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 "GrTypes.h"
11 #include "SkMathPriv.h"
12 #include "SkPointPriv.h"
13
14 static const SkScalar gMinCurveTol = 0.0001f;
15
scaleToleranceToSrc(SkScalar devTol,const SkMatrix & viewM,const SkRect & pathBounds)16 SkScalar 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 = SkMaxScalar(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 = SkTMax(pathBounds.width(), pathBounds.height());
39 } else {
40 srcTol = devTol / stretch;
41 }
42 if (srcTol < gMinCurveTol) {
43 srcTol = gMinCurveTol;
44 }
45 return srcTol;
46 }
47
quadraticPointCount(const SkPoint points[],SkScalar tol)48 uint32_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 SkTMin(pow2, kMaxPointsPerCurve);
75 }
76 }
77 }
78
generateQuadraticPoints(const SkPoint & p0,const SkPoint & p1,const SkPoint & p2,SkScalar tolSqd,SkPoint ** points,uint32_t pointsLeft)79 uint32_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
cubicPointCount(const SkPoint points[],SkScalar tol)104 uint32_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 = SkTMax(
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 SkTMin(pow2, kMaxPointsPerCurve);
131 }
132 }
133 }
134
generateCubicPoints(const SkPoint & p0,const SkPoint & p1,const SkPoint & p2,const SkPoint & p3,SkScalar tolSqd,SkPoint ** points,uint32_t pointsLeft)135 uint32_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
worstCasePointCount(const SkPath & path,int * subpaths,SkScalar tol)165 int 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, false)) != 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 }
192 case SkPath::kQuad_Verb:
193 pointCount += quadraticPointCount(pts, tol);
194 break;
195 case SkPath::kCubic_Verb:
196 pointCount += cubicPointCount(pts, tol);
197 break;
198 case SkPath::kMove_Verb:
199 pointCount += 1;
200 if (!first) {
201 ++(*subpaths);
202 }
203 break;
204 default:
205 break;
206 }
207 first = false;
208 }
209 return pointCount;
210 }
211
set(const SkPoint qPts[3])212 void GrPathUtils::QuadUVMatrix::set(const SkPoint qPts[3]) {
213 SkMatrix m;
214 // We want M such that M * xy_pt = uv_pt
215 // We know M * control_pts = [0 1/2 1]
216 // [0 0 1]
217 // [1 1 1]
218 // And control_pts = [x0 x1 x2]
219 // [y0 y1 y2]
220 // [1 1 1 ]
221 // We invert the control pt matrix and post concat to both sides to get M.
222 // Using the known form of the control point matrix and the result, we can
223 // optimize and improve precision.
224
225 double x0 = qPts[0].fX;
226 double y0 = qPts[0].fY;
227 double x1 = qPts[1].fX;
228 double y1 = qPts[1].fY;
229 double x2 = qPts[2].fX;
230 double y2 = qPts[2].fY;
231 double det = x0*y1 - y0*x1 + x2*y0 - y2*x0 + x1*y2 - y1*x2;
232
233 if (!sk_float_isfinite(det)
234 || SkScalarNearlyZero((float)det, SK_ScalarNearlyZero * SK_ScalarNearlyZero)) {
235 // The quad is degenerate. Hopefully this is rare. Find the pts that are
236 // farthest apart to compute a line (unless it is really a pt).
237 SkScalar maxD = SkPointPriv::DistanceToSqd(qPts[0], qPts[1]);
238 int maxEdge = 0;
239 SkScalar d = SkPointPriv::DistanceToSqd(qPts[1], qPts[2]);
240 if (d > maxD) {
241 maxD = d;
242 maxEdge = 1;
243 }
244 d = SkPointPriv::DistanceToSqd(qPts[2], qPts[0]);
245 if (d > maxD) {
246 maxD = d;
247 maxEdge = 2;
248 }
249 // We could have a tolerance here, not sure if it would improve anything
250 if (maxD > 0) {
251 // Set the matrix to give (u = 0, v = distance_to_line)
252 SkVector lineVec = qPts[(maxEdge + 1)%3] - qPts[maxEdge];
253 // when looking from the point 0 down the line we want positive
254 // distances to be to the left. This matches the non-degenerate
255 // case.
256 lineVec = SkPointPriv::MakeOrthog(lineVec, SkPointPriv::kLeft_Side);
257 // first row
258 fM[0] = 0;
259 fM[1] = 0;
260 fM[2] = 0;
261 // second row
262 fM[3] = lineVec.fX;
263 fM[4] = lineVec.fY;
264 fM[5] = -lineVec.dot(qPts[maxEdge]);
265 } else {
266 // It's a point. It should cover zero area. Just set the matrix such
267 // that (u, v) will always be far away from the quad.
268 fM[0] = 0; fM[1] = 0; fM[2] = 100.f;
269 fM[3] = 0; fM[4] = 0; fM[5] = 100.f;
270 }
271 } else {
272 double scale = 1.0/det;
273
274 // compute adjugate matrix
275 double a2, a3, a4, a5, a6, a7, a8;
276 a2 = x1*y2-x2*y1;
277
278 a3 = y2-y0;
279 a4 = x0-x2;
280 a5 = x2*y0-x0*y2;
281
282 a6 = y0-y1;
283 a7 = x1-x0;
284 a8 = x0*y1-x1*y0;
285
286 // this performs the uv_pts*adjugate(control_pts) multiply,
287 // then does the scale by 1/det afterwards to improve precision
288 m[SkMatrix::kMScaleX] = (float)((0.5*a3 + a6)*scale);
289 m[SkMatrix::kMSkewX] = (float)((0.5*a4 + a7)*scale);
290 m[SkMatrix::kMTransX] = (float)((0.5*a5 + a8)*scale);
291
292 m[SkMatrix::kMSkewY] = (float)(a6*scale);
293 m[SkMatrix::kMScaleY] = (float)(a7*scale);
294 m[SkMatrix::kMTransY] = (float)(a8*scale);
295
296 // kMPersp0 & kMPersp1 should algebraically be zero
297 m[SkMatrix::kMPersp0] = 0.0f;
298 m[SkMatrix::kMPersp1] = 0.0f;
299 m[SkMatrix::kMPersp2] = (float)((a2 + a5 + a8)*scale);
300
301 // It may not be normalized to have 1.0 in the bottom right
302 float m33 = m.get(SkMatrix::kMPersp2);
303 if (1.f != m33) {
304 m33 = 1.f / m33;
305 fM[0] = m33 * m.get(SkMatrix::kMScaleX);
306 fM[1] = m33 * m.get(SkMatrix::kMSkewX);
307 fM[2] = m33 * m.get(SkMatrix::kMTransX);
308 fM[3] = m33 * m.get(SkMatrix::kMSkewY);
309 fM[4] = m33 * m.get(SkMatrix::kMScaleY);
310 fM[5] = m33 * m.get(SkMatrix::kMTransY);
311 } else {
312 fM[0] = m.get(SkMatrix::kMScaleX);
313 fM[1] = m.get(SkMatrix::kMSkewX);
314 fM[2] = m.get(SkMatrix::kMTransX);
315 fM[3] = m.get(SkMatrix::kMSkewY);
316 fM[4] = m.get(SkMatrix::kMScaleY);
317 fM[5] = m.get(SkMatrix::kMTransY);
318 }
319 }
320 }
321
322 ////////////////////////////////////////////////////////////////////////////////
323
324 // k = (y2 - y0, x0 - x2, x2*y0 - x0*y2)
325 // l = (y1 - y0, x0 - x1, x1*y0 - x0*y1) * 2*w
326 // m = (y2 - y1, x1 - x2, x2*y1 - x1*y2) * 2*w
getConicKLM(const SkPoint p[3],const SkScalar weight,SkMatrix * out)327 void GrPathUtils::getConicKLM(const SkPoint p[3], const SkScalar weight, SkMatrix* out) {
328 SkMatrix& klm = *out;
329 const SkScalar w2 = 2.f * weight;
330 klm[0] = p[2].fY - p[0].fY;
331 klm[1] = p[0].fX - p[2].fX;
332 klm[2] = p[2].fX * p[0].fY - p[0].fX * p[2].fY;
333
334 klm[3] = w2 * (p[1].fY - p[0].fY);
335 klm[4] = w2 * (p[0].fX - p[1].fX);
336 klm[5] = w2 * (p[1].fX * p[0].fY - p[0].fX * p[1].fY);
337
338 klm[6] = w2 * (p[2].fY - p[1].fY);
339 klm[7] = w2 * (p[1].fX - p[2].fX);
340 klm[8] = w2 * (p[2].fX * p[1].fY - p[1].fX * p[2].fY);
341
342 // scale the max absolute value of coeffs to 10
343 SkScalar scale = 0.f;
344 for (int i = 0; i < 9; ++i) {
345 scale = SkMaxScalar(scale, SkScalarAbs(klm[i]));
346 }
347 SkASSERT(scale > 0.f);
348 scale = 10.f / scale;
349 for (int i = 0; i < 9; ++i) {
350 klm[i] *= scale;
351 }
352 }
353
354 ////////////////////////////////////////////////////////////////////////////////
355
356 namespace {
357
358 // a is the first control point of the cubic.
359 // ab is the vector from a to the second control point.
360 // dc is the vector from the fourth to the third control point.
361 // d is the fourth control point.
362 // p is the candidate quadratic control point.
363 // this assumes that the cubic doesn't inflect and is simple
is_point_within_cubic_tangents(const SkPoint & a,const SkVector & ab,const SkVector & dc,const SkPoint & d,SkPathPriv::FirstDirection dir,const SkPoint p)364 bool is_point_within_cubic_tangents(const SkPoint& a,
365 const SkVector& ab,
366 const SkVector& dc,
367 const SkPoint& d,
368 SkPathPriv::FirstDirection dir,
369 const SkPoint p) {
370 SkVector ap = p - a;
371 SkScalar apXab = ap.cross(ab);
372 if (SkPathPriv::kCW_FirstDirection == dir) {
373 if (apXab > 0) {
374 return false;
375 }
376 } else {
377 SkASSERT(SkPathPriv::kCCW_FirstDirection == dir);
378 if (apXab < 0) {
379 return false;
380 }
381 }
382
383 SkVector dp = p - d;
384 SkScalar dpXdc = dp.cross(dc);
385 if (SkPathPriv::kCW_FirstDirection == dir) {
386 if (dpXdc < 0) {
387 return false;
388 }
389 } else {
390 SkASSERT(SkPathPriv::kCCW_FirstDirection == dir);
391 if (dpXdc > 0) {
392 return false;
393 }
394 }
395 return true;
396 }
397
convert_noninflect_cubic_to_quads(const SkPoint p[4],SkScalar toleranceSqd,SkTArray<SkPoint,true> * quads,int sublevel=0,bool preserveFirstTangent=true,bool preserveLastTangent=true)398 void convert_noninflect_cubic_to_quads(const SkPoint p[4],
399 SkScalar toleranceSqd,
400 SkTArray<SkPoint, true>* quads,
401 int sublevel = 0,
402 bool preserveFirstTangent = true,
403 bool preserveLastTangent = true) {
404 // Notation: Point a is always p[0]. Point b is p[1] unless p[1] == p[0], in which case it is
405 // 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].
406 SkVector ab = p[1] - p[0];
407 SkVector dc = p[2] - p[3];
408
409 if (SkPointPriv::LengthSqd(ab) < SK_ScalarNearlyZero) {
410 if (SkPointPriv::LengthSqd(dc) < SK_ScalarNearlyZero) {
411 SkPoint* degQuad = quads->push_back_n(3);
412 degQuad[0] = p[0];
413 degQuad[1] = p[0];
414 degQuad[2] = p[3];
415 return;
416 }
417 ab = p[2] - p[0];
418 }
419 if (SkPointPriv::LengthSqd(dc) < SK_ScalarNearlyZero) {
420 dc = p[1] - p[3];
421 }
422
423 static const SkScalar kLengthScale = 3 * SK_Scalar1 / 2;
424 static const int kMaxSubdivs = 10;
425
426 ab.scale(kLengthScale);
427 dc.scale(kLengthScale);
428
429 // c0 and c1 are extrapolations along vectors ab and dc.
430 SkPoint c0 = p[0] + ab;
431 SkPoint c1 = p[3] + dc;
432
433 SkScalar dSqd = sublevel > kMaxSubdivs ? 0 : SkPointPriv::DistanceToSqd(c0, c1);
434 if (dSqd < toleranceSqd) {
435 SkPoint newC;
436 if (preserveFirstTangent == preserveLastTangent) {
437 // We used to force a split when both tangents need to be preserved and c0 != c1.
438 // This introduced a large performance regression for tiny paths for no noticeable
439 // quality improvement. However, we aren't quite fulfilling our contract of guaranteeing
440 // the two tangent vectors and this could introduce a missed pixel in
441 // GrAAHairlinePathRenderer.
442 newC = (c0 + c1) * 0.5f;
443 } else if (preserveFirstTangent) {
444 newC = c0;
445 } else {
446 newC = c1;
447 }
448
449 SkPoint* pts = quads->push_back_n(3);
450 pts[0] = p[0];
451 pts[1] = newC;
452 pts[2] = p[3];
453 return;
454 }
455 SkPoint choppedPts[7];
456 SkChopCubicAtHalf(p, choppedPts);
457 convert_noninflect_cubic_to_quads(
458 choppedPts + 0, toleranceSqd, quads, sublevel + 1, preserveFirstTangent, false);
459 convert_noninflect_cubic_to_quads(
460 choppedPts + 3, toleranceSqd, quads, sublevel + 1, false, preserveLastTangent);
461 }
462
convert_noninflect_cubic_to_quads_with_constraint(const SkPoint p[4],SkScalar toleranceSqd,SkPathPriv::FirstDirection dir,SkTArray<SkPoint,true> * quads,int sublevel=0)463 void convert_noninflect_cubic_to_quads_with_constraint(const SkPoint p[4],
464 SkScalar toleranceSqd,
465 SkPathPriv::FirstDirection dir,
466 SkTArray<SkPoint, true>* quads,
467 int sublevel = 0) {
468 // Notation: Point a is always p[0]. Point b is p[1] unless p[1] == p[0], in which case it is
469 // 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].
470
471 SkVector ab = p[1] - p[0];
472 SkVector dc = p[2] - p[3];
473
474 if (SkPointPriv::LengthSqd(ab) < SK_ScalarNearlyZero) {
475 if (SkPointPriv::LengthSqd(dc) < SK_ScalarNearlyZero) {
476 SkPoint* degQuad = quads->push_back_n(3);
477 degQuad[0] = p[0];
478 degQuad[1] = p[0];
479 degQuad[2] = p[3];
480 return;
481 }
482 ab = p[2] - p[0];
483 }
484 if (SkPointPriv::LengthSqd(dc) < SK_ScalarNearlyZero) {
485 dc = p[1] - p[3];
486 }
487
488 // When the ab and cd tangents are degenerate or nearly parallel with vector from d to a the
489 // constraint that the quad point falls between the tangents becomes hard to enforce and we are
490 // likely to hit the max subdivision count. However, in this case the cubic is approaching a
491 // line and the accuracy of the quad point isn't so important. We check if the two middle cubic
492 // control points are very close to the baseline vector. If so then we just pick quadratic
493 // points on the control polygon.
494
495 SkVector da = p[0] - p[3];
496 bool doQuads = SkPointPriv::LengthSqd(dc) < SK_ScalarNearlyZero ||
497 SkPointPriv::LengthSqd(ab) < SK_ScalarNearlyZero;
498 if (!doQuads) {
499 SkScalar invDALengthSqd = SkPointPriv::LengthSqd(da);
500 if (invDALengthSqd > SK_ScalarNearlyZero) {
501 invDALengthSqd = SkScalarInvert(invDALengthSqd);
502 // cross(ab, da)^2/length(da)^2 == sqd distance from b to line from d to a.
503 // same goes for point c using vector cd.
504 SkScalar detABSqd = ab.cross(da);
505 detABSqd = SkScalarSquare(detABSqd);
506 SkScalar detDCSqd = dc.cross(da);
507 detDCSqd = SkScalarSquare(detDCSqd);
508 if (detABSqd * invDALengthSqd < toleranceSqd &&
509 detDCSqd * invDALengthSqd < toleranceSqd) {
510 doQuads = true;
511 }
512 }
513 }
514 if (doQuads) {
515 SkPoint b = p[0] + ab;
516 SkPoint c = p[3] + dc;
517 SkPoint mid = b + c;
518 mid.scale(SK_ScalarHalf);
519 // Insert two quadratics to cover the case when ab points away from d and/or dc
520 // points away from a.
521 if (SkVector::DotProduct(da, dc) < 0 || SkVector::DotProduct(ab, da) > 0) {
522 SkPoint* qpts = quads->push_back_n(6);
523 qpts[0] = p[0];
524 qpts[1] = b;
525 qpts[2] = mid;
526 qpts[3] = mid;
527 qpts[4] = c;
528 qpts[5] = p[3];
529 } else {
530 SkPoint* qpts = quads->push_back_n(3);
531 qpts[0] = p[0];
532 qpts[1] = mid;
533 qpts[2] = p[3];
534 }
535 return;
536 }
537
538 static const SkScalar kLengthScale = 3 * SK_Scalar1 / 2;
539 static const int kMaxSubdivs = 10;
540
541 ab.scale(kLengthScale);
542 dc.scale(kLengthScale);
543
544 // c0 and c1 are extrapolations along vectors ab and dc.
545 SkVector c0 = p[0] + ab;
546 SkVector c1 = p[3] + dc;
547
548 SkScalar dSqd = sublevel > kMaxSubdivs ? 0 : SkPointPriv::DistanceToSqd(c0, c1);
549 if (dSqd < toleranceSqd) {
550 SkPoint cAvg = (c0 + c1) * 0.5f;
551 bool subdivide = false;
552
553 if (!is_point_within_cubic_tangents(p[0], ab, dc, p[3], dir, cAvg)) {
554 // choose a new cAvg that is the intersection of the two tangent lines.
555 ab = SkPointPriv::MakeOrthog(ab);
556 SkScalar z0 = -ab.dot(p[0]);
557 dc = SkPointPriv::MakeOrthog(dc);
558 SkScalar z1 = -dc.dot(p[3]);
559 cAvg.fX = ab.fY * z1 - z0 * dc.fY;
560 cAvg.fY = z0 * dc.fX - ab.fX * z1;
561 SkScalar z = ab.fX * dc.fY - ab.fY * dc.fX;
562 z = SkScalarInvert(z);
563 cAvg.fX *= z;
564 cAvg.fY *= z;
565 if (sublevel <= kMaxSubdivs) {
566 SkScalar d0Sqd = SkPointPriv::DistanceToSqd(c0, cAvg);
567 SkScalar d1Sqd = SkPointPriv::DistanceToSqd(c1, cAvg);
568 // We need to subdivide if d0 + d1 > tolerance but we have the sqd values. We know
569 // the distances and tolerance can't be negative.
570 // (d0 + d1)^2 > toleranceSqd
571 // d0Sqd + 2*d0*d1 + d1Sqd > toleranceSqd
572 SkScalar d0d1 = SkScalarSqrt(d0Sqd * d1Sqd);
573 subdivide = 2 * d0d1 + d0Sqd + d1Sqd > toleranceSqd;
574 }
575 }
576 if (!subdivide) {
577 SkPoint* pts = quads->push_back_n(3);
578 pts[0] = p[0];
579 pts[1] = cAvg;
580 pts[2] = p[3];
581 return;
582 }
583 }
584 SkPoint choppedPts[7];
585 SkChopCubicAtHalf(p, choppedPts);
586 convert_noninflect_cubic_to_quads_with_constraint(
587 choppedPts + 0, toleranceSqd, dir, quads, sublevel + 1);
588 convert_noninflect_cubic_to_quads_with_constraint(
589 choppedPts + 3, toleranceSqd, dir, quads, sublevel + 1);
590 }
591 }
592
convertCubicToQuads(const SkPoint p[4],SkScalar tolScale,SkTArray<SkPoint,true> * quads)593 void GrPathUtils::convertCubicToQuads(const SkPoint p[4],
594 SkScalar tolScale,
595 SkTArray<SkPoint, true>* quads) {
596 if (!p[0].isFinite() || !p[1].isFinite() || !p[2].isFinite() || !p[3].isFinite()) {
597 return;
598 }
599 if (!SkScalarIsFinite(tolScale)) {
600 return;
601 }
602 SkPoint chopped[10];
603 int count = SkChopCubicAtInflections(p, chopped);
604
605 const SkScalar tolSqd = SkScalarSquare(tolScale);
606
607 for (int i = 0; i < count; ++i) {
608 SkPoint* cubic = chopped + 3*i;
609 convert_noninflect_cubic_to_quads(cubic, tolSqd, quads);
610 }
611 }
612
convertCubicToQuadsConstrainToTangents(const SkPoint p[4],SkScalar tolScale,SkPathPriv::FirstDirection dir,SkTArray<SkPoint,true> * quads)613 void GrPathUtils::convertCubicToQuadsConstrainToTangents(const SkPoint p[4],
614 SkScalar tolScale,
615 SkPathPriv::FirstDirection dir,
616 SkTArray<SkPoint, true>* quads) {
617 if (!p[0].isFinite() || !p[1].isFinite() || !p[2].isFinite() || !p[3].isFinite()) {
618 return;
619 }
620 if (!SkScalarIsFinite(tolScale)) {
621 return;
622 }
623 SkPoint chopped[10];
624 int count = SkChopCubicAtInflections(p, chopped);
625
626 const SkScalar tolSqd = SkScalarSquare(tolScale);
627
628 for (int i = 0; i < count; ++i) {
629 SkPoint* cubic = chopped + 3*i;
630 convert_noninflect_cubic_to_quads_with_constraint(cubic, tolSqd, dir, quads);
631 }
632 }
633
634 ////////////////////////////////////////////////////////////////////////////////
635
636 using ExcludedTerm = GrPathUtils::ExcludedTerm;
637
calcCubicInverseTransposePowerBasisMatrix(const SkPoint p[4],SkMatrix * out)638 ExcludedTerm GrPathUtils::calcCubicInverseTransposePowerBasisMatrix(const SkPoint p[4],
639 SkMatrix* out) {
640 GR_STATIC_ASSERT(SK_SCALAR_IS_FLOAT);
641
642 // First convert the bezier coordinates p[0..3] to power basis coefficients X,Y(,W=[0 0 0 1]).
643 // M3 is the matrix that does this conversion. The homogeneous equation for the cubic becomes:
644 //
645 // | X Y 0 |
646 // C(t,s) = [t^3 t^2*s t*s^2 s^3] * | . . 0 |
647 // | . . 0 |
648 // | . . 1 |
649 //
650 const Sk4f M3[3] = {Sk4f(-1, 3, -3, 1),
651 Sk4f(3, -6, 3, 0),
652 Sk4f(-3, 3, 0, 0)};
653 // 4th col of M3 = Sk4f(1, 0, 0, 0)};
654 Sk4f X(p[3].x(), 0, 0, 0);
655 Sk4f Y(p[3].y(), 0, 0, 0);
656 for (int i = 2; i >= 0; --i) {
657 X += M3[i] * p[i].x();
658 Y += M3[i] * p[i].y();
659 }
660
661 // The matrix is 3x4. In order to invert it, we first need to make it square by throwing out one
662 // of the middle two rows. We toss the row that leaves us with the largest absolute determinant.
663 // Since the right column will be [0 0 1], the respective determinants reduce to x0*y2 - y0*x2
664 // and x0*y1 - y0*x1.
665 SkScalar dets[4];
666 Sk4f D = SkNx_shuffle<0,0,2,1>(X) * SkNx_shuffle<2,1,0,0>(Y);
667 D -= SkNx_shuffle<2,3,0,1>(D);
668 D.store(dets);
669 ExcludedTerm skipTerm = SkScalarAbs(dets[0]) > SkScalarAbs(dets[1]) ?
670 ExcludedTerm::kQuadraticTerm : ExcludedTerm::kLinearTerm;
671 SkScalar det = dets[ExcludedTerm::kQuadraticTerm == skipTerm ? 0 : 1];
672 if (0 == det) {
673 return ExcludedTerm::kNonInvertible;
674 }
675 SkScalar rdet = 1 / det;
676
677 // Compute the inverse-transpose of the power basis matrix with the 'skipRow'th row removed.
678 // Since W=[0 0 0 1], it follows that our corresponding solution will be equal to:
679 //
680 // | y1 -x1 x1*y2 - y1*x2 |
681 // 1/det * | -y0 x0 -x0*y2 + y0*x2 |
682 // | 0 0 det |
683 //
684 SkScalar x[4], y[4], z[4];
685 X.store(x);
686 Y.store(y);
687 (X * SkNx_shuffle<3,3,3,3>(Y) - Y * SkNx_shuffle<3,3,3,3>(X)).store(z);
688
689 int middleRow = ExcludedTerm::kQuadraticTerm == skipTerm ? 2 : 1;
690 out->setAll( y[middleRow] * rdet, -x[middleRow] * rdet, z[middleRow] * rdet,
691 -y[0] * rdet, x[0] * rdet, -z[0] * rdet,
692 0, 0, 1);
693
694 return skipTerm;
695 }
696
calc_serp_kcoeffs(SkScalar tl,SkScalar sl,SkScalar tm,SkScalar sm,ExcludedTerm skipTerm,SkScalar outCoeffs[3])697 inline static void calc_serp_kcoeffs(SkScalar tl, SkScalar sl, SkScalar tm, SkScalar sm,
698 ExcludedTerm skipTerm, SkScalar outCoeffs[3]) {
699 SkASSERT(ExcludedTerm::kQuadraticTerm == skipTerm || ExcludedTerm::kLinearTerm == skipTerm);
700 outCoeffs[0] = 0;
701 outCoeffs[1] = (ExcludedTerm::kLinearTerm == skipTerm) ? sl*sm : -tl*sm - tm*sl;
702 outCoeffs[2] = tl*tm;
703 }
704
calc_serp_lmcoeffs(SkScalar t,SkScalar s,ExcludedTerm skipTerm,SkScalar outCoeffs[3])705 inline static void calc_serp_lmcoeffs(SkScalar t, SkScalar s, ExcludedTerm skipTerm,
706 SkScalar outCoeffs[3]) {
707 SkASSERT(ExcludedTerm::kQuadraticTerm == skipTerm || ExcludedTerm::kLinearTerm == skipTerm);
708 outCoeffs[0] = -s*s*s;
709 outCoeffs[1] = (ExcludedTerm::kLinearTerm == skipTerm) ? 3*s*s*t : -3*s*t*t;
710 outCoeffs[2] = t*t*t;
711 }
712
calc_loop_kcoeffs(SkScalar td,SkScalar sd,SkScalar te,SkScalar se,SkScalar tdse,SkScalar tesd,ExcludedTerm skipTerm,SkScalar outCoeffs[3])713 inline static void calc_loop_kcoeffs(SkScalar td, SkScalar sd, SkScalar te, SkScalar se,
714 SkScalar tdse, SkScalar tesd, ExcludedTerm skipTerm,
715 SkScalar outCoeffs[3]) {
716 SkASSERT(ExcludedTerm::kQuadraticTerm == skipTerm || ExcludedTerm::kLinearTerm == skipTerm);
717 outCoeffs[0] = 0;
718 outCoeffs[1] = (ExcludedTerm::kLinearTerm == skipTerm) ? sd*se : -tdse - tesd;
719 outCoeffs[2] = td*te;
720 }
721
calc_loop_lmcoeffs(SkScalar t2,SkScalar s2,SkScalar t1,SkScalar s1,SkScalar t2s1,SkScalar t1s2,ExcludedTerm skipTerm,SkScalar outCoeffs[3])722 inline static void calc_loop_lmcoeffs(SkScalar t2, SkScalar s2, SkScalar t1, SkScalar s1,
723 SkScalar t2s1, SkScalar t1s2, ExcludedTerm skipTerm,
724 SkScalar outCoeffs[3]) {
725 SkASSERT(ExcludedTerm::kQuadraticTerm == skipTerm || ExcludedTerm::kLinearTerm == skipTerm);
726 outCoeffs[0] = -s2*s2*s1;
727 outCoeffs[1] = (ExcludedTerm::kLinearTerm == skipTerm) ? s2 * (2*t2s1 + t1s2)
728 : -t2 * (t2s1 + 2*t1s2);
729 outCoeffs[2] = t2*t2*t1;
730 }
731
732 // For the case when a cubic bezier is actually a quadratic. We duplicate k in l so that the
733 // implicit becomes:
734 //
735 // k^3 - l*m == k^3 - l*k == k * (k^2 - l)
736 //
737 // In the quadratic case we can simply assign fixed values at each control point:
738 //
739 // | ..K.. | | pts[0] pts[1] pts[2] pts[3] | | 0 1/3 2/3 1 |
740 // | ..L.. | * | . . . . | == | 0 0 1/3 1 |
741 // | ..K.. | | 1 1 1 1 | | 0 1/3 2/3 1 |
742 //
calc_quadratic_klm(const SkPoint pts[4],double d3,SkMatrix * klm)743 static void calc_quadratic_klm(const SkPoint pts[4], double d3, SkMatrix* klm) {
744 SkMatrix klmAtPts;
745 klmAtPts.setAll(0, 1.f/3, 1,
746 0, 0, 1,
747 0, 1.f/3, 1);
748
749 SkMatrix inversePts;
750 inversePts.setAll(pts[0].x(), pts[1].x(), pts[3].x(),
751 pts[0].y(), pts[1].y(), pts[3].y(),
752 1, 1, 1);
753 SkAssertResult(inversePts.invert(&inversePts));
754
755 klm->setConcat(klmAtPts, inversePts);
756
757 // If d3 > 0 we need to flip the orientation of our curve
758 // This is done by negating the k and l values
759 if (d3 > 0) {
760 klm->postScale(-1, -1);
761 }
762 }
763
764 // For the case when a cubic bezier is actually a line. We set K=0, L=1, M=-line, which results in
765 // the following implicit:
766 //
767 // k^3 - l*m == 0^3 - 1*(-line) == -(-line) == line
768 //
calc_line_klm(const SkPoint pts[4],SkMatrix * klm)769 static void calc_line_klm(const SkPoint pts[4], SkMatrix* klm) {
770 SkScalar ny = pts[0].x() - pts[3].x();
771 SkScalar nx = pts[3].y() - pts[0].y();
772 SkScalar k = nx * pts[0].x() + ny * pts[0].y();
773 klm->setAll( 0, 0, 0,
774 0, 0, 1,
775 -nx, -ny, k);
776 }
777
getCubicKLM(const SkPoint src[4],SkMatrix * klm,double tt[2],double ss[2])778 SkCubicType GrPathUtils::getCubicKLM(const SkPoint src[4], SkMatrix* klm, double tt[2],
779 double ss[2]) {
780 double d[4];
781 SkCubicType type = SkClassifyCubic(src, tt, ss, d);
782
783 if (SkCubicType::kLineOrPoint == type) {
784 calc_line_klm(src, klm);
785 return SkCubicType::kLineOrPoint;
786 }
787
788 if (SkCubicType::kQuadratic == type) {
789 calc_quadratic_klm(src, d[3], klm);
790 return SkCubicType::kQuadratic;
791 }
792
793 SkMatrix CIT;
794 ExcludedTerm skipTerm = calcCubicInverseTransposePowerBasisMatrix(src, &CIT);
795 if (ExcludedTerm::kNonInvertible == skipTerm) {
796 // This could technically also happen if the curve were quadratic, but SkClassifyCubic
797 // should have detected that case already with tolerance.
798 calc_line_klm(src, klm);
799 return SkCubicType::kLineOrPoint;
800 }
801
802 const SkScalar t0 = static_cast<SkScalar>(tt[0]), t1 = static_cast<SkScalar>(tt[1]),
803 s0 = static_cast<SkScalar>(ss[0]), s1 = static_cast<SkScalar>(ss[1]);
804
805 SkMatrix klmCoeffs;
806 switch (type) {
807 case SkCubicType::kCuspAtInfinity:
808 SkASSERT(1 == t1 && 0 == s1); // Infinity.
809 // fallthru.
810 case SkCubicType::kLocalCusp:
811 case SkCubicType::kSerpentine:
812 calc_serp_kcoeffs(t0, s0, t1, s1, skipTerm, &klmCoeffs[0]);
813 calc_serp_lmcoeffs(t0, s0, skipTerm, &klmCoeffs[3]);
814 calc_serp_lmcoeffs(t1, s1, skipTerm, &klmCoeffs[6]);
815 break;
816 case SkCubicType::kLoop: {
817 const SkScalar tdse = t0 * s1;
818 const SkScalar tesd = t1 * s0;
819 calc_loop_kcoeffs(t0, s0, t1, s1, tdse, tesd, skipTerm, &klmCoeffs[0]);
820 calc_loop_lmcoeffs(t0, s0, t1, s1, tdse, tesd, skipTerm, &klmCoeffs[3]);
821 calc_loop_lmcoeffs(t1, s1, t0, s0, tesd, tdse, skipTerm, &klmCoeffs[6]);
822 break;
823 }
824 default:
825 SK_ABORT("Unexpected cubic type.");
826 break;
827 }
828
829 klm->setConcat(klmCoeffs, CIT);
830 return type;
831 }
832
chopCubicAtLoopIntersection(const SkPoint src[4],SkPoint dst[10],SkMatrix * klm,int * loopIndex)833 int GrPathUtils::chopCubicAtLoopIntersection(const SkPoint src[4], SkPoint dst[10], SkMatrix* klm,
834 int* loopIndex) {
835 SkSTArray<2, SkScalar> chops;
836 *loopIndex = -1;
837
838 double t[2], s[2];
839 if (SkCubicType::kLoop == GrPathUtils::getCubicKLM(src, klm, t, s)) {
840 SkScalar t0 = static_cast<SkScalar>(t[0] / s[0]);
841 SkScalar t1 = static_cast<SkScalar>(t[1] / s[1]);
842 SkASSERT(t0 <= t1); // Technically t0 != t1 in a loop, but there may be FP error.
843
844 if (t0 < 1 && t1 > 0) {
845 *loopIndex = 0;
846 if (t0 > 0) {
847 chops.push_back(t0);
848 *loopIndex = 1;
849 }
850 if (t1 < 1) {
851 chops.push_back(t1);
852 *loopIndex = chops.count() - 1;
853 }
854 }
855 }
856
857 SkChopCubicAt(src, dst, chops.begin(), chops.count());
858 return chops.count() + 1;
859 }
860