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comparison: gfx/skia/trunk/src/gpu/GrPathUtils.cpp

gfx/skia/trunk/src/gpu/GrPathUtils.cpp

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1 /*
2 * Copyright 2011 Google Inc.
3 *
4 * Use of this source code is governed by a BSD-style license that can be
5 * found in the LICENSE file.
6 */
7
8 #include "GrPathUtils.h"
9
10 #include "GrPoint.h"
11 #include "SkGeometry.h"
12
13 SkScalar GrPathUtils::scaleToleranceToSrc(SkScalar devTol,
14 const SkMatrix& viewM,
15 const SkRect& pathBounds) {
16 // In order to tesselate the path we get a bound on how much the matrix can
17 // stretch when mapping to screen coordinates.
18 SkScalar stretch = viewM.getMaxStretch();
19 SkScalar srcTol = devTol;
20
21 if (stretch < 0) {
22 // take worst case mapRadius amoung four corners.
23 // (less than perfect)
24 for (int i = 0; i < 4; ++i) {
25 SkMatrix mat;
26 mat.setTranslate((i % 2) ? pathBounds.fLeft : pathBounds.fRight,
27 (i < 2) ? pathBounds.fTop : pathBounds.fBottom);
28 mat.postConcat(viewM);
29 stretch = SkMaxScalar(stretch, mat.mapRadius(SK_Scalar1));
30 }
31 }
32 srcTol = SkScalarDiv(srcTol, stretch);
33 return srcTol;
34 }
35
36 static const int MAX_POINTS_PER_CURVE = 1 << 10;
37 static const SkScalar gMinCurveTol = 0.0001f;
38
39 uint32_t GrPathUtils::quadraticPointCount(const GrPoint points[],
40 SkScalar tol) {
41 if (tol < gMinCurveTol) {
42 tol = gMinCurveTol;
43 }
44 SkASSERT(tol > 0);
45
46 SkScalar d = points[1].distanceToLineSegmentBetween(points[0], points[2]);
47 if (d <= tol) {
48 return 1;
49 } else {
50 // Each time we subdivide, d should be cut in 4. So we need to
51 // subdivide x = log4(d/tol) times. x subdivisions creates 2^(x)
52 // points.
53 // 2^(log4(x)) = sqrt(x);
54 int temp = SkScalarCeilToInt(SkScalarSqrt(SkScalarDiv(d, tol)));
55 int pow2 = GrNextPow2(temp);
56 // Because of NaNs & INFs we can wind up with a degenerate temp
57 // such that pow2 comes out negative. Also, our point generator
58 // will always output at least one pt.
59 if (pow2 < 1) {
60 pow2 = 1;
61 }
62 return GrMin(pow2, MAX_POINTS_PER_CURVE);
63 }
64 }
65
66 uint32_t GrPathUtils::generateQuadraticPoints(const GrPoint& p0,
67 const GrPoint& p1,
68 const GrPoint& p2,
69 SkScalar tolSqd,
70 GrPoint** points,
71 uint32_t pointsLeft) {
72 if (pointsLeft < 2 ||
73 (p1.distanceToLineSegmentBetweenSqd(p0, p2)) < tolSqd) {
74 (*points)[0] = p2;
75 *points += 1;
76 return 1;
77 }
78
79 GrPoint q[] = {
80 { SkScalarAve(p0.fX, p1.fX), SkScalarAve(p0.fY, p1.fY) },
81 { SkScalarAve(p1.fX, p2.fX), SkScalarAve(p1.fY, p2.fY) },
82 };
83 GrPoint r = { SkScalarAve(q[0].fX, q[1].fX), SkScalarAve(q[0].fY, q[1].fY) };
84
85 pointsLeft >>= 1;
86 uint32_t a = generateQuadraticPoints(p0, q[0], r, tolSqd, points, pointsLeft);
87 uint32_t b = generateQuadraticPoints(r, q[1], p2, tolSqd, points, pointsLeft);
88 return a + b;
89 }
90
91 uint32_t GrPathUtils::cubicPointCount(const GrPoint points[],
92 SkScalar tol) {
93 if (tol < gMinCurveTol) {
94 tol = gMinCurveTol;
95 }
96 SkASSERT(tol > 0);
97
98 SkScalar d = GrMax(
99 points[1].distanceToLineSegmentBetweenSqd(points[0], points[3]),
100 points[2].distanceToLineSegmentBetweenSqd(points[0], points[3]));
101 d = SkScalarSqrt(d);
102 if (d <= tol) {
103 return 1;
104 } else {
105 int temp = SkScalarCeilToInt(SkScalarSqrt(SkScalarDiv(d, tol)));
106 int pow2 = GrNextPow2(temp);
107 // Because of NaNs & INFs we can wind up with a degenerate temp
108 // such that pow2 comes out negative. Also, our point generator
109 // will always output at least one pt.
110 if (pow2 < 1) {
111 pow2 = 1;
112 }
113 return GrMin(pow2, MAX_POINTS_PER_CURVE);
114 }
115 }
116
117 uint32_t GrPathUtils::generateCubicPoints(const GrPoint& p0,
118 const GrPoint& p1,
119 const GrPoint& p2,
120 const GrPoint& p3,
121 SkScalar tolSqd,
122 GrPoint** points,
123 uint32_t pointsLeft) {
124 if (pointsLeft < 2 ||
125 (p1.distanceToLineSegmentBetweenSqd(p0, p3) < tolSqd &&
126 p2.distanceToLineSegmentBetweenSqd(p0, p3) < tolSqd)) {
127 (*points)[0] = p3;
128 *points += 1;
129 return 1;
130 }
131 GrPoint q[] = {
132 { SkScalarAve(p0.fX, p1.fX), SkScalarAve(p0.fY, p1.fY) },
133 { SkScalarAve(p1.fX, p2.fX), SkScalarAve(p1.fY, p2.fY) },
134 { SkScalarAve(p2.fX, p3.fX), SkScalarAve(p2.fY, p3.fY) }
135 };
136 GrPoint r[] = {
137 { SkScalarAve(q[0].fX, q[1].fX), SkScalarAve(q[0].fY, q[1].fY) },
138 { SkScalarAve(q[1].fX, q[2].fX), SkScalarAve(q[1].fY, q[2].fY) }
139 };
140 GrPoint s = { SkScalarAve(r[0].fX, r[1].fX), SkScalarAve(r[0].fY, r[1].fY) };
141 pointsLeft >>= 1;
142 uint32_t a = generateCubicPoints(p0, q[0], r[0], s, tolSqd, points, pointsLeft);
143 uint32_t b = generateCubicPoints(s, r[1], q[2], p3, tolSqd, points, pointsLeft);
144 return a + b;
145 }
146
147 int GrPathUtils::worstCasePointCount(const SkPath& path, int* subpaths,
148 SkScalar tol) {
149 if (tol < gMinCurveTol) {
150 tol = gMinCurveTol;
151 }
152 SkASSERT(tol > 0);
153
154 int pointCount = 0;
155 *subpaths = 1;
156
157 bool first = true;
158
159 SkPath::Iter iter(path, false);
160 SkPath::Verb verb;
161
162 GrPoint pts[4];
163 while ((verb = iter.next(pts)) != SkPath::kDone_Verb) {
164
165 switch (verb) {
166 case SkPath::kLine_Verb:
167 pointCount += 1;
168 break;
169 case SkPath::kQuad_Verb:
170 pointCount += quadraticPointCount(pts, tol);
171 break;
172 case SkPath::kCubic_Verb:
173 pointCount += cubicPointCount(pts, tol);
174 break;
175 case SkPath::kMove_Verb:
176 pointCount += 1;
177 if (!first) {
178 ++(*subpaths);
179 }
180 break;
181 default:
182 break;
183 }
184 first = false;
185 }
186 return pointCount;
187 }
188
189 void GrPathUtils::QuadUVMatrix::set(const GrPoint qPts[3]) {
190 SkMatrix m;
191 // We want M such that M * xy_pt = uv_pt
192 // We know M * control_pts = [0 1/2 1]
193 // [0 0 1]
194 // [1 1 1]
195 // And control_pts = [x0 x1 x2]
196 // [y0 y1 y2]
197 // [1 1 1 ]
198 // We invert the control pt matrix and post concat to both sides to get M.
199 // Using the known form of the control point matrix and the result, we can
200 // optimize and improve precision.
201
202 double x0 = qPts[0].fX;
203 double y0 = qPts[0].fY;
204 double x1 = qPts[1].fX;
205 double y1 = qPts[1].fY;
206 double x2 = qPts[2].fX;
207 double y2 = qPts[2].fY;
208 double det = x0*y1 - y0*x1 + x2*y0 - y2*x0 + x1*y2 - y1*x2;
209
210 if (!sk_float_isfinite(det)
211 || SkScalarNearlyZero((float)det, SK_ScalarNearlyZero * SK_ScalarNearlyZero)) {
212 // The quad is degenerate. Hopefully this is rare. Find the pts that are
213 // farthest apart to compute a line (unless it is really a pt).
214 SkScalar maxD = qPts[0].distanceToSqd(qPts[1]);
215 int maxEdge = 0;
216 SkScalar d = qPts[1].distanceToSqd(qPts[2]);
217 if (d > maxD) {
218 maxD = d;
219 maxEdge = 1;
220 }
221 d = qPts[2].distanceToSqd(qPts[0]);
222 if (d > maxD) {
223 maxD = d;
224 maxEdge = 2;
225 }
226 // We could have a tolerance here, not sure if it would improve anything
227 if (maxD > 0) {
228 // Set the matrix to give (u = 0, v = distance_to_line)
229 GrVec lineVec = qPts[(maxEdge + 1)%3] - qPts[maxEdge];
230 // when looking from the point 0 down the line we want positive
231 // distances to be to the left. This matches the non-degenerate
232 // case.
233 lineVec.setOrthog(lineVec, GrPoint::kLeft_Side);
234 lineVec.dot(qPts[0]);
235 // first row
236 fM[0] = 0;
237 fM[1] = 0;
238 fM[2] = 0;
239 // second row
240 fM[3] = lineVec.fX;
241 fM[4] = lineVec.fY;
242 fM[5] = -lineVec.dot(qPts[maxEdge]);
243 } else {
244 // It's a point. It should cover zero area. Just set the matrix such
245 // that (u, v) will always be far away from the quad.
246 fM[0] = 0; fM[1] = 0; fM[2] = 100.f;
247 fM[3] = 0; fM[4] = 0; fM[5] = 100.f;
248 }
249 } else {
250 double scale = 1.0/det;
251
252 // compute adjugate matrix
253 double a0, a1, a2, a3, a4, a5, a6, a7, a8;
254 a0 = y1-y2;
255 a1 = x2-x1;
256 a2 = x1*y2-x2*y1;
257
258 a3 = y2-y0;
259 a4 = x0-x2;
260 a5 = x2*y0-x0*y2;
261
262 a6 = y0-y1;
263 a7 = x1-x0;
264 a8 = x0*y1-x1*y0;
265
266 // this performs the uv_pts*adjugate(control_pts) multiply,
267 // then does the scale by 1/det afterwards to improve precision
268 m[SkMatrix::kMScaleX] = (float)((0.5*a3 + a6)*scale);
269 m[SkMatrix::kMSkewX] = (float)((0.5*a4 + a7)*scale);
270 m[SkMatrix::kMTransX] = (float)((0.5*a5 + a8)*scale);
271
272 m[SkMatrix::kMSkewY] = (float)(a6*scale);
273 m[SkMatrix::kMScaleY] = (float)(a7*scale);
274 m[SkMatrix::kMTransY] = (float)(a8*scale);
275
276 m[SkMatrix::kMPersp0] = (float)((a0 + a3 + a6)*scale);
277 m[SkMatrix::kMPersp1] = (float)((a1 + a4 + a7)*scale);
278 m[SkMatrix::kMPersp2] = (float)((a2 + a5 + a8)*scale);
279
280 // The matrix should not have perspective.
281 SkDEBUGCODE(static const SkScalar gTOL = 1.f / 100.f);
282 SkASSERT(SkScalarAbs(m.get(SkMatrix::kMPersp0)) < gTOL);
283 SkASSERT(SkScalarAbs(m.get(SkMatrix::kMPersp1)) < gTOL);
284
285 // It may not be normalized to have 1.0 in the bottom right
286 float m33 = m.get(SkMatrix::kMPersp2);
287 if (1.f != m33) {
288 m33 = 1.f / m33;
289 fM[0] = m33 * m.get(SkMatrix::kMScaleX);
290 fM[1] = m33 * m.get(SkMatrix::kMSkewX);
291 fM[2] = m33 * m.get(SkMatrix::kMTransX);
292 fM[3] = m33 * m.get(SkMatrix::kMSkewY);
293 fM[4] = m33 * m.get(SkMatrix::kMScaleY);
294 fM[5] = m33 * m.get(SkMatrix::kMTransY);
295 } else {
296 fM[0] = m.get(SkMatrix::kMScaleX);
297 fM[1] = m.get(SkMatrix::kMSkewX);
298 fM[2] = m.get(SkMatrix::kMTransX);
299 fM[3] = m.get(SkMatrix::kMSkewY);
300 fM[4] = m.get(SkMatrix::kMScaleY);
301 fM[5] = m.get(SkMatrix::kMTransY);
302 }
303 }
304 }
305
306 ////////////////////////////////////////////////////////////////////////////////
307
308 // k = (y2 - y0, x0 - x2, (x2 - x0)*y0 - (y2 - y0)*x0 )
309 // l = (2*w * (y1 - y0), 2*w * (x0 - x1), 2*w * (x1*y0 - x0*y1))
310 // m = (2*w * (y2 - y1), 2*w * (x1 - x2), 2*w * (x2*y1 - x1*y2))
311 void GrPathUtils::getConicKLM(const SkPoint p[3], const SkScalar weight, SkScalar klm[9]) {
312 const SkScalar w2 = 2.f * weight;
313 klm[0] = p[2].fY - p[0].fY;
314 klm[1] = p[0].fX - p[2].fX;
315 klm[2] = (p[2].fX - p[0].fX) * p[0].fY - (p[2].fY - p[0].fY) * p[0].fX;
316
317 klm[3] = w2 * (p[1].fY - p[0].fY);
318 klm[4] = w2 * (p[0].fX - p[1].fX);
319 klm[5] = w2 * (p[1].fX * p[0].fY - p[0].fX * p[1].fY);
320
321 klm[6] = w2 * (p[2].fY - p[1].fY);
322 klm[7] = w2 * (p[1].fX - p[2].fX);
323 klm[8] = w2 * (p[2].fX * p[1].fY - p[1].fX * p[2].fY);
324
325 // scale the max absolute value of coeffs to 10
326 SkScalar scale = 0.f;
327 for (int i = 0; i < 9; ++i) {
328 scale = SkMaxScalar(scale, SkScalarAbs(klm[i]));
329 }
330 SkASSERT(scale > 0.f);
331 scale = 10.f / scale;
332 for (int i = 0; i < 9; ++i) {
333 klm[i] *= scale;
334 }
335 }
336
337 ////////////////////////////////////////////////////////////////////////////////
338
339 namespace {
340
341 // a is the first control point of the cubic.
342 // ab is the vector from a to the second control point.
343 // dc is the vector from the fourth to the third control point.
344 // d is the fourth control point.
345 // p is the candidate quadratic control point.
346 // this assumes that the cubic doesn't inflect and is simple
347 bool is_point_within_cubic_tangents(const SkPoint& a,
348 const SkVector& ab,
349 const SkVector& dc,
350 const SkPoint& d,
351 SkPath::Direction dir,
352 const SkPoint p) {
353 SkVector ap = p - a;
354 SkScalar apXab = ap.cross(ab);
355 if (SkPath::kCW_Direction == dir) {
356 if (apXab > 0) {
357 return false;
358 }
359 } else {
360 SkASSERT(SkPath::kCCW_Direction == dir);
361 if (apXab < 0) {
362 return false;
363 }
364 }
365
366 SkVector dp = p - d;
367 SkScalar dpXdc = dp.cross(dc);
368 if (SkPath::kCW_Direction == dir) {
369 if (dpXdc < 0) {
370 return false;
371 }
372 } else {
373 SkASSERT(SkPath::kCCW_Direction == dir);
374 if (dpXdc > 0) {
375 return false;
376 }
377 }
378 return true;
379 }
380
381 void convert_noninflect_cubic_to_quads(const SkPoint p[4],
382 SkScalar toleranceSqd,
383 bool constrainWithinTangents,
384 SkPath::Direction dir,
385 SkTArray<SkPoint, true>* quads,
386 int sublevel = 0) {
387
388 // Notation: Point a is always p[0]. Point b is p[1] unless p[1] == p[0], in which case it is
389 // p[2]. Point d is always p[3]. Point c is p[2] unless p[2] == p[3], in which case it is p[1].
390
391 SkVector ab = p[1] - p[0];
392 SkVector dc = p[2] - p[3];
393
394 if (ab.isZero()) {
395 if (dc.isZero()) {
396 SkPoint* degQuad = quads->push_back_n(3);
397 degQuad[0] = p[0];
398 degQuad[1] = p[0];
399 degQuad[2] = p[3];
400 return;
401 }
402 ab = p[2] - p[0];
403 }
404 if (dc.isZero()) {
405 dc = p[1] - p[3];
406 }
407
408 // When the ab and cd tangents are nearly parallel with vector from d to a the constraint that
409 // the quad point falls between the tangents becomes hard to enforce and we are likely to hit
410 // the max subdivision count. However, in this case the cubic is approaching a line and the
411 // accuracy of the quad point isn't so important. We check if the two middle cubic control
412 // points are very close to the baseline vector. If so then we just pick quadratic points on the
413 // control polygon.
414
415 if (constrainWithinTangents) {
416 SkVector da = p[0] - p[3];
417 SkScalar invDALengthSqd = da.lengthSqd();
418 if (invDALengthSqd > SK_ScalarNearlyZero) {
419 invDALengthSqd = SkScalarInvert(invDALengthSqd);
420 // cross(ab, da)^2/length(da)^2 == sqd distance from b to line from d to a.
421 // same goed for point c using vector cd.
422 SkScalar detABSqd = ab.cross(da);
423 detABSqd = SkScalarSquare(detABSqd);
424 SkScalar detDCSqd = dc.cross(da);
425 detDCSqd = SkScalarSquare(detDCSqd);
426 if (SkScalarMul(detABSqd, invDALengthSqd) < toleranceSqd &&
427 SkScalarMul(detDCSqd, invDALengthSqd) < toleranceSqd) {
428 SkPoint b = p[0] + ab;
429 SkPoint c = p[3] + dc;
430 SkPoint mid = b + c;
431 mid.scale(SK_ScalarHalf);
432 // Insert two quadratics to cover the case when ab points away from d and/or dc
433 // points away from a.
434 if (SkVector::DotProduct(da, dc) < 0 || SkVector::DotProduct(ab,da) > 0) {
435 SkPoint* qpts = quads->push_back_n(6);
436 qpts[0] = p[0];
437 qpts[1] = b;
438 qpts[2] = mid;
439 qpts[3] = mid;
440 qpts[4] = c;
441 qpts[5] = p[3];
442 } else {
443 SkPoint* qpts = quads->push_back_n(3);
444 qpts[0] = p[0];
445 qpts[1] = mid;
446 qpts[2] = p[3];
447 }
448 return;
449 }
450 }
451 }
452
453 static const SkScalar kLengthScale = 3 * SK_Scalar1 / 2;
454 static const int kMaxSubdivs = 10;
455
456 ab.scale(kLengthScale);
457 dc.scale(kLengthScale);
458
459 // e0 and e1 are extrapolations along vectors ab and dc.
460 SkVector c0 = p[0];
461 c0 += ab;
462 SkVector c1 = p[3];
463 c1 += dc;
464
465 SkScalar dSqd = sublevel > kMaxSubdivs ? 0 : c0.distanceToSqd(c1);
466 if (dSqd < toleranceSqd) {
467 SkPoint cAvg = c0;
468 cAvg += c1;
469 cAvg.scale(SK_ScalarHalf);
470
471 bool subdivide = false;
472
473 if (constrainWithinTangents &&
474 !is_point_within_cubic_tangents(p[0], ab, dc, p[3], dir, cAvg)) {
475 // choose a new cAvg that is the intersection of the two tangent lines.
476 ab.setOrthog(ab);
477 SkScalar z0 = -ab.dot(p[0]);
478 dc.setOrthog(dc);
479 SkScalar z1 = -dc.dot(p[3]);
480 cAvg.fX = SkScalarMul(ab.fY, z1) - SkScalarMul(z0, dc.fY);
481 cAvg.fY = SkScalarMul(z0, dc.fX) - SkScalarMul(ab.fX, z1);
482 SkScalar z = SkScalarMul(ab.fX, dc.fY) - SkScalarMul(ab.fY, dc.fX);
483 z = SkScalarInvert(z);
484 cAvg.fX *= z;
485 cAvg.fY *= z;
486 if (sublevel <= kMaxSubdivs) {
487 SkScalar d0Sqd = c0.distanceToSqd(cAvg);
488 SkScalar d1Sqd = c1.distanceToSqd(cAvg);
489 // We need to subdivide if d0 + d1 > tolerance but we have the sqd values. We know
490 // the distances and tolerance can't be negative.
491 // (d0 + d1)^2 > toleranceSqd
492 // d0Sqd + 2*d0*d1 + d1Sqd > toleranceSqd
493 SkScalar d0d1 = SkScalarSqrt(SkScalarMul(d0Sqd, d1Sqd));
494 subdivide = 2 * d0d1 + d0Sqd + d1Sqd > toleranceSqd;
495 }
496 }
497 if (!subdivide) {
498 SkPoint* pts = quads->push_back_n(3);
499 pts[0] = p[0];
500 pts[1] = cAvg;
501 pts[2] = p[3];
502 return;
503 }
504 }
505 SkPoint choppedPts[7];
506 SkChopCubicAtHalf(p, choppedPts);
507 convert_noninflect_cubic_to_quads(choppedPts + 0,
508 toleranceSqd,
509 constrainWithinTangents,
510 dir,
511 quads,
512 sublevel + 1);
513 convert_noninflect_cubic_to_quads(choppedPts + 3,
514 toleranceSqd,
515 constrainWithinTangents,
516 dir,
517 quads,
518 sublevel + 1);
519 }
520 }
521
522 void GrPathUtils::convertCubicToQuads(const GrPoint p[4],
523 SkScalar tolScale,
524 bool constrainWithinTangents,
525 SkPath::Direction dir,
526 SkTArray<SkPoint, true>* quads) {
527 SkPoint chopped[10];
528 int count = SkChopCubicAtInflections(p, chopped);
529
530 // base tolerance is 1 pixel.
531 static const SkScalar kTolerance = SK_Scalar1;
532 const SkScalar tolSqd = SkScalarSquare(SkScalarMul(tolScale, kTolerance));
533
534 for (int i = 0; i < count; ++i) {
535 SkPoint* cubic = chopped + 3*i;
536 convert_noninflect_cubic_to_quads(cubic, tolSqd, constrainWithinTangents, dir, quads);
537 }
538
539 }
540
541 ////////////////////////////////////////////////////////////////////////////////
542
543 enum CubicType {
544 kSerpentine_CubicType,
545 kCusp_CubicType,
546 kLoop_CubicType,
547 kQuadratic_CubicType,
548 kLine_CubicType,
549 kPoint_CubicType
550 };
551
552 // discr(I) = d0^2 * (3*d1^2 - 4*d0*d2)
553 // Classification:
554 // discr(I) > 0 Serpentine
555 // discr(I) = 0 Cusp
556 // discr(I) < 0 Loop
557 // d0 = d1 = 0 Quadratic
558 // d0 = d1 = d2 = 0 Line
559 // p0 = p1 = p2 = p3 Point
560 static CubicType classify_cubic(const SkPoint p[4], const SkScalar d[3]) {
561 if (p[0] == p[1] && p[0] == p[2] && p[0] == p[3]) {
562 return kPoint_CubicType;
563 }
564 const SkScalar discr = d[0] * d[0] * (3.f * d[1] * d[1] - 4.f * d[0] * d[2]);
565 if (discr > SK_ScalarNearlyZero) {
566 return kSerpentine_CubicType;
567 } else if (discr < -SK_ScalarNearlyZero) {
568 return kLoop_CubicType;
569 } else {
570 if (0.f == d[0] && 0.f == d[1]) {
571 return (0.f == d[2] ? kLine_CubicType : kQuadratic_CubicType);
572 } else {
573 return kCusp_CubicType;
574 }
575 }
576 }
577
578 // Assumes the third component of points is 1.
579 // Calcs p0 . (p1 x p2)
580 static SkScalar calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) {
581 const SkScalar xComp = p0.fX * (p1.fY - p2.fY);
582 const SkScalar yComp = p0.fY * (p2.fX - p1.fX);
583 const SkScalar wComp = p1.fX * p2.fY - p1.fY * p2.fX;
584 return (xComp + yComp + wComp);
585 }
586
587 // Solves linear system to extract klm
588 // P.K = k (similarly for l, m)
589 // Where P is matrix of control points
590 // K is coefficients for the line K
591 // k is vector of values of K evaluated at the control points
592 // Solving for K, thus K = P^(-1) . k
593 static void calc_cubic_klm(const SkPoint p[4], const SkScalar controlK[4],
594 const SkScalar controlL[4], const SkScalar controlM[4],
595 SkScalar k[3], SkScalar l[3], SkScalar m[3]) {
596 SkMatrix matrix;
597 matrix.setAll(p[0].fX, p[0].fY, 1.f,
598 p[1].fX, p[1].fY, 1.f,
599 p[2].fX, p[2].fY, 1.f);
600 SkMatrix inverse;
601 if (matrix.invert(&inverse)) {
602 inverse.mapHomogeneousPoints(k, controlK, 1);
603 inverse.mapHomogeneousPoints(l, controlL, 1);
604 inverse.mapHomogeneousPoints(m, controlM, 1);
605 }
606
607 }
608
609 static void set_serp_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) {
610 SkScalar tempSqrt = SkScalarSqrt(9.f * d[1] * d[1] - 12.f * d[0] * d[2]);
611 SkScalar ls = 3.f * d[1] - tempSqrt;
612 SkScalar lt = 6.f * d[0];
613 SkScalar ms = 3.f * d[1] + tempSqrt;
614 SkScalar mt = 6.f * d[0];
615
616 k[0] = ls * ms;
617 k[1] = (3.f * ls * ms - ls * mt - lt * ms) / 3.f;
618 k[2] = (lt * (mt - 2.f * ms) + ls * (3.f * ms - 2.f * mt)) / 3.f;
619 k[3] = (lt - ls) * (mt - ms);
620
621 l[0] = ls * ls * ls;
622 const SkScalar lt_ls = lt - ls;
623 l[1] = ls * ls * lt_ls * -1.f;
624 l[2] = lt_ls * lt_ls * ls;
625 l[3] = -1.f * lt_ls * lt_ls * lt_ls;
626
627 m[0] = ms * ms * ms;
628 const SkScalar mt_ms = mt - ms;
629 m[1] = ms * ms * mt_ms * -1.f;
630 m[2] = mt_ms * mt_ms * ms;
631 m[3] = -1.f * mt_ms * mt_ms * mt_ms;
632
633 // If d0 < 0 we need to flip the orientation of our curve
634 // This is done by negating the k and l values
635 // We want negative distance values to be on the inside
636 if ( d[0] > 0) {
637 for (int i = 0; i < 4; ++i) {
638 k[i] = -k[i];
639 l[i] = -l[i];
640 }
641 }
642 }
643
644 static void set_loop_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) {
645 SkScalar tempSqrt = SkScalarSqrt(4.f * d[0] * d[2] - 3.f * d[1] * d[1]);
646 SkScalar ls = d[1] - tempSqrt;
647 SkScalar lt = 2.f * d[0];
648 SkScalar ms = d[1] + tempSqrt;
649 SkScalar mt = 2.f * d[0];
650
651 k[0] = ls * ms;
652 k[1] = (3.f * ls*ms - ls * mt - lt * ms) / 3.f;
653 k[2] = (lt * (mt - 2.f * ms) + ls * (3.f * ms - 2.f * mt)) / 3.f;
654 k[3] = (lt - ls) * (mt - ms);
655
656 l[0] = ls * ls * ms;
657 l[1] = (ls * (ls * (mt - 3.f * ms) + 2.f * lt * ms))/-3.f;
658 l[2] = ((lt - ls) * (ls * (2.f * mt - 3.f * ms) + lt * ms))/3.f;
659 l[3] = -1.f * (lt - ls) * (lt - ls) * (mt - ms);
660
661 m[0] = ls * ms * ms;
662 m[1] = (ms * (ls * (2.f * mt - 3.f * ms) + lt * ms))/-3.f;
663 m[2] = ((mt - ms) * (ls * (mt - 3.f * ms) + 2.f * lt * ms))/3.f;
664 m[3] = -1.f * (lt - ls) * (mt - ms) * (mt - ms);
665
666
667 // If (d0 < 0 && sign(k1) > 0) || (d0 > 0 && sign(k1) < 0),
668 // we need to flip the orientation of our curve.
669 // This is done by negating the k and l values
670 if ( (d[0] < 0 && k[1] > 0) || (d[0] > 0 && k[1] < 0)) {
671 for (int i = 0; i < 4; ++i) {
672 k[i] = -k[i];
673 l[i] = -l[i];
674 }
675 }
676 }
677
678 static void set_cusp_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) {
679 const SkScalar ls = d[2];
680 const SkScalar lt = 3.f * d[1];
681
682 k[0] = ls;
683 k[1] = ls - lt / 3.f;
684 k[2] = ls - 2.f * lt / 3.f;
685 k[3] = ls - lt;
686
687 l[0] = ls * ls * ls;
688 const SkScalar ls_lt = ls - lt;
689 l[1] = ls * ls * ls_lt;
690 l[2] = ls_lt * ls_lt * ls;
691 l[3] = ls_lt * ls_lt * ls_lt;
692
693 m[0] = 1.f;
694 m[1] = 1.f;
695 m[2] = 1.f;
696 m[3] = 1.f;
697 }
698
699 // For the case when a cubic is actually a quadratic
700 // M =
701 // 0 0 0
702 // 1/3 0 1/3
703 // 2/3 1/3 2/3
704 // 1 1 1
705 static void set_quadratic_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) {
706 k[0] = 0.f;
707 k[1] = 1.f/3.f;
708 k[2] = 2.f/3.f;
709 k[3] = 1.f;
710
711 l[0] = 0.f;
712 l[1] = 0.f;
713 l[2] = 1.f/3.f;
714 l[3] = 1.f;
715
716 m[0] = 0.f;
717 m[1] = 1.f/3.f;
718 m[2] = 2.f/3.f;
719 m[3] = 1.f;
720
721 // If d2 < 0 we need to flip the orientation of our curve
722 // This is done by negating the k and l values
723 if ( d[2] > 0) {
724 for (int i = 0; i < 4; ++i) {
725 k[i] = -k[i];
726 l[i] = -l[i];
727 }
728 }
729 }
730
731 // Calc coefficients of I(s,t) where roots of I are inflection points of curve
732 // I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2)
733 // d0 = a1 - 2*a2+3*a3
734 // d1 = -a2 + 3*a3
735 // d2 = 3*a3
736 // a1 = p0 . (p3 x p2)
737 // a2 = p1 . (p0 x p3)
738 // a3 = p2 . (p1 x p0)
739 // Places the values of d1, d2, d3 in array d passed in
740 static void calc_cubic_inflection_func(const SkPoint p[4], SkScalar d[3]) {
741 SkScalar a1 = calc_dot_cross_cubic(p[0], p[3], p[2]);
742 SkScalar a2 = calc_dot_cross_cubic(p[1], p[0], p[3]);
743 SkScalar a3 = calc_dot_cross_cubic(p[2], p[1], p[0]);
744
745 // need to scale a's or values in later calculations will grow to high
746 SkScalar max = SkScalarAbs(a1);
747 max = SkMaxScalar(max, SkScalarAbs(a2));
748 max = SkMaxScalar(max, SkScalarAbs(a3));
749 max = 1.f/max;
750 a1 = a1 * max;
751 a2 = a2 * max;
752 a3 = a3 * max;
753
754 d[2] = 3.f * a3;
755 d[1] = d[2] - a2;
756 d[0] = d[1] - a2 + a1;
757 }
758
759 int GrPathUtils::chopCubicAtLoopIntersection(const SkPoint src[4], SkPoint dst[10], SkScalar klm[9],
760 SkScalar klm_rev[3]) {
761 // Variable to store the two parametric values at the loop double point
762 SkScalar smallS = 0.f;
763 SkScalar largeS = 0.f;
764
765 SkScalar d[3];
766 calc_cubic_inflection_func(src, d);
767
768 CubicType cType = classify_cubic(src, d);
769
770 int chop_count = 0;
771 if (kLoop_CubicType == cType) {
772 SkScalar tempSqrt = SkScalarSqrt(4.f * d[0] * d[2] - 3.f * d[1] * d[1]);
773 SkScalar ls = d[1] - tempSqrt;
774 SkScalar lt = 2.f * d[0];
775 SkScalar ms = d[1] + tempSqrt;
776 SkScalar mt = 2.f * d[0];
777 ls = ls / lt;
778 ms = ms / mt;
779 // need to have t values sorted since this is what is expected by SkChopCubicAt
780 if (ls <= ms) {
781 smallS = ls;
782 largeS = ms;
783 } else {
784 smallS = ms;
785 largeS = ls;
786 }
787
788 SkScalar chop_ts[2];
789 if (smallS > 0.f && smallS < 1.f) {
790 chop_ts[chop_count++] = smallS;
791 }
792 if (largeS > 0.f && largeS < 1.f) {
793 chop_ts[chop_count++] = largeS;
794 }
795 if(dst) {
796 SkChopCubicAt(src, dst, chop_ts, chop_count);
797 }
798 } else {
799 if (dst) {
800 memcpy(dst, src, sizeof(SkPoint) * 4);
801 }
802 }
803
804 if (klm && klm_rev) {
805 // Set klm_rev to to match the sub_section of cubic that needs to have its orientation
806 // flipped. This will always be the section that is the "loop"
807 if (2 == chop_count) {
808 klm_rev[0] = 1.f;
809 klm_rev[1] = -1.f;
810 klm_rev[2] = 1.f;
811 } else if (1 == chop_count) {
812 if (smallS < 0.f) {
813 klm_rev[0] = -1.f;
814 klm_rev[1] = 1.f;
815 } else {
816 klm_rev[0] = 1.f;
817 klm_rev[1] = -1.f;
818 }
819 } else {
820 if (smallS < 0.f && largeS > 1.f) {
821 klm_rev[0] = -1.f;
822 } else {
823 klm_rev[0] = 1.f;
824 }
825 }
826 SkScalar controlK[4];
827 SkScalar controlL[4];
828 SkScalar controlM[4];
829
830 if (kSerpentine_CubicType == cType || (kCusp_CubicType == cType && 0.f != d[0])) {
831 set_serp_klm(d, controlK, controlL, controlM);
832 } else if (kLoop_CubicType == cType) {
833 set_loop_klm(d, controlK, controlL, controlM);
834 } else if (kCusp_CubicType == cType) {
835 SkASSERT(0.f == d[0]);
836 set_cusp_klm(d, controlK, controlL, controlM);
837 } else if (kQuadratic_CubicType == cType) {
838 set_quadratic_klm(d, controlK, controlL, controlM);
839 }
840
841 calc_cubic_klm(src, controlK, controlL, controlM, klm, &klm[3], &klm[6]);
842 }
843 return chop_count + 1;
844 }
845
846 void GrPathUtils::getCubicKLM(const SkPoint p[4], SkScalar klm[9]) {
847 SkScalar d[3];
848 calc_cubic_inflection_func(p, d);
849
850 CubicType cType = classify_cubic(p, d);
851
852 SkScalar controlK[4];
853 SkScalar controlL[4];
854 SkScalar controlM[4];
855
856 if (kSerpentine_CubicType == cType || (kCusp_CubicType == cType && 0.f != d[0])) {
857 set_serp_klm(d, controlK, controlL, controlM);
858 } else if (kLoop_CubicType == cType) {
859 set_loop_klm(d, controlK, controlL, controlM);
860 } else if (kCusp_CubicType == cType) {
861 SkASSERT(0.f == d[0]);
862 set_cusp_klm(d, controlK, controlL, controlM);
863 } else if (kQuadratic_CubicType == cType) {
864 set_quadratic_klm(d, controlK, controlL, controlM);
865 }
866
867 calc_cubic_klm(p, controlK, controlL, controlM, klm, &klm[3], &klm[6]);
868 }

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