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 "SkGeometry.h" 12 13SkScalar 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 // scale when mapping to screen coordinates. 18 SkScalar stretch = viewM.getMaxScale(); 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 36static const int MAX_POINTS_PER_CURVE = 1 << 10; 37static const SkScalar gMinCurveTol = 0.0001f; 38 39uint32_t GrPathUtils::quadraticPointCount(const SkPoint 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 SkTMin(pow2, MAX_POINTS_PER_CURVE); 63 } 64} 65 66uint32_t GrPathUtils::generateQuadraticPoints(const SkPoint& p0, 67 const SkPoint& p1, 68 const SkPoint& p2, 69 SkScalar tolSqd, 70 SkPoint** 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 SkPoint 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 SkPoint 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 91uint32_t GrPathUtils::cubicPointCount(const SkPoint points[], 92 SkScalar tol) { 93 if (tol < gMinCurveTol) { 94 tol = gMinCurveTol; 95 } 96 SkASSERT(tol > 0); 97 98 SkScalar d = SkTMax( 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 SkTMin(pow2, MAX_POINTS_PER_CURVE); 114 } 115} 116 117uint32_t GrPathUtils::generateCubicPoints(const SkPoint& p0, 118 const SkPoint& p1, 119 const SkPoint& p2, 120 const SkPoint& p3, 121 SkScalar tolSqd, 122 SkPoint** 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 SkPoint 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 SkPoint 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 SkPoint 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 147int 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 SkPoint 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 189void GrPathUtils::QuadUVMatrix::set(const SkPoint 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 SkVector 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, SkPoint::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)) 311void 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 339namespace { 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 347bool 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 381void 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 degenerate or nearly parallel with vector from d to a the 409 // constraint that the quad point falls between the tangents becomes hard to enforce and we are 410 // likely to hit the max subdivision count. However, in this case the cubic is approaching a 411 // line and the accuracy of the quad point isn't so important. We check if the two middle cubic 412 // control points are very close to the baseline vector. If so then we just pick quadratic 413 // points on the control polygon. 414 415 if (constrainWithinTangents) { 416 SkVector da = p[0] - p[3]; 417 bool doQuads = dc.lengthSqd() < SK_ScalarNearlyZero || 418 ab.lengthSqd() < SK_ScalarNearlyZero; 419 if (!doQuads) { 420 SkScalar invDALengthSqd = da.lengthSqd(); 421 if (invDALengthSqd > SK_ScalarNearlyZero) { 422 invDALengthSqd = SkScalarInvert(invDALengthSqd); 423 // cross(ab, da)^2/length(da)^2 == sqd distance from b to line from d to a. 424 // same goes for point c using vector cd. 425 SkScalar detABSqd = ab.cross(da); 426 detABSqd = SkScalarSquare(detABSqd); 427 SkScalar detDCSqd = dc.cross(da); 428 detDCSqd = SkScalarSquare(detDCSqd); 429 if (SkScalarMul(detABSqd, invDALengthSqd) < toleranceSqd && 430 SkScalarMul(detDCSqd, invDALengthSqd) < toleranceSqd) { 431 doQuads = true; 432 } 433 } 434 } 435 if (doQuads) { 436 SkPoint b = p[0] + ab; 437 SkPoint c = p[3] + dc; 438 SkPoint mid = b + c; 439 mid.scale(SK_ScalarHalf); 440 // Insert two quadratics to cover the case when ab points away from d and/or dc 441 // points away from a. 442 if (SkVector::DotProduct(da, dc) < 0 || SkVector::DotProduct(ab,da) > 0) { 443 SkPoint* qpts = quads->push_back_n(6); 444 qpts[0] = p[0]; 445 qpts[1] = b; 446 qpts[2] = mid; 447 qpts[3] = mid; 448 qpts[4] = c; 449 qpts[5] = p[3]; 450 } else { 451 SkPoint* qpts = quads->push_back_n(3); 452 qpts[0] = p[0]; 453 qpts[1] = mid; 454 qpts[2] = p[3]; 455 } 456 return; 457 } 458 } 459 460 static const SkScalar kLengthScale = 3 * SK_Scalar1 / 2; 461 static const int kMaxSubdivs = 10; 462 463 ab.scale(kLengthScale); 464 dc.scale(kLengthScale); 465 466 // e0 and e1 are extrapolations along vectors ab and dc. 467 SkVector c0 = p[0]; 468 c0 += ab; 469 SkVector c1 = p[3]; 470 c1 += dc; 471 472 SkScalar dSqd = sublevel > kMaxSubdivs ? 0 : c0.distanceToSqd(c1); 473 if (dSqd < toleranceSqd) { 474 SkPoint cAvg = c0; 475 cAvg += c1; 476 cAvg.scale(SK_ScalarHalf); 477 478 bool subdivide = false; 479 480 if (constrainWithinTangents && 481 !is_point_within_cubic_tangents(p[0], ab, dc, p[3], dir, cAvg)) { 482 // choose a new cAvg that is the intersection of the two tangent lines. 483 ab.setOrthog(ab); 484 SkScalar z0 = -ab.dot(p[0]); 485 dc.setOrthog(dc); 486 SkScalar z1 = -dc.dot(p[3]); 487 cAvg.fX = SkScalarMul(ab.fY, z1) - SkScalarMul(z0, dc.fY); 488 cAvg.fY = SkScalarMul(z0, dc.fX) - SkScalarMul(ab.fX, z1); 489 SkScalar z = SkScalarMul(ab.fX, dc.fY) - SkScalarMul(ab.fY, dc.fX); 490 z = SkScalarInvert(z); 491 cAvg.fX *= z; 492 cAvg.fY *= z; 493 if (sublevel <= kMaxSubdivs) { 494 SkScalar d0Sqd = c0.distanceToSqd(cAvg); 495 SkScalar d1Sqd = c1.distanceToSqd(cAvg); 496 // We need to subdivide if d0 + d1 > tolerance but we have the sqd values. We know 497 // the distances and tolerance can't be negative. 498 // (d0 + d1)^2 > toleranceSqd 499 // d0Sqd + 2*d0*d1 + d1Sqd > toleranceSqd 500 SkScalar d0d1 = SkScalarSqrt(SkScalarMul(d0Sqd, d1Sqd)); 501 subdivide = 2 * d0d1 + d0Sqd + d1Sqd > toleranceSqd; 502 } 503 } 504 if (!subdivide) { 505 SkPoint* pts = quads->push_back_n(3); 506 pts[0] = p[0]; 507 pts[1] = cAvg; 508 pts[2] = p[3]; 509 return; 510 } 511 } 512 SkPoint choppedPts[7]; 513 SkChopCubicAtHalf(p, choppedPts); 514 convert_noninflect_cubic_to_quads(choppedPts + 0, 515 toleranceSqd, 516 constrainWithinTangents, 517 dir, 518 quads, 519 sublevel + 1); 520 convert_noninflect_cubic_to_quads(choppedPts + 3, 521 toleranceSqd, 522 constrainWithinTangents, 523 dir, 524 quads, 525 sublevel + 1); 526} 527} 528 529void GrPathUtils::convertCubicToQuads(const SkPoint p[4], 530 SkScalar tolScale, 531 bool constrainWithinTangents, 532 SkPath::Direction dir, 533 SkTArray<SkPoint, true>* quads) { 534 SkPoint chopped[10]; 535 int count = SkChopCubicAtInflections(p, chopped); 536 537 // base tolerance is 1 pixel. 538 static const SkScalar kTolerance = SK_Scalar1; 539 const SkScalar tolSqd = SkScalarSquare(SkScalarMul(tolScale, kTolerance)); 540 541 for (int i = 0; i < count; ++i) { 542 SkPoint* cubic = chopped + 3*i; 543 convert_noninflect_cubic_to_quads(cubic, tolSqd, constrainWithinTangents, dir, quads); 544 } 545 546} 547 548//////////////////////////////////////////////////////////////////////////////// 549 550enum CubicType { 551 kSerpentine_CubicType, 552 kCusp_CubicType, 553 kLoop_CubicType, 554 kQuadratic_CubicType, 555 kLine_CubicType, 556 kPoint_CubicType 557}; 558 559// discr(I) = d0^2 * (3*d1^2 - 4*d0*d2) 560// Classification: 561// discr(I) > 0 Serpentine 562// discr(I) = 0 Cusp 563// discr(I) < 0 Loop 564// d0 = d1 = 0 Quadratic 565// d0 = d1 = d2 = 0 Line 566// p0 = p1 = p2 = p3 Point 567static CubicType classify_cubic(const SkPoint p[4], const SkScalar d[3]) { 568 if (p[0] == p[1] && p[0] == p[2] && p[0] == p[3]) { 569 return kPoint_CubicType; 570 } 571 const SkScalar discr = d[0] * d[0] * (3.f * d[1] * d[1] - 4.f * d[0] * d[2]); 572 if (discr > SK_ScalarNearlyZero) { 573 return kSerpentine_CubicType; 574 } else if (discr < -SK_ScalarNearlyZero) { 575 return kLoop_CubicType; 576 } else { 577 if (0.f == d[0] && 0.f == d[1]) { 578 return (0.f == d[2] ? kLine_CubicType : kQuadratic_CubicType); 579 } else { 580 return kCusp_CubicType; 581 } 582 } 583} 584 585// Assumes the third component of points is 1. 586// Calcs p0 . (p1 x p2) 587static SkScalar calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) { 588 const SkScalar xComp = p0.fX * (p1.fY - p2.fY); 589 const SkScalar yComp = p0.fY * (p2.fX - p1.fX); 590 const SkScalar wComp = p1.fX * p2.fY - p1.fY * p2.fX; 591 return (xComp + yComp + wComp); 592} 593 594// Solves linear system to extract klm 595// P.K = k (similarly for l, m) 596// Where P is matrix of control points 597// K is coefficients for the line K 598// k is vector of values of K evaluated at the control points 599// Solving for K, thus K = P^(-1) . k 600static void calc_cubic_klm(const SkPoint p[4], const SkScalar controlK[4], 601 const SkScalar controlL[4], const SkScalar controlM[4], 602 SkScalar k[3], SkScalar l[3], SkScalar m[3]) { 603 SkMatrix matrix; 604 matrix.setAll(p[0].fX, p[0].fY, 1.f, 605 p[1].fX, p[1].fY, 1.f, 606 p[2].fX, p[2].fY, 1.f); 607 SkMatrix inverse; 608 if (matrix.invert(&inverse)) { 609 inverse.mapHomogeneousPoints(k, controlK, 1); 610 inverse.mapHomogeneousPoints(l, controlL, 1); 611 inverse.mapHomogeneousPoints(m, controlM, 1); 612 } 613 614} 615 616static void set_serp_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { 617 SkScalar tempSqrt = SkScalarSqrt(9.f * d[1] * d[1] - 12.f * d[0] * d[2]); 618 SkScalar ls = 3.f * d[1] - tempSqrt; 619 SkScalar lt = 6.f * d[0]; 620 SkScalar ms = 3.f * d[1] + tempSqrt; 621 SkScalar mt = 6.f * d[0]; 622 623 k[0] = ls * ms; 624 k[1] = (3.f * ls * ms - ls * mt - lt * ms) / 3.f; 625 k[2] = (lt * (mt - 2.f * ms) + ls * (3.f * ms - 2.f * mt)) / 3.f; 626 k[3] = (lt - ls) * (mt - ms); 627 628 l[0] = ls * ls * ls; 629 const SkScalar lt_ls = lt - ls; 630 l[1] = ls * ls * lt_ls * -1.f; 631 l[2] = lt_ls * lt_ls * ls; 632 l[3] = -1.f * lt_ls * lt_ls * lt_ls; 633 634 m[0] = ms * ms * ms; 635 const SkScalar mt_ms = mt - ms; 636 m[1] = ms * ms * mt_ms * -1.f; 637 m[2] = mt_ms * mt_ms * ms; 638 m[3] = -1.f * mt_ms * mt_ms * mt_ms; 639 640 // If d0 < 0 we need to flip the orientation of our curve 641 // This is done by negating the k and l values 642 // We want negative distance values to be on the inside 643 if ( d[0] > 0) { 644 for (int i = 0; i < 4; ++i) { 645 k[i] = -k[i]; 646 l[i] = -l[i]; 647 } 648 } 649} 650 651static void set_loop_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { 652 SkScalar tempSqrt = SkScalarSqrt(4.f * d[0] * d[2] - 3.f * d[1] * d[1]); 653 SkScalar ls = d[1] - tempSqrt; 654 SkScalar lt = 2.f * d[0]; 655 SkScalar ms = d[1] + tempSqrt; 656 SkScalar mt = 2.f * d[0]; 657 658 k[0] = ls * ms; 659 k[1] = (3.f * ls*ms - ls * mt - lt * ms) / 3.f; 660 k[2] = (lt * (mt - 2.f * ms) + ls * (3.f * ms - 2.f * mt)) / 3.f; 661 k[3] = (lt - ls) * (mt - ms); 662 663 l[0] = ls * ls * ms; 664 l[1] = (ls * (ls * (mt - 3.f * ms) + 2.f * lt * ms))/-3.f; 665 l[2] = ((lt - ls) * (ls * (2.f * mt - 3.f * ms) + lt * ms))/3.f; 666 l[3] = -1.f * (lt - ls) * (lt - ls) * (mt - ms); 667 668 m[0] = ls * ms * ms; 669 m[1] = (ms * (ls * (2.f * mt - 3.f * ms) + lt * ms))/-3.f; 670 m[2] = ((mt - ms) * (ls * (mt - 3.f * ms) + 2.f * lt * ms))/3.f; 671 m[3] = -1.f * (lt - ls) * (mt - ms) * (mt - ms); 672 673 674 // If (d0 < 0 && sign(k1) > 0) || (d0 > 0 && sign(k1) < 0), 675 // we need to flip the orientation of our curve. 676 // This is done by negating the k and l values 677 if ( (d[0] < 0 && k[1] > 0) || (d[0] > 0 && k[1] < 0)) { 678 for (int i = 0; i < 4; ++i) { 679 k[i] = -k[i]; 680 l[i] = -l[i]; 681 } 682 } 683} 684 685static void set_cusp_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { 686 const SkScalar ls = d[2]; 687 const SkScalar lt = 3.f * d[1]; 688 689 k[0] = ls; 690 k[1] = ls - lt / 3.f; 691 k[2] = ls - 2.f * lt / 3.f; 692 k[3] = ls - lt; 693 694 l[0] = ls * ls * ls; 695 const SkScalar ls_lt = ls - lt; 696 l[1] = ls * ls * ls_lt; 697 l[2] = ls_lt * ls_lt * ls; 698 l[3] = ls_lt * ls_lt * ls_lt; 699 700 m[0] = 1.f; 701 m[1] = 1.f; 702 m[2] = 1.f; 703 m[3] = 1.f; 704} 705 706// For the case when a cubic is actually a quadratic 707// M = 708// 0 0 0 709// 1/3 0 1/3 710// 2/3 1/3 2/3 711// 1 1 1 712static void set_quadratic_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { 713 k[0] = 0.f; 714 k[1] = 1.f/3.f; 715 k[2] = 2.f/3.f; 716 k[3] = 1.f; 717 718 l[0] = 0.f; 719 l[1] = 0.f; 720 l[2] = 1.f/3.f; 721 l[3] = 1.f; 722 723 m[0] = 0.f; 724 m[1] = 1.f/3.f; 725 m[2] = 2.f/3.f; 726 m[3] = 1.f; 727 728 // If d2 < 0 we need to flip the orientation of our curve 729 // This is done by negating the k and l values 730 if ( d[2] > 0) { 731 for (int i = 0; i < 4; ++i) { 732 k[i] = -k[i]; 733 l[i] = -l[i]; 734 } 735 } 736} 737 738// Calc coefficients of I(s,t) where roots of I are inflection points of curve 739// I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2) 740// d0 = a1 - 2*a2+3*a3 741// d1 = -a2 + 3*a3 742// d2 = 3*a3 743// a1 = p0 . (p3 x p2) 744// a2 = p1 . (p0 x p3) 745// a3 = p2 . (p1 x p0) 746// Places the values of d1, d2, d3 in array d passed in 747static void calc_cubic_inflection_func(const SkPoint p[4], SkScalar d[3]) { 748 SkScalar a1 = calc_dot_cross_cubic(p[0], p[3], p[2]); 749 SkScalar a2 = calc_dot_cross_cubic(p[1], p[0], p[3]); 750 SkScalar a3 = calc_dot_cross_cubic(p[2], p[1], p[0]); 751 752 // need to scale a's or values in later calculations will grow to high 753 SkScalar max = SkScalarAbs(a1); 754 max = SkMaxScalar(max, SkScalarAbs(a2)); 755 max = SkMaxScalar(max, SkScalarAbs(a3)); 756 max = 1.f/max; 757 a1 = a1 * max; 758 a2 = a2 * max; 759 a3 = a3 * max; 760 761 d[2] = 3.f * a3; 762 d[1] = d[2] - a2; 763 d[0] = d[1] - a2 + a1; 764} 765 766int GrPathUtils::chopCubicAtLoopIntersection(const SkPoint src[4], SkPoint dst[10], SkScalar klm[9], 767 SkScalar klm_rev[3]) { 768 // Variable to store the two parametric values at the loop double point 769 SkScalar smallS = 0.f; 770 SkScalar largeS = 0.f; 771 772 SkScalar d[3]; 773 calc_cubic_inflection_func(src, d); 774 775 CubicType cType = classify_cubic(src, d); 776 777 int chop_count = 0; 778 if (kLoop_CubicType == cType) { 779 SkScalar tempSqrt = SkScalarSqrt(4.f * d[0] * d[2] - 3.f * d[1] * d[1]); 780 SkScalar ls = d[1] - tempSqrt; 781 SkScalar lt = 2.f * d[0]; 782 SkScalar ms = d[1] + tempSqrt; 783 SkScalar mt = 2.f * d[0]; 784 ls = ls / lt; 785 ms = ms / mt; 786 // need to have t values sorted since this is what is expected by SkChopCubicAt 787 if (ls <= ms) { 788 smallS = ls; 789 largeS = ms; 790 } else { 791 smallS = ms; 792 largeS = ls; 793 } 794 795 SkScalar chop_ts[2]; 796 if (smallS > 0.f && smallS < 1.f) { 797 chop_ts[chop_count++] = smallS; 798 } 799 if (largeS > 0.f && largeS < 1.f) { 800 chop_ts[chop_count++] = largeS; 801 } 802 if(dst) { 803 SkChopCubicAt(src, dst, chop_ts, chop_count); 804 } 805 } else { 806 if (dst) { 807 memcpy(dst, src, sizeof(SkPoint) * 4); 808 } 809 } 810 811 if (klm && klm_rev) { 812 // Set klm_rev to to match the sub_section of cubic that needs to have its orientation 813 // flipped. This will always be the section that is the "loop" 814 if (2 == chop_count) { 815 klm_rev[0] = 1.f; 816 klm_rev[1] = -1.f; 817 klm_rev[2] = 1.f; 818 } else if (1 == chop_count) { 819 if (smallS < 0.f) { 820 klm_rev[0] = -1.f; 821 klm_rev[1] = 1.f; 822 } else { 823 klm_rev[0] = 1.f; 824 klm_rev[1] = -1.f; 825 } 826 } else { 827 if (smallS < 0.f && largeS > 1.f) { 828 klm_rev[0] = -1.f; 829 } else { 830 klm_rev[0] = 1.f; 831 } 832 } 833 SkScalar controlK[4]; 834 SkScalar controlL[4]; 835 SkScalar controlM[4]; 836 837 if (kSerpentine_CubicType == cType || (kCusp_CubicType == cType && 0.f != d[0])) { 838 set_serp_klm(d, controlK, controlL, controlM); 839 } else if (kLoop_CubicType == cType) { 840 set_loop_klm(d, controlK, controlL, controlM); 841 } else if (kCusp_CubicType == cType) { 842 SkASSERT(0.f == d[0]); 843 set_cusp_klm(d, controlK, controlL, controlM); 844 } else if (kQuadratic_CubicType == cType) { 845 set_quadratic_klm(d, controlK, controlL, controlM); 846 } 847 848 calc_cubic_klm(src, controlK, controlL, controlM, klm, &klm[3], &klm[6]); 849 } 850 return chop_count + 1; 851} 852 853void GrPathUtils::getCubicKLM(const SkPoint p[4], SkScalar klm[9]) { 854 SkScalar d[3]; 855 calc_cubic_inflection_func(p, d); 856 857 CubicType cType = classify_cubic(p, d); 858 859 SkScalar controlK[4]; 860 SkScalar controlL[4]; 861 SkScalar controlM[4]; 862 863 if (kSerpentine_CubicType == cType || (kCusp_CubicType == cType && 0.f != d[0])) { 864 set_serp_klm(d, controlK, controlL, controlM); 865 } else if (kLoop_CubicType == cType) { 866 set_loop_klm(d, controlK, controlL, controlM); 867 } else if (kCusp_CubicType == cType) { 868 SkASSERT(0.f == d[0]); 869 set_cusp_klm(d, controlK, controlL, controlM); 870 } else if (kQuadratic_CubicType == cType) { 871 set_quadratic_klm(d, controlK, controlL, controlM); 872 } 873 874 calc_cubic_klm(p, controlK, controlL, controlM, klm, &klm[3], &klm[6]); 875} 876