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 13 static const SkScalar gMinCurveTol = 0.0001f; 14 15 SkScalar GrPathUtils::scaleToleranceToSrc(SkScalar devTol, 16 const SkMatrix& viewM, 17 const SkRect& pathBounds) { 18 // In order to tesselate the path we get a bound on how much the matrix can 19 // scale when mapping to screen coordinates. 20 SkScalar stretch = viewM.getMaxScale(); 21 22 if (stretch < 0) { 23 // take worst case mapRadius amoung four corners. 24 // (less than perfect) 25 for (int i = 0; i < 4; ++i) { 26 SkMatrix mat; 27 mat.setTranslate((i % 2) ? pathBounds.fLeft : pathBounds.fRight, 28 (i < 2) ? pathBounds.fTop : pathBounds.fBottom); 29 mat.postConcat(viewM); 30 stretch = SkMaxScalar(stretch, mat.mapRadius(SK_Scalar1)); 31 } 32 } 33 SkScalar srcTol = devTol / stretch; 34 if (srcTol < gMinCurveTol) { 35 srcTol = gMinCurveTol; 36 } 37 return srcTol; 38 } 39 40 uint32_t GrPathUtils::quadraticPointCount(const SkPoint points[], SkScalar tol) { 41 // You should have called scaleToleranceToSrc, which guarantees this 42 SkASSERT(tol >= gMinCurveTol); 43 44 SkScalar d = points[1].distanceToLineSegmentBetween(points[0], points[2]); 45 if (!SkScalarIsFinite(d)) { 46 return kMaxPointsPerCurve; 47 } else 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 SkScalar divSqrt = SkScalarSqrt(d / tol); 55 if (((SkScalar)SK_MaxS32) <= divSqrt) { 56 return kMaxPointsPerCurve; 57 } else { 58 int temp = SkScalarCeilToInt(divSqrt); 59 int pow2 = GrNextPow2(temp); 60 // Because of NaNs & INFs we can wind up with a degenerate temp 61 // such that pow2 comes out negative. Also, our point generator 62 // will always output at least one pt. 63 if (pow2 < 1) { 64 pow2 = 1; 65 } 66 return SkTMin(pow2, kMaxPointsPerCurve); 67 } 68 } 69 } 70 71 uint32_t GrPathUtils::generateQuadraticPoints(const SkPoint& p0, 72 const SkPoint& p1, 73 const SkPoint& p2, 74 SkScalar tolSqd, 75 SkPoint** points, 76 uint32_t pointsLeft) { 77 if (pointsLeft < 2 || 78 (p1.distanceToLineSegmentBetweenSqd(p0, p2)) < tolSqd) { 79 (*points)[0] = p2; 80 *points += 1; 81 return 1; 82 } 83 84 SkPoint q[] = { 85 { SkScalarAve(p0.fX, p1.fX), SkScalarAve(p0.fY, p1.fY) }, 86 { SkScalarAve(p1.fX, p2.fX), SkScalarAve(p1.fY, p2.fY) }, 87 }; 88 SkPoint r = { SkScalarAve(q[0].fX, q[1].fX), SkScalarAve(q[0].fY, q[1].fY) }; 89 90 pointsLeft >>= 1; 91 uint32_t a = generateQuadraticPoints(p0, q[0], r, tolSqd, points, pointsLeft); 92 uint32_t b = generateQuadraticPoints(r, q[1], p2, tolSqd, points, pointsLeft); 93 return a + b; 94 } 95 96 uint32_t GrPathUtils::cubicPointCount(const SkPoint points[], 97 SkScalar tol) { 98 // You should have called scaleToleranceToSrc, which guarantees this 99 SkASSERT(tol >= gMinCurveTol); 100 101 SkScalar d = SkTMax( 102 points[1].distanceToLineSegmentBetweenSqd(points[0], points[3]), 103 points[2].distanceToLineSegmentBetweenSqd(points[0], points[3])); 104 d = SkScalarSqrt(d); 105 if (!SkScalarIsFinite(d)) { 106 return kMaxPointsPerCurve; 107 } else if (d <= tol) { 108 return 1; 109 } else { 110 SkScalar divSqrt = SkScalarSqrt(d / tol); 111 if (((SkScalar)SK_MaxS32) <= divSqrt) { 112 return kMaxPointsPerCurve; 113 } else { 114 int temp = SkScalarCeilToInt(SkScalarSqrt(d / tol)); 115 int pow2 = GrNextPow2(temp); 116 // Because of NaNs & INFs we can wind up with a degenerate temp 117 // such that pow2 comes out negative. Also, our point generator 118 // will always output at least one pt. 119 if (pow2 < 1) { 120 pow2 = 1; 121 } 122 return SkTMin(pow2, kMaxPointsPerCurve); 123 } 124 } 125 } 126 127 uint32_t GrPathUtils::generateCubicPoints(const SkPoint& p0, 128 const SkPoint& p1, 129 const SkPoint& p2, 130 const SkPoint& p3, 131 SkScalar tolSqd, 132 SkPoint** points, 133 uint32_t pointsLeft) { 134 if (pointsLeft < 2 || 135 (p1.distanceToLineSegmentBetweenSqd(p0, p3) < tolSqd && 136 p2.distanceToLineSegmentBetweenSqd(p0, p3) < tolSqd)) { 137 (*points)[0] = p3; 138 *points += 1; 139 return 1; 140 } 141 SkPoint q[] = { 142 { SkScalarAve(p0.fX, p1.fX), SkScalarAve(p0.fY, p1.fY) }, 143 { SkScalarAve(p1.fX, p2.fX), SkScalarAve(p1.fY, p2.fY) }, 144 { SkScalarAve(p2.fX, p3.fX), SkScalarAve(p2.fY, p3.fY) } 145 }; 146 SkPoint r[] = { 147 { SkScalarAve(q[0].fX, q[1].fX), SkScalarAve(q[0].fY, q[1].fY) }, 148 { SkScalarAve(q[1].fX, q[2].fX), SkScalarAve(q[1].fY, q[2].fY) } 149 }; 150 SkPoint s = { SkScalarAve(r[0].fX, r[1].fX), SkScalarAve(r[0].fY, r[1].fY) }; 151 pointsLeft >>= 1; 152 uint32_t a = generateCubicPoints(p0, q[0], r[0], s, tolSqd, points, pointsLeft); 153 uint32_t b = generateCubicPoints(s, r[1], q[2], p3, tolSqd, points, pointsLeft); 154 return a + b; 155 } 156 157 int GrPathUtils::worstCasePointCount(const SkPath& path, int* subpaths, SkScalar tol) { 158 // You should have called scaleToleranceToSrc, which guarantees this 159 SkASSERT(tol >= gMinCurveTol); 160 161 int pointCount = 0; 162 *subpaths = 1; 163 164 bool first = true; 165 166 SkPath::Iter iter(path, false); 167 SkPath::Verb verb; 168 169 SkPoint pts[4]; 170 while ((verb = iter.next(pts)) != SkPath::kDone_Verb) { 171 172 switch (verb) { 173 case SkPath::kLine_Verb: 174 pointCount += 1; 175 break; 176 case SkPath::kConic_Verb: { 177 SkScalar weight = iter.conicWeight(); 178 SkAutoConicToQuads converter; 179 const SkPoint* quadPts = converter.computeQuads(pts, weight, tol); 180 for (int i = 0; i < converter.countQuads(); ++i) { 181 pointCount += quadraticPointCount(quadPts + 2*i, tol); 182 } 183 } 184 case SkPath::kQuad_Verb: 185 pointCount += quadraticPointCount(pts, tol); 186 break; 187 case SkPath::kCubic_Verb: 188 pointCount += cubicPointCount(pts, tol); 189 break; 190 case SkPath::kMove_Verb: 191 pointCount += 1; 192 if (!first) { 193 ++(*subpaths); 194 } 195 break; 196 default: 197 break; 198 } 199 first = false; 200 } 201 return pointCount; 202 } 203 204 void GrPathUtils::QuadUVMatrix::set(const SkPoint qPts[3]) { 205 SkMatrix m; 206 // We want M such that M * xy_pt = uv_pt 207 // We know M * control_pts = [0 1/2 1] 208 // [0 0 1] 209 // [1 1 1] 210 // And control_pts = [x0 x1 x2] 211 // [y0 y1 y2] 212 // [1 1 1 ] 213 // We invert the control pt matrix and post concat to both sides to get M. 214 // Using the known form of the control point matrix and the result, we can 215 // optimize and improve precision. 216 217 double x0 = qPts[0].fX; 218 double y0 = qPts[0].fY; 219 double x1 = qPts[1].fX; 220 double y1 = qPts[1].fY; 221 double x2 = qPts[2].fX; 222 double y2 = qPts[2].fY; 223 double det = x0*y1 - y0*x1 + x2*y0 - y2*x0 + x1*y2 - y1*x2; 224 225 if (!sk_float_isfinite(det) 226 || SkScalarNearlyZero((float)det, SK_ScalarNearlyZero * SK_ScalarNearlyZero)) { 227 // The quad is degenerate. Hopefully this is rare. Find the pts that are 228 // farthest apart to compute a line (unless it is really a pt). 229 SkScalar maxD = qPts[0].distanceToSqd(qPts[1]); 230 int maxEdge = 0; 231 SkScalar d = qPts[1].distanceToSqd(qPts[2]); 232 if (d > maxD) { 233 maxD = d; 234 maxEdge = 1; 235 } 236 d = qPts[2].distanceToSqd(qPts[0]); 237 if (d > maxD) { 238 maxD = d; 239 maxEdge = 2; 240 } 241 // We could have a tolerance here, not sure if it would improve anything 242 if (maxD > 0) { 243 // Set the matrix to give (u = 0, v = distance_to_line) 244 SkVector lineVec = qPts[(maxEdge + 1)%3] - qPts[maxEdge]; 245 // when looking from the point 0 down the line we want positive 246 // distances to be to the left. This matches the non-degenerate 247 // case. 248 lineVec.setOrthog(lineVec, SkPoint::kLeft_Side); 249 // first row 250 fM[0] = 0; 251 fM[1] = 0; 252 fM[2] = 0; 253 // second row 254 fM[3] = lineVec.fX; 255 fM[4] = lineVec.fY; 256 fM[5] = -lineVec.dot(qPts[maxEdge]); 257 } else { 258 // It's a point. It should cover zero area. Just set the matrix such 259 // that (u, v) will always be far away from the quad. 260 fM[0] = 0; fM[1] = 0; fM[2] = 100.f; 261 fM[3] = 0; fM[4] = 0; fM[5] = 100.f; 262 } 263 } else { 264 double scale = 1.0/det; 265 266 // compute adjugate matrix 267 double a2, a3, a4, a5, a6, a7, a8; 268 a2 = x1*y2-x2*y1; 269 270 a3 = y2-y0; 271 a4 = x0-x2; 272 a5 = x2*y0-x0*y2; 273 274 a6 = y0-y1; 275 a7 = x1-x0; 276 a8 = x0*y1-x1*y0; 277 278 // this performs the uv_pts*adjugate(control_pts) multiply, 279 // then does the scale by 1/det afterwards to improve precision 280 m[SkMatrix::kMScaleX] = (float)((0.5*a3 + a6)*scale); 281 m[SkMatrix::kMSkewX] = (float)((0.5*a4 + a7)*scale); 282 m[SkMatrix::kMTransX] = (float)((0.5*a5 + a8)*scale); 283 284 m[SkMatrix::kMSkewY] = (float)(a6*scale); 285 m[SkMatrix::kMScaleY] = (float)(a7*scale); 286 m[SkMatrix::kMTransY] = (float)(a8*scale); 287 288 // kMPersp0 & kMPersp1 should algebraically be zero 289 m[SkMatrix::kMPersp0] = 0.0f; 290 m[SkMatrix::kMPersp1] = 0.0f; 291 m[SkMatrix::kMPersp2] = (float)((a2 + a5 + a8)*scale); 292 293 // It may not be normalized to have 1.0 in the bottom right 294 float m33 = m.get(SkMatrix::kMPersp2); 295 if (1.f != m33) { 296 m33 = 1.f / m33; 297 fM[0] = m33 * m.get(SkMatrix::kMScaleX); 298 fM[1] = m33 * m.get(SkMatrix::kMSkewX); 299 fM[2] = m33 * m.get(SkMatrix::kMTransX); 300 fM[3] = m33 * m.get(SkMatrix::kMSkewY); 301 fM[4] = m33 * m.get(SkMatrix::kMScaleY); 302 fM[5] = m33 * m.get(SkMatrix::kMTransY); 303 } else { 304 fM[0] = m.get(SkMatrix::kMScaleX); 305 fM[1] = m.get(SkMatrix::kMSkewX); 306 fM[2] = m.get(SkMatrix::kMTransX); 307 fM[3] = m.get(SkMatrix::kMSkewY); 308 fM[4] = m.get(SkMatrix::kMScaleY); 309 fM[5] = m.get(SkMatrix::kMTransY); 310 } 311 } 312 } 313 314 //////////////////////////////////////////////////////////////////////////////// 315 316 // k = (y2 - y0, x0 - x2, x2*y0 - x0*y2) 317 // l = (y1 - y0, x0 - x1, x1*y0 - x0*y1) * 2*w 318 // m = (y2 - y1, x1 - x2, x2*y1 - x1*y2) * 2*w 319 void GrPathUtils::getConicKLM(const SkPoint p[3], const SkScalar weight, SkMatrix* out) { 320 SkMatrix& klm = *out; 321 const SkScalar w2 = 2.f * weight; 322 klm[0] = p[2].fY - p[0].fY; 323 klm[1] = p[0].fX - p[2].fX; 324 klm[2] = p[2].fX * p[0].fY - p[0].fX * p[2].fY; 325 326 klm[3] = w2 * (p[1].fY - p[0].fY); 327 klm[4] = w2 * (p[0].fX - p[1].fX); 328 klm[5] = w2 * (p[1].fX * p[0].fY - p[0].fX * p[1].fY); 329 330 klm[6] = w2 * (p[2].fY - p[1].fY); 331 klm[7] = w2 * (p[1].fX - p[2].fX); 332 klm[8] = w2 * (p[2].fX * p[1].fY - p[1].fX * p[2].fY); 333 334 // scale the max absolute value of coeffs to 10 335 SkScalar scale = 0.f; 336 for (int i = 0; i < 9; ++i) { 337 scale = SkMaxScalar(scale, SkScalarAbs(klm[i])); 338 } 339 SkASSERT(scale > 0.f); 340 scale = 10.f / scale; 341 for (int i = 0; i < 9; ++i) { 342 klm[i] *= scale; 343 } 344 } 345 346 //////////////////////////////////////////////////////////////////////////////// 347 348 namespace { 349 350 // a is the first control point of the cubic. 351 // ab is the vector from a to the second control point. 352 // dc is the vector from the fourth to the third control point. 353 // d is the fourth control point. 354 // p is the candidate quadratic control point. 355 // this assumes that the cubic doesn't inflect and is simple 356 bool is_point_within_cubic_tangents(const SkPoint& a, 357 const SkVector& ab, 358 const SkVector& dc, 359 const SkPoint& d, 360 SkPathPriv::FirstDirection dir, 361 const SkPoint p) { 362 SkVector ap = p - a; 363 SkScalar apXab = ap.cross(ab); 364 if (SkPathPriv::kCW_FirstDirection == dir) { 365 if (apXab > 0) { 366 return false; 367 } 368 } else { 369 SkASSERT(SkPathPriv::kCCW_FirstDirection == dir); 370 if (apXab < 0) { 371 return false; 372 } 373 } 374 375 SkVector dp = p - d; 376 SkScalar dpXdc = dp.cross(dc); 377 if (SkPathPriv::kCW_FirstDirection == dir) { 378 if (dpXdc < 0) { 379 return false; 380 } 381 } else { 382 SkASSERT(SkPathPriv::kCCW_FirstDirection == dir); 383 if (dpXdc > 0) { 384 return false; 385 } 386 } 387 return true; 388 } 389 390 void convert_noninflect_cubic_to_quads(const SkPoint p[4], 391 SkScalar toleranceSqd, 392 bool constrainWithinTangents, 393 SkPathPriv::FirstDirection dir, 394 SkTArray<SkPoint, true>* quads, 395 int sublevel = 0) { 396 397 // Notation: Point a is always p[0]. Point b is p[1] unless p[1] == p[0], in which case it is 398 // 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]. 399 400 SkVector ab = p[1] - p[0]; 401 SkVector dc = p[2] - p[3]; 402 403 if (ab.lengthSqd() < SK_ScalarNearlyZero) { 404 if (dc.lengthSqd() < SK_ScalarNearlyZero) { 405 SkPoint* degQuad = quads->push_back_n(3); 406 degQuad[0] = p[0]; 407 degQuad[1] = p[0]; 408 degQuad[2] = p[3]; 409 return; 410 } 411 ab = p[2] - p[0]; 412 } 413 if (dc.lengthSqd() < SK_ScalarNearlyZero) { 414 dc = p[1] - p[3]; 415 } 416 417 // When the ab and cd tangents are degenerate or nearly parallel with vector from d to a the 418 // constraint that the quad point falls between the tangents becomes hard to enforce and we are 419 // likely to hit the max subdivision count. However, in this case the cubic is approaching a 420 // line and the accuracy of the quad point isn't so important. We check if the two middle cubic 421 // control points are very close to the baseline vector. If so then we just pick quadratic 422 // points on the control polygon. 423 424 if (constrainWithinTangents) { 425 SkVector da = p[0] - p[3]; 426 bool doQuads = dc.lengthSqd() < SK_ScalarNearlyZero || 427 ab.lengthSqd() < SK_ScalarNearlyZero; 428 if (!doQuads) { 429 SkScalar invDALengthSqd = da.lengthSqd(); 430 if (invDALengthSqd > SK_ScalarNearlyZero) { 431 invDALengthSqd = SkScalarInvert(invDALengthSqd); 432 // cross(ab, da)^2/length(da)^2 == sqd distance from b to line from d to a. 433 // same goes for point c using vector cd. 434 SkScalar detABSqd = ab.cross(da); 435 detABSqd = SkScalarSquare(detABSqd); 436 SkScalar detDCSqd = dc.cross(da); 437 detDCSqd = SkScalarSquare(detDCSqd); 438 if (detABSqd * invDALengthSqd < toleranceSqd && 439 detDCSqd * invDALengthSqd < toleranceSqd) 440 { 441 doQuads = true; 442 } 443 } 444 } 445 if (doQuads) { 446 SkPoint b = p[0] + ab; 447 SkPoint c = p[3] + dc; 448 SkPoint mid = b + c; 449 mid.scale(SK_ScalarHalf); 450 // Insert two quadratics to cover the case when ab points away from d and/or dc 451 // points away from a. 452 if (SkVector::DotProduct(da, dc) < 0 || SkVector::DotProduct(ab,da) > 0) { 453 SkPoint* qpts = quads->push_back_n(6); 454 qpts[0] = p[0]; 455 qpts[1] = b; 456 qpts[2] = mid; 457 qpts[3] = mid; 458 qpts[4] = c; 459 qpts[5] = p[3]; 460 } else { 461 SkPoint* qpts = quads->push_back_n(3); 462 qpts[0] = p[0]; 463 qpts[1] = mid; 464 qpts[2] = p[3]; 465 } 466 return; 467 } 468 } 469 470 static const SkScalar kLengthScale = 3 * SK_Scalar1 / 2; 471 static const int kMaxSubdivs = 10; 472 473 ab.scale(kLengthScale); 474 dc.scale(kLengthScale); 475 476 // e0 and e1 are extrapolations along vectors ab and dc. 477 SkVector c0 = p[0]; 478 c0 += ab; 479 SkVector c1 = p[3]; 480 c1 += dc; 481 482 SkScalar dSqd = sublevel > kMaxSubdivs ? 0 : c0.distanceToSqd(c1); 483 if (dSqd < toleranceSqd) { 484 SkPoint cAvg = c0; 485 cAvg += c1; 486 cAvg.scale(SK_ScalarHalf); 487 488 bool subdivide = false; 489 490 if (constrainWithinTangents && 491 !is_point_within_cubic_tangents(p[0], ab, dc, p[3], dir, cAvg)) { 492 // choose a new cAvg that is the intersection of the two tangent lines. 493 ab.setOrthog(ab); 494 SkScalar z0 = -ab.dot(p[0]); 495 dc.setOrthog(dc); 496 SkScalar z1 = -dc.dot(p[3]); 497 cAvg.fX = ab.fY * z1 - z0 * dc.fY; 498 cAvg.fY = z0 * dc.fX - ab.fX * z1; 499 SkScalar z = ab.fX * dc.fY - ab.fY * dc.fX; 500 z = SkScalarInvert(z); 501 cAvg.fX *= z; 502 cAvg.fY *= z; 503 if (sublevel <= kMaxSubdivs) { 504 SkScalar d0Sqd = c0.distanceToSqd(cAvg); 505 SkScalar d1Sqd = c1.distanceToSqd(cAvg); 506 // We need to subdivide if d0 + d1 > tolerance but we have the sqd values. We know 507 // the distances and tolerance can't be negative. 508 // (d0 + d1)^2 > toleranceSqd 509 // d0Sqd + 2*d0*d1 + d1Sqd > toleranceSqd 510 SkScalar d0d1 = SkScalarSqrt(d0Sqd * d1Sqd); 511 subdivide = 2 * d0d1 + d0Sqd + d1Sqd > toleranceSqd; 512 } 513 } 514 if (!subdivide) { 515 SkPoint* pts = quads->push_back_n(3); 516 pts[0] = p[0]; 517 pts[1] = cAvg; 518 pts[2] = p[3]; 519 return; 520 } 521 } 522 SkPoint choppedPts[7]; 523 SkChopCubicAtHalf(p, choppedPts); 524 convert_noninflect_cubic_to_quads(choppedPts + 0, 525 toleranceSqd, 526 constrainWithinTangents, 527 dir, 528 quads, 529 sublevel + 1); 530 convert_noninflect_cubic_to_quads(choppedPts + 3, 531 toleranceSqd, 532 constrainWithinTangents, 533 dir, 534 quads, 535 sublevel + 1); 536 } 537 } 538 539 void GrPathUtils::convertCubicToQuads(const SkPoint p[4], 540 SkScalar tolScale, 541 SkTArray<SkPoint, true>* quads) { 542 SkPoint chopped[10]; 543 int count = SkChopCubicAtInflections(p, chopped); 544 545 const SkScalar tolSqd = SkScalarSquare(tolScale); 546 547 for (int i = 0; i < count; ++i) { 548 SkPoint* cubic = chopped + 3*i; 549 // The direction param is ignored if the third param is false. 550 convert_noninflect_cubic_to_quads(cubic, tolSqd, false, 551 SkPathPriv::kCCW_FirstDirection, quads); 552 } 553 } 554 555 void GrPathUtils::convertCubicToQuadsConstrainToTangents(const SkPoint p[4], 556 SkScalar tolScale, 557 SkPathPriv::FirstDirection dir, 558 SkTArray<SkPoint, true>* quads) { 559 SkPoint chopped[10]; 560 int count = SkChopCubicAtInflections(p, chopped); 561 562 const SkScalar tolSqd = SkScalarSquare(tolScale); 563 564 for (int i = 0; i < count; ++i) { 565 SkPoint* cubic = chopped + 3*i; 566 convert_noninflect_cubic_to_quads(cubic, tolSqd, true, dir, quads); 567 } 568 } 569 570 //////////////////////////////////////////////////////////////////////////////// 571 572 /** 573 * Computes an SkMatrix that can find the cubic KLM functionals as follows: 574 * 575 * | ..K.. | | ..kcoeffs.. | 576 * | ..L.. | = | ..lcoeffs.. | * inverse_transpose_power_basis_matrix 577 * | ..M.. | | ..mcoeffs.. | 578 * 579 * 'kcoeffs' are the power basis coefficients to a scalar valued cubic function that returns the 580 * signed distance to line K from a given point on the curve: 581 * 582 * k(t,s) = C(t,s) * K [C(t,s) is defined in the following comment] 583 * 584 * The same applies for lcoeffs and mcoeffs. These are found separately, depending on the type of 585 * curve. There are 4 coefficients but 3 rows in the matrix, so in order to do this calculation the 586 * caller must first remove a specific column of coefficients. 587 * 588 * @return which column of klm coefficients to exclude from the calculation. 589 */ 590 static int calc_inverse_transpose_power_basis_matrix(const SkPoint pts[4], SkMatrix* out) { 591 using SkScalar4 = SkNx<4, SkScalar>; 592 593 // First we convert the bezier coordinates 'pts' to power basis coefficients X,Y,W=[0 0 0 1]. 594 // M3 is the matrix that does this conversion. The homogeneous equation for the cubic becomes: 595 // 596 // | X Y 0 | 597 // C(t,s) = [t^3 t^2*s t*s^2 s^3] * | . . 0 | 598 // | . . 0 | 599 // | . . 1 | 600 // 601 const SkScalar4 M3[3] = {SkScalar4(-1, 3, -3, 1), 602 SkScalar4(3, -6, 3, 0), 603 SkScalar4(-3, 3, 0, 0)}; 604 // 4th column of M3 = SkScalar4(1, 0, 0, 0)}; 605 SkScalar4 X(pts[3].x(), 0, 0, 0); 606 SkScalar4 Y(pts[3].y(), 0, 0, 0); 607 for (int i = 2; i >= 0; --i) { 608 X += M3[i] * pts[i].x(); 609 Y += M3[i] * pts[i].y(); 610 } 611 612 // The matrix is 3x4. In order to invert it, we first need to make it square by throwing out one 613 // of the top three rows. We toss the row that leaves us with the largest absolute determinant. 614 // Since the right column will be [0 0 1], the determinant reduces to x0*y1 - y0*x1. 615 SkScalar absDet[4]; 616 const SkScalar4 DETX1 = SkNx_shuffle<1,0,0,3>(X), DETY1 = SkNx_shuffle<1,0,0,3>(Y); 617 const SkScalar4 DETX2 = SkNx_shuffle<2,2,1,3>(X), DETY2 = SkNx_shuffle<2,2,1,3>(Y); 618 const SkScalar4 DET = DETX1 * DETY2 - DETY1 * DETX2; 619 DET.abs().store(absDet); 620 const int skipRow = absDet[0] > absDet[2] ? (absDet[0] > absDet[1] ? 0 : 1) 621 : (absDet[1] > absDet[2] ? 1 : 2); 622 const SkScalar rdet = 1 / DET[skipRow]; 623 const int row0 = (0 != skipRow) ? 0 : 1; 624 const int row1 = (2 == skipRow) ? 1 : 2; 625 626 // Compute the inverse-transpose of the power basis matrix with the 'skipRow'th row removed. 627 // Since W=[0 0 0 1], it follows that our corresponding solution will be equal to: 628 // 629 // | y1 -x1 x1*y2 - y1*x2 | 630 // 1/det * | -y0 x0 -x0*y2 + y0*x2 | 631 // | 0 0 det | 632 // 633 const SkScalar4 R(rdet, rdet, rdet, 1); 634 X *= R; 635 Y *= R; 636 637 SkScalar x[4], y[4], z[4]; 638 X.store(x); 639 Y.store(y); 640 (X * SkNx_shuffle<3,3,3,3>(Y) - Y * SkNx_shuffle<3,3,3,3>(X)).store(z); 641 642 out->setAll( y[row1], -x[row1], z[row1], 643 -y[row0], x[row0], -z[row0], 644 0, 0, 1); 645 646 return skipRow; 647 } 648 649 static void calc_serp_klm(const SkPoint pts[4], SkScalar tl, SkScalar sl, SkScalar tm, SkScalar sm, 650 SkMatrix* klm) { 651 SkMatrix CIT; 652 int skipCol = calc_inverse_transpose_power_basis_matrix(pts, &CIT); 653 654 SkMatrix klmCoeffs; 655 int col = 0; 656 if (0 != skipCol) { 657 klmCoeffs[0] = 0; 658 klmCoeffs[3] = -sl * sl * sl; 659 klmCoeffs[6] = -sm * sm * sm; 660 ++col; 661 } 662 if (1 != skipCol) { 663 klmCoeffs[col + 0] = sl * sm; 664 klmCoeffs[col + 3] = 3 * sl * sl * tl; 665 klmCoeffs[col + 6] = 3 * sm * sm * tm; 666 ++col; 667 } 668 if (2 != skipCol) { 669 klmCoeffs[col + 0] = -tl * sm - tm * sl; 670 klmCoeffs[col + 3] = -3 * sl * tl * tl; 671 klmCoeffs[col + 6] = -3 * sm * tm * tm; 672 ++col; 673 } 674 675 SkASSERT(2 == col); 676 klmCoeffs[2] = tl * tm; 677 klmCoeffs[5] = tl * tl * tl; 678 klmCoeffs[8] = tm * tm * tm; 679 680 klm->setConcat(klmCoeffs, CIT); 681 } 682 683 static void calc_loop_klm(const SkPoint pts[4], SkScalar td, SkScalar sd, SkScalar te, SkScalar se, 684 SkMatrix* klm) { 685 SkMatrix CIT; 686 int skipCol = calc_inverse_transpose_power_basis_matrix(pts, &CIT); 687 688 const SkScalar tdse = td * se; 689 const SkScalar tesd = te * sd; 690 691 SkMatrix klmCoeffs; 692 int col = 0; 693 if (0 != skipCol) { 694 klmCoeffs[0] = 0; 695 klmCoeffs[3] = -sd * sd * se; 696 klmCoeffs[6] = -se * se * sd; 697 ++col; 698 } 699 if (1 != skipCol) { 700 klmCoeffs[col + 0] = sd * se; 701 klmCoeffs[col + 3] = sd * (2 * tdse + tesd); 702 klmCoeffs[col + 6] = se * (2 * tesd + tdse); 703 ++col; 704 } 705 if (2 != skipCol) { 706 klmCoeffs[col + 0] = -tdse - tesd; 707 klmCoeffs[col + 3] = -td * (tdse + 2 * tesd); 708 klmCoeffs[col + 6] = -te * (tesd + 2 * tdse); 709 ++col; 710 } 711 712 SkASSERT(2 == col); 713 klmCoeffs[2] = td * te; 714 klmCoeffs[5] = td * td * te; 715 klmCoeffs[8] = te * te * td; 716 717 klm->setConcat(klmCoeffs, CIT); 718 } 719 720 // For the case when we have a cusp at a parameter value of infinity (discr == 0, d1 == 0). 721 static void calc_inf_cusp_klm(const SkPoint pts[4], SkScalar tn, SkScalar sn, SkMatrix* klm) { 722 SkMatrix CIT; 723 int skipCol = calc_inverse_transpose_power_basis_matrix(pts, &CIT); 724 725 SkMatrix klmCoeffs; 726 int col = 0; 727 if (0 != skipCol) { 728 klmCoeffs[0] = 0; 729 klmCoeffs[3] = -sn * sn * sn; 730 ++col; 731 } 732 if (1 != skipCol) { 733 klmCoeffs[col + 0] = 0; 734 klmCoeffs[col + 3] = 3 * sn * sn * tn; 735 ++col; 736 } 737 if (2 != skipCol) { 738 klmCoeffs[col + 0] = -sn; 739 klmCoeffs[col + 3] = -3 * sn * tn * tn; 740 ++col; 741 } 742 743 SkASSERT(2 == col); 744 klmCoeffs[2] = tn; 745 klmCoeffs[5] = tn * tn * tn; 746 747 klmCoeffs[6] = 0; 748 klmCoeffs[7] = 0; 749 klmCoeffs[8] = 1; 750 751 klm->setConcat(klmCoeffs, CIT); 752 } 753 754 // For the case when a cubic bezier is actually a quadratic. We duplicate k in l so that the 755 // implicit becomes: 756 // 757 // k^3 - l*m == k^3 - l*k == k * (k^2 - l) 758 // 759 // In the quadratic case we can simply assign fixed values at each control point: 760 // 761 // | ..K.. | | pts[0] pts[1] pts[2] pts[3] | | 0 1/3 2/3 1 | 762 // | ..L.. | * | . . . . | == | 0 0 1/3 1 | 763 // | ..K.. | | 1 1 1 1 | | 0 1/3 2/3 1 | 764 // 765 static void calc_quadratic_klm(const SkPoint pts[4], double d3, SkMatrix* klm) { 766 SkMatrix klmAtPts; 767 klmAtPts.setAll(0, 1.f/3, 1, 768 0, 0, 1, 769 0, 1.f/3, 1); 770 771 SkMatrix inversePts; 772 inversePts.setAll(pts[0].x(), pts[1].x(), pts[3].x(), 773 pts[0].y(), pts[1].y(), pts[3].y(), 774 1, 1, 1); 775 SkAssertResult(inversePts.invert(&inversePts)); 776 777 klm->setConcat(klmAtPts, inversePts); 778 779 // If d3 > 0 we need to flip the orientation of our curve 780 // This is done by negating the k and l values 781 if (d3 > 0) { 782 klm->postScale(-1, -1); 783 } 784 } 785 786 // For the case when a cubic bezier is actually a line. We set K=0, L=1, M=-line, which results in 787 // the following implicit: 788 // 789 // k^3 - l*m == 0^3 - 1*(-line) == -(-line) == line 790 // 791 static void calc_line_klm(const SkPoint pts[4], SkMatrix* klm) { 792 SkScalar ny = pts[0].x() - pts[3].x(); 793 SkScalar nx = pts[3].y() - pts[0].y(); 794 SkScalar k = nx * pts[0].x() + ny * pts[0].y(); 795 klm->setAll( 0, 0, 0, 796 0, 0, 1, 797 -nx, -ny, k); 798 } 799 800 SkCubicType GrPathUtils::getCubicKLM(const SkPoint src[4], SkMatrix* klm, double t[2], 801 double s[2]) { 802 double d[4]; 803 SkCubicType type = SkClassifyCubic(src, t, s, d); 804 805 const SkScalar tt[2] = {static_cast<SkScalar>(t[0]), static_cast<SkScalar>(t[1])}; 806 const SkScalar ss[2] = {static_cast<SkScalar>(s[0]), static_cast<SkScalar>(s[1])}; 807 808 switch (type) { 809 case SkCubicType::kSerpentine: 810 calc_serp_klm(src, tt[0], ss[0], tt[1], ss[1], klm); 811 break; 812 case SkCubicType::kLoop: 813 calc_loop_klm(src, tt[0], ss[0], tt[1], ss[1], klm); 814 break; 815 case SkCubicType::kLocalCusp: 816 calc_serp_klm(src, tt[0], ss[0], tt[1], ss[1], klm); 817 break; 818 case SkCubicType::kCuspAtInfinity: 819 calc_inf_cusp_klm(src, tt[0], ss[0], klm); 820 break; 821 case SkCubicType::kQuadratic: 822 calc_quadratic_klm(src, d[3], klm); 823 break; 824 case SkCubicType::kLineOrPoint: 825 calc_line_klm(src, klm); 826 break; 827 } 828 829 return type; 830 } 831 832 int GrPathUtils::chopCubicAtLoopIntersection(const SkPoint src[4], SkPoint dst[10], SkMatrix* klm, 833 int* loopIndex) { 834 SkSTArray<2, SkScalar> chops; 835 *loopIndex = -1; 836 837 double t[2], s[2]; 838 if (SkCubicType::kLoop == GrPathUtils::getCubicKLM(src, klm, t, s)) { 839 SkScalar t0 = static_cast<SkScalar>(t[0] / s[0]); 840 SkScalar t1 = static_cast<SkScalar>(t[1] / s[1]); 841 SkASSERT(t0 <= t1); // Technically t0 != t1 in a loop, but there may be FP error. 842 843 if (t0 < 1 && t1 > 0) { 844 *loopIndex = 0; 845 if (t0 > 0) { 846 chops.push_back(t0); 847 *loopIndex = 1; 848 } 849 if (t1 < 1) { 850 chops.push_back(t1); 851 *loopIndex = chops.count() - 1; 852 } 853 } 854 } 855 856 SkChopCubicAt(src, dst, chops.begin(), chops.count()); 857 return chops.count() + 1; 858 } 859