forked from therion/therion
-
Notifications
You must be signed in to change notification settings - Fork 0
/
thbezier.cxx
1485 lines (1297 loc) · 44.6 KB
/
thbezier.cxx
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
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
/**
* @file thbezier.cxx
*/
/*
* Bezier interpolation for inkscape drawing code.
* Original code published in:
* An Algorithm for Automatically Fitting Digitized Curves
* by Philip J. Schneider
* "Graphics Gems", Academic Press, 1990
*
* Authors:
* Philip J. Schneider
* Lauris Kaplinski <lauris@kaplinski.com>
* Peter Moulder <pmoulder@mail.csse.monash.edu.au>
*
* Copyright (C) 1990 Philip J. Schneider
* Copyright (C) 2001 Lauris Kaplinski
* Copyright (C) 2001 Ximian, Inc.
* Copyright (C) 2003,2004 Monash University
*
*/
/* Copyright (C) 2007 Stacho Mudrak
*
* $Date: $
* $RCSfile: $
* $Revision: $
*
* --------------------------------------------------------------------
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
* --------------------------------------------------------------------
*/
#include "thbezier.h"
#include <cassert>
#include <cstring>
#include <cstdio>
#include <cmath>
#define __SP_BEZIER_UTILS_C__
#define SP_HUGE 1e5
#define noBEZIER_DEBUG
#ifdef HAVE_CONFIG_H
# include <config.h>
#endif
#ifdef HAVE_IEEEFP_H
# include <ieeefp.h>
#endif
//#include <glib/gmessages.h>
//#include <glib/gmem.h>
//#include <glib/gtypes.h>
//#include <libnr/nr-forward.h>
//#include <libnr/nr-point-fns.h>
#include <math.h>
#ifdef THMSVC
#include <float.h>
#define hypot _hypot
#endif
#define g_return_val_if_fail(C,V) if (!(C)) return (V);
#define g_return_if_fail(C) if (!(C)) return;
#define g_new(T,L) new T [L];
#define g_free(V) delete [] V
#define g_assert(X) assert(X)
/* You might try changing the above to <cmath> if you have problems.
* Whether you use math.h or cmath, you may need to edit the .cpp file
* and/or other .h files to use the same header file.
*/
#if defined(__isnan)
# define isNaN(_a) (__isnan(_a)) /* MacOSX/Darwin definition < 10.4 */
#elif defined(WIN32) || defined(_isnan)
# define isNaN(_a) (_isnan(_a)) /* Win32 definition */
#elif defined(isnan) || defined(__FreeBSD__) || defined(__osf__)
# define isNaN(_a) (isnan(_a)) /* GNU definition */
#else
# define isNaN(_a) (std::isnan(_a))
#endif
/* If the above doesn't work, then try (a != a).
* Also, please report a bug as per http://www.inkscape.org/report_bugs.php,
* giving information about what platform and compiler version you're using.
*/
#if defined(__isfinite)
# define isFinite(_a) (__isfinite(_a)) /* MacOSX/Darwin definition < 10.4 */
#elif defined(__sgi)
# define isFinite(_a) (_isfinite(_a))
#elif defined(isfinite)
# define isFinite(_a) (isfinite(_a))
#elif defined(__osf__)
# define isFinite(_a) (finite(_a) && !isNaN(_a))
#elif defined(THMSVC)
# define isFinite(_a) (_finite(_a) && !isNaN(_a))
#else
# define isFinite(_a) (std::isfinite(_a))
#endif
/* If the above doesn't work, then try (finite(_a) && !isNaN(_a)) or (!isNaN((_a) - (_a))).
* Also, please report a bug as per http://www.inkscape.org/report_bugs.php,
* giving information about what platform and compiler version you're using.
*/
class NR_Point {
private:
double _pt[2];
public:
inline NR_Point()
{ _pt[0] = _pt[1] = 0; }
inline NR_Point(double x, double y) {
_pt[0] = x;
_pt[1] = y;
}
inline NR_Point(NR_Point const &p) {
for (unsigned i = 0; i < 2; ++i) {
_pt[i] = p._pt[i];
}
}
inline NR_Point &operator=(NR_Point const &p) {
for (unsigned i = 0; i < 2; ++i) {
_pt[i] = p._pt[i];
}
return *this;
}
inline double operator[](unsigned i) const {
return _pt[i];
}
inline double &operator[](unsigned i) {
return _pt[i];
}
//double operator[](size_t d) const throw() { return _pt[d]; }
//double &operator[](size_t d) throw() { return _pt[d]; }
/** Return a point like this point but rotated -90 degrees.
(If the y axis grows downwards and the x axis grows to the
right, then this is 90 degrees counter-clockwise.)
**/
NR_Point ccw() const {
return NR_Point(_pt[1], -_pt[0]);
}
/** Return a point like this point but rotated +90 degrees.
(If the y axis grows downwards and the x axis grows to the
right, then this is 90 degrees clockwise.)
**/
NR_Point cw() const {
return NR_Point(-_pt[1], _pt[0]);
}
void normalize();
inline NR_Point &operator+=(NR_Point const &o) {
for ( unsigned i = 0 ; i < 2 ; ++i ) {
_pt[i] += o._pt[i];
}
return *this;
}
inline NR_Point &operator-=(NR_Point const &o) {
for ( unsigned i = 0 ; i < 2 ; ++i ) {
_pt[i] -= o._pt[i];
}
return *this;
}
inline NR_Point &operator/=(double const s) {
for ( unsigned i = 0 ; i < 2 ; ++i ) {
_pt[i] /= s;
}
return *this;
}
inline NR_Point &operator*=(double const s) {
for ( unsigned i = 0 ; i < 2 ; ++i ) {
_pt[i] *= s;
}
return *this;
}
//inline bool operator == (const NR_Point &in_pnt) const {
// return ((_pt[0] == in_pnt[0]) && (_pt[1] == in_pnt[1]));
//}
};
inline NR_Point rot90(NR_Point const &p)
{
return NR_Point(-p[1], p[0]);
}
inline NR_Point operator+(NR_Point const &a, NR_Point const &b)
{
NR_Point ret;
for (int i = 0; i < 2; i++) {
ret[i] = a[i] + b[i];
}
return ret;
}
inline double dot(NR_Point const &a, NR_Point const &b)
{
double ret = 0;
for ( int i = 0 ; i < 2 ; i++ ) {
ret += a[i] * b[i];
}
return ret;
}
void NR_Point::normalize() {
double len = hypot(_pt[0], _pt[1]);
g_return_if_fail(len != 0);
g_return_if_fail(!isNaN(len));
static double const inf = 1e300;
if(len != inf) {
*this /= len;
} else {
unsigned n_inf_coords = 0;
/* Delay updating pt in case neither coord is infinite. */
NR_Point tmp;
for ( unsigned i = 0 ; i < 2 ; ++i ) {
if ( _pt[i] == inf ) {
++n_inf_coords;
tmp[i] = 1.0;
} else if ( _pt[i] == -inf ) {
++n_inf_coords;
tmp[i] = -1.0;
} else {
tmp[i] = 0.0;
}
}
switch (n_inf_coords) {
case 0:
/* Can happen if both coords are near +/-DBL_MAX. */
*this /= 4.0;
len = hypot(_pt[0], _pt[1]);
g_assert(len != inf);
*this /= len;
break;
case 1:
*this = tmp;
break;
case 2:
*this = NR_Point(sqrt(0.5) * tmp[0], sqrt(0.5) * tmp[1]);
break;
}
}
}
inline NR_Point operator-(NR_Point const &a, NR_Point const &b)
{
NR_Point ret;
for (int i = 0; i < 2; i++) {
ret[i] = a[i] - b[i];
}
return ret;
}
/** This is a rotation (sort of). */
inline NR_Point operator^(NR_Point const &a, NR_Point const &b)
{
NR_Point const ret(a[0] * b[0] - a[1] * b[1],
a[1] * b[0] + a[0] * b[1]);
return ret;
}
inline NR_Point operator-(NR_Point const &a)
{
NR_Point ret;
for(unsigned i = 0; i < 2; i++) {
ret[i] = -a[i];
}
return ret;
}
inline NR_Point operator*(double const s, NR_Point const &b)
{
NR_Point ret;
for(int i = 0; i < 2; i++) {
ret[i] = s * b[i];
}
return ret;
}
inline NR_Point operator/(NR_Point const &b, double const d)
{
NR_Point ret;
for(int i = 0; i < 2; i++) {
ret[i] = b[i] / d;
}
return ret;
}
inline bool operator==(NR_Point const &a, NR_Point const &b)
{
return ( ( a[0] == b[0] ) && ( a[1] == b[1] ) );
}
inline bool operator!=(NR_Point const &a, NR_Point const &b)
{
return ( ( a[0] != b[0] ) || ( a[1] != b[1] ) );
}
inline double L2(NR_Point const &p) {
return hypot(p[0], p[1]);
}
bool is_zero(NR_Point const &p)
{
return ( p[0] == 0 &&
p[1] == 0 );
}
NR_Point unit_vector(NR_Point const &a)
{
NR_Point ret(a);
ret.normalize();
return ret;
}
NR_Point bezier_pt(unsigned degree, NR_Point const V[], double t);
long sp_bezier_fit_cubic(NR_Point bezier[], NR_Point const data[], long len, double error);
long sp_bezier_fit_cubic_r(NR_Point bezier[], NR_Point const data[], long len, double error,
unsigned max_beziers);
long sp_bezier_fit_cubic_full(NR_Point bezier[], int split_points[], NR_Point const data[], long len,
NR_Point const &tHat1, NR_Point const &tHat2,
double error, unsigned max_beziers);
NR_Point sp_darray_left_tangent(NR_Point const d[], unsigned const len);
NR_Point sp_darray_left_tangent(NR_Point const d[], unsigned const len, double const tolerance_sq);
NR_Point sp_darray_right_tangent(NR_Point const d[], unsigned const length, double const tolerance_sq);
/*
* Temporary fix for various misdefinitions of isnan().
* isnan() is becoming undef'd in some .h files.
* #include this last in your .cpp file to get it right.
*
* The problem is that isnan and isfinite are part of C99 but aren't part of
* the C++ standard (which predates C99).
*
* Authors:
* Inkscape groupies and obsessive-compulsives
*
* Copyright (C) 2004 authors
*
* Released under GNU GPL, read the file 'COPYING' for more information
*
* 2005 modification hereby placed in public domain. Probably supercedes the 2004 copyright
* for the code itself.
*/
typedef NR_Point BezierCurve[];
/* Forward declarations */
static void generate_bezier(NR_Point b[], NR_Point const d[], double const u[], unsigned len,
NR_Point const &tHat1, NR_Point const &tHat2, double tolerance_sq);
static void estimate_lengths(NR_Point bezier[],
NR_Point const data[], double const u[], unsigned len,
NR_Point const &tHat1, NR_Point const &tHat2);
static void estimate_bi(NR_Point b[4], unsigned ei,
NR_Point const data[], double const u[], unsigned len);
static void reparameterize(NR_Point const d[], unsigned len, double u[], BezierCurve const bezCurve);
static double NewtonRaphsonRootFind(BezierCurve const Q, NR_Point const &P, double u);
static NR_Point sp_darray_center_tangent(NR_Point const d[], unsigned center, unsigned length);
static NR_Point sp_darray_right_tangent(NR_Point const d[], unsigned const len);
static unsigned copy_without_nans_or_adjacent_duplicates(NR_Point const src[], unsigned src_len, NR_Point dest[]);
static void chord_length_parameterize(NR_Point const d[], double u[], unsigned len);
static double compute_max_error_ratio(NR_Point const d[], double const u[], unsigned len,
BezierCurve const bezCurve, double tolerance,
unsigned *splitPoint);
static double compute_hook(NR_Point const &a, NR_Point const &b, double const u, BezierCurve const bezCurve,
double const tolerance);
static NR_Point const unconstrained_tangent(0, 0);
/*
* B0, B1, B2, B3 : Bezier multipliers
*/
#define B0(u) ( ( 1.0 - u ) * ( 1.0 - u ) * ( 1.0 - u ) )
#define B1(u) ( 3 * u * ( 1.0 - u ) * ( 1.0 - u ) )
#define B2(u) ( 3 * u * u * ( 1.0 - u ) )
#define B3(u) ( u * u * u )
#ifdef BEZIER_DEBUG
# define DOUBLE_ASSERT(x) g_assert( ( (x) > -SP_HUGE ) && ( (x) < SP_HUGE ) )
# define BEZIER_ASSERT(b) do { \
DOUBLE_ASSERT((b)[0][0]); DOUBLE_ASSERT((b)[0][1]); \
DOUBLE_ASSERT((b)[1][0]); DOUBLE_ASSERT((b)[1][1]); \
DOUBLE_ASSERT((b)[2][0]); DOUBLE_ASSERT((b)[2][1]); \
DOUBLE_ASSERT((b)[3][0]); DOUBLE_ASSERT((b)[3][1]); \
} while(0)
#else
# define DOUBLE_ASSERT(x) do { } while(0)
# define BEZIER_ASSERT(b) do { } while(0)
#endif
/**
* Fit a single-segment Bezier curve to a set of digitized points.
*
* \return Number of segments generated, or -1 on error.
*/
long
sp_bezier_fit_cubic(NR_Point *bezier, NR_Point const *data, long len, double error)
{
return sp_bezier_fit_cubic_r(bezier, data, len, error, 1);
}
/**
* Fit a multi-segment Bezier curve to a set of digitized points, with
* possible weedout of identical points and NaNs.
*
* \param max_beziers Maximum number of generated segments
* \param Result array, must be large enough for n. segments * 4 elements.
*
* \return Number of segments generated, or -1 on error.
*/
long
sp_bezier_fit_cubic_r(NR_Point bezier[], NR_Point const data[], long const len, double const error, unsigned const max_beziers)
{
g_return_val_if_fail(bezier != NULL, -1);
g_return_val_if_fail(data != NULL, -1);
g_return_val_if_fail(len > 0, -1);
g_return_val_if_fail(max_beziers < (1ul << (31 - 2 - 1 - 3)), -1);
NR_Point *uniqued_data = g_new(NR_Point, len);
unsigned uniqued_len = copy_without_nans_or_adjacent_duplicates(data, len, uniqued_data);
if ( uniqued_len < 2 ) {
g_free(uniqued_data);
return 0;
}
/* Call fit-cubic function with recursion. */
long const ret = sp_bezier_fit_cubic_full(bezier, NULL, uniqued_data, uniqued_len,
unconstrained_tangent, unconstrained_tangent,
error, max_beziers);
g_free(uniqued_data);
return ret;
}
/**
* Copy points from src to dest, filter out points containing NaN and
* adjacent points with equal x and y.
* \return length of dest
*/
static unsigned
copy_without_nans_or_adjacent_duplicates(NR_Point const src[], unsigned src_len, NR_Point dest[])
{
unsigned si = 0;
for (;;) {
if ( si == src_len ) {
return 0;
}
if (!isNaN(src[si][0]) &&
!isNaN(src[si][1])) {
dest[0] = NR_Point(src[si]);
++si;
break;
}
}
unsigned di = 0;
for (; si < src_len; ++si) {
NR_Point const src_pt = NR_Point(src[si]);
if ( src_pt != dest[di]
&& !isNaN(src_pt[0])
&& !isNaN(src_pt[1])) {
dest[++di] = src_pt;
}
}
unsigned dest_len = di + 1;
g_assert( dest_len <= src_len );
return dest_len;
}
/**
* Fit a multi-segment Bezier curve to a set of digitized points, without
* possible weedout of identical points and NaNs.
*
* \pre data is uniqued, i.e. not exist i: data[i] == data[i + 1].
* \param max_beziers Maximum number of generated segments
* \param Result array, must be large enough for n. segments * 4 elements.
*/
long
sp_bezier_fit_cubic_full(NR_Point bezier[], int split_points[],
NR_Point const data[], long const len,
NR_Point const &tHat1, NR_Point const &tHat2,
double const error, unsigned const max_beziers)
{
int const maxIterations = 4; /* Max times to try iterating */
g_return_val_if_fail(bezier != NULL, -1);
g_return_val_if_fail(data != NULL, -1);
g_return_val_if_fail(len > 0, -1);
g_return_val_if_fail(max_beziers >= 1, -1);
g_return_val_if_fail(error >= 0.0, -1);
if ( len < 2 ) return 0;
if ( len == 2 ) {
/* We have 2 points, which can be fitted trivially. */
bezier[0] = data[0];
bezier[3] = data[len - 1];
double const dist = ( L2( data[len - 1]
- data[0] )
/ 3.0 );
if (isNaN(dist)) {
/* Numerical problem, fall back to straight line segment. */
bezier[1] = bezier[0];
bezier[2] = bezier[3];
} else {
bezier[1] = ( is_zero(tHat1)
? ( 2 * bezier[0] + bezier[3] ) / 3.
: bezier[0] + dist * tHat1 );
bezier[2] = ( is_zero(tHat2)
? ( bezier[0] + 2 * bezier[3] ) / 3.
: bezier[3] + dist * tHat2 );
}
BEZIER_ASSERT(bezier);
return 1;
}
/* Parameterize points, and attempt to fit curve */
unsigned splitPoint; /* Point to split point set at. */
bool is_corner;
{
double *u = g_new(double, len);
chord_length_parameterize(data, u, len);
if ( u[len - 1] == 0.0 ) {
/* Zero-length path: every point in data[] is the same.
*
* (Clients aren't allowed to pass such data; handling the case is defensive
* programming.)
*/
g_free(u);
return 0;
}
generate_bezier(bezier, data, u, len, tHat1, tHat2, error);
reparameterize(data, len, u, bezier);
/* Find max deviation of points to fitted curve. */
double const tolerance = sqrt(error + 1e-9);
double maxErrorRatio = compute_max_error_ratio(data, u, len, bezier, tolerance, &splitPoint);
if ( fabs(maxErrorRatio) <= 1.0 ) {
BEZIER_ASSERT(bezier);
g_free(u);
return 1;
}
/* If error not too large, then try some reparameterization and iteration. */
if ( 0.0 <= maxErrorRatio && maxErrorRatio <= 3.0 ) {
for (int i = 0; i < maxIterations; i++) {
generate_bezier(bezier, data, u, len, tHat1, tHat2, error);
reparameterize(data, len, u, bezier);
maxErrorRatio = compute_max_error_ratio(data, u, len, bezier, tolerance, &splitPoint);
if ( fabs(maxErrorRatio) <= 1.0 ) {
BEZIER_ASSERT(bezier);
g_free(u);
return 1;
}
}
}
g_free(u);
is_corner = (maxErrorRatio < 0);
}
if (is_corner) {
g_assert(splitPoint < unsigned(len));
if (splitPoint == 0) {
if (is_zero(tHat1)) {
/* Got spike even with unconstrained initial tangent. */
++splitPoint;
} else {
return sp_bezier_fit_cubic_full(bezier, split_points, data, len, unconstrained_tangent, tHat2,
error, max_beziers);
}
} else if (splitPoint == unsigned(len - 1)) {
if (is_zero(tHat2)) {
/* Got spike even with unconstrained final tangent. */
--splitPoint;
} else {
return sp_bezier_fit_cubic_full(bezier, split_points, data, len, tHat1, unconstrained_tangent,
error, max_beziers);
}
}
}
if ( 1 < max_beziers ) {
/*
* Fitting failed -- split at max error point and fit recursively
*/
unsigned const rec_max_beziers1 = max_beziers - 1;
NR_Point recTHat2, recTHat1;
if (is_corner) {
g_return_val_if_fail(0 < splitPoint && splitPoint < unsigned(len - 1), -1);
recTHat1 = recTHat2 = unconstrained_tangent;
} else {
/* Unit tangent vector at splitPoint. */
recTHat2 = sp_darray_center_tangent(data, splitPoint, len);
recTHat1 = -recTHat2;
}
long const nsegs1 = sp_bezier_fit_cubic_full(bezier, split_points, data, splitPoint + 1,
tHat1, recTHat2, error, rec_max_beziers1);
if ( nsegs1 < 0 ) {
#ifdef BEZIER_DEBUG
g_print("fit_cubic[1]: recursive call failed\n");
#endif
return -1;
}
g_assert( nsegs1 != 0 );
if (split_points != NULL) {
split_points[nsegs1 - 1] = splitPoint;
}
unsigned const rec_max_beziers2 = max_beziers - nsegs1;
long const nsegs2 = sp_bezier_fit_cubic_full(bezier + nsegs1*4,
( split_points == NULL
? NULL
: split_points + nsegs1 ),
data + splitPoint, len - splitPoint,
recTHat1, tHat2, error, rec_max_beziers2);
if ( nsegs2 < 0 ) {
#ifdef BEZIER_DEBUG
g_print("fit_cubic[2]: recursive call failed\n");
#endif
return -1;
}
#ifdef BEZIER_DEBUG
g_print("fit_cubic: success[nsegs: %d+%d=%d] on max_beziers:%u\n",
nsegs1, nsegs2, nsegs1 + nsegs2, max_beziers);
#endif
return nsegs1 + nsegs2;
} else {
return -1;
}
}
/**
* Fill in \a bezier[] based on the given data and tangent requirements, using
* a least-squares fit.
*
* Each of tHat1 and tHat2 should be either a zero vector or a unit vector.
* If it is zero, then bezier[1 or 2] is estimated without constraint; otherwise,
* it bezier[1 or 2] is placed in the specified direction from bezier[0 or 3].
*
* \param tolerance_sq Used only for an initial guess as to tangent directions
* when \a tHat1 or \a tHat2 is zero.
*/
static void
generate_bezier(NR_Point bezier[],
NR_Point const data[], double const u[], unsigned const len,
NR_Point const &tHat1, NR_Point const &tHat2,
double const tolerance_sq)
{
bool const est1 = is_zero(tHat1);
bool const est2 = is_zero(tHat2);
NR_Point est_tHat1( est1
? sp_darray_left_tangent(data, len, tolerance_sq)
: tHat1 );
NR_Point est_tHat2( est2
? sp_darray_right_tangent(data, len, tolerance_sq)
: tHat2 );
estimate_lengths(bezier, data, u, len, est_tHat1, est_tHat2);
/* We find that sp_darray_right_tangent tends to produce better results
for our current freehand tool than full estimation. */
if (est1) {
estimate_bi(bezier, 1, data, u, len);
if (bezier[1] != bezier[0]) {
est_tHat1 = unit_vector(bezier[1] - bezier[0]);
}
estimate_lengths(bezier, data, u, len, est_tHat1, est_tHat2);
}
}
static void
estimate_lengths(NR_Point bezier[],
NR_Point const data[], double const uPrime[], unsigned const len,
NR_Point const &tHat1, NR_Point const &tHat2)
{
double C[2][2]; /* Matrix C. */
double X[2]; /* Matrix X. */
/* Create the C and X matrices. */
C[0][0] = 0.0;
C[0][1] = 0.0;
C[1][0] = 0.0;
C[1][1] = 0.0;
X[0] = 0.0;
X[1] = 0.0;
/* First and last control points of the Bezier curve are positioned exactly at the first and
last data points. */
bezier[0] = data[0];
bezier[3] = data[len - 1];
for (unsigned i = 0; i < len; i++) {
/* Bezier control point coefficients. */
double const b0 = B0(uPrime[i]);
double const b1 = B1(uPrime[i]);
double const b2 = B2(uPrime[i]);
double const b3 = B3(uPrime[i]);
/* rhs for eqn */
NR_Point const a1 = b1 * tHat1;
NR_Point const a2 = b2 * tHat2;
C[0][0] += dot(a1, a1);
C[0][1] += dot(a1, a2);
C[1][0] = C[0][1];
C[1][1] += dot(a2, a2);
/* Additional offset to the data point from the predicted point if we were to set bezier[1]
to bezier[0] and bezier[2] to bezier[3]. */
NR_Point const shortfall
= ( data[i]
- ( ( b0 + b1 ) * bezier[0] )
- ( ( b2 + b3 ) * bezier[3] ) );
X[0] += dot(a1, shortfall);
X[1] += dot(a2, shortfall);
}
/* We've constructed a pair of equations in the form of a matrix product C * alpha = X.
Now solve for alpha. */
double alpha_l, alpha_r;
/* Compute the determinants of C and X. */
double const det_C0_C1 = C[0][0] * C[1][1] - C[1][0] * C[0][1];
if ( det_C0_C1 != 0 ) {
/* Apparently Kramer's rule. */
double const det_C0_X = C[0][0] * X[1] - C[0][1] * X[0];
double const det_X_C1 = X[0] * C[1][1] - X[1] * C[0][1];
alpha_l = det_X_C1 / det_C0_C1;
alpha_r = det_C0_X / det_C0_C1;
} else {
/* The matrix is under-determined. Try requiring alpha_l == alpha_r.
*
* One way of implementing the constraint alpha_l == alpha_r is to treat them as the same
* variable in the equations. We can do this by adding the columns of C to form a single
* column, to be multiplied by alpha to give the column vector X.
*
* We try each row in turn.
*/
double const c0 = C[0][0] + C[0][1];
if (c0 != 0) {
alpha_l = alpha_r = X[0] / c0;
} else {
double const c1 = C[1][0] + C[1][1];
if (c1 != 0) {
alpha_l = alpha_r = X[1] / c1;
} else {
/* Let the below code handle this. */
alpha_l = alpha_r = 0.;
}
}
}
/* If alpha negative, use the Wu/Barsky heuristic (see text). (If alpha is 0, you get
coincident control points that lead to divide by zero in any subsequent
NewtonRaphsonRootFind() call.) */
/// \todo Check whether this special-casing is necessary now that
/// NewtonRaphsonRootFind handles non-positive denominator.
if ( alpha_l < 1.0e-6 ||
alpha_r < 1.0e-6 )
{
alpha_l = alpha_r = ( L2( data[len - 1]
- data[0] )
/ 3.0 );
}
/* Control points 1 and 2 are positioned an alpha distance out on the tangent vectors, left and
right, respectively. */
bezier[1] = alpha_l * tHat1 + bezier[0];
bezier[2] = alpha_r * tHat2 + bezier[3];
return;
}
static double lensq(NR_Point const p) {
return dot(p, p);
}
static void
estimate_bi(NR_Point bezier[4], unsigned const ei,
NR_Point const data[], double const u[], unsigned const len)
{
g_return_if_fail(1 <= ei && ei <= 2);
unsigned const oi = 3 - ei;
double num[2] = {0., 0.};
double den = 0.;
for (unsigned i = 0; i < len; ++i) {
double const ui = u[i];
double const b[4] = {
B0(ui),
B1(ui),
B2(ui),
B3(ui)
};
for (unsigned d = 0; d < 2; ++d) {
num[d] += b[ei] * (b[0] * bezier[0][d] +
b[oi] * bezier[oi][d] +
b[3] * bezier[3][d] +
- data[i][d]);
}
den -= b[ei] * b[ei];
}
if (den != 0.) {
for (unsigned d = 0; d < 2; ++d) {
bezier[ei][d] = num[d] / den;
}
} else {
bezier[ei] = ( oi * bezier[0] + ei * bezier[3] ) / 3.;
}
}
/**
* Given set of points and their parameterization, try to find a better assignment of parameter
* values for the points.
*
* \param d Array of digitized points.
* \param u Current parameter values.
* \param bezCurve Current fitted curve.
* \param len Number of values in both d and u arrays.
* Also the size of the array that is allocated for return.
*/
static void
reparameterize(NR_Point const d[],
unsigned const len,
double u[],
BezierCurve const bezCurve)
{
g_assert( 2 <= len );
unsigned const last = len - 1;
g_assert( bezCurve[0] == d[0] );
g_assert( bezCurve[3] == d[last] );
g_assert( u[0] == 0.0 );
g_assert( u[last] == 1.0 );
/* Otherwise, consider including 0 and last in the below loop. */
for (unsigned i = 1; i < last; i++) {
u[i] = NewtonRaphsonRootFind(bezCurve, d[i], u[i]);
}
}
/**
* Use Newton-Raphson iteration to find better root.
*
* \param Q Current fitted curve
* \param P Digitized point
* \param u Parameter value for "P"
*
* \return Improved u
*/
static double
NewtonRaphsonRootFind(BezierCurve const Q, NR_Point const &P, double const u)
{
g_assert( 0.0 <= u );
g_assert( u <= 1.0 );
/* Generate control vertices for Q'. */
NR_Point Q1[3];
for (unsigned i = 0; i < 3; i++) {
Q1[i] = 3.0 * ( Q[i+1] - Q[i] );
}
/* Generate control vertices for Q''. */
NR_Point Q2[2];
for (unsigned i = 0; i < 2; i++) {
Q2[i] = 2.0 * ( Q1[i+1] - Q1[i] );
}
/* Compute Q(u), Q'(u) and Q''(u). */
NR_Point const Q_u = bezier_pt(3, Q, u);
NR_Point const Q1_u = bezier_pt(2, Q1, u);
NR_Point const Q2_u = bezier_pt(1, Q2, u);
/* Compute f(u)/f'(u), where f is the derivative wrt u of distsq(u) = 0.5 * the square of the
distance from P to Q(u). Here we're using Newton-Raphson to find a stationary point in the
distsq(u), hopefully corresponding to a local minimum in distsq (and hence a local minimum
distance from P to Q(u)). */
NR_Point const diff = Q_u - P;
double numerator = dot(diff, Q1_u);
double denominator = dot(Q1_u, Q1_u) + dot(diff, Q2_u);
double improved_u;
if ( denominator > 0. ) {
/* One iteration of Newton-Raphson:
improved_u = u - f(u)/f'(u) */
improved_u = u - ( numerator / denominator );
} else {
/* Using Newton-Raphson would move in the wrong direction (towards a local maximum rather
than local minimum), so we move an arbitrary amount in the right direction. */
if ( numerator > 0. ) {
improved_u = u * .98 - .01;
} else if ( numerator < 0. ) {
/* Deliberately asymmetrical, to reduce the chance of cycling. */
improved_u = .031 + u * .98;
} else {
improved_u = u;
}
}
if (!isFinite(improved_u)) {
improved_u = u;
} else if ( improved_u < 0.0 ) {
improved_u = 0.0;
} else if ( improved_u > 1.0 ) {
improved_u = 1.0;
}
/* Ensure that improved_u isn't actually worse. */
{
double const diff_lensq = lensq(diff);
for (double proportion = .125; ; proportion += .125) {
if ( lensq( bezier_pt(3, Q, improved_u) - P ) > diff_lensq ) {
if ( proportion > 1.0 ) {
//g_warning("found proportion %g", proportion);
improved_u = u;
break;
}
improved_u = ( ( 1 - proportion ) * improved_u +
proportion * u );
} else {
break;
}
}
}
DOUBLE_ASSERT(improved_u);
return improved_u;
}
/**
* Evaluate a Bezier curve at parameter value \a t.
*
* \param degree The degree of the Bezier curve: 3 for cubic, 2 for quadratic etc.
* \param V The control points for the Bezier curve. Must have (\a degree+1)
* elements.
* \param t The "parameter" value, specifying whereabouts along the curve to
* evaluate. Typically in the range [0.0, 1.0].
*
* Let s = 1 - t.
* BezierII(1, V) gives (s, t) * V, i.e. t of the way
* from V[0] to V[1].
* BezierII(2, V) gives (s**2, 2*s*t, t**2) * V.
* BezierII(3, V) gives (s**3, 3 s**2 t, 3s t**2, t**3) * V.