1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
|
-- File: BSplSLib.cdl
-- Created: Mon Aug 26 07:44:24 1991
-- Author: JCV
---Copyright: Matra Datavision 1991
package BSplSLib
--- Purpose : BSplSLib B-spline surface Library
-- This package provides an implementation of geometric
-- functions for rational and non rational, periodic and non
-- periodic B-spline surface computation.
--
-- this package uses the multi-dimensions splines methods
-- provided in the package BSplCLib.
--
-- In this package the B-spline surface is defined with :
-- . its control points : Array2OfPnt Poles
-- . its weights : Array2OfReal Weights
-- . its knots and their multiplicity in the two parametric
-- direction U and V : Array1OfReal UKnots, VKnots and
-- Array1OfInteger UMults, VMults.
-- . the degree of the normalized Spline functions :
-- UDegree, VDegree
--
-- . the Booleans URational, VRational to know if the weights
-- are constant in the U or V direction.
--
-- . the Booleans UPeriodic, VRational to know if the the
-- surface is periodic in the U or V direction.
--
-- Warnings : The bounds of UKnots and UMults should be the
-- same, the bounds of VKnots and VMults should be the same,
-- the bounds of Poles and Weights shoud be the same.
--
-- The Control points representation is :
-- Poles(Uorigin,Vorigin) ...................Poles(Uorigin,Vend)
-- . .
-- . .
-- Poles(Uend, Vorigin) .....................Poles(Uend, Vend)
--
-- For the double array the row indice corresponds to the
-- parametric U direction and the columns indice corresponds
-- to the parametric V direction.
--
-- KeyWords :
-- B-spline surface, Functions, Library
--
-- References :
-- . A survey of curve and surface methods in CADG Wolfgang BOHM
-- CAGD 1 (1984)
-- . On de Boor-like algorithms and blossoming Wolfgang BOEHM
-- cagd 5 (1988)
-- . Blossoming and knot insertion algorithms for B-spline curves
-- Ronald N. GOLDMAN
-- . Modelisation des surfaces en CAO, Henri GIAUME Peugeot SA
-- . Curves and Surfaces for Computer Aided Geometric Design,
-- a practical guide Gerald Farin
uses TColStd, gp, TColgp
is
imported EvaluatorFunction ;
---Purpose:
-- this is a one dimensional function
-- typedef void (*EvaluatorFunction) (
-- Standard_Integer // Derivative Request
-- Standard_Real * // StartEnd[2][2]
-- // [0] = U
-- // [1] = V
-- // [0] = start
-- // [1] = end
-- Standard_Real // UParameter
-- Standard_Real // VParamerer
-- Standard_Real & // Result
-- Standard_Integer &) ;// Error Code
-- serves to multiply a given vectorial BSpline by a function
-------------------------------------------------------------
-------------------------------------------------------------
---------- -----------
---------- Surface Evaluations -----------
---------- -----------
-------------------------------------------------------------
-------------------------------------------------------------
RationalDerivative(UDeg,VDeg : Integer;
N,M : Integer;
Ders : in out Real;
RDers : in out Real;
All : Boolean = Standard_True);
---Purpose: Computes the derivatives of a ratio of
-- two-variables functions x(u,v) / w(u,v) at orders
-- <N,M>, x(u,v) is a vector in dimension
-- <3>.
--
-- <Ders> is an array containing the values of the
-- input derivatives from 0 to Min(<N>,<UDeg>), 0 to
-- Min(<M>,<VDeg>). For orders higher than
-- <UDeg,VDeg> the input derivatives are assumed to
-- be 0.
--
-- The <Ders> is a 2d array and the dimension of the
-- lines is always (<VDeg>+1) * (<3>+1), even
-- if <N> is smaller than <Udeg> (the derivatives
-- higher than <N> are not used).
--
-- Content of <Ders> :
--
-- x(i,j)[k] means : the composant k of x derivated
-- (i) times in u and (j) times in v.
--
-- ... First line ...
--
-- x[1],x[2],...,x[3],w
-- x(0,1)[1],...,x(0,1)[3],w(1,0)
-- ...
-- x(0,VDeg)[1],...,x(0,VDeg)[3],w(0,VDeg)
--
-- ... Then second line ...
--
-- x(1,0)[1],...,x(1,0)[3],w(1,0)
-- x(1,1)[1],...,x(1,1)[3],w(1,1)
-- ...
-- x(1,VDeg)[1],...,x(1,VDeg)[3],w(1,VDeg)
--
-- ...
--
-- ... Last line ...
--
-- x(UDeg,0)[1],...,x(UDeg,0)[3],w(UDeg,0)
-- x(UDeg,1)[1],...,x(UDeg,1)[3],w(UDeg,1)
-- ...
-- x(Udeg,VDeg)[1],...,x(UDeg,VDeg)[3],w(Udeg,VDeg)
--
--
--
-- If <All> is false, only the derivative at order
-- <N,M> is computed. <RDers> is an array of length
-- 3 which will contain the result :
--
-- x(1)/w , x(2)/w , ... derivated <N> <M> times
--
-- If <All> is true multiples derivatives are
-- computed. All the derivatives (i,j) with 0 <= i+j
-- <= Max(N,M) are computed. <RDers> is an array of
-- length 3 * (<N>+1) * (<M>+1) which will
-- contains :
--
-- x(1)/w , x(2)/w , ...
-- x(1)/w , x(2)/w , ... derivated <0,1> times
-- x(1)/w , x(2)/w , ... derivated <0,2> times
-- ...
-- x(1)/w , x(2)/w , ... derivated <0,N> times
--
-- x(1)/w , x(2)/w , ... derivated <1,0> times
-- x(1)/w , x(2)/w , ... derivated <1,1> times
-- ...
-- x(1)/w , x(2)/w , ... derivated <1,N> times
--
-- x(1)/w , x(2)/w , ... derivated <N,0> times
-- ....
-- Warning: <RDers> must be dimensionned properly.
D0 (U, V : in Real;
UIndex, VIndex : in Integer;
Poles : in Array2OfPnt from TColgp;
Weights : in Array2OfReal from TColStd;
UKnots, VKnots : in Array1OfReal from TColStd;
UMults, VMults : in Array1OfInteger from TColStd;
UDegree, VDegree : in Integer;
URat,VRat : in Boolean;
UPer,VPer : in Boolean;
P : out Pnt from gp);
D1 (U, V : in Real;
UIndex, VIndex : in Integer;
Poles : in Array2OfPnt from TColgp;
Weights : in Array2OfReal from TColStd;
UKnots, VKnots : in Array1OfReal from TColStd;
UMults, VMults : in Array1OfInteger from TColStd;
Degree, VDegree : in Integer;
URat,VRat : in Boolean;
UPer,VPer : in Boolean;
P : out Pnt from gp;
Vu, Vv : out Vec from gp);
D2 (U, V : in Real;
UIndex, VIndex : in Integer;
Poles : in Array2OfPnt from TColgp;
Weights : in Array2OfReal from TColStd;
UKnots, VKnots : in Array1OfReal from TColStd;
UMults, VMults : in Array1OfInteger from TColStd;
UDegree, VDegree : in Integer;
URat,VRat : in Boolean;
UPer,VPer : in Boolean;
P : out Pnt from gp;
Vu, Vv : out Vec from gp;
Vuu, Vvv, Vuv : out Vec from gp);
D3 (U, V : in Real;
UIndex, VIndex : in Integer;
Poles : in Array2OfPnt from TColgp;
Weights : in Array2OfReal from TColStd;
UKnots, VKnots : in Array1OfReal from TColStd;
UMults, VMults : in Array1OfInteger from TColStd;
UDegree, VDegree : in Integer;
URat,VRat : in Boolean;
UPer,VPer : in Boolean;
P : out Pnt from gp;
Vu, Vv : out Vec from gp;
Vuu, Vvv, Vuv : out Vec from gp;
Vuuu, Vvvv, Vuuv, Vuvv : out Vec from gp);
DN (U, V : in Real;
Nu, Nv : in Integer;
UIndex, VIndex : in Integer;
Poles : in Array2OfPnt from TColgp;
Weights : in Array2OfReal from TColStd;
UKnots, VKnots : in Array1OfReal from TColStd;
UMults, VMults : in Array1OfInteger from TColStd;
UDegree, VDegree : in Integer;
URat,VRat : in Boolean;
UPer,VPer : in Boolean;
Vn : out Vec from gp);
Iso (Param : in Real;
IsU : in Boolean;
Poles : in Array2OfPnt from TColgp;
Weights : in Array2OfReal from TColStd;
Knots : in Array1OfReal from TColStd;
Mults : in Array1OfInteger from TColStd;
Degree : in Integer;
Periodic : in Boolean;
CPoles : out Array1OfPnt from TColgp;
CWeights : out Array1OfReal from TColStd);
---Purpose: Computes the poles and weights of an isoparametric
-- curve at parameter <Param> (UIso if <IsU> is True,
-- VIso else).
Reverse (Poles : in out Array2OfPnt from TColgp;
Last : Integer from Standard;
UDirection : Boolean from Standard);
---Purpose: Reverses the array of poles. Last is the Index of
-- the new first Row( Col) of Poles.
-- On a non periodic surface Last is
-- Poles.Upper().
-- On a periodic curve last is
-- (number of flat knots - degree - 1)
-- or
-- (sum of multiplicities(but for the last) + degree
-- - 1)
HomogeneousD0 (U, V : in Real;
UIndex, VIndex : in Integer;
Poles : in Array2OfPnt from TColgp;
Weights : in Array2OfReal from TColStd;
UKnots, VKnots : in Array1OfReal from TColStd;
UMults, VMults : in Array1OfInteger from TColStd;
UDegree, VDegree : in Integer;
URat,VRat : in Boolean;
UPer,VPer : in Boolean;
W : out Real ;
P : out Pnt from gp);
---Purpose: Makes an homogeneous evaluation of Poles and Weights
-- any and returns in P the Numerator value and
-- in W the Denominator value if Weights are present
-- otherwise returns 1.0e0
--
HomogeneousD1 (U, V : in Real;
UIndex, VIndex : in Integer;
Poles : in Array2OfPnt from TColgp;
Weights : in Array2OfReal from TColStd;
UKnots, VKnots : in Array1OfReal from TColStd;
UMults, VMults : in Array1OfInteger from TColStd;
UDegree, VDegree : in Integer;
URat,VRat : in Boolean;
UPer,VPer : in Boolean;
N : out Pnt from gp;
Nu : out Vec from gp;
Nv : out Vec from gp;
D : out Real ;
Du : out Real ;
Dv : out Real) ;
---Purpose: Makes an homogeneous evaluation of Poles and Weights
-- any and returns in P the Numerator value and
-- in W the Denominator value if Weights are present
-- otherwise returns 1.0e0
--
Reverse (Weights : in out Array2OfReal from TColStd;
Last : Integer from Standard;
UDirection : Boolean from Standard);
---Purpose: Reverses the array of weights.
IsRational(Weights : Array2OfReal from TColStd;
I1,I2 : Integer from Standard;
J1,J2 : Integer from Standard;
Epsilon : Real = 0.0) returns Boolean;
---Purpose:
-- Returns False if all the weights of the array <Weights>
-- in the area [I1,I2] * [J1,J2] are identic.
-- Epsilon is used for comparing weights.
-- If Epsilon is 0. the Epsilon of the first weight is used.
SetPoles(Poles : Array2OfPnt from TColgp;
FP : out Array1OfReal from TColStd;
UDirection : Boolean from Standard);
---Purpose: Copy in FP the coordinates of the poles.
SetPoles(Poles : Array2OfPnt from TColgp;
Weights : Array2OfReal from TColStd;
FP : out Array1OfReal from TColStd;
UDirection : Boolean from Standard);
---Purpose: Copy in FP the coordinates of the poles.
GetPoles(FP : Array1OfReal from TColStd;
Poles : out Array2OfPnt from TColgp;
UDirection : Boolean from Standard);
---Purpose: Get from FP the coordinates of the poles.
GetPoles(FP : Array1OfReal from TColStd;
Poles : out Array2OfPnt from TColgp;
Weights : out Array2OfReal from TColStd;
UDirection : Boolean from Standard);
---Purpose: Get from FP the coordinates of the poles.
MovePoint(U, V : Real; -- parameters of the point
Displ : Vec from gp; -- translation vector of the point
UIndex1 : Integer; -- first movable pole in U
UIndex2 : Integer; -- last movable pole in U
VIndex1 : Integer; -- first movable pole in V
VIndex2 : Integer; -- last movable pole in V
UDegree : Integer;
VDegree : Integer;
Rational : Boolean;
Poles : Array2OfPnt from TColgp;
Weights : Array2OfReal from TColStd;
UFlatKnots : Array1OfReal from TColStd;
VFlatKnots : Array1OfReal from TColStd;
UFirstIndex : in out Integer; -- first pole modified in U
ULastIndex : in out Integer; -- last pole modified in U
VFirstIndex : in out Integer; -- first pole modified in V
VLastIndex : in out Integer; -- last pole modified in V
NewPoles : in out Array2OfPnt from TColgp); -- new poles
---Purpose: Find the new poles which allows an old point (with a
-- given u,v as parameters) to reach a new position
-- UIndex1,UIndex2 indicate the range of poles we can
-- move for U
-- (1, UNbPoles-1) or (2, UNbPoles) -> no constraint
-- for one side in U
-- (2, UNbPoles-1) -> the ends are enforced for U
-- don't enter (1,NbPoles) and (1,VNbPoles)
-- -> error: rigid move
-- if problem in BSplineBasis calculation, no change
-- for the curve and
-- UFirstIndex, VLastIndex = 0
-- VFirstIndex, VLastIndex = 0
InsertKnots(UDirection : in Boolean from Standard;
Degree : in Integer from Standard;
Periodic : in Boolean from Standard;
Poles : in Array2OfPnt from TColgp;
Weights : in Array2OfReal from TColStd;
Knots : in Array1OfReal from TColStd;
Mults : in Array1OfInteger from TColStd;
AddKnots : in Array1OfReal from TColStd;
AddMults : in Array1OfInteger from TColStd;
NewPoles : out Array2OfPnt from TColgp;
NewWeights : out Array2OfReal from TColStd;
NewKnots : out Array1OfReal from TColStd;
NewMults : out Array1OfInteger from TColStd;
Epsilon : in Real from Standard;
Add : in Boolean from Standard = Standard_True);
RemoveKnot(UDirection : in Boolean from Standard;
Index : in Integer from Standard;
Mult : in Integer from Standard;
Degree : in Integer from Standard;
Periodic : in Boolean from Standard;
Poles : in Array2OfPnt from TColgp;
Weights : in Array2OfReal from TColStd;
Knots : in Array1OfReal from TColStd;
Mults : in Array1OfInteger from TColStd;
NewPoles : out Array2OfPnt from TColgp;
NewWeights : out Array2OfReal from TColStd;
NewKnots : out Array1OfReal from TColStd;
NewMults : out Array1OfInteger from TColStd;
Tolerance : in Real from Standard)
returns Boolean from Standard;
IncreaseDegree(UDirection : in Boolean from Standard;
Degree : in Integer from Standard;
NewDegree : in Integer from Standard;
Periodic : in Boolean from Standard;
Poles : in Array2OfPnt from TColgp;
Weights : in Array2OfReal from TColStd;
Knots : in Array1OfReal from TColStd;
Mults : in Array1OfInteger from TColStd;
NewPoles : out Array2OfPnt from TColgp;
NewWeights : out Array2OfReal from TColStd;
NewKnots : out Array1OfReal from TColStd;
NewMults : out Array1OfInteger from TColStd);
Unperiodize(UDirection : in Boolean from Standard;
Degree : in Integer from Standard;
Mults : in Array1OfInteger from TColStd;
Knots : in Array1OfReal from TColStd;
Poles : in Array2OfPnt from TColgp;
Weights : in Array2OfReal from TColStd;
NewMults : out Array1OfInteger from TColStd;
NewKnots : out Array1OfReal from TColStd;
NewPoles : out Array2OfPnt from TColgp;
NewWeights : out Array2OfReal from TColStd);
NoWeights returns Array2OfReal from TColStd;
---Purpose: Used as argument for a non rational curve.
--
---C++: return &
---C++: inline
BuildCache(U,V : Real;
USpanDomain,VSpanDomain : Real;
UPeriodicFlag,VPeriodicFlag : Boolean ;
UDegree,VDegree : Integer;
UIndex, VIndex : Integer;
UFlatKnots,VFlatKnots : Array1OfReal from TColStd ;
Poles : Array2OfPnt from TColgp;
Weights : Array2OfReal from TColStd ;
CachePoles : in out Array2OfPnt from TColgp;
CacheWeights : in out Array2OfReal from TColStd);
---Purpose: Perform the evaluation of the Taylor expansion
-- of the Bspline normalized between 0 and 1.
-- If rational computes the homogeneous Taylor expension
-- for the numerator and stores it in CachePoles
--
--
CacheD0(U,V : Real;
UDegree,VDegree : Integer;
UCacheParameter,VCacheParameter : Real;
USpanLenght,VSpanLength : Real;
Poles : Array2OfPnt from TColgp ;
Weights : Array2OfReal from TColStd ;
Point : out Pnt from gp) ;
---Purpose: Perform the evaluation of the of the cache
-- the parameter must be normalized between
-- the 0 and 1 for the span.
-- The Cache must be valid when calling this
-- routine. Geom Package will insure that.
-- and then multiplies by the weights
-- this just evaluates the current point
-- the CacheParameter is where the Cache was
-- constructed the SpanLength is to normalize
-- the polynomial in the cache to avoid bad conditioning
-- effects
--
CoefsD0(U,V : Real;
Poles : Array2OfPnt from TColgp ;
Weights : Array2OfReal from TColStd ;
Point : out Pnt from gp) ;
---Purpose: Calls CacheD0 for Bezier Surfaces Arrays computed with
-- the method PolesCoefficients.
-- Warning: To be used for BezierSurfaces ONLY!!!
---C++: inline
CacheD1(U,V : Real;
UDegree,VDegree : Integer;
UCacheParameter,VCacheParameter : Real;
USpanLenght,VSpanLength : Real;
Poles : Array2OfPnt from TColgp ;
Weights : Array2OfReal from TColStd ;
Point : out Pnt from gp;
VecU, VecV : out Vec from gp) ;
---Purpose: Perform the evaluation of the of the cache
-- the parameter must be normalized between
-- the 0 and 1 for the span.
-- The Cache must be valid when calling this
-- routine. Geom Package will insure that.
-- and then multiplies by the weights
-- this just evaluates the current point
-- the CacheParameter is where the Cache was
-- constructed the SpanLength is to normalize
-- the polynomial in the cache to avoid bad conditioning
-- effects
--
CoefsD1(U,V : Real;
Poles : Array2OfPnt from TColgp;
Weights : Array2OfReal from TColStd;
Point : out Pnt from gp;
VecU, VecV : out Vec from gp) ;
---Purpose: Calls CacheD0 for Bezier Surfaces Arrays computed with
-- the method PolesCoefficients.
-- Warning: To be used for BezierSurfaces ONLY!!!
---C++: inline
CacheD2(U,V : Real;
UDegree,VDegree : Integer;
UCacheParameter,VCacheParameter : Real;
USpanLenght,VSpanLength : Real;
Poles : Array2OfPnt from TColgp ;
Weights : Array2OfReal from TColStd ;
Point : out Pnt from gp;
VecU, VecV, VecUU, VecUV, VecVV : out Vec from gp) ;
---Purpose: Perform the evaluation of the of the cache
-- the parameter must be normalized between
-- the 0 and 1 for the span.
-- The Cache must be valid when calling this
-- routine. Geom Package will insure that.
-- and then multiplies by the weights
-- this just evaluates the current point
-- the CacheParameter is where the Cache was
-- constructed the SpanLength is to normalize
-- the polynomial in the cache to avoid bad conditioning
-- effects
--
CoefsD2(U,V : Real;
Poles : Array2OfPnt from TColgp ;
Weights : Array2OfReal from TColStd ;
Point : out Pnt from gp;
VecU, VecV, VecUU, VecUV, VecVV : out Vec from gp) ;
---Purpose: Calls CacheD0 for Bezier Surfaces Arrays computed with
-- the method PolesCoefficients.
-- Warning: To be used for BezierSurfaces ONLY!!!
---C++: inline
PolesCoefficients(Poles : Array2OfPnt from TColgp;
CachePoles : in out Array2OfPnt from TColgp);
---Purpose: Warning! To be used for BezierSurfaces ONLY!!!
---C++: inline
PolesCoefficients(Poles : Array2OfPnt from TColgp;
Weights : Array2OfReal from TColStd ;
CachePoles : in out Array2OfPnt from TColgp;
CacheWeights : in out Array2OfReal from TColStd) ;
---Purpose: Encapsulation of BuildCache to perform the
-- evaluation of the Taylor expansion for beziersurfaces
-- at parameters 0.,0.;
-- Warning: To be used for BezierSurfaces ONLY!!!
--
Resolution(Poles : in Array2OfPnt from TColgp ;
Weights : in Array2OfReal from TColStd;
UKnots, VKnots : in Array1OfReal from TColStd;
UMults, VMults : in Array1OfInteger from TColStd;
UDegree, VDegree : in Integer;
URat,VRat : in Boolean;
UPer,VPer : in Boolean;
Tolerance3D : in Real from Standard ;
UTolerance : in out Real from Standard ;
VTolerance : in out Real from Standard) ;
---Purpose: Given a tolerance in 3D space returns two
-- tolerances, one in U one in V such that for
-- all (u1,v1) and (u0,v0) in the domain of
-- the surface f(u,v) we have :
-- | u1 - u0 | < UTolerance and
-- | v1 - v0 | < VTolerance
-- we have |f (u1,v1) - f (u0,v0)| < Tolerance3D
Interpolate(UDegree, VDegree : Integer ;
UFlatKnots , VFlatKnots : Array1OfReal from TColStd ;
UParameters, VParameters : Array1OfReal from TColStd ;
Poles : in out Array2OfPnt from TColgp ;
Weights : in out Array2OfReal from TColStd ;
InversionProblem : out Integer) ;
---Purpose: Performs the interpolation of the data points given in
-- the Poles array in the form
-- [1,...,RL][1,...,RC][1...PolesDimension] . The
-- ColLength CL and the Length of UParameters must be the
-- same. The length of VFlatKnots is VDegree + CL + 1.
--
-- The RowLength RL and the Length of VParameters must be
-- the same. The length of VFlatKnots is Degree + RL + 1.
--
-- Warning: the method used to do that interpolation
-- is gauss elimination WITHOUT pivoting. Thus if the
-- diagonal is not dominant there is no guarantee that
-- the algorithm will work. Nevertheless for Cubic
-- interpolation at knots or interpolation at Scheonberg
-- points the method will work. The InversionProblem
-- will report 0 if there was no problem else it will
-- give the index of the faulty pivot
--
Interpolate(UDegree, VDegree : Integer ;
UFlatKnots , VFlatKnots : Array1OfReal from TColStd ;
UParameters, VParameters : Array1OfReal from TColStd ;
Poles : in out Array2OfPnt from TColgp ;
InversionProblem : out Integer) ;
---Purpose: Performs the interpolation of the data points given in
-- the Poles array.
-- The ColLength CL and the Length of UParameters must be
-- the same. The length of VFlatKnots is VDegree + CL + 1.
--
-- The RowLength RL and the Length of VParameters must be
-- the same. The length of VFlatKnots is Degree + RL + 1.
--
-- Warning: the method used to do that interpolation
-- is gauss elimination WITHOUT pivoting. Thus if the
-- diagonal is not dominant there is no guarantee that
-- the algorithm will work. Nevertheless for Cubic
-- interpolation at knots or interpolation at Scheonberg
-- points the method will work. The InversionProblem
-- will report 0 if there was no problem else it will
-- give the index of the faulty pivot
--
FunctionMultiply(
Function : EvaluatorFunction from BSplSLib ;
UBSplineDegree : Integer ;
VBSplineDegree : Integer ;
UBSplineKnots : Array1OfReal from TColStd ;
VBSplineKnots : Array1OfReal from TColStd ;
UMults : Array1OfInteger from TColStd ;
VMults : Array1OfInteger from TColStd ;
Poles : Array2OfPnt from TColgp ;
Weights : Array2OfReal from TColStd ;
UFlatKnots : Array1OfReal from TColStd ;
VFlatKnots : Array1OfReal from TColStd ;
UNewDegree : Integer ;
VNewDegree : Integer ;
NewNumerator : in out Array2OfPnt from TColgp ;
NewDenominator : in out Array2OfReal from TColStd ;
Status : in out Integer) ;
---Purpose: this will multiply a given BSpline numerator N(u,v)
-- and denominator D(u,v) defined by its
-- U/VBSplineDegree and U/VBSplineKnots, and
-- U/VMults. Its Poles and Weights are arrays which are
-- coded as array2 of the form
-- [1..UNumPoles][1..VNumPoles] by a function a(u,v)
-- which is assumed to satisfy the following : 1.
-- a(u,v) * N(u,v) and a(u,v) * D(u,v) is a polynomial
-- BSpline that can be expressed exactly as a BSpline of
-- degree U/VNewDegree on the knots U/VFlatKnots 2. the range
-- of a(u,v) is the same as the range of N(u,v)
-- or D(u,v)
-- ---Warning: it is the caller's responsability to
-- insure that conditions 1. and 2. above are satisfied
-- : no check whatsoever is made in this method --
-- Status will return 0 if OK else it will return the
-- pivot index -- of the matrix that was inverted to
-- compute the multiplied -- BSpline : the method used
-- is interpolation at Schoenenberg -- points of
-- a(u,v)* N(u,v) and a(u,v) * D(u,v)
-- Status will return 0 if OK else it will return the pivot index
-- of the matrix that was inverted to compute the multiplied
-- BSpline : the method used is interpolation at Schoenenberg
-- points of a(u,v)*F(u,v)
-- --
--
end BSplSLib;
|
