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
|
//----------------------------------------------------------------------------
// Anti-Grain Geometry - Version 2.4
// Copyright (C) 2002-2005 Maxim Shemanarev (http://www.antigrain.com)
//
// Permission to copy, use, modify, sell and distribute this software
// is granted provided this copyright notice appears in all copies.
// This software is provided "as is" without express or implied
// warranty, and with no claim as to its suitability for any purpose.
//
//----------------------------------------------------------------------------
// Contact: mcseem@antigrain.com
// mcseemagg@yahoo.com
// http://www.antigrain.com
//----------------------------------------------------------------------------
//
// Perspective 2D transformations
//
//----------------------------------------------------------------------------
#ifndef AGG_TRANS_PERSPECTIVE_INCLUDED
#define AGG_TRANS_PERSPECTIVE_INCLUDED
#include "agg_trans_affine.h"
namespace agg
{
//=======================================================trans_perspective
struct trans_perspective
{
double sx, shy, w0, shx, sy, w1, tx, ty, w2;
//------------------------------------------------------- Construction
// Identity matrix
trans_perspective() :
sx (1), shy(0), w0(0),
shx(0), sy (1), w1(0),
tx (0), ty (0), w2(1) {}
// Custom matrix
trans_perspective(double v0, double v1, double v2,
double v3, double v4, double v5,
double v6, double v7, double v8) :
sx (v0), shy(v1), w0(v2),
shx(v3), sy (v4), w1(v5),
tx (v6), ty (v7), w2(v8) {}
// Custom matrix from m[9]
explicit trans_perspective(const double* m) :
sx (m[0]), shy(m[1]), w0(m[2]),
shx(m[3]), sy (m[4]), w1(m[5]),
tx (m[6]), ty (m[7]), w2(m[8]) {}
// From affine
explicit trans_perspective(const trans_affine& a) :
sx (a.sx ), shy(a.shy), w0(0),
shx(a.shx), sy (a.sy ), w1(0),
tx (a.tx ), ty (a.ty ), w2(1) {}
// Rectangle to quadrilateral
trans_perspective(double x1, double y1, double x2, double y2,
const double* quad);
// Quadrilateral to rectangle
trans_perspective(const double* quad,
double x1, double y1, double x2, double y2);
// Arbitrary quadrilateral transformations
trans_perspective(const double* src, const double* dst);
//-------------------------------------- Quadrilateral transformations
// The arguments are double[8] that are mapped to quadrilaterals:
// x1,y1, x2,y2, x3,y3, x4,y4
bool quad_to_quad(const double* qs, const double* qd);
bool rect_to_quad(double x1, double y1,
double x2, double y2,
const double* q);
bool quad_to_rect(const double* q,
double x1, double y1,
double x2, double y2);
// Map square (0,0,1,1) to the quadrilateral and vice versa
bool square_to_quad(const double* q);
bool quad_to_square(const double* q);
//--------------------------------------------------------- Operations
// Reset - load an identity matrix
const trans_perspective& reset();
// Invert matrix. Returns false in degenerate case
bool invert();
// Direct transformations operations
const trans_perspective& translate(double x, double y);
const trans_perspective& rotate(double a);
const trans_perspective& scale(double s);
const trans_perspective& scale(double x, double y);
// Multiply the matrix by another one
const trans_perspective& multiply(const trans_perspective& m);
// Multiply "m" by "this" and assign the result to "this"
const trans_perspective& premultiply(const trans_perspective& m);
// Multiply matrix to inverse of another one
const trans_perspective& multiply_inv(const trans_perspective& m);
// Multiply inverse of "m" by "this" and assign the result to "this"
const trans_perspective& premultiply_inv(const trans_perspective& m);
// Multiply the matrix by another one
const trans_perspective& multiply(const trans_affine& m);
// Multiply "m" by "this" and assign the result to "this"
const trans_perspective& premultiply(const trans_affine& m);
// Multiply the matrix by inverse of another one
const trans_perspective& multiply_inv(const trans_affine& m);
// Multiply inverse of "m" by "this" and assign the result to "this"
const trans_perspective& premultiply_inv(const trans_affine& m);
//--------------------------------------------------------- Load/Store
void store_to(double* m) const;
const trans_perspective& load_from(const double* m);
//---------------------------------------------------------- Operators
// Multiply the matrix by another one
const trans_perspective& operator *= (const trans_perspective& m)
{
return multiply(m);
}
const trans_perspective& operator *= (const trans_affine& m)
{
return multiply(m);
}
// Multiply the matrix by inverse of another one
const trans_perspective& operator /= (const trans_perspective& m)
{
return multiply_inv(m);
}
const trans_perspective& operator /= (const trans_affine& m)
{
return multiply_inv(m);
}
// Multiply the matrix by another one and return
// the result in a separete matrix.
trans_perspective operator * (const trans_perspective& m) const
{
return trans_perspective(*this).multiply(m);
}
trans_perspective operator * (const trans_affine& m) const
{
return trans_perspective(*this).multiply(m);
}
// Multiply the matrix by inverse of another one
// and return the result in a separete matrix.
trans_perspective operator / (const trans_perspective& m) const
{
return trans_perspective(*this).multiply_inv(m);
}
trans_perspective operator / (const trans_affine& m) const
{
return trans_perspective(*this).multiply_inv(m);
}
// Calculate and return the inverse matrix
trans_perspective operator ~ () const
{
trans_perspective ret = *this;
ret.invert();
return ret;
}
// Equal operator with default epsilon
bool operator == (const trans_perspective& m) const
{
return is_equal(m, affine_epsilon);
}
// Not Equal operator with default epsilon
bool operator != (const trans_perspective& m) const
{
return !is_equal(m, affine_epsilon);
}
//---------------------------------------------------- Transformations
// Direct transformation of x and y
void transform(double* x, double* y) const;
// Direct transformation of x and y, affine part only
void transform_affine(double* x, double* y) const;
// Direct transformation of x and y, 2x2 matrix only, no translation
void transform_2x2(double* x, double* y) const;
// Inverse transformation of x and y. It works slow because
// it explicitly inverts the matrix on every call. For massive
// operations it's better to invert() the matrix and then use
// direct transformations.
void inverse_transform(double* x, double* y) const;
//---------------------------------------------------------- Auxiliary
const trans_perspective& from_affine(const trans_affine& a);
double determinant() const;
double determinant_reciprocal() const;
bool is_valid(double epsilon = affine_epsilon) const;
bool is_identity(double epsilon = affine_epsilon) const;
bool is_equal(const trans_perspective& m,
double epsilon = affine_epsilon) const;
// Determine the major affine parameters. Use with caution
// considering possible degenerate cases.
double scale() const;
double rotation() const;
void translation(double* dx, double* dy) const;
void scaling(double* x, double* y) const;
void scaling_abs(double* x, double* y) const;
//--------------------------------------------------------------------
class iterator_x
{
double den;
double den_step;
double nom_x;
double nom_x_step;
double nom_y;
double nom_y_step;
public:
double x;
double y;
iterator_x() {}
iterator_x(double px, double py, double step, const trans_perspective& m) :
den(px * m.w0 + py * m.w1 + m.w2),
den_step(m.w0 * step),
nom_x(px * m.sx + py * m.shx + m.tx),
nom_x_step(step * m.sx),
nom_y(px * m.shy + py * m.sy + m.ty),
nom_y_step(step * m.shy),
x(nom_x / den),
y(nom_y / den)
{}
void operator ++ ()
{
den += den_step;
nom_x += nom_x_step;
nom_y += nom_y_step;
double d = 1.0 / den;
x = nom_x * d;
y = nom_y * d;
}
};
//--------------------------------------------------------------------
iterator_x begin(double x, double y, double step) const
{
return iterator_x(x, y, step, *this);
}
};
//------------------------------------------------------------------------
inline bool trans_perspective::square_to_quad(const double* q)
{
double dx = q[0] - q[2] + q[4] - q[6];
double dy = q[1] - q[3] + q[5] - q[7];
if(dx == 0.0 && dy == 0.0)
{
// Affine case (parallelogram)
//---------------
sx = q[2] - q[0];
shy = q[3] - q[1];
w0 = 0.0;
shx = q[4] - q[2];
sy = q[5] - q[3];
w1 = 0.0;
tx = q[0];
ty = q[1];
w2 = 1.0;
}
else
{
double dx1 = q[2] - q[4];
double dy1 = q[3] - q[5];
double dx2 = q[6] - q[4];
double dy2 = q[7] - q[5];
double den = dx1 * dy2 - dx2 * dy1;
if(den == 0.0)
{
// Singular case
//---------------
sx = shy = w0 = shx = sy = w1 = tx = ty = w2 = 0.0;
return false;
}
// General case
//---------------
double u = (dx * dy2 - dy * dx2) / den;
double v = (dy * dx1 - dx * dy1) / den;
sx = q[2] - q[0] + u * q[2];
shy = q[3] - q[1] + u * q[3];
w0 = u;
shx = q[6] - q[0] + v * q[6];
sy = q[7] - q[1] + v * q[7];
w1 = v;
tx = q[0];
ty = q[1];
w2 = 1.0;
}
return true;
}
//------------------------------------------------------------------------
inline bool trans_perspective::invert()
{
double d0 = sy * w2 - w1 * ty;
double d1 = w0 * ty - shy * w2;
double d2 = shy * w1 - w0 * sy;
double d = sx * d0 + shx * d1 + tx * d2;
if(d == 0.0)
{
sx = shy = w0 = shx = sy = w1 = tx = ty = w2 = 0.0;
return false;
}
d = 1.0 / d;
trans_perspective a = *this;
sx = d * d0;
shy = d * d1;
w0 = d * d2;
shx = d * (a.w1 *a.tx - a.shx*a.w2);
sy = d * (a.sx *a.w2 - a.w0 *a.tx);
w1 = d * (a.w0 *a.shx - a.sx *a.w1);
tx = d * (a.shx*a.ty - a.sy *a.tx);
ty = d * (a.shy*a.tx - a.sx *a.ty);
w2 = d * (a.sx *a.sy - a.shy*a.shx);
return true;
}
//------------------------------------------------------------------------
inline bool trans_perspective::quad_to_square(const double* q)
{
if(!square_to_quad(q)) return false;
invert();
return true;
}
//------------------------------------------------------------------------
inline bool trans_perspective::quad_to_quad(const double* qs,
const double* qd)
{
trans_perspective p;
if(! quad_to_square(qs)) return false;
if(!p.square_to_quad(qd)) return false;
multiply(p);
return true;
}
//------------------------------------------------------------------------
inline bool trans_perspective::rect_to_quad(double x1, double y1,
double x2, double y2,
const double* q)
{
double r[8];
r[0] = r[6] = x1;
r[2] = r[4] = x2;
r[1] = r[3] = y1;
r[5] = r[7] = y2;
return quad_to_quad(r, q);
}
//------------------------------------------------------------------------
inline bool trans_perspective::quad_to_rect(const double* q,
double x1, double y1,
double x2, double y2)
{
double r[8];
r[0] = r[6] = x1;
r[2] = r[4] = x2;
r[1] = r[3] = y1;
r[5] = r[7] = y2;
return quad_to_quad(q, r);
}
//------------------------------------------------------------------------
inline trans_perspective::trans_perspective(double x1, double y1,
double x2, double y2,
const double* quad)
{
rect_to_quad(x1, y1, x2, y2, quad);
}
//------------------------------------------------------------------------
inline trans_perspective::trans_perspective(const double* quad,
double x1, double y1,
double x2, double y2)
{
quad_to_rect(quad, x1, y1, x2, y2);
}
//------------------------------------------------------------------------
inline trans_perspective::trans_perspective(const double* src,
const double* dst)
{
quad_to_quad(src, dst);
}
//------------------------------------------------------------------------
inline const trans_perspective& trans_perspective::reset()
{
sx = 1; shy = 0; w0 = 0;
shx = 0; sy = 1; w1 = 0;
tx = 0; ty = 0; w2 = 1;
return *this;
}
//------------------------------------------------------------------------
inline const trans_perspective&
trans_perspective::multiply(const trans_perspective& a)
{
trans_perspective b = *this;
sx = a.sx *b.sx + a.shx*b.shy + a.tx*b.w0;
shx = a.sx *b.shx + a.shx*b.sy + a.tx*b.w1;
tx = a.sx *b.tx + a.shx*b.ty + a.tx*b.w2;
shy = a.shy*b.sx + a.sy *b.shy + a.ty*b.w0;
sy = a.shy*b.shx + a.sy *b.sy + a.ty*b.w1;
ty = a.shy*b.tx + a.sy *b.ty + a.ty*b.w2;
w0 = a.w0 *b.sx + a.w1 *b.shy + a.w2*b.w0;
w1 = a.w0 *b.shx + a.w1 *b.sy + a.w2*b.w1;
w2 = a.w0 *b.tx + a.w1 *b.ty + a.w2*b.w2;
return *this;
}
//------------------------------------------------------------------------
inline const trans_perspective&
trans_perspective::multiply(const trans_affine& a)
{
trans_perspective b = *this;
sx = a.sx *b.sx + a.shx*b.shy + a.tx*b.w0;
shx = a.sx *b.shx + a.shx*b.sy + a.tx*b.w1;
tx = a.sx *b.tx + a.shx*b.ty + a.tx*b.w2;
shy = a.shy*b.sx + a.sy *b.shy + a.ty*b.w0;
sy = a.shy*b.shx + a.sy *b.sy + a.ty*b.w1;
ty = a.shy*b.tx + a.sy *b.ty + a.ty*b.w2;
return *this;
}
//------------------------------------------------------------------------
inline const trans_perspective&
trans_perspective::premultiply(const trans_perspective& b)
{
trans_perspective a = *this;
sx = a.sx *b.sx + a.shx*b.shy + a.tx*b.w0;
shx = a.sx *b.shx + a.shx*b.sy + a.tx*b.w1;
tx = a.sx *b.tx + a.shx*b.ty + a.tx*b.w2;
shy = a.shy*b.sx + a.sy *b.shy + a.ty*b.w0;
sy = a.shy*b.shx + a.sy *b.sy + a.ty*b.w1;
ty = a.shy*b.tx + a.sy *b.ty + a.ty*b.w2;
w0 = a.w0 *b.sx + a.w1 *b.shy + a.w2*b.w0;
w1 = a.w0 *b.shx + a.w1 *b.sy + a.w2*b.w1;
w2 = a.w0 *b.tx + a.w1 *b.ty + a.w2*b.w2;
return *this;
}
//------------------------------------------------------------------------
inline const trans_perspective&
trans_perspective::premultiply(const trans_affine& b)
{
trans_perspective a = *this;
sx = a.sx *b.sx + a.shx*b.shy;
shx = a.sx *b.shx + a.shx*b.sy;
tx = a.sx *b.tx + a.shx*b.ty + a.tx;
shy = a.shy*b.sx + a.sy *b.shy;
sy = a.shy*b.shx + a.sy *b.sy;
ty = a.shy*b.tx + a.sy *b.ty + a.ty;
w0 = a.w0 *b.sx + a.w1 *b.shy;
w1 = a.w0 *b.shx + a.w1 *b.sy;
w2 = a.w0 *b.tx + a.w1 *b.ty + a.w2;
return *this;
}
//------------------------------------------------------------------------
const trans_perspective&
trans_perspective::multiply_inv(const trans_perspective& m)
{
trans_perspective t = m;
t.invert();
return multiply(t);
}
//------------------------------------------------------------------------
const trans_perspective&
trans_perspective::multiply_inv(const trans_affine& m)
{
trans_affine t = m;
t.invert();
return multiply(t);
}
//------------------------------------------------------------------------
const trans_perspective&
trans_perspective::premultiply_inv(const trans_perspective& m)
{
trans_perspective t = m;
t.invert();
return *this = t.multiply(*this);
}
//------------------------------------------------------------------------
const trans_perspective&
trans_perspective::premultiply_inv(const trans_affine& m)
{
trans_perspective t(m);
t.invert();
return *this = t.multiply(*this);
}
//------------------------------------------------------------------------
inline const trans_perspective&
trans_perspective::translate(double x, double y)
{
tx += x;
ty += y;
return *this;
}
//------------------------------------------------------------------------
inline const trans_perspective& trans_perspective::rotate(double a)
{
multiply(trans_affine_rotation(a));
return *this;
}
//------------------------------------------------------------------------
inline const trans_perspective& trans_perspective::scale(double s)
{
multiply(trans_affine_scaling(s));
return *this;
}
//------------------------------------------------------------------------
inline const trans_perspective& trans_perspective::scale(double x, double y)
{
multiply(trans_affine_scaling(x, y));
return *this;
}
//------------------------------------------------------------------------
inline void trans_perspective::transform(double* px, double* py) const
{
double x = *px;
double y = *py;
double m = 1.0 / (x*w0 + y*w1 + w2);
*px = m * (x*sx + y*shx + tx);
*py = m * (x*shy + y*sy + ty);
}
//------------------------------------------------------------------------
inline void trans_perspective::transform_affine(double* x, double* y) const
{
double tmp = *x;
*x = tmp * sx + *y * shx + tx;
*y = tmp * shy + *y * sy + ty;
}
//------------------------------------------------------------------------
inline void trans_perspective::transform_2x2(double* x, double* y) const
{
double tmp = *x;
*x = tmp * sx + *y * shx;
*y = tmp * shy + *y * sy;
}
//------------------------------------------------------------------------
inline void trans_perspective::inverse_transform(double* x, double* y) const
{
trans_perspective t(*this);
if(t.invert()) t.transform(x, y);
}
//------------------------------------------------------------------------
inline void trans_perspective::store_to(double* m) const
{
*m++ = sx; *m++ = shy; *m++ = w0;
*m++ = shx; *m++ = sy; *m++ = w1;
*m++ = tx; *m++ = ty; *m++ = w2;
}
//------------------------------------------------------------------------
inline const trans_perspective& trans_perspective::load_from(const double* m)
{
sx = *m++; shy = *m++; w0 = *m++;
shx = *m++; sy = *m++; w1 = *m++;
tx = *m++; ty = *m++; w2 = *m++;
return *this;
}
//------------------------------------------------------------------------
inline const trans_perspective&
trans_perspective::from_affine(const trans_affine& a)
{
sx = a.sx; shy = a.shy; w0 = 0;
shx = a.shx; sy = a.sy; w1 = 0;
tx = a.tx; ty = a.ty; w2 = 1;
return *this;
}
//------------------------------------------------------------------------
inline double trans_perspective::determinant() const
{
return sx * (sy * w2 - ty * w1) +
shx * (ty * w0 - shy * w2) +
tx * (shy * w1 - sy * w0);
}
//------------------------------------------------------------------------
inline double trans_perspective::determinant_reciprocal() const
{
return 1.0 / determinant();
}
//------------------------------------------------------------------------
inline bool trans_perspective::is_valid(double epsilon) const
{
return fabs(sx) > epsilon && fabs(sy) > epsilon && fabs(w2) > epsilon;
}
//------------------------------------------------------------------------
inline bool trans_perspective::is_identity(double epsilon) const
{
return is_equal_eps(sx, 1.0, epsilon) &&
is_equal_eps(shy, 0.0, epsilon) &&
is_equal_eps(w0, 0.0, epsilon) &&
is_equal_eps(shx, 0.0, epsilon) &&
is_equal_eps(sy, 1.0, epsilon) &&
is_equal_eps(w1, 0.0, epsilon) &&
is_equal_eps(tx, 0.0, epsilon) &&
is_equal_eps(ty, 0.0, epsilon) &&
is_equal_eps(w2, 1.0, epsilon);
}
//------------------------------------------------------------------------
inline bool trans_perspective::is_equal(const trans_perspective& m,
double epsilon) const
{
return is_equal_eps(sx, m.sx, epsilon) &&
is_equal_eps(shy, m.shy, epsilon) &&
is_equal_eps(w0, m.w0, epsilon) &&
is_equal_eps(shx, m.shx, epsilon) &&
is_equal_eps(sy, m.sy, epsilon) &&
is_equal_eps(w1, m.w1, epsilon) &&
is_equal_eps(tx, m.tx, epsilon) &&
is_equal_eps(ty, m.ty, epsilon) &&
is_equal_eps(w2, m.w2, epsilon);
}
//------------------------------------------------------------------------
inline double trans_perspective::scale() const
{
double x = 0.707106781 * sx + 0.707106781 * shx;
double y = 0.707106781 * shy + 0.707106781 * sy;
return sqrt(x*x + y*y);
}
//------------------------------------------------------------------------
inline double trans_perspective::rotation() const
{
double x1 = 0.0;
double y1 = 0.0;
double x2 = 1.0;
double y2 = 0.0;
transform(&x1, &y1);
transform(&x2, &y2);
return atan2(y2-y1, x2-x1);
}
//------------------------------------------------------------------------
void trans_perspective::translation(double* dx, double* dy) const
{
*dx = tx;
*dy = ty;
}
//------------------------------------------------------------------------
void trans_perspective::scaling(double* x, double* y) const
{
double x1 = 0.0;
double y1 = 0.0;
double x2 = 1.0;
double y2 = 1.0;
trans_perspective t(*this);
t *= trans_affine_rotation(-rotation());
t.transform(&x1, &y1);
t.transform(&x2, &y2);
*x = x2 - x1;
*y = y2 - y1;
}
//------------------------------------------------------------------------
void trans_perspective::scaling_abs(double* x, double* y) const
{
*x = sqrt(sx * sx + shx * shx);
*y = sqrt(shy * shy + sy * sy);
}
}
#endif
|