.6 CATEGORIES OF LIGHT 332 8.6.1
EMISSION 332 8.6.2 AMBIENT 332 8.6.3 DIFFUSE 334 8.6-4 SPECULAR 338 8.7
COMBINED LIGHTING EQUATION 343 8.8 LIGHTING AND SHADING 348 8.8.1
FLAT-SHADED LIGHTING 349 8.8.2 PER-VERTEX LIGHTING 350 8.8.3
PER-FRAGMENT LIGHTING 354 8.9 TEXTURES AND LIGHTING 358 8.G.1 BASIC
MODULATION 359 8.G.2 SPECULAR LIGHTING AND TEXTURES 360 8.9.3 TEXTURES
AS MATERIALS 362 8.10 ADVANCED LIGHTING 363 8.10.1 NORMAL MAPPING 363
8.11 REFLECTIVE OBJECTS 366 8.12 SHADOWS 367 8.13 CHAPTER SUMMARY 368
CHAPTER 9 RASTERIZATION 9.1 INTRODUCTION 369 9.2 DISPLAYS AND
FRAMEBUFFERS 370 9.3 CONCEPTUAL RASTERIZATION PIPELINE 371 9.3.1
RASTERIZATION STAGES 372 9-4 DETERMINING THE FRAGMENTS: PIXELS COVERED
BY A TRIANGLE 373 9-4.1 FRAGMENTS 373 9.4.2 DEPTH COMPLEXITY 373 9-4.3
CONVERTING TRIANGLES TO FRAGMENTS 375 G-4-4 HANDLING PARTIAL FRAGMENTS
376 9.5 DETERMINING VISIBLE GEOMETRY 378 9.5.1 DEPTH BUFFERING 378 9.5.2
DEPTH BUFFERING IN PRACTICE 387 9.6 COMPUTING FRAGMENT SHADER INPUTS 388
G.6.1 UNIFORM VALUES 389 G.6.2 PER-VERTCX ATTRIBUTES 389 XVI CONTENTS
9.6.3 INTERPOLATING TEXTURE COORDINATES 392 9.604 OTHER SOURCES OF
TEXTURE COORDINATES 394 9.7 EVALUATING THE FRAGMENT SHADER 395 9.8
RASTERIZING TEXTURES 395 9.8.1 TEXTURE COORDINATE REVIEW 396 9.8.2
MAPPING A COORDINATE TO A TEXEL 396 9.8.3 MIPMAPPING 404 9.9 FROM
FRAGMENTS TO PIXELS 415 9.9.1 PIXEL BLENDING 416 9.9.2 ANTIALIASING 420
9.9.3 ANTIALIASING IN PRACTICE 427 9.10 CHAPTER SUMMARY 428 CHAPTER 10
INTERPOLATION 10.1 INTRODUCTION 431 10.2 INTERPOLATION OF POSITION 433
10.2.1 GENERAL DEFINITIONS 433 10.2.2 LINEAR INTERPOLATION 435 10.2.3
HERMITE CURVES 438 10.204 CATMULL-ROM SPLINES 448 10.2.5
KOCHANEK-BARTELS SPLINES 450 10.2.6 BEZIER CURVES 452 10.2.7 OTHER CURVE
TYPES 456 10.3 INTERPOLATION OF ORIENTATION 458 10.3.1 GENERAL
DISCUSSION 458 10.3.2 LINEAR INTERPOLATION 461 10.3.3 SPHERICAL LINEAR
INTERPOLATION 465 10.304 PERFORMANCE IMPROVEMENTS 469 10-4 SAMPLING
CURVES 470 10+1 FORWARU DIFFERENCING 471 1004.2 MIDPOINT SUBDIVISION 473
10+3 COMPUTING ARC LENGTH 476 10.5 CONTROLLING SPEED ALONG A CURVE 480
10.5.1 MOVING AT CONSTANT SPEED 480 10.5.2 MOVING AT VARIABLE SPEED 485
10.6 CAMERA CONTROL 488 10.7 CHAPTER SUMMARY 491 CHAPTER 11 RANDOM
NUMBERS 11.1 INTRODUCTION 493 431 493 CONTENTS XVLL 11.2 PROBABILITY 493
11.2.1 BASIC PROBABILITY 494 11.2.2 RANDOM VARIABLES 497 11.2.3 MEAN AND
STANDARD DEVIATION 501 11.2-4 SPECIAL PROBABIIITY DISTRIBUTIONS 502 11.3
DETENNINING RANDOMNESS SOS 11.3.1 CHI-SQUARE TEST 506 11.3.2 SPECTRAL
TEST 512 11.4 RANDOM NUMBER GENERATORS SB 11.4.1 LINEAR CONGRUENTIAL
METHODS 516 11.4.2 LAGGED FIBONACCI METHODS 520 11.4.3 CARRY METHODS 521
11.4-4 MERSENNE TWISTER 523 11.4.5 CONCLUSIONS 526 11.5 SPECIAL
APPLICATIONS 527 11.5.1 INTEGERS AND RANGES OF INTEGERS 527 11.5.2
FLOATING-POINT NUMBCRS 528 11.5.3 NONUNIFORM DISTRIBUTIONS 528 11.5.4
SPHERICAL SAMPLING 530 11.5.5 DISC SAMPLING 532 11.5.6 NOISE AND
TURBULENCE 534 11.6 CHAPTER SUMMARY 538 CHAPTER 12 INTERSECTION TESTING
541 12.1 INTRODUCTION 541 12.2 CLOSEST POINT AND DISTANCE TESTS 542
12.2.1 CLOSEST POINT ON LINE TO POINT 542 12.2.2 LINE-POINT DISTANCE 544
12.2.3 CLOSEST POINT ON LINE SEGMENT TO POINT 545 12.2-4 LINE
SEGMENT-POINT DISTANCE 546 12.2.5 CLOSEST POINTS BETWEEN TWO LINES 548
12.2.6 LINE-LINE DISTANCE 550 12.2.7 CLOSEST POINTS BETWEEN TWO LINE
SEGMENTS 551 12.2.8 LINE SEGMENT-LINE SEGMENT DISTANCE 553 12.2.G
GENERAL LINEAR COMPONENTS 554 12.3 OBJECT INTERSECTION 554 12.3.1
SPHERES 556 12.3.2 AXIS-ALIGNED BOUNDING BOXES 563 12.3.3 SWEPT SPHERES
571 12.3-4 OBJECT-ORIENTED BOXES 576 12.3.5 TRIANGLES 583 XVIII
C01ZTENTS 12 -4 A SIMPLE COLLISION SYSTEM 588 12+ 1 CHOOSING A BASE
PRIMITIVE 589 12-4.2 BOUNDING HIERARCHIES 590 12+3 DYNAMIC OBJECTS 591
12-4-4 PERFORMANCE IMPROVEMENTS 593 12-4.5 RELATED SYSTEMS 596 12+6
SECTION SUMMARY 599 12.5 CHAPTER SUMMARY 599 CHAPTER 13 RIGID BODY
DYNAMICS 13.1 INTRODUCTION 601 13.2 LINEAR DYNAMICS 602 13.2.1 MOVING
WITH CONSTANT ACCELERATION 602 13.2.2 FORCES 605 13.2.3 LINEAR MOMENTUM
606 13.2-4 MOVING WITH VARIABLE ACCELERATION 607 13.3 NUMERICAL
INTEGRATION 609 13.3.1 DEFINITION 609 13.3.2 EULER S METHOD 611 13.3.3
RUNGE-KUTTA METHODS 614 13.3-4 VERLET INTEGRATION 616 13.3.5 IMPLICIT
METHODS 619 13.3.6 SEMI-IMPLICIT METHODS 621 13.4 ROTATIONAL DYNAMICS
622 13+ 1 DEFINITION 622 13+2 ORIENTATION AND ANGULAR VELOCITY 622 13+3
TORQUE 625 13.4-4 ANGULAR MOMENTUM AND INERTIA TENSOR 626 13+5
INTEGRATING ROTATIONAL QUANTITIES 628 13.5 COLLISION RESPONSE 630 13.5.1
CONTACT GENERATION 630 13.5.2 LINEAR COLLISION RESPONSE 634 13.5.3
ROTATIONAL COLLISION RESPONSE 638 13.5-4 EXTENDING THE SYSTEM 640 13.6
EFFICIENCY 643 13.7 CHAPTER SUMMARY 645 BIBLIOGRAPHY INDEX TRADEMARKS
AEBOUT TH E CD-ROM 601 647 655 671 672
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| adam_txt |
ESSENTIAL MATHEMATICS FOR GAMES AND INTERACTIVE APPLICATIONS A
PROGRAMMER'S GUIDE SECOND EDITION JAMES M. VAN VERTH LARS M. BISHOP
AMSTERDAM * BOSTON. HEIDELBERG * LONDON NEW YORK * OXFORD * PARIS. SAN
DIEGO SAN FRANCISCO * SINGAPORE * SYDNEY * TOKYO 14 ~~ ELSEVIER MORGAN
KAUHNANN PUBLISHERS IS AN IMPRINT OF ELSEVIER HORGAN KAUFMANN PUBLISHERS
CONTENTS PREFACE INTRODUCTION XIX XXIII CHAPTER 1 REAL-WORLD COMPUTER
NUMBER REPRESENTATION 1 1. 1 INTRODUCTION 1 1.2 REPRESENTING REAL
NUMBERS 2 1.2.1 APPROXIMATIONS 2 1.2.2 PRECISION AND ERROR 3 1.3
FLOATING-POINT NUMBERS 4 1.3.1 REVIEW: SCIENTIFIC NOTATION 4 1.3.2 A
RESTRICTED SCIENTIFIC NOTATION 5 1.4 BINARY "SCIENTIFIC NOTATION" 6 1.5
IEEE 754 FLOATING-POINT STANDARD 9 1.5.1 BASIC REPRESENTATION 9 1.5.2
RANGE AND PRECISION 11 1.5.3 ARITHMETIC OPERATIONS 13 1.5-4 SPECIAL
VALUES 16 1.5.5 VERY SMALL VALUES 19 1.5.6 CATASTROPHIC CANCELATION 22
1.5.7 DOUBLE PRECISION 24 1.6 REAL-WORLD FLOATING-POINT 25 1.6.1
INTERNAL FPU PRECISION 25 1.6.2 PERFORMANCE 26 1.6.3 IEEE SPECIFICATION
COMPLIANCE 29 1.6-4 GRAPHICS PROCESSING UNITS AND HALF-PRECISION
FLOATING-POINT FORMATS 31 1.7 CODE 32 1.8 CHAPTER SUMMARY 33 IX X
CONTENTS CHAPTER 2 VECTORS AND POINTS 2.1 INTRODUCTION 35 2.2 VECTORS 36
2.2.1 GEOMETRIE VECTORS 36 2.2.2 LINEAR COMBINATIONS 39 2.2.3 VECTOR
REPRESENTATION 40 2.2-4 BASIC VECTOR CLASS IMPLEMENTATION 42 2.2.5
VECTOR LENGTH 44 2.2.6 DOT PRODUCT 47 2.2.7 GRAM-SCHMIDT
ORTHOGONALIZATION 51 2.2.8 CROSS PRODUCT 53 2.2.9 TRIPIE PRODUCTS 56
2.2.10 REAL VECTOR SPACES 59 2.2.11 BASIS VECTORS 62 2.3 POINTS 63 2.3.1
POINTS AS GEOMETRY 64 2.3.2 AFFINE SPACES 66 2.3.3 AFFINE COMBINATIONS
68 2.3-4 POINT IMPLEMENTATION 70 2.3.5 POLAR AND SPHERICAL COORDINATES
72 2-4 LINES 75 2-4.1 DEFINITION 75 2-4.2 PARAMETERIZED LINES 76 2-4.3
GENERALIZED LINE EQUATION 77 2-4-4 COLLINEAR POINTS 79 2.5 PLANES 80
2.5.1 PARAMETERIZED PLANES 80 2.5.2 GENERALIZED PLANE EQUATION 80 2.5.3
COPLANAR POINTS 82 2.6 POLYGONS AND TRIANGLES 82 2.7 CHAPTER SUMMARY 86
CHAPTER 3 MATRICES AND LINEAR TRANSFORMATIONS 3.1 INTRODUCTION 87 3.2
MATRICES 88 3.2.1 INTRODUCTION TO MATRICES 88 3.2.2 SIMPLE OPERATIONS 90
3.2.3 VECTOR REPRESENTATION 92 3.2-4 BLOCK MATRICES 92 3.2.5 MATRIX
PRODUCT 94 3.2.6 IDENTITY MATRIX 96 35 87 CONTENTS XL 3.2.7 PERFORMING
VECTOR OPERATIONS WITH MATRICES 97 3.2.8 IMPLEMENTATION 98 3.3 LINEAR
TRANSFORMATIONS 101 3.3.1 DEFINITIONS 101 3.3.2 NULL SPACE AND RANGE 103
3.3.3 LINEAR TRANSFORMATIONS AND BASIS VECTORS 104 3.3-4 MATRICES AND
LINEAR TRANSFORMATIONS 106 3.3.5 COMBINING LINEAR TRANSFORMATIONS 108
3-4 SYSTEMS OF LINEAR EQUATIONS 110 3+1 DEFINITION 110 3-4.2 SOLVING
LINEAR SYSTEMS 112 3-4.3 GAUSSIAN ELIMINATION 113 3.5 MATRIX INVERSE 117
3.5.1 DEFINITION 117 3.5.2 SIMPLE INVERSES 120 3.6 DETERMINANT 121 3.6.1
DEFINITION 121 3.6.2 COMPUTING THE DETERMINANT 123 3.6.3 DETERMINANTS
AND ELEMENTARY ROW OPERATIONS 126 3.6-4 ADJOINT MATRIX AND INVERSE 128
3.7 EIGENVALUES AND EIGENVECTORS 129 3.8 CHAPTER SUMMARY 130 CHAPTER 4
AFFINE TRANSFORMATIONS 4.1 INTRODUCTION 133 4.2 AFFINE TRANSFORMATIONS
134 4.2.1 MATRIX DEFINITION 134 4.2.2 FORMAL DEFINITION 136 4.2.3 FORMAL
REPRESENTATION 138 4.3 STANDARD AFFINE TRANSFORMATIONS 139 4.3.1
TRANSLATION 139 4.3.2 ROTATION 141 4.3.3 SCALING 150 4.3-4 REFLECTION
151 4.3.5 SHEAR 154 4.3.6 APPLYING AN AFFINE TRANSFORMATION AROUND AN
ARBITRARY POINT 156 4.3.7 TRANSFORMING PLANE NORMALS 158 4-4 USING
AFFINE TRANSFORMATIONS 159 4+ 1 MANIPULATION OF GAME OBJECTS 159 4-4.2
MATRIX DECOMPOSITION 164 4-4.3 AVOIDING MATRIX DECOMPOSITION 166 133 XII
CONTENTS 4.5 OBJECT HIERARCHIES 169 4.6 CHAPTER SUMMARY 171 CHAPTER 5
ORIENTATION REPRESENTATION 5.1 INTRODUCTION 173 5.2 ROTATION MATRICES
174 5.3 FIXED AND EULER ANGLES 174 5.3.1 DEFINITION 174 5.3.2 FORMAT
CONVERSION 177 5.3.3 CONCATENATION 178 5.3-4 VECTOR ROTATION 178 5.3.5
OTHER ISSUES 179 5.4 AXIS-ANGLE REPRESENTATION 181 5 -4. 1 DEFINITION
181 5-4.2 FORMAT CONVERSION 182 5-4-3 CONCATCNATION 184 5-4.4 VECTOR
ROTATION 184 5.4.5 AXIS-ANGLE SUMMARY 185 5.5 QUATEMIONS 185 5.5.1
DEFINITION 185 5.5.2 QUATERNIONS AS ROTATIONS 186 5.5.3 ADDITION AND
SCALAR MULTIPLICATION 187 5.5-4 NEGATION 188 5.5.5 MAGNITUDE AND
NORMALIZATION 188 5.5.6 DOTPRODUCT 189 5.5.7 FORMAT CONVERSION 189 5.5.8
CONCATENATION 193 5.5.9 IDENTITY AND INVERSE 195 5.5.10 VECTOR ROTATION
197 5.5.11 SHORTEST PATH OF ROTATION 199 5.5.12 QUATERNIONS AND
TRANSFORMATIONS 200 5.6 CHAPTER SUMMARY 201 173 CHAPTER 6 VIEWING AND
PROJECTION 203 6.1 INTRODUCTION 203 6.2 VIEW FRAME AND VIEW
TRANSFORMATION 205 6.2.1 DEFINING A VIRTUAL CAMERA 205 6.2.2
CONSTRUCTING THE VIEW-TO-WORLD TRANSFORMATION 206 6.2.3 CONTROLLING THE
CAMERA 208 6.2-4 CONSTRUCTING THE WORLD-TO-VIEW TRANSFORMATION 211
CONTENTS XIII 6.3 PROJECTIVE TRANSFONNATION 212 6.3.1 DEFINITION 212
6.3.2 NORMALIZED DEVICE COORDINATES 216 6.3.3 VIEW FRUSTUM 216 6.3.4
HOMOGENEOUS COORDINATES 220 6.3.5 PERSPECTIVE PROJECTION 221 6.3.6
OBLIQUE PERSPECTIVE 228 6.3.7 ORTHOGRAPHIE PARALLEL PROJECTION 231 6.3.8
OBLIQUE PARALLEL PROJECTION 232 6-4 CULLING AND CLIPPING 235 6+ 1 WHY
CULL OR CLIP? 235 6+2 CULLING 238 6+3 GENERAL PLANE CLIPPING 239 6+4
HOMOGENEOUS CLIPPING 244 6.5 SCREEN TRANSFONNATION 246 6.5.1 PIXEL
ASPECT RATIO 248 6.6 PICKING 249 6.7 MANAGEMENT OF VIEWING
TRANSFONNATIONS 252 6.8 CHAPTER SUMMARY 254 CHAPTER 7 GEOMETRY AND
PROGRAMMABLE SHADING 255 7.1 INTRODUCTION 255 7.2 COLOR REPRESENTATION
257 7.2.1 RGB COLOR MODEL 257 7.2.2 COLORS AS "VECTORS" 257 7.2.3 COLOR
RANGE LIMITATION 258 7.2-4 OPERATIONS ON COLORS 259 7.2.5 ALPHA VALUES
260 7.2.6 COLOR STORAGE FORMATS 264 7.3 POINTS AND VERTICES 266 7.3.1
PER-VERTEX ATTRIBUTES 266 7.3.2 AN OBJECT'S VERTICES 267 7-4 SURFACE
REPRESENTATION 270 7,{.1 VERTICES AND SURFACE AMBIGUITY 270 7+2
TRIANGLES 271 7+3 CONNECTING VERTICES INTO TRIANGLES 271 7-4-4 DRAWING
GEOMETRY 274 7.5 RENDERING PIPELINE 275 7.5.1 FIXED-FUNCTION VERSUS
PROGRAMMABLE PIPELINES 277 7.6 SHADERS 278 7.6.1 USING SHADERS TO MOVE
FROM VERTEX TO TRIANGLE TO FRAGMENT 278 }CI" ~ORLTELLTS 7.6.2 SHADER
INPUT AND OUTPUT VALUES 279 7.6.3 SHADER OPERATIONS AND LANGUAGE
CONSTRUCTS 280 7.7 VERTEX SHADERS 280 7.7.1 VERTEX SHADER INPUTS 280
7.7.2 VERTEX SHADER OUTPUTS 281 7.7.3 BASIC VERTEX SHADERS 282 7.704
LINKING VERTEX AND FRAGMENT SHADERS 282 7.8 FRAGMENT SHADERS 283 7.8.1
FRAGMENT SHADER INPUTS 283 7.8.2 FRAGMENT SHADER OUTPUTS 284 7.8.3
COMPILING, LINKING, AND USING SHADERS 284 7.804 SETTING UNIFORM VALUES
286 7.9 BASIC COLORING METHODS 287 7.9.1 PER-OBJECTCOLORS 288 7.9.2
PER-VERTEX COLORS 288 7.9.3 PER-TRIANGLE COLORS 290 7.904 SHARP EDGES
AND VERTEX COLORS 290 7.9.5 MORE ABOUT BASIC SHADING 291 7.9.6
LIMITATIONS OF BASIC SHADING METHODS 292 7.10 TEXTURE MAPPING 292 7.10.1
INTRODUCTION 292 7.10.2 SHADING VIA IMAGE LOOKUP 293 7.10.3 TEXTURE
IMAGES 294 7.1004 TEXTURE SAMPIERS 297 7.11 TEXTURE COORDINATES 297
7.11.1 MAPPING TEXTURE COORDINATES ONTO OBJECTS 298 7.11. 2 GENERATING
TEXTURE COORDINATES 300 7.11.3 TEXTURE COORDINATE DISCONTINUITIES 301
7.11-4 MAPPING OUTSIDE THE UNIT SQUARE 302 7.11.5 TEXTURE SAMPIERS IN
SHADER CODE 309 7.12 THE STEPS OFTEXTURING 309 7.12.1 OTHER FORMS OF
TEXTURE COORDINATES 310 7.12.2 FROM TEXTURE COORDINATES TO A TEXTURE
SAMPIE COLOR 311 7.13 LIMITATIONS OF STATIC SHADING 312 7.14 CHAPTER
SUMMARY 313 CHAPTER 8 LIGHTING 8.1 INTRODUCTION 315 8.2 BASICS OF LIGHT
APPROXIMATION 316 8.2.1 MEASURINGLIGHT 317 8.2.2 LIGHT AS A RAY 318 8.3
A SIMPLE APPROXIMATION OF LIGHTING 318 315 369 CONTENTS XV 8-4 TYPES OF
LIGHT SOURCES 319 8-4.1 DIRECTIONAL LIGHTS 320 8-4.2 POINT LIGHTS 321
8-4.3 SPOTLIGHTS 327 8-4-4 OTHER TYPES UF LIGHT SOURCES 330 8.5 SURFACE
MATERIALS AND LIGHT INTERACTION 331 8.6 CATEGORIES OF LIGHT 332 8.6.1
EMISSION 332 8.6.2 AMBIENT 332 8.6.3 DIFFUSE 334 8.6-4 SPECULAR 338 8.7
COMBINED LIGHTING EQUATION 343 8.8 LIGHTING AND SHADING 348 8.8.1
FLAT-SHADED LIGHTING 349 8.8.2 PER-VERTEX LIGHTING 350 8.8.3
PER-FRAGMENT LIGHTING 354 8.9 TEXTURES AND LIGHTING 358 8.G.1 BASIC
MODULATION 359 8.G.2 SPECULAR LIGHTING AND TEXTURES 360 8.9.3 TEXTURES
AS MATERIALS 362 8.10 ADVANCED LIGHTING 363 8.10.1 NORMAL MAPPING 363
8.11 REFLECTIVE OBJECTS 366 8.12 SHADOWS 367 8.13 CHAPTER SUMMARY 368
CHAPTER 9 RASTERIZATION 9.1 INTRODUCTION 369 9.2 DISPLAYS AND
FRAMEBUFFERS 370 9.3 CONCEPTUAL RASTERIZATION PIPELINE 371 9.3.1
RASTERIZATION STAGES 372 9-4 DETERMINING THE FRAGMENTS: PIXELS COVERED
BY A TRIANGLE 373 9-4.1 FRAGMENTS 373 9.4.2 DEPTH COMPLEXITY 373 9-4.3
CONVERTING TRIANGLES TO FRAGMENTS 375 G-4-4 HANDLING PARTIAL FRAGMENTS
376 9.5 DETERMINING VISIBLE GEOMETRY 378 9.5.1 DEPTH BUFFERING 378 9.5.2
DEPTH BUFFERING IN PRACTICE 387 9.6 COMPUTING FRAGMENT SHADER INPUTS 388
G.6.1 UNIFORM VALUES 389 G.6.2 PER-VERTCX ATTRIBUTES 389 XVI CONTENTS
9.6.3 INTERPOLATING TEXTURE COORDINATES 392 9.604 OTHER SOURCES OF
TEXTURE COORDINATES 394 9.7 EVALUATING THE FRAGMENT SHADER 395 9.8
RASTERIZING TEXTURES 395 9.8.1 TEXTURE COORDINATE REVIEW 396 9.8.2
MAPPING A COORDINATE TO A TEXEL 396 9.8.3 MIPMAPPING 404 9.9 FROM
FRAGMENTS TO PIXELS 415 9.9.1 PIXEL BLENDING 416 9.9.2 ANTIALIASING 420
9.9.3 ANTIALIASING IN PRACTICE 427 9.10 CHAPTER SUMMARY 428 CHAPTER 10
INTERPOLATION 10.1 INTRODUCTION 431 10.2 INTERPOLATION OF POSITION 433
10.2.1 GENERAL DEFINITIONS 433 10.2.2 LINEAR INTERPOLATION 435 10.2.3
HERMITE CURVES 438 10.204 CATMULL-ROM SPLINES 448 10.2.5
KOCHANEK-BARTELS SPLINES 450 10.2.6 BEZIER CURVES 452 10.2.7 OTHER CURVE
TYPES 456 10.3 INTERPOLATION OF ORIENTATION 458 10.3.1 GENERAL
DISCUSSION 458 10.3.2 LINEAR INTERPOLATION 461 10.3.3 SPHERICAL LINEAR
INTERPOLATION 465 10.304 PERFORMANCE IMPROVEMENTS 469 10-4 SAMPLING
CURVES 470 10+1 FORWARU DIFFERENCING 471 1004.2 MIDPOINT SUBDIVISION 473
10+3 COMPUTING ARC LENGTH 476 10.5 CONTROLLING SPEED ALONG A CURVE 480
10.5.1 MOVING AT CONSTANT SPEED 480 10.5.2 MOVING AT VARIABLE SPEED 485
10.6 CAMERA CONTROL 488 10.7 CHAPTER SUMMARY 491 CHAPTER 11 RANDOM
NUMBERS 11.1 INTRODUCTION 493 431 493 CONTENTS XVLL 11.2 PROBABILITY 493
11.2.1 BASIC PROBABILITY 494 11.2.2 RANDOM VARIABLES 497 11.2.3 MEAN AND
STANDARD DEVIATION 501 11.2-4 SPECIAL PROBABIIITY DISTRIBUTIONS 502 11.3
DETENNINING RANDOMNESS SOS 11.3.1 CHI-SQUARE TEST 506 11.3.2 SPECTRAL
TEST 512 11.4 RANDOM NUMBER GENERATORS SB 11.4.1 LINEAR CONGRUENTIAL
METHODS 516 11.4.2 LAGGED FIBONACCI METHODS 520 11.4.3 CARRY METHODS 521
11.4-4 MERSENNE TWISTER 523 11.4.5 CONCLUSIONS 526 11.5 SPECIAL
APPLICATIONS 527 11.5.1 INTEGERS AND RANGES OF INTEGERS 527 11.5.2
FLOATING-POINT NUMBCRS 528 11.5.3 NONUNIFORM DISTRIBUTIONS 528 11.5.4
SPHERICAL SAMPLING 530 11.5.5 DISC SAMPLING 532 11.5.6 NOISE AND
TURBULENCE 534 11.6 CHAPTER SUMMARY 538 CHAPTER 12 INTERSECTION TESTING
541 12.1 INTRODUCTION 541 12.2 CLOSEST POINT AND DISTANCE TESTS 542
12.2.1 CLOSEST POINT ON LINE TO POINT 542 12.2.2 LINE-POINT DISTANCE 544
12.2.3 CLOSEST POINT ON LINE SEGMENT TO POINT 545 12.2-4 LINE
SEGMENT-POINT DISTANCE 546 12.2.5 CLOSEST POINTS BETWEEN TWO LINES 548
12.2.6 LINE-LINE DISTANCE 550 12.2.7 CLOSEST POINTS BETWEEN TWO LINE
SEGMENTS 551 12.2.8 LINE SEGMENT-LINE SEGMENT DISTANCE 553 12.2.G
GENERAL LINEAR COMPONENTS 554 12.3 OBJECT INTERSECTION 554 12.3.1
SPHERES 556 12.3.2 AXIS-ALIGNED BOUNDING BOXES 563 12.3.3 SWEPT SPHERES
571 12.3-4 OBJECT-ORIENTED BOXES 576 12.3.5 TRIANGLES 583 XVIII
C01ZTENTS 12 -4 A SIMPLE COLLISION SYSTEM 588 12+ 1 CHOOSING A BASE
PRIMITIVE 589 12-4.2 BOUNDING HIERARCHIES 590 12+3 DYNAMIC OBJECTS 591
12-4-4 PERFORMANCE IMPROVEMENTS 593 12-4.5 RELATED SYSTEMS 596 12+6
SECTION SUMMARY 599 12.5 CHAPTER SUMMARY 599 CHAPTER 13 RIGID BODY
DYNAMICS 13.1 INTRODUCTION 601 13.2 LINEAR DYNAMICS 602 13.2.1 MOVING
WITH CONSTANT ACCELERATION 602 13.2.2 FORCES 605 13.2.3 LINEAR MOMENTUM
606 13.2-4 MOVING WITH VARIABLE ACCELERATION 607 13.3 NUMERICAL
INTEGRATION 609 13.3.1 DEFINITION 609 13.3.2 EULER'S METHOD 611 13.3.3
RUNGE-KUTTA METHODS 614 13.3-4 VERLET INTEGRATION 616 13.3.5 IMPLICIT
METHODS 619 13.3.6 SEMI-IMPLICIT METHODS 621 13.4 ROTATIONAL DYNAMICS
622 13+ 1 DEFINITION 622 13+2 ORIENTATION AND ANGULAR VELOCITY 622 13+3
TORQUE 625 13.4-4 ANGULAR MOMENTUM AND INERTIA TENSOR 626 13+5
INTEGRATING ROTATIONAL QUANTITIES 628 13.5 COLLISION RESPONSE 630 13.5.1
CONTACT GENERATION 630 13.5.2 LINEAR COLLISION RESPONSE 634 13.5.3
ROTATIONAL COLLISION RESPONSE 638 13.5-4 EXTENDING THE SYSTEM 640 13.6
EFFICIENCY 643 13.7 CHAPTER SUMMARY 645 BIBLIOGRAPHY INDEX TRADEMARKS
AEBOUT TH E CD-ROM 601 647 655 671 672 |
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| author | Van Verth, James M. Bishop, Lars M. |
| author_GND | (DE-588)1109808593 |
| author_facet | Van Verth, James M. Bishop, Lars M. |
| author_role | aut aut |
| author_sort | Van Verth, James M. |
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| building | Verbundindex |
| bvnumber | BV023380877 |
| callnumber-first | Q - Science |
| callnumber-label | QA76 |
| callnumber-raw | QA76.76.C672 |
| callnumber-search | QA76.76.C672 |
| callnumber-sort | QA 276.76 C672 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | ST 320 ST 323 |
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| ctrlnum | (OCoLC)222152271 (DE-599)BVBBV023380877 |
| dewey-full | 794.8151 |
| dewey-hundreds | 700 - The arts |
| dewey-ones | 794 - Indoor games of skill |
| dewey-raw | 794.8151 |
| dewey-search | 794.8151 |
| dewey-sort | 3794.8151 |
| dewey-tens | 790 - Recreational and performing arts |
| discipline | Sport Informatik |
| discipline_str_mv | Sport Informatik |
| edition | 2. ed. |
| format | Book |
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| illustrated | Illustrated |
| index_date | 2024-07-02T21:16:20Z |
| indexdate | 2024-07-09T21:17:18Z |
| institution | BVB |
| isbn | 9780123742971 0123742978 9780123742988 |
| language | English |
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| physical | XXX, 670 S. Ill., graph. Darst. CD-ROM (12 cm) |
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| spelling | Van Verth, James M. Verfasser (DE-588)1109808593 aut Essential mathematics for games and interactive applications a programmer's guide James M. van Verth ; Lars M. Bishop 2. ed. Amsterdam [u.a.] Elsevier, Morgan Kaufmann Publ. 2008 XXX, 670 S. Ill., graph. Darst. CD-ROM (12 cm) txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Mathematik Computer games Programming Three-dimensional display systems Mathematics Programmierung (DE-588)4076370-5 gnd rswk-swf Dreidimensionale Computergrafik (DE-588)4133691-4 gnd rswk-swf Computerspiel (DE-588)4010457-6 gnd rswk-swf Computerspiel (DE-588)4010457-6 s Dreidimensionale Computergrafik (DE-588)4133691-4 s Programmierung (DE-588)4076370-5 s DE-604 Bishop, Lars M. Verfasser aut http://www.loc.gov/catdir/description/els051/2003028267.html Publisher description http://www.loc.gov/catdir/toc/ecip0413/2003028267.html Table of contents GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016563977&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Van Verth, James M. Bishop, Lars M. Essential mathematics for games and interactive applications a programmer's guide Mathematik Computer games Programming Three-dimensional display systems Mathematics Programmierung (DE-588)4076370-5 gnd Dreidimensionale Computergrafik (DE-588)4133691-4 gnd Computerspiel (DE-588)4010457-6 gnd |
| subject_GND | (DE-588)4076370-5 (DE-588)4133691-4 (DE-588)4010457-6 |
| title | Essential mathematics for games and interactive applications a programmer's guide |
| title_auth | Essential mathematics for games and interactive applications a programmer's guide |
| title_exact_search | Essential mathematics for games and interactive applications a programmer's guide |
| title_exact_search_txtP | Essential mathematics for games and interactive applications a programmer's guide |
| title_full | Essential mathematics for games and interactive applications a programmer's guide James M. van Verth ; Lars M. Bishop |
| title_fullStr | Essential mathematics for games and interactive applications a programmer's guide James M. van Verth ; Lars M. Bishop |
| title_full_unstemmed | Essential mathematics for games and interactive applications a programmer's guide James M. van Verth ; Lars M. Bishop |
| title_short | Essential mathematics for games and interactive applications |
| title_sort | essential mathematics for games and interactive applications a programmer s guide |
| title_sub | a programmer's guide |
| topic | Mathematik Computer games Programming Three-dimensional display systems Mathematics Programmierung (DE-588)4076370-5 gnd Dreidimensionale Computergrafik (DE-588)4133691-4 gnd Computerspiel (DE-588)4010457-6 gnd |
| topic_facet | Mathematik Computer games Programming Three-dimensional display systems Mathematics Programmierung Dreidimensionale Computergrafik Computerspiel |
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