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 "src/gpu/geometry/GrPathUtils.h"
9
10#include "include/gpu/GrTypes.h"
11#include "src/core/SkMathPriv.h"
12#include "src/core/SkPointPriv.h"
13
14static const SkScalar gMinCurveTol = 0.0001f;
15
16SkScalar GrPathUtils::scaleToleranceToSrc(SkScalar devTol,
17 const SkMatrix& viewM,
18 const SkRect& pathBounds) {
19 // In order to tesselate the path we get a bound on how much the matrix can
20 // scale when mapping to screen coordinates.
21 SkScalar stretch = viewM.getMaxScale();
22
23 if (stretch < 0) {
24 // take worst case mapRadius amoung four corners.
25 // (less than perfect)
26 for (int i = 0; i < 4; ++i) {
27 SkMatrix mat;
28 mat.setTranslate((i % 2) ? pathBounds.fLeft : pathBounds.fRight,
29 (i < 2) ? pathBounds.fTop : pathBounds.fBottom);
30 mat.postConcat(viewM);
31 stretch = std::max(stretch, mat.mapRadius(SK_Scalar1));
32 }
33 }
34 SkScalar srcTol = 0;
35 if (stretch <= 0) {
36 // We have degenerate bounds or some degenerate matrix. Thus we set the tolerance to be the
37 // max of the path pathBounds width and height.
38 srcTol = std::max(pathBounds.width(), pathBounds.height());
39 } else {
40 srcTol = devTol / stretch;
41 }
42 if (srcTol < gMinCurveTol) {
43 srcTol = gMinCurveTol;
44 }
45 return srcTol;
46}
47
48uint32_t GrPathUtils::quadraticPointCount(const SkPoint points[], SkScalar tol) {
49 // You should have called scaleToleranceToSrc, which guarantees this
50 SkASSERT(tol >= gMinCurveTol);
51
52 SkScalar d = SkPointPriv::DistanceToLineSegmentBetween(points[1], points[0], points[2]);
53 if (!SkScalarIsFinite(d)) {
54 return kMaxPointsPerCurve;
55 } else if (d <= tol) {
56 return 1;
57 } else {
58 // Each time we subdivide, d should be cut in 4. So we need to
59 // subdivide x = log4(d/tol) times. x subdivisions creates 2^(x)
60 // points.
61 // 2^(log4(x)) = sqrt(x);
62 SkScalar divSqrt = SkScalarSqrt(d / tol);
63 if (((SkScalar)SK_MaxS32) <= divSqrt) {
64 return kMaxPointsPerCurve;
65 } else {
66 int temp = SkScalarCeilToInt(divSqrt);
67 int pow2 = GrNextPow2(temp);
68 // Because of NaNs & INFs we can wind up with a degenerate temp
69 // such that pow2 comes out negative. Also, our point generator
70 // will always output at least one pt.
71 if (pow2 < 1) {
72 pow2 = 1;
73 }
74 return std::min(pow2, kMaxPointsPerCurve);
75 }
76 }
77}
78
79uint32_t GrPathUtils::generateQuadraticPoints(const SkPoint& p0,
80 const SkPoint& p1,
81 const SkPoint& p2,
82 SkScalar tolSqd,
83 SkPoint** points,
84 uint32_t pointsLeft) {
85 if (pointsLeft < 2 ||
86 (SkPointPriv::DistanceToLineSegmentBetweenSqd(p1, p0, p2)) < tolSqd) {
87 (*points)[0] = p2;
88 *points += 1;
89 return 1;
90 }
91
92 SkPoint q[] = {
93 { SkScalarAve(p0.fX, p1.fX), SkScalarAve(p0.fY, p1.fY) },
94 { SkScalarAve(p1.fX, p2.fX), SkScalarAve(p1.fY, p2.fY) },
95 };
96 SkPoint r = { SkScalarAve(q[0].fX, q[1].fX), SkScalarAve(q[0].fY, q[1].fY) };
97
98 pointsLeft >>= 1;
99 uint32_t a = generateQuadraticPoints(p0, q[0], r, tolSqd, points, pointsLeft);
100 uint32_t b = generateQuadraticPoints(r, q[1], p2, tolSqd, points, pointsLeft);
101 return a + b;
102}
103
104uint32_t GrPathUtils::cubicPointCount(const SkPoint points[],
105 SkScalar tol) {
106 // You should have called scaleToleranceToSrc, which guarantees this
107 SkASSERT(tol >= gMinCurveTol);
108
109 SkScalar d = std::max(
110 SkPointPriv::DistanceToLineSegmentBetweenSqd(points[1], points[0], points[3]),
111 SkPointPriv::DistanceToLineSegmentBetweenSqd(points[2], points[0], points[3]));
112 d = SkScalarSqrt(d);
113 if (!SkScalarIsFinite(d)) {
114 return kMaxPointsPerCurve;
115 } else if (d <= tol) {
116 return 1;
117 } else {
118 SkScalar divSqrt = SkScalarSqrt(d / tol);
119 if (((SkScalar)SK_MaxS32) <= divSqrt) {
120 return kMaxPointsPerCurve;
121 } else {
122 int temp = SkScalarCeilToInt(SkScalarSqrt(d / tol));
123 int pow2 = GrNextPow2(temp);
124 // Because of NaNs & INFs we can wind up with a degenerate temp
125 // such that pow2 comes out negative. Also, our point generator
126 // will always output at least one pt.
127 if (pow2 < 1) {
128 pow2 = 1;
129 }
130 return std::min(pow2, kMaxPointsPerCurve);
131 }
132 }
133}
134
135uint32_t GrPathUtils::generateCubicPoints(const SkPoint& p0,
136 const SkPoint& p1,
137 const SkPoint& p2,
138 const SkPoint& p3,
139 SkScalar tolSqd,
140 SkPoint** points,
141 uint32_t pointsLeft) {
142 if (pointsLeft < 2 ||
143 (SkPointPriv::DistanceToLineSegmentBetweenSqd(p1, p0, p3) < tolSqd &&
144 SkPointPriv::DistanceToLineSegmentBetweenSqd(p2, p0, p3) < tolSqd)) {
145 (*points)[0] = p3;
146 *points += 1;
147 return 1;
148 }
149 SkPoint q[] = {
150 { SkScalarAve(p0.fX, p1.fX), SkScalarAve(p0.fY, p1.fY) },
151 { SkScalarAve(p1.fX, p2.fX), SkScalarAve(p1.fY, p2.fY) },
152 { SkScalarAve(p2.fX, p3.fX), SkScalarAve(p2.fY, p3.fY) }
153 };
154 SkPoint r[] = {
155 { SkScalarAve(q[0].fX, q[1].fX), SkScalarAve(q[0].fY, q[1].fY) },
156 { SkScalarAve(q[1].fX, q[2].fX), SkScalarAve(q[1].fY, q[2].fY) }
157 };
158 SkPoint s = { SkScalarAve(r[0].fX, r[1].fX), SkScalarAve(r[0].fY, r[1].fY) };
159 pointsLeft >>= 1;
160 uint32_t a = generateCubicPoints(p0, q[0], r[0], s, tolSqd, points, pointsLeft);
161 uint32_t b = generateCubicPoints(s, r[1], q[2], p3, tolSqd, points, pointsLeft);
162 return a + b;
163}
164
165int GrPathUtils::worstCasePointCount(const SkPath& path, int* subpaths, SkScalar tol) {
166 // You should have called scaleToleranceToSrc, which guarantees this
167 SkASSERT(tol >= gMinCurveTol);
168
169 int pointCount = 0;
170 *subpaths = 1;
171
172 bool first = true;
173
174 SkPath::Iter iter(path, false);
175 SkPath::Verb verb;
176
177 SkPoint pts[4];
178 while ((verb = iter.next(pts)) != SkPath::kDone_Verb) {
179
180 switch (verb) {
181 case SkPath::kLine_Verb:
182 pointCount += 1;
183 break;
184 case SkPath::kConic_Verb: {
185 SkScalar weight = iter.conicWeight();
186 SkAutoConicToQuads converter;
187 const SkPoint* quadPts = converter.computeQuads(pts, weight, tol);
188 for (int i = 0; i < converter.countQuads(); ++i) {
189 pointCount += quadraticPointCount(quadPts + 2*i, tol);
190 }
191 [[fallthrough]];
192 }
193 case SkPath::kQuad_Verb:
194 pointCount += quadraticPointCount(pts, tol);
195 break;
196 case SkPath::kCubic_Verb:
197 pointCount += cubicPointCount(pts, tol);
198 break;
199 case SkPath::kMove_Verb:
200 pointCount += 1;
201 if (!first) {
202 ++(*subpaths);
203 }
204 break;
205 default:
206 break;
207 }
208 first = false;
209 }
210 return pointCount;
211}
212
213void GrPathUtils::QuadUVMatrix::set(const SkPoint qPts[3]) {
214 SkMatrix m;
215 // We want M such that M * xy_pt = uv_pt
216 // We know M * control_pts = [0 1/2 1]
217 // [0 0 1]
218 // [1 1 1]
219 // And control_pts = [x0 x1 x2]
220 // [y0 y1 y2]
221 // [1 1 1 ]
222 // We invert the control pt matrix and post concat to both sides to get M.
223 // Using the known form of the control point matrix and the result, we can
224 // optimize and improve precision.
225
226 double x0 = qPts[0].fX;
227 double y0 = qPts[0].fY;
228 double x1 = qPts[1].fX;
229 double y1 = qPts[1].fY;
230 double x2 = qPts[2].fX;
231 double y2 = qPts[2].fY;
232 double det = x0*y1 - y0*x1 + x2*y0 - y2*x0 + x1*y2 - y1*x2;
233
234 if (!sk_float_isfinite(det)
235 || SkScalarNearlyZero((float)det, SK_ScalarNearlyZero * SK_ScalarNearlyZero)) {
236 // The quad is degenerate. Hopefully this is rare. Find the pts that are
237 // farthest apart to compute a line (unless it is really a pt).
238 SkScalar maxD = SkPointPriv::DistanceToSqd(qPts[0], qPts[1]);
239 int maxEdge = 0;
240 SkScalar d = SkPointPriv::DistanceToSqd(qPts[1], qPts[2]);
241 if (d > maxD) {
242 maxD = d;
243 maxEdge = 1;
244 }
245 d = SkPointPriv::DistanceToSqd(qPts[2], qPts[0]);
246 if (d > maxD) {
247 maxD = d;
248 maxEdge = 2;
249 }
250 // We could have a tolerance here, not sure if it would improve anything
251 if (maxD > 0) {
252 // Set the matrix to give (u = 0, v = distance_to_line)
253 SkVector lineVec = qPts[(maxEdge + 1)%3] - qPts[maxEdge];
254 // when looking from the point 0 down the line we want positive
255 // distances to be to the left. This matches the non-degenerate
256 // case.
257 lineVec = SkPointPriv::MakeOrthog(lineVec, SkPointPriv::kLeft_Side);
258 // first row
259 fM[0] = 0;
260 fM[1] = 0;
261 fM[2] = 0;
262 // second row
263 fM[3] = lineVec.fX;
264 fM[4] = lineVec.fY;
265 fM[5] = -lineVec.dot(qPts[maxEdge]);
266 } else {
267 // It's a point. It should cover zero area. Just set the matrix such
268 // that (u, v) will always be far away from the quad.
269 fM[0] = 0; fM[1] = 0; fM[2] = 100.f;
270 fM[3] = 0; fM[4] = 0; fM[5] = 100.f;
271 }
272 } else {
273 double scale = 1.0/det;
274
275 // compute adjugate matrix
276 double a2, a3, a4, a5, a6, a7, a8;
277 a2 = x1*y2-x2*y1;
278
279 a3 = y2-y0;
280 a4 = x0-x2;
281 a5 = x2*y0-x0*y2;
282
283 a6 = y0-y1;
284 a7 = x1-x0;
285 a8 = x0*y1-x1*y0;
286
287 // this performs the uv_pts*adjugate(control_pts) multiply,
288 // then does the scale by 1/det afterwards to improve precision
289 m[SkMatrix::kMScaleX] = (float)((0.5*a3 + a6)*scale);
290 m[SkMatrix::kMSkewX] = (float)((0.5*a4 + a7)*scale);
291 m[SkMatrix::kMTransX] = (float)((0.5*a5 + a8)*scale);
292
293 m[SkMatrix::kMSkewY] = (float)(a6*scale);
294 m[SkMatrix::kMScaleY] = (float)(a7*scale);
295 m[SkMatrix::kMTransY] = (float)(a8*scale);
296
297 // kMPersp0 & kMPersp1 should algebraically be zero
298 m[SkMatrix::kMPersp0] = 0.0f;
299 m[SkMatrix::kMPersp1] = 0.0f;
300 m[SkMatrix::kMPersp2] = (float)((a2 + a5 + a8)*scale);
301
302 // It may not be normalized to have 1.0 in the bottom right
303 float m33 = m.get(SkMatrix::kMPersp2);
304 if (1.f != m33) {
305 m33 = 1.f / m33;
306 fM[0] = m33 * m.get(SkMatrix::kMScaleX);
307 fM[1] = m33 * m.get(SkMatrix::kMSkewX);
308 fM[2] = m33 * m.get(SkMatrix::kMTransX);
309 fM[3] = m33 * m.get(SkMatrix::kMSkewY);
310 fM[4] = m33 * m.get(SkMatrix::kMScaleY);
311 fM[5] = m33 * m.get(SkMatrix::kMTransY);
312 } else {
313 fM[0] = m.get(SkMatrix::kMScaleX);
314 fM[1] = m.get(SkMatrix::kMSkewX);
315 fM[2] = m.get(SkMatrix::kMTransX);
316 fM[3] = m.get(SkMatrix::kMSkewY);
317 fM[4] = m.get(SkMatrix::kMScaleY);
318 fM[5] = m.get(SkMatrix::kMTransY);
319 }
320 }
321}
322
323////////////////////////////////////////////////////////////////////////////////
324
325// k = (y2 - y0, x0 - x2, x2*y0 - x0*y2)
326// l = (y1 - y0, x0 - x1, x1*y0 - x0*y1) * 2*w
327// m = (y2 - y1, x1 - x2, x2*y1 - x1*y2) * 2*w
328void GrPathUtils::getConicKLM(const SkPoint p[3], const SkScalar weight, SkMatrix* out) {
329 SkMatrix& klm = *out;
330 const SkScalar w2 = 2.f * weight;
331 klm[0] = p[2].fY - p[0].fY;
332 klm[1] = p[0].fX - p[2].fX;
333 klm[2] = p[2].fX * p[0].fY - p[0].fX * p[2].fY;
334
335 klm[3] = w2 * (p[1].fY - p[0].fY);
336 klm[4] = w2 * (p[0].fX - p[1].fX);
337 klm[5] = w2 * (p[1].fX * p[0].fY - p[0].fX * p[1].fY);
338
339 klm[6] = w2 * (p[2].fY - p[1].fY);
340 klm[7] = w2 * (p[1].fX - p[2].fX);
341 klm[8] = w2 * (p[2].fX * p[1].fY - p[1].fX * p[2].fY);
342
343 // scale the max absolute value of coeffs to 10
344 SkScalar scale = 0.f;
345 for (int i = 0; i < 9; ++i) {
346 scale = std::max(scale, SkScalarAbs(klm[i]));
347 }
348 SkASSERT(scale > 0.f);
349 scale = 10.f / scale;
350 for (int i = 0; i < 9; ++i) {
351 klm[i] *= scale;
352 }
353}
354
355////////////////////////////////////////////////////////////////////////////////
356
357namespace {
358
359// a is the first control point of the cubic.
360// ab is the vector from a to the second control point.
361// dc is the vector from the fourth to the third control point.
362// d is the fourth control point.
363// p is the candidate quadratic control point.
364// this assumes that the cubic doesn't inflect and is simple
365bool is_point_within_cubic_tangents(const SkPoint& a,
366 const SkVector& ab,
367 const SkVector& dc,
368 const SkPoint& d,
369 SkPathPriv::FirstDirection dir,
370 const SkPoint p) {
371 SkVector ap = p - a;
372 SkScalar apXab = ap.cross(ab);
373 if (SkPathPriv::kCW_FirstDirection == dir) {
374 if (apXab > 0) {
375 return false;
376 }
377 } else {
378 SkASSERT(SkPathPriv::kCCW_FirstDirection == dir);
379 if (apXab < 0) {
380 return false;
381 }
382 }
383
384 SkVector dp = p - d;
385 SkScalar dpXdc = dp.cross(dc);
386 if (SkPathPriv::kCW_FirstDirection == dir) {
387 if (dpXdc < 0) {
388 return false;
389 }
390 } else {
391 SkASSERT(SkPathPriv::kCCW_FirstDirection == dir);
392 if (dpXdc > 0) {
393 return false;
394 }
395 }
396 return true;
397}
398
399void convert_noninflect_cubic_to_quads(const SkPoint p[4],
400 SkScalar toleranceSqd,
401 SkTArray<SkPoint, true>* quads,
402 int sublevel = 0,
403 bool preserveFirstTangent = true,
404 bool preserveLastTangent = true) {
405 // Notation: Point a is always p[0]. Point b is p[1] unless p[1] == p[0], in which case it is
406 // 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].
407 SkVector ab = p[1] - p[0];
408 SkVector dc = p[2] - p[3];
409
410 if (SkPointPriv::LengthSqd(ab) < SK_ScalarNearlyZero) {
411 if (SkPointPriv::LengthSqd(dc) < SK_ScalarNearlyZero) {
412 SkPoint* degQuad = quads->push_back_n(3);
413 degQuad[0] = p[0];
414 degQuad[1] = p[0];
415 degQuad[2] = p[3];
416 return;
417 }
418 ab = p[2] - p[0];
419 }
420 if (SkPointPriv::LengthSqd(dc) < SK_ScalarNearlyZero) {
421 dc = p[1] - p[3];
422 }
423
424 static const SkScalar kLengthScale = 3 * SK_Scalar1 / 2;
425 static const int kMaxSubdivs = 10;
426
427 ab.scale(kLengthScale);
428 dc.scale(kLengthScale);
429
430 // c0 and c1 are extrapolations along vectors ab and dc.
431 SkPoint c0 = p[0] + ab;
432 SkPoint c1 = p[3] + dc;
433
434 SkScalar dSqd = sublevel > kMaxSubdivs ? 0 : SkPointPriv::DistanceToSqd(c0, c1);
435 if (dSqd < toleranceSqd) {
436 SkPoint newC;
437 if (preserveFirstTangent == preserveLastTangent) {
438 // We used to force a split when both tangents need to be preserved and c0 != c1.
439 // This introduced a large performance regression for tiny paths for no noticeable
440 // quality improvement. However, we aren't quite fulfilling our contract of guaranteeing
441 // the two tangent vectors and this could introduce a missed pixel in
442 // GrAAHairlinePathRenderer.
443 newC = (c0 + c1) * 0.5f;
444 } else if (preserveFirstTangent) {
445 newC = c0;
446 } else {
447 newC = c1;
448 }
449
450 SkPoint* pts = quads->push_back_n(3);
451 pts[0] = p[0];
452 pts[1] = newC;
453 pts[2] = p[3];
454 return;
455 }
456 SkPoint choppedPts[7];
457 SkChopCubicAtHalf(p, choppedPts);
458 convert_noninflect_cubic_to_quads(
459 choppedPts + 0, toleranceSqd, quads, sublevel + 1, preserveFirstTangent, false);
460 convert_noninflect_cubic_to_quads(
461 choppedPts + 3, toleranceSqd, quads, sublevel + 1, false, preserveLastTangent);
462}
463
464void convert_noninflect_cubic_to_quads_with_constraint(const SkPoint p[4],
465 SkScalar toleranceSqd,
466 SkPathPriv::FirstDirection dir,
467 SkTArray<SkPoint, true>* quads,
468 int sublevel = 0) {
469 // Notation: Point a is always p[0]. Point b is p[1] unless p[1] == p[0], in which case it is
470 // 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].
471
472 SkVector ab = p[1] - p[0];
473 SkVector dc = p[2] - p[3];
474
475 if (SkPointPriv::LengthSqd(ab) < SK_ScalarNearlyZero) {
476 if (SkPointPriv::LengthSqd(dc) < SK_ScalarNearlyZero) {
477 SkPoint* degQuad = quads->push_back_n(3);
478 degQuad[0] = p[0];
479 degQuad[1] = p[0];
480 degQuad[2] = p[3];
481 return;
482 }
483 ab = p[2] - p[0];
484 }
485 if (SkPointPriv::LengthSqd(dc) < SK_ScalarNearlyZero) {
486 dc = p[1] - p[3];
487 }
488
489 // When the ab and cd tangents are degenerate or nearly parallel with vector from d to a the
490 // constraint that the quad point falls between the tangents becomes hard to enforce and we are
491 // likely to hit the max subdivision count. However, in this case the cubic is approaching a
492 // line and the accuracy of the quad point isn't so important. We check if the two middle cubic
493 // control points are very close to the baseline vector. If so then we just pick quadratic
494 // points on the control polygon.
495
496 SkVector da = p[0] - p[3];
497 bool doQuads = SkPointPriv::LengthSqd(dc) < SK_ScalarNearlyZero ||
498 SkPointPriv::LengthSqd(ab) < SK_ScalarNearlyZero;
499 if (!doQuads) {
500 SkScalar invDALengthSqd = SkPointPriv::LengthSqd(da);
501 if (invDALengthSqd > SK_ScalarNearlyZero) {
502 invDALengthSqd = SkScalarInvert(invDALengthSqd);
503 // cross(ab, da)^2/length(da)^2 == sqd distance from b to line from d to a.
504 // same goes for point c using vector cd.
505 SkScalar detABSqd = ab.cross(da);
506 detABSqd = SkScalarSquare(detABSqd);
507 SkScalar detDCSqd = dc.cross(da);
508 detDCSqd = SkScalarSquare(detDCSqd);
509 if (detABSqd * invDALengthSqd < toleranceSqd &&
510 detDCSqd * invDALengthSqd < toleranceSqd) {
511 doQuads = true;
512 }
513 }
514 }
515 if (doQuads) {
516 SkPoint b = p[0] + ab;
517 SkPoint c = p[3] + dc;
518 SkPoint mid = b + c;
519 mid.scale(SK_ScalarHalf);
520 // Insert two quadratics to cover the case when ab points away from d and/or dc
521 // points away from a.
522 if (SkVector::DotProduct(da, dc) < 0 || SkVector::DotProduct(ab, da) > 0) {
523 SkPoint* qpts = quads->push_back_n(6);
524 qpts[0] = p[0];
525 qpts[1] = b;
526 qpts[2] = mid;
527 qpts[3] = mid;
528 qpts[4] = c;
529 qpts[5] = p[3];
530 } else {
531 SkPoint* qpts = quads->push_back_n(3);
532 qpts[0] = p[0];
533 qpts[1] = mid;
534 qpts[2] = p[3];
535 }
536 return;
537 }
538
539 static const SkScalar kLengthScale = 3 * SK_Scalar1 / 2;
540 static const int kMaxSubdivs = 10;
541
542 ab.scale(kLengthScale);
543 dc.scale(kLengthScale);
544
545 // c0 and c1 are extrapolations along vectors ab and dc.
546 SkVector c0 = p[0] + ab;
547 SkVector c1 = p[3] + dc;
548
549 SkScalar dSqd = sublevel > kMaxSubdivs ? 0 : SkPointPriv::DistanceToSqd(c0, c1);
550 if (dSqd < toleranceSqd) {
551 SkPoint cAvg = (c0 + c1) * 0.5f;
552 bool subdivide = false;
553
554 if (!is_point_within_cubic_tangents(p[0], ab, dc, p[3], dir, cAvg)) {
555 // choose a new cAvg that is the intersection of the two tangent lines.
556 ab = SkPointPriv::MakeOrthog(ab);
557 SkScalar z0 = -ab.dot(p[0]);
558 dc = SkPointPriv::MakeOrthog(dc);
559 SkScalar z1 = -dc.dot(p[3]);
560 cAvg.fX = ab.fY * z1 - z0 * dc.fY;
561 cAvg.fY = z0 * dc.fX - ab.fX * z1;
562 SkScalar z = ab.fX * dc.fY - ab.fY * dc.fX;
563 z = SkScalarInvert(z);
564 cAvg.fX *= z;
565 cAvg.fY *= z;
566 if (sublevel <= kMaxSubdivs) {
567 SkScalar d0Sqd = SkPointPriv::DistanceToSqd(c0, cAvg);
568 SkScalar d1Sqd = SkPointPriv::DistanceToSqd(c1, cAvg);
569 // We need to subdivide if d0 + d1 > tolerance but we have the sqd values. We know
570 // the distances and tolerance can't be negative.
571 // (d0 + d1)^2 > toleranceSqd
572 // d0Sqd + 2*d0*d1 + d1Sqd > toleranceSqd
573 SkScalar d0d1 = SkScalarSqrt(d0Sqd * d1Sqd);
574 subdivide = 2 * d0d1 + d0Sqd + d1Sqd > toleranceSqd;
575 }
576 }
577 if (!subdivide) {
578 SkPoint* pts = quads->push_back_n(3);
579 pts[0] = p[0];
580 pts[1] = cAvg;
581 pts[2] = p[3];
582 return;
583 }
584 }
585 SkPoint choppedPts[7];
586 SkChopCubicAtHalf(p, choppedPts);
587 convert_noninflect_cubic_to_quads_with_constraint(
588 choppedPts + 0, toleranceSqd, dir, quads, sublevel + 1);
589 convert_noninflect_cubic_to_quads_with_constraint(
590 choppedPts + 3, toleranceSqd, dir, quads, sublevel + 1);
591}
592} // namespace
593
594void GrPathUtils::convertCubicToQuads(const SkPoint p[4],
595 SkScalar tolScale,
596 SkTArray<SkPoint, true>* quads) {
597 if (!p[0].isFinite() || !p[1].isFinite() || !p[2].isFinite() || !p[3].isFinite()) {
598 return;
599 }
600 if (!SkScalarIsFinite(tolScale)) {
601 return;
602 }
603 SkPoint chopped[10];
604 int count = SkChopCubicAtInflections(p, chopped);
605
606 const SkScalar tolSqd = SkScalarSquare(tolScale);
607
608 for (int i = 0; i < count; ++i) {
609 SkPoint* cubic = chopped + 3*i;
610 convert_noninflect_cubic_to_quads(cubic, tolSqd, quads);
611 }
612}
613
614void GrPathUtils::convertCubicToQuadsConstrainToTangents(const SkPoint p[4],
615 SkScalar tolScale,
616 SkPathPriv::FirstDirection dir,
617 SkTArray<SkPoint, true>* quads) {
618 if (!p[0].isFinite() || !p[1].isFinite() || !p[2].isFinite() || !p[3].isFinite()) {
619 return;
620 }
621 if (!SkScalarIsFinite(tolScale)) {
622 return;
623 }
624 SkPoint chopped[10];
625 int count = SkChopCubicAtInflections(p, chopped);
626
627 const SkScalar tolSqd = SkScalarSquare(tolScale);
628
629 for (int i = 0; i < count; ++i) {
630 SkPoint* cubic = chopped + 3*i;
631 convert_noninflect_cubic_to_quads_with_constraint(cubic, tolSqd, dir, quads);
632 }
633}
634
635////////////////////////////////////////////////////////////////////////////////
636
637using ExcludedTerm = GrPathUtils::ExcludedTerm;
638
639ExcludedTerm GrPathUtils::calcCubicInverseTransposePowerBasisMatrix(const SkPoint p[4],
640 SkMatrix* out) {
641 static_assert(SK_SCALAR_IS_FLOAT);
642
643 // First convert the bezier coordinates p[0..3] to power basis coefficients X,Y(,W=[0 0 0 1]).
644 // M3 is the matrix that does this conversion. The homogeneous equation for the cubic becomes:
645 //
646 // | X Y 0 |
647 // C(t,s) = [t^3 t^2*s t*s^2 s^3] * | . . 0 |
648 // | . . 0 |
649 // | . . 1 |
650 //
651 const Sk4f M3[3] = {Sk4f(-1, 3, -3, 1),
652 Sk4f(3, -6, 3, 0),
653 Sk4f(-3, 3, 0, 0)};
654 // 4th col of M3 = Sk4f(1, 0, 0, 0)};
655 Sk4f X(p[3].x(), 0, 0, 0);
656 Sk4f Y(p[3].y(), 0, 0, 0);
657 for (int i = 2; i >= 0; --i) {
658 X += M3[i] * p[i].x();
659 Y += M3[i] * p[i].y();
660 }
661
662 // The matrix is 3x4. In order to invert it, we first need to make it square by throwing out one
663 // of the middle two rows. We toss the row that leaves us with the largest absolute determinant.
664 // Since the right column will be [0 0 1], the respective determinants reduce to x0*y2 - y0*x2
665 // and x0*y1 - y0*x1.
666 SkScalar dets[4];
667 Sk4f D = SkNx_shuffle<0,0,2,1>(X) * SkNx_shuffle<2,1,0,0>(Y);
668 D -= SkNx_shuffle<2,3,0,1>(D);
669 D.store(dets);
670 ExcludedTerm skipTerm = SkScalarAbs(dets[0]) > SkScalarAbs(dets[1]) ?
671 ExcludedTerm::kQuadraticTerm : ExcludedTerm::kLinearTerm;
672 SkScalar det = dets[ExcludedTerm::kQuadraticTerm == skipTerm ? 0 : 1];
673 if (0 == det) {
674 return ExcludedTerm::kNonInvertible;
675 }
676 SkScalar rdet = 1 / det;
677
678 // Compute the inverse-transpose of the power basis matrix with the 'skipRow'th row removed.
679 // Since W=[0 0 0 1], it follows that our corresponding solution will be equal to:
680 //
681 // | y1 -x1 x1*y2 - y1*x2 |
682 // 1/det * | -y0 x0 -x0*y2 + y0*x2 |
683 // | 0 0 det |
684 //
685 SkScalar x[4], y[4], z[4];
686 X.store(x);
687 Y.store(y);
688 (X * SkNx_shuffle<3,3,3,3>(Y) - Y * SkNx_shuffle<3,3,3,3>(X)).store(z);
689
690 int middleRow = ExcludedTerm::kQuadraticTerm == skipTerm ? 2 : 1;
691 out->setAll( y[middleRow] * rdet, -x[middleRow] * rdet, z[middleRow] * rdet,
692 -y[0] * rdet, x[0] * rdet, -z[0] * rdet,
693 0, 0, 1);
694
695 return skipTerm;
696}
697
698inline static void calc_serp_kcoeffs(SkScalar tl, SkScalar sl, SkScalar tm, SkScalar sm,
699 ExcludedTerm skipTerm, SkScalar outCoeffs[3]) {
700 SkASSERT(ExcludedTerm::kQuadraticTerm == skipTerm || ExcludedTerm::kLinearTerm == skipTerm);
701 outCoeffs[0] = 0;
702 outCoeffs[1] = (ExcludedTerm::kLinearTerm == skipTerm) ? sl*sm : -tl*sm - tm*sl;
703 outCoeffs[2] = tl*tm;
704}
705
706inline static void calc_serp_lmcoeffs(SkScalar t, SkScalar s, ExcludedTerm skipTerm,
707 SkScalar outCoeffs[3]) {
708 SkASSERT(ExcludedTerm::kQuadraticTerm == skipTerm || ExcludedTerm::kLinearTerm == skipTerm);
709 outCoeffs[0] = -s*s*s;
710 outCoeffs[1] = (ExcludedTerm::kLinearTerm == skipTerm) ? 3*s*s*t : -3*s*t*t;
711 outCoeffs[2] = t*t*t;
712}
713
714inline static void calc_loop_kcoeffs(SkScalar td, SkScalar sd, SkScalar te, SkScalar se,
715 SkScalar tdse, SkScalar tesd, ExcludedTerm skipTerm,
716 SkScalar outCoeffs[3]) {
717 SkASSERT(ExcludedTerm::kQuadraticTerm == skipTerm || ExcludedTerm::kLinearTerm == skipTerm);
718 outCoeffs[0] = 0;
719 outCoeffs[1] = (ExcludedTerm::kLinearTerm == skipTerm) ? sd*se : -tdse - tesd;
720 outCoeffs[2] = td*te;
721}
722
723inline static void calc_loop_lmcoeffs(SkScalar t2, SkScalar s2, SkScalar t1, SkScalar s1,
724 SkScalar t2s1, SkScalar t1s2, ExcludedTerm skipTerm,
725 SkScalar outCoeffs[3]) {
726 SkASSERT(ExcludedTerm::kQuadraticTerm == skipTerm || ExcludedTerm::kLinearTerm == skipTerm);
727 outCoeffs[0] = -s2*s2*s1;
728 outCoeffs[1] = (ExcludedTerm::kLinearTerm == skipTerm) ? s2 * (2*t2s1 + t1s2)
729 : -t2 * (t2s1 + 2*t1s2);
730 outCoeffs[2] = t2*t2*t1;
731}
732
733// For the case when a cubic bezier is actually a quadratic. We duplicate k in l so that the
734// implicit becomes:
735//
736// k^3 - l*m == k^3 - l*k == k * (k^2 - l)
737//
738// In the quadratic case we can simply assign fixed values at each control point:
739//
740// | ..K.. | | pts[0] pts[1] pts[2] pts[3] | | 0 1/3 2/3 1 |
741// | ..L.. | * | . . . . | == | 0 0 1/3 1 |
742// | ..K.. | | 1 1 1 1 | | 0 1/3 2/3 1 |
743//
744static void calc_quadratic_klm(const SkPoint pts[4], double d3, SkMatrix* klm) {
745 SkMatrix klmAtPts;
746 klmAtPts.setAll(0, 1.f/3, 1,
747 0, 0, 1,
748 0, 1.f/3, 1);
749
750 SkMatrix inversePts;
751 inversePts.setAll(pts[0].x(), pts[1].x(), pts[3].x(),
752 pts[0].y(), pts[1].y(), pts[3].y(),
753 1, 1, 1);
754 SkAssertResult(inversePts.invert(&inversePts));
755
756 klm->setConcat(klmAtPts, inversePts);
757
758 // If d3 > 0 we need to flip the orientation of our curve
759 // This is done by negating the k and l values
760 if (d3 > 0) {
761 klm->postScale(-1, -1);
762 }
763}
764
765// For the case when a cubic bezier is actually a line. We set K=0, L=1, M=-line, which results in
766// the following implicit:
767//
768// k^3 - l*m == 0^3 - 1*(-line) == -(-line) == line
769//
770static void calc_line_klm(const SkPoint pts[4], SkMatrix* klm) {
771 SkScalar ny = pts[0].x() - pts[3].x();
772 SkScalar nx = pts[3].y() - pts[0].y();
773 SkScalar k = nx * pts[0].x() + ny * pts[0].y();
774 klm->setAll( 0, 0, 0,
775 0, 0, 1,
776 -nx, -ny, k);
777}
778
779SkCubicType GrPathUtils::getCubicKLM(const SkPoint src[4], SkMatrix* klm, double tt[2],
780 double ss[2]) {
781 double d[4];
782 SkCubicType type = SkClassifyCubic(src, tt, ss, d);
783
784 if (SkCubicType::kLineOrPoint == type) {
785 calc_line_klm(src, klm);
786 return SkCubicType::kLineOrPoint;
787 }
788
789 if (SkCubicType::kQuadratic == type) {
790 calc_quadratic_klm(src, d[3], klm);
791 return SkCubicType::kQuadratic;
792 }
793
794 SkMatrix CIT;
795 ExcludedTerm skipTerm = calcCubicInverseTransposePowerBasisMatrix(src, &CIT);
796 if (ExcludedTerm::kNonInvertible == skipTerm) {
797 // This could technically also happen if the curve were quadratic, but SkClassifyCubic
798 // should have detected that case already with tolerance.
799 calc_line_klm(src, klm);
800 return SkCubicType::kLineOrPoint;
801 }
802
803 const SkScalar t0 = static_cast<SkScalar>(tt[0]), t1 = static_cast<SkScalar>(tt[1]),
804 s0 = static_cast<SkScalar>(ss[0]), s1 = static_cast<SkScalar>(ss[1]);
805
806 SkMatrix klmCoeffs;
807 switch (type) {
808 case SkCubicType::kCuspAtInfinity:
809 SkASSERT(1 == t1 && 0 == s1); // Infinity.
810 [[fallthrough]];
811 case SkCubicType::kLocalCusp:
812 case SkCubicType::kSerpentine:
813 calc_serp_kcoeffs(t0, s0, t1, s1, skipTerm, &klmCoeffs[0]);
814 calc_serp_lmcoeffs(t0, s0, skipTerm, &klmCoeffs[3]);
815 calc_serp_lmcoeffs(t1, s1, skipTerm, &klmCoeffs[6]);
816 break;
817 case SkCubicType::kLoop: {
818 const SkScalar tdse = t0 * s1;
819 const SkScalar tesd = t1 * s0;
820 calc_loop_kcoeffs(t0, s0, t1, s1, tdse, tesd, skipTerm, &klmCoeffs[0]);
821 calc_loop_lmcoeffs(t0, s0, t1, s1, tdse, tesd, skipTerm, &klmCoeffs[3]);
822 calc_loop_lmcoeffs(t1, s1, t0, s0, tesd, tdse, skipTerm, &klmCoeffs[6]);
823 break;
824 }
825 default:
826 SK_ABORT("Unexpected cubic type.");
827 break;
828 }
829
830 klm->setConcat(klmCoeffs, CIT);
831 return type;
832}
833
834int GrPathUtils::chopCubicAtLoopIntersection(const SkPoint src[4], SkPoint dst[10], SkMatrix* klm,
835 int* loopIndex) {
836 SkSTArray<2, SkScalar> chops;
837 *loopIndex = -1;
838
839 double t[2], s[2];
840 if (SkCubicType::kLoop == GrPathUtils::getCubicKLM(src, klm, t, s)) {
841 SkScalar t0 = static_cast<SkScalar>(t[0] / s[0]);
842 SkScalar t1 = static_cast<SkScalar>(t[1] / s[1]);
843 SkASSERT(t0 <= t1); // Technically t0 != t1 in a loop, but there may be FP error.
844
845 if (t0 < 1 && t1 > 0) {
846 *loopIndex = 0;
847 if (t0 > 0) {
848 chops.push_back(t0);
849 *loopIndex = 1;
850 }
851 if (t1 < 1) {
852 chops.push_back(t1);
853 *loopIndex = chops.count() - 1;
854 }
855 }
856 }
857
858 SkChopCubicAt(src, dst, chops.begin(), chops.count());
859 return chops.count() + 1;
860}
861