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Piegl L.A.,, Tiller W. — The NURBS Book (Monographs in Visual Communication)
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Название: The NURBS Book (Monographs in Visual Communication)
Авторы: Piegl L.A.,, Tiller W.
Аннотация: The second, revised edition of this book covers all aspects of non-uniform rational B-splines necessary to design geometry in a computer-aided environment. Basic B-spline features, curve and surface algorithms, and state-of-the-art geometry tools are all discussed. Detailed code for design algorithms and computational tricks are covered, too, in a lucid, easy-to-understand style, with a minimum of mathematics and using numerous worked examples. The book is a must for students, researchers, and implementors whose work involves the use of splines.
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Рубрика: Computer science /
Статус предметного указателя: Готов указатель с номерами страниц
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Год издания: 1996
Количество страниц: 646
Добавлена в каталог: 03.06.2005
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Предметный указатель
Affine transformation 83 105 118 130
Algorithm, Bezier to power basis conversion matrix 269
Algorithm, bicubically blended Coons surface 502
Algorithm, compute knot removal error bound curve 428
Algorithm, constraint-based surface modification 564
Algorithm, control points of B-spline derivative curves 98
Algorithm, conversion from NURBS to other spline forms 590
Algorithm, curve approximation, global, to points and derivatives 417
Algorithm, curve approximation, global, to points, error bounded 431
Algorithm, curve approximation, local, error bounded cubic 448
Algorithm, curve approximation, local, error bounded quadratic 440
Algorithm, curve global interpolation, cubic spline to points 373
Algorithm, curve global interpolation, to points 369
Algorithm, decompose B-spline, curve into piecewise Bezier segments 173
Algorithm, decompose B-spline, surface into piecewise Bezier patches 177
Algorithm, degree elevation, curve 206 223
Algorithm, degree elevation, surface 209
Algorithm, derivatives of rational B-spline, curve 127
Algorithm, derivatives of rational B-spline, surface 137
Algorithm, find knot span 68
Algorithm, Gordon surface 494
Algorithm, knot insertion, curve 151
Algorithm, knot insertion, surface 155
Algorithm, knot refinement, curve 164
Algorithm, knot refinement, surface 167
Algorithm, knot removal curve 185
Algorithm, NURBS, circle 308
Algorithm, NURBS, open conic arc 317
Algorithm, NURBS, surface of revolution 346
Algorithm, NURBS, three-sided patch on sphere 359
Algorithm, point and derivatives of B-spline, curve 93 99
Algorithm, point and derivatives of B-spline, surface 111 115
Algorithm, point on B-spline curve 82
Algorithm, point on B-spline surface 103
Algorithm, point on Bezier curve 22 24
Algorithm, point on Bezier surface 39
Algorithm, point on power basis curve 7
Algorithm, point on power basis surface 36
Algorithm, point on rational B-spline curve 124 153
Algorithm, point on rational B-spline surface 134
Algorithm, power basis to Bezier conversion matrix 275
Algorithm, quadratic rational Bezier arc 314
Algorithm, remove knots from curve, error bounded 429
Algorithm, surface approximation global to points 422
Algorithm, surface fitting parameter values 377
Algorithm, surface interpolation, global to points 380
Algorithm, surface interpolation, local to points 404
Algorithm, surfaces 114
Algorithm, swept surface, trajectory interpolated 475
Algorithm, swept surface, trajectory not interpolated 477
Algorithm, unclamp a B-spline curve 577
Algorithm, value of Bernstein polynomial 20
Algorithm, value of one B-spline basis function 74
Algorithm, value of one B-spline basis function and derivatives 76
Algorithm, values of nonzero B-spline basis functions 70
Algorithm, values of nonzero B-spline basis functions and derivatives 72
Algorithm, values of nth-degree Bernstein polynomials 20 21
Approximation 361
B-spline basis functions 50. See also Basis functions
B-spline curve 81. See also Curve B-spline
B-spline programming 593. See also Software
B-spline surface 100. See also Surface B-spline
Barycentric coordinates 528
Basis functions, B-spline, computational algorithms 67—78
Basis functions, B-spline, continuity 57 58
Basis functions, B-spline, definition and examples 50—55
Basis functions, B-spline, derivatives 57—63 78
Basis functions, B-spline, properties 55—58 63—67
Basis functions, curve, Bezier 10 15—17
Basis functions, curve, power basis 6
Basis functions, curve, rational B-spline 118
Basis functions, curve, rational Bezier 27
Basis functions, products of B-splines 102 104
Basis functions, surface, Bezier 37
Basis functions, surface, power basis 35
Basis functions, surface, rational B-spline 128—130
Basis functions, surface, rational Bezier 40
Bending 547
Bending, circular 550
Bending, cylindrical 551
Bending, parabolic 550
Bernstein polynomials 10 52 104
Bezier comparison with power basis form 24 25
Bezier curve 9 382
Bezier curve, definition 9
Bezier curve, degree elevation 202—205
Bezier curve, degree reduction 214 219—221
Bezier curve, derivatives 22
Bezier curve, properties 12—14
Bezier surface, definition 37
Bezier surface, properties 38
Blending functions 6. See also Basis functions
Breakpoint 47 51 588
circle 1 26 28 32 45 244 254 279 295 296
Circle, construction of 298—309
Circle, higher degree rational Bezier 326—330
Circle, undamped 578
Clamped curves 576
Clamped knots 571
Composite B-spline curve 181 192
Conic form, conversion and type classification 320—325
Conic form, implicit equation 282 320
Conic form, implicit equation, standard position 283
Conic form, maximum inscribed area 289 320
Conic form, NURBS 321
Conic form, rational Bezier 291 321
Conic form, rational power basis 288 321
Conic form, shape factor 261. See also Shape factor
Conic form, shoulder point 294 394 439
Conic, construction of 310—320
Conic, eccentricity 281
Constraint-based modification, curves 555
Constraint-based modification, surfaces 562
Constraints 361
Continuity in skinning 469
Continuity of B-spline basis functions 57 58
Continuity of B-spline curves 88
Continuity of B-spline surfaces 109
Continuity of closed B-spline curves 575
Continuity of piecewise polynomial and rational curves 47
Continuity, order of 47 588
Continuity, parametric (C) vs geometric (G) 382 588
Control net, Bezier 37
Control net, data structure 600
Control point, B-spline 81
Control point, Bezier 10
Control point, data structure 597
Control point, infinite 295. See also Infinite control point
Control point, multiple 90
Control point, rational B-spline 117
Control point, repositioning 84 108 511 516 533 555 562
Control point, repositioning, minimum length solution 557
Control point, weighted 30 120
Control polygon, B-spline 82
Control polygon, Bezier 11
Control polygon, data structure 598
Control polygon, rational B-spline 117
Conversion, B-spline to/from piecewise power basis polynomials 180 265—278
Conversion, Bezier to/from power basis polynomials 265 272
Conversion, of conies 320
Convex hull property 229
Convex hull property, B-spline 83 105
Convex hull property, Bezier 12 38
Convex hull property, rational B-spline 118 130
Coons surface, bicubically blended 497
Coons surface, bilinearly blended 496
Cross ratio 524 547
Cross-boundary derivatives 497
Curve, B-spline, definition and properties 81—91
Curve, B-spline, derivatives and continuity 88 180
Curve, Bezier 9 82 119
Curve, Bezier, degree elevation 202—205
Curve, Bezier, degree reduction 214 219—221
Curve, clamped 576
Curve, cubic 8 12
Curve, data structure 596 599
Curve, derivative of 2 6
Curve, evaluation of points and derivatives by knot insertion 151
Curve, fitting approximation 410 413 427 438 441
Curve, fitting interpolation 364 370 371 373 388 392 395
Curve, implicit form 1
Curve, parametric form 1
Curve, piecewise, Bezier 47 48
Curve, piecewise, polynomial 47—49 82
Curve, piecewise, rational 47 50
Curve, power basis form 6
Curve, rational B-spline, definition and properties 117—120
Curve, rational B-spline, derivatives 125—127
Curve, rational Bezier, definition 27
Curve, rational Bezier, homogeneous form 30
Curve, rational Bezier, properties 27
Curve, reversal 263
Curve, splitting / subdivision by knot insertion 151
Curve, unclamped 576
Cusp 8 13
Cylinder 40
Data exchange 571 580 583 586
deCasteljau Algorithm 24 32 38 145 314 396 446
Decomposition of B-splines into piecewise, Bezier 169—179
Decomposition of B-splines into piecewise, polynomials 162 168
Degree elevation 188—212 326
Degree reduction 212—227
Derivatives of B-spline basis functions 57—63
Derivatives of B-spline curves 88 91—100
Derivatives of B-spline curves, with respect to the weights 520
Derivatives of B-spline surfaces 109—115
Derivatives of Bernstein polynomials 17
Derivatives of Bezier curves 22
Derivatives of composition of two functions 247
Derivatives of parametric curves 2
Derivatives of parametric surfaces 4
Derivatives of power basis curves 6
Derivatives of power basis surfaces 36
Derivatives of rational B-spline curves 125—127
Derivatives of rational B-spline surfaces 136—138
Derivatives of rational basis functions 555
ellipse 26 282 290 293
Ellipse, construction of full 320
Fitting, choosing knots 365 370 371 374 389 398 412
Fitting, choosing parameters 364 365 376 386 389 398
Flattening 542
Frenet frame 479
Global fitting algorithms 363 405
Gordon surface 485
Homogeneous coordinates 29
Homogeneous representation of circular arc 33
Homogeneous representation of NURBS curve 120
Homogeneous representation of NURBS surface 132
Horner's method, curve 6 9
Horner's method, surface 36
Hyperbola 26 282 290 293
IGES 580
IGES, parametric splines 270 277
Implicit equation, compared with parametric form 4 5
Implicit equation, of curve 1
Implicit equation, of surface 2
Infinite control point 295 298 309 312
Inflection point 8
Interpolation 361
Inverse knot insertion 154 512 525 532
Isoparametric curve 35
Isoparametric curve on B-spline surface 109
Isoparametric curve on Bezier surface 37
Isoparametric curve on rational B-spline surface 135
Isoparametric curve, extraction by knot insertion 161
Knot insertion 141—161 309 326
Knot refinement 162—179 265 533
Knot removal 179—188 272 427 533 576
Knots, clamped open (nonperiodic) 66 572
Knots, clamped vs undamped 571
Knots, data structure 598
Knots, definition 50 51
Knots, merging 162
Knots, multiplicity 57 58 64
Knots, nonperiodic (clamped open) 66 572
Knots, partitioning of the control polygon 147
Knots, periodic vs nonperiodic 574
Knots, span 51
Knots, uniform and nonuniform 66 571
Least squares approximation 410
Line 7 10
Local fitting algorithms 363 382 437
Local support 49 55 118 128
Lofted surface 457. See also Skinned surface
Loop 8 13
Newton iteration 230
Node, B-spline 512 516 530
NURBS representation of bilinear surface 333
NURBS representation of Coons surface 496
NURBS representation of ellipsoid 349
NURBS representation of elliptic paraboloid 349
NURBS representation of general cone 338
NURBS representation of general cylinder 334
NURBS representation of Gordon surface 485
NURBS representation of hyperbolic paraboloid 334
NURBS representation of planar surface 334
NURBS representation of ruled surface 337
NURBS representation of skinned surface 457
NURBS representation of sphere 344
NURBS representation of surface of revolution 340
NURBS representation of swept surface 472
NURBS representation of swung surface 455
NURBS representation of three-sided patch on sphere 351
NURBS representation of torus 344
Offsetting 475 479 552 588
Order 50
parabola 7 11 26 282 290 293
Parametric error 221
Parametric form 1 4
Parametric form of curve 1
Parametric form of surface 2
Parametric form, comparison with implicit equation form 4 5
Parametric form, derivative of 2 4
Parametric form, surface normal vector 4
Partition of unity 16 27 38 57 118 128
Periodic vs nonperiodic curves 575
Perspective map 30 121
PHIGS 585
Point inversion 229. See also Point projection)
Point projection to curve 229—232
Point projection to surface 232—234
Polynomial 6
Power basis form 5 9 24
Power basis form of curve 5
Power basis form of surface 35
Power basis form, derivative of 6
Programming, B-spline 593. See also Software
Projections of curves and surfaces, parallel 236
Projections of curves and surfaces, perspective 237
Rational B-spline curve 117. See also Curve rational
Rational B-spline surface 128. See also Surface rational
Rational B-spline, basis functions 118 534.
Rational Bezier curve definition 27
Rational Bezier surface definition 40
Rational functions 26 *
Reparameterization 44 241—263 265 269 271 589
Reparameterization with a linear function 250 263
Reparameterization with a linear rational function 255
Reparameterization with a piecewise polynomial function 250
Reparameterization with a piecewise rational function 251
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