source: trunk/src/gui/painting/qbezier.cpp@ 1034

Last change on this file since 1034 was 846, checked in by Dmitry A. Kuminov, 14 years ago

trunk: Merged in qt 4.7.2 sources from branches/vendor/nokia/qt.

File size: 19.3 KB
Line 
1/****************************************************************************
2**
3** Copyright (C) 2011 Nokia Corporation and/or its subsidiary(-ies).
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5** Contact: Nokia Corporation (qt-info@nokia.com)
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7** This file is part of the QtGui module of the Qt Toolkit.
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40****************************************************************************/
41
42#include "qbezier_p.h"
43#include <qdebug.h>
44#include <qline.h>
45#include <qpolygon.h>
46#include <qvector.h>
47#include <qlist.h>
48#include <qmath.h>
49
50#include <private/qnumeric_p.h>
51#include <private/qmath_p.h>
52
53QT_BEGIN_NAMESPACE
54
55//#define QDEBUG_BEZIER
56
57#ifdef FLOAT_ACCURACY
58#define INV_EPS (1L<<23)
59#else
60/* The value of 1.0 / (1L<<14) is enough for most applications */
61#define INV_EPS (1L<<14)
62#endif
63
64#ifndef M_SQRT2
65#define M_SQRT2 1.41421356237309504880
66#endif
67
68#define log2(x) (qLn(x)/qLn(2.))
69
70static inline qreal log4(qreal x)
71{
72 return qreal(0.5) * log2(x);
73}
74
75/*!
76 \internal
77*/
78QBezier QBezier::fromPoints(const QPointF &p1, const QPointF &p2,
79 const QPointF &p3, const QPointF &p4)
80{
81 QBezier b;
82 b.x1 = p1.x();
83 b.y1 = p1.y();
84 b.x2 = p2.x();
85 b.y2 = p2.y();
86 b.x3 = p3.x();
87 b.y3 = p3.y();
88 b.x4 = p4.x();
89 b.y4 = p4.y();
90 return b;
91}
92
93/*!
94 \internal
95*/
96QPolygonF QBezier::toPolygon(qreal bezier_flattening_threshold) const
97{
98 // flattening is done by splitting the bezier until we can replace the segment by a straight
99 // line. We split further until the control points are close enough to the line connecting the
100 // boundary points.
101 //
102 // the Distance of a point p from a line given by the points (a,b) is given by:
103 //
104 // d = abs( (bx - ax)(ay - py) - (by - ay)(ax - px) ) / line_length
105 //
106 // We can stop splitting if both control points are close enough to the line.
107 // To make the algorithm faster we use the manhattan length of the line.
108
109 QPolygonF polygon;
110 polygon.append(QPointF(x1, y1));
111 addToPolygon(&polygon, bezier_flattening_threshold);
112 return polygon;
113}
114
115QBezier QBezier::mapBy(const QTransform &transform) const
116{
117 return QBezier::fromPoints(transform.map(pt1()), transform.map(pt2()), transform.map(pt3()), transform.map(pt4()));
118}
119
120QBezier QBezier::getSubRange(qreal t0, qreal t1) const
121{
122 QBezier result;
123 QBezier temp;
124
125 // cut at t1
126 if (qFuzzyIsNull(t1 - qreal(1.))) {
127 result = *this;
128 } else {
129 temp = *this;
130 temp.parameterSplitLeft(t1, &result);
131 }
132
133 // cut at t0
134 if (!qFuzzyIsNull(t0))
135 result.parameterSplitLeft(t0 / t1, &temp);
136
137 return result;
138}
139
140static inline int quadraticRoots(qreal a, qreal b, qreal c,
141 qreal *x1, qreal *x2)
142{
143 if (qFuzzyIsNull(a)) {
144 if (qFuzzyIsNull(b))
145 return 0;
146 *x1 = *x2 = (-c / b);
147 return 1;
148 } else {
149 const qreal det = b * b - 4 * a * c;
150 if (qFuzzyIsNull(det)) {
151 *x1 = *x2 = -b / (2 * a);
152 return 1;
153 }
154 if (det > 0) {
155 if (qFuzzyIsNull(b)) {
156 *x2 = qSqrt(-c / a);
157 *x1 = -(*x2);
158 return 2;
159 }
160 const qreal stableA = b / (2 * a);
161 const qreal stableB = c / (a * stableA * stableA);
162 const qreal stableC = -1 - qSqrt(1 - stableB);
163 *x2 = stableA * stableC;
164 *x1 = (stableA * stableB) / stableC;
165 return 2;
166 } else
167 return 0;
168 }
169}
170
171static inline bool findInflections(qreal a, qreal b, qreal c,
172 qreal *t1 , qreal *t2, qreal *tCups)
173{
174 qreal r1 = 0, r2 = 0;
175
176 short rootsCount = quadraticRoots(a, b, c, &r1, &r2);
177
178 if (rootsCount >= 1) {
179 if (r1 < r2) {
180 *t1 = r1;
181 *t2 = r2;
182 } else {
183 *t1 = r2;
184 *t2 = r1;
185 }
186 if (!qFuzzyIsNull(a))
187 *tCups = 0.5 * (-b / a);
188 else
189 *tCups = 2;
190
191 return true;
192 }
193
194 return false;
195}
196
197
198void QBezier::addToPolygon(QPolygonF *polygon, qreal bezier_flattening_threshold) const
199{
200 QBezier beziers[32];
201 beziers[0] = *this;
202 QBezier *b = beziers;
203
204 while (b >= beziers) {
205 // check if we can pop the top bezier curve from the stack
206 qreal y4y1 = b->y4 - b->y1;
207 qreal x4x1 = b->x4 - b->x1;
208 qreal l = qAbs(x4x1) + qAbs(y4y1);
209 qreal d;
210 if (l > 1.) {
211 d = qAbs( (x4x1)*(b->y1 - b->y2) - (y4y1)*(b->x1 - b->x2) )
212 + qAbs( (x4x1)*(b->y1 - b->y3) - (y4y1)*(b->x1 - b->x3) );
213 } else {
214 d = qAbs(b->x1 - b->x2) + qAbs(b->y1 - b->y2) +
215 qAbs(b->x1 - b->x3) + qAbs(b->y1 - b->y3);
216 l = 1.;
217 }
218 if (d < bezier_flattening_threshold*l || b == beziers + 31) {
219 // good enough, we pop it off and add the endpoint
220 polygon->append(QPointF(b->x4, b->y4));
221 --b;
222 } else {
223 // split, second half of the polygon goes lower into the stack
224 b->split(b+1, b);
225 ++b;
226 }
227 }
228}
229
230QRectF QBezier::bounds() const
231{
232 qreal xmin = x1;
233 qreal xmax = x1;
234 if (x2 < xmin)
235 xmin = x2;
236 else if (x2 > xmax)
237 xmax = x2;
238 if (x3 < xmin)
239 xmin = x3;
240 else if (x3 > xmax)
241 xmax = x3;
242 if (x4 < xmin)
243 xmin = x4;
244 else if (x4 > xmax)
245 xmax = x4;
246
247 qreal ymin = y1;
248 qreal ymax = y1;
249 if (y2 < ymin)
250 ymin = y2;
251 else if (y2 > ymax)
252 ymax = y2;
253 if (y3 < ymin)
254 ymin = y3;
255 else if (y3 > ymax)
256 ymax = y3;
257 if (y4 < ymin)
258 ymin = y4;
259 else if (y4 > ymax)
260 ymax = y4;
261 return QRectF(xmin, ymin, xmax-xmin, ymax-ymin);
262}
263
264
265enum ShiftResult {
266 Ok,
267 Discard,
268 Split,
269 Circle
270};
271
272static ShiftResult good_offset(const QBezier *b1, const QBezier *b2, qreal offset, qreal threshold)
273{
274 const qreal o2 = offset*offset;
275 const qreal max_dist_line = threshold*offset*offset;
276 const qreal max_dist_normal = threshold*offset;
277 const qreal spacing = 0.25;
278 for (qreal i = spacing; i < 0.99; i += spacing) {
279 QPointF p1 = b1->pointAt(i);
280 QPointF p2 = b2->pointAt(i);
281 qreal d = (p1.x() - p2.x())*(p1.x() - p2.x()) + (p1.y() - p2.y())*(p1.y() - p2.y());
282 if (qAbs(d - o2) > max_dist_line)
283 return Split;
284
285 QPointF normalPoint = b1->normalVector(i);
286 qreal l = qAbs(normalPoint.x()) + qAbs(normalPoint.y());
287 if (l != 0.) {
288 d = qAbs( normalPoint.x()*(p1.y() - p2.y()) - normalPoint.y()*(p1.x() - p2.x()) ) / l;
289 if (d > max_dist_normal)
290 return Split;
291 }
292 }
293 return Ok;
294}
295
296static ShiftResult shift(const QBezier *orig, QBezier *shifted, qreal offset, qreal threshold)
297{
298 int map[4];
299 bool p1_p2_equal = (orig->x1 == orig->x2 && orig->y1 == orig->y2);
300 bool p2_p3_equal = (orig->x2 == orig->x3 && orig->y2 == orig->y3);
301 bool p3_p4_equal = (orig->x3 == orig->x4 && orig->y3 == orig->y4);
302
303 QPointF points[4];
304 int np = 0;
305 points[np] = QPointF(orig->x1, orig->y1);
306 map[0] = 0;
307 ++np;
308 if (!p1_p2_equal) {
309 points[np] = QPointF(orig->x2, orig->y2);
310 ++np;
311 }
312 map[1] = np - 1;
313 if (!p2_p3_equal) {
314 points[np] = QPointF(orig->x3, orig->y3);
315 ++np;
316 }
317 map[2] = np - 1;
318 if (!p3_p4_equal) {
319 points[np] = QPointF(orig->x4, orig->y4);
320 ++np;
321 }
322 map[3] = np - 1;
323 if (np == 1)
324 return Discard;
325
326 QRectF b = orig->bounds();
327 if (np == 4 && b.width() < .1*offset && b.height() < .1*offset) {
328 qreal l = (orig->x1 - orig->x2)*(orig->x1 - orig->x2) +
329 (orig->y1 - orig->y2)*(orig->y1 - orig->y1) *
330 (orig->x3 - orig->x4)*(orig->x3 - orig->x4) +
331 (orig->y3 - orig->y4)*(orig->y3 - orig->y4);
332 qreal dot = (orig->x1 - orig->x2)*(orig->x3 - orig->x4) +
333 (orig->y1 - orig->y2)*(orig->y3 - orig->y4);
334 if (dot < 0 && dot*dot < 0.8*l)
335 // the points are close and reverse dirction. Approximate the whole
336 // thing by a semi circle
337 return Circle;
338 }
339
340 QPointF points_shifted[4];
341
342 QLineF prev = QLineF(QPointF(), points[1] - points[0]);
343 QPointF prev_normal = prev.normalVector().unitVector().p2();
344
345 points_shifted[0] = points[0] + offset * prev_normal;
346
347 for (int i = 1; i < np - 1; ++i) {
348 QLineF next = QLineF(QPointF(), points[i + 1] - points[i]);
349 QPointF next_normal = next.normalVector().unitVector().p2();
350
351 QPointF normal_sum = prev_normal + next_normal;
352
353 qreal r = 1.0 + prev_normal.x() * next_normal.x()
354 + prev_normal.y() * next_normal.y();
355
356 if (qFuzzyIsNull(r)) {
357 points_shifted[i] = points[i] + offset * prev_normal;
358 } else {
359 qreal k = offset / r;
360 points_shifted[i] = points[i] + k * normal_sum;
361 }
362
363 prev_normal = next_normal;
364 }
365
366 points_shifted[np - 1] = points[np - 1] + offset * prev_normal;
367
368 *shifted = QBezier::fromPoints(points_shifted[map[0]], points_shifted[map[1]],
369 points_shifted[map[2]], points_shifted[map[3]]);
370
371 return good_offset(orig, shifted, offset, threshold);
372}
373
374// This value is used to determine the length of control point vectors
375// when approximating arc segments as curves. The factor is multiplied
376// with the radius of the circle.
377#define KAPPA 0.5522847498
378
379
380static bool addCircle(const QBezier *b, qreal offset, QBezier *o)
381{
382 QPointF normals[3];
383
384 normals[0] = QPointF(b->y2 - b->y1, b->x1 - b->x2);
385 qreal dist = qSqrt(normals[0].x()*normals[0].x() + normals[0].y()*normals[0].y());
386 if (qFuzzyIsNull(dist))
387 return false;
388 normals[0] /= dist;
389 normals[2] = QPointF(b->y4 - b->y3, b->x3 - b->x4);
390 dist = qSqrt(normals[2].x()*normals[2].x() + normals[2].y()*normals[2].y());
391 if (qFuzzyIsNull(dist))
392 return false;
393 normals[2] /= dist;
394
395 normals[1] = QPointF(b->x1 - b->x2 - b->x3 + b->x4, b->y1 - b->y2 - b->y3 + b->y4);
396 normals[1] /= -1*qSqrt(normals[1].x()*normals[1].x() + normals[1].y()*normals[1].y());
397
398 qreal angles[2];
399 qreal sign = 1.;
400 for (int i = 0; i < 2; ++i) {
401 qreal cos_a = normals[i].x()*normals[i+1].x() + normals[i].y()*normals[i+1].y();
402 if (cos_a > 1.)
403 cos_a = 1.;
404 if (cos_a < -1.)
405 cos_a = -1;
406 angles[i] = qAcos(cos_a)/Q_PI;
407 }
408
409 if (angles[0] + angles[1] > 1.) {
410 // more than 180 degrees
411 normals[1] = -normals[1];
412 angles[0] = 1. - angles[0];
413 angles[1] = 1. - angles[1];
414 sign = -1.;
415
416 }
417
418 QPointF circle[3];
419 circle[0] = QPointF(b->x1, b->y1) + normals[0]*offset;
420 circle[1] = QPointF(0.5*(b->x1 + b->x4), 0.5*(b->y1 + b->y4)) + normals[1]*offset;
421 circle[2] = QPointF(b->x4, b->y4) + normals[2]*offset;
422
423 for (int i = 0; i < 2; ++i) {
424 qreal kappa = 2.*KAPPA * sign * offset * angles[i];
425
426 o->x1 = circle[i].x();
427 o->y1 = circle[i].y();
428 o->x2 = circle[i].x() - normals[i].y()*kappa;
429 o->y2 = circle[i].y() + normals[i].x()*kappa;
430 o->x3 = circle[i+1].x() + normals[i+1].y()*kappa;
431 o->y3 = circle[i+1].y() - normals[i+1].x()*kappa;
432 o->x4 = circle[i+1].x();
433 o->y4 = circle[i+1].y();
434
435 ++o;
436 }
437 return true;
438}
439
440int QBezier::shifted(QBezier *curveSegments, int maxSegments, qreal offset, float threshold) const
441{
442 Q_ASSERT(curveSegments);
443 Q_ASSERT(maxSegments > 0);
444
445 if (x1 == x2 && x1 == x3 && x1 == x4 &&
446 y1 == y2 && y1 == y3 && y1 == y4)
447 return 0;
448
449 --maxSegments;
450 QBezier beziers[10];
451redo:
452 beziers[0] = *this;
453 QBezier *b = beziers;
454 QBezier *o = curveSegments;
455
456 while (b >= beziers) {
457 int stack_segments = b - beziers + 1;
458 if ((stack_segments == 10) || (o - curveSegments == maxSegments - stack_segments)) {
459 threshold *= 1.5;
460 if (threshold > 2.)
461 goto give_up;
462 goto redo;
463 }
464 ShiftResult res = shift(b, o, offset, threshold);
465 if (res == Discard) {
466 --b;
467 } else if (res == Ok) {
468 ++o;
469 --b;
470 continue;
471 } else if (res == Circle && maxSegments - (o - curveSegments) >= 2) {
472 // add semi circle
473 if (addCircle(b, offset, o))
474 o += 2;
475 --b;
476 } else {
477 b->split(b+1, b);
478 ++b;
479 }
480 }
481
482give_up:
483 while (b >= beziers) {
484 ShiftResult res = shift(b, o, offset, threshold);
485
486 // if res isn't Ok or Split then *o is undefined
487 if (res == Ok || res == Split)
488 ++o;
489
490 --b;
491 }
492
493 Q_ASSERT(o - curveSegments <= maxSegments);
494 return o - curveSegments;
495}
496
497#ifdef QDEBUG_BEZIER
498static QDebug operator<<(QDebug dbg, const QBezier &bz)
499{
500 dbg << '[' << bz.x1<< ", " << bz.y1 << "], "
501 << '[' << bz.x2 <<", " << bz.y2 << "], "
502 << '[' << bz.x3 <<", " << bz.y3 << "], "
503 << '[' << bz.x4 <<", " << bz.y4 << ']';
504 return dbg;
505}
506#endif
507
508static inline void splitBezierAt(const QBezier &bez, qreal t,
509 QBezier *left, QBezier *right)
510{
511 left->x1 = bez.x1;
512 left->y1 = bez.y1;
513
514 left->x2 = bez.x1 + t * ( bez.x2 - bez.x1 );
515 left->y2 = bez.y1 + t * ( bez.y2 - bez.y1 );
516
517 left->x3 = bez.x2 + t * ( bez.x3 - bez.x2 ); // temporary holding spot
518 left->y3 = bez.y2 + t * ( bez.y3 - bez.y2 ); // temporary holding spot
519
520 right->x3 = bez.x3 + t * ( bez.x4 - bez.x3 );
521 right->y3 = bez.y3 + t * ( bez.y4 - bez.y3 );
522
523 right->x2 = left->x3 + t * ( right->x3 - left->x3);
524 right->y2 = left->y3 + t * ( right->y3 - left->y3);
525
526 left->x3 = left->x2 + t * ( left->x3 - left->x2 );
527 left->y3 = left->y2 + t * ( left->y3 - left->y2 );
528
529 left->x4 = right->x1 = left->x3 + t * (right->x2 - left->x3);
530 left->y4 = right->y1 = left->y3 + t * (right->y2 - left->y3);
531
532 right->x4 = bez.x4;
533 right->y4 = bez.y4;
534}
535
536qreal QBezier::length(qreal error) const
537{
538 qreal length = 0.0;
539
540 addIfClose(&length, error);
541
542 return length;
543}
544
545void QBezier::addIfClose(qreal *length, qreal error) const
546{
547 QBezier left, right; /* bez poly splits */
548
549 qreal len = 0.0; /* arc length */
550 qreal chord; /* chord length */
551
552 len = len + QLineF(QPointF(x1, y1),QPointF(x2, y2)).length();
553 len = len + QLineF(QPointF(x2, y2),QPointF(x3, y3)).length();
554 len = len + QLineF(QPointF(x3, y3),QPointF(x4, y4)).length();
555
556 chord = QLineF(QPointF(x1, y1),QPointF(x4, y4)).length();
557
558 if((len-chord) > error) {
559 split(&left, &right); /* split in two */
560 left.addIfClose(length, error); /* try left side */
561 right.addIfClose(length, error); /* try right side */
562 return;
563 }
564
565 *length = *length + len;
566
567 return;
568}
569
570qreal QBezier::tForY(qreal t0, qreal t1, qreal y) const
571{
572 qreal py0 = pointAt(t0).y();
573 qreal py1 = pointAt(t1).y();
574
575 if (py0 > py1) {
576 qSwap(py0, py1);
577 qSwap(t0, t1);
578 }
579
580 Q_ASSERT(py0 <= py1);
581
582 if (py0 >= y)
583 return t0;
584 else if (py1 <= y)
585 return t1;
586
587 Q_ASSERT(py0 < y && y < py1);
588
589 qreal lt = t0;
590 qreal dt;
591 do {
592 qreal t = 0.5 * (t0 + t1);
593
594 qreal a, b, c, d;
595 QBezier::coefficients(t, a, b, c, d);
596 qreal yt = a * y1 + b * y2 + c * y3 + d * y4;
597
598 if (yt < y) {
599 t0 = t;
600 py0 = yt;
601 } else {
602 t1 = t;
603 py1 = yt;
604 }
605 dt = lt - t;
606 lt = t;
607 } while (qAbs(dt) > 1e-7);
608
609 return t0;
610}
611
612int QBezier::stationaryYPoints(qreal &t0, qreal &t1) const
613{
614 // y(t) = (1 - t)^3 * y1 + 3 * (1 - t)^2 * t * y2 + 3 * (1 - t) * t^2 * y3 + t^3 * y4
615 // y'(t) = 3 * (-(1-2t+t^2) * y1 + (1 - 4 * t + 3 * t^2) * y2 + (2 * t - 3 * t^2) * y3 + t^2 * y4)
616 // y'(t) = 3 * ((-y1 + 3 * y2 - 3 * y3 + y4)t^2 + (2 * y1 - 4 * y2 + 2 * y3)t + (-y1 + y2))
617
618 const qreal a = -y1 + 3 * y2 - 3 * y3 + y4;
619 const qreal b = 2 * y1 - 4 * y2 + 2 * y3;
620 const qreal c = -y1 + y2;
621
622 if (qFuzzyIsNull(a)) {
623 if (qFuzzyIsNull(b))
624 return 0;
625
626 t0 = -c / b;
627 return t0 > 0 && t0 < 1;
628 }
629
630 qreal reciprocal = b * b - 4 * a * c;
631
632 if (qFuzzyIsNull(reciprocal)) {
633 t0 = -b / (2 * a);
634 return t0 > 0 && t0 < 1;
635 } else if (reciprocal > 0) {
636 qreal temp = qSqrt(reciprocal);
637
638 t0 = (-b - temp)/(2*a);
639 t1 = (-b + temp)/(2*a);
640
641 if (t1 < t0)
642 qSwap(t0, t1);
643
644 int count = 0;
645 qreal t[2] = { 0, 1 };
646
647 if (t0 > 0 && t0 < 1)
648 t[count++] = t0;
649 if (t1 > 0 && t1 < 1)
650 t[count++] = t1;
651
652 t0 = t[0];
653 t1 = t[1];
654
655 return count;
656 }
657
658 return 0;
659}
660
661qreal QBezier::tAtLength(qreal l) const
662{
663 qreal len = length();
664 qreal t = 1.0;
665 const qreal error = (qreal)0.01;
666 if (l > len || qFuzzyCompare(l, len))
667 return t;
668
669 t *= 0.5;
670 //int iters = 0;
671 //qDebug()<<"LEN is "<<l<<len;
672 qreal lastBigger = 1.;
673 while (1) {
674 //qDebug()<<"\tt is "<<t;
675 QBezier right = *this;
676 QBezier left;
677 right.parameterSplitLeft(t, &left);
678 qreal lLen = left.length();
679 if (qAbs(lLen - l) < error)
680 break;
681
682 if (lLen < l) {
683 t += (lastBigger - t)*.5;
684 } else {
685 lastBigger = t;
686 t -= t*.5;
687 }
688 //++iters;
689 }
690 //qDebug()<<"number of iters is "<<iters;
691 return t;
692}
693
694QBezier QBezier::bezierOnInterval(qreal t0, qreal t1) const
695{
696 if (t0 == 0 && t1 == 1)
697 return *this;
698
699 QBezier bezier = *this;
700
701 QBezier result;
702 bezier.parameterSplitLeft(t0, &result);
703 qreal trueT = (t1-t0)/(1-t0);
704 bezier.parameterSplitLeft(trueT, &result);
705
706 return result;
707}
708
709QT_END_NAMESPACE
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