GrPathUtils.cpp revision 858638d8a5bef8f9940ccec2346a9bcc5f804979
1/* 2 * Copyright 2011 Google Inc. 3 * 4 * Use of this source code is governed by a BSD-style license that can be 5 * found in the LICENSE file. 6 */ 7 8#include "GrPathUtils.h" 9 10#include "GrPoint.h" 11#include "SkGeometry.h" 12 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 // stretch when mapping to screen coordinates. 18 SkScalar stretch = viewM.getMaxStretch(); 19 SkScalar srcTol = devTol; 20 21 if (stretch < 0) { 22 // take worst case mapRadius amoung four corners. 23 // (less than perfect) 24 for (int i = 0; i < 4; ++i) { 25 SkMatrix mat; 26 mat.setTranslate((i % 2) ? pathBounds.fLeft : pathBounds.fRight, 27 (i < 2) ? pathBounds.fTop : pathBounds.fBottom); 28 mat.postConcat(viewM); 29 stretch = SkMaxScalar(stretch, mat.mapRadius(SK_Scalar1)); 30 } 31 } 32 srcTol = SkScalarDiv(srcTol, stretch); 33 return srcTol; 34} 35 36static const int MAX_POINTS_PER_CURVE = 1 << 10; 37static const SkScalar gMinCurveTol = SkFloatToScalar(0.0001f); 38 39uint32_t GrPathUtils::quadraticPointCount(const GrPoint points[], 40 SkScalar tol) { 41 if (tol < gMinCurveTol) { 42 tol = gMinCurveTol; 43 } 44 SkASSERT(tol > 0); 45 46 SkScalar d = points[1].distanceToLineSegmentBetween(points[0], points[2]); 47 if (d <= tol) { 48 return 1; 49 } else { 50 // Each time we subdivide, d should be cut in 4. So we need to 51 // subdivide x = log4(d/tol) times. x subdivisions creates 2^(x) 52 // points. 53 // 2^(log4(x)) = sqrt(x); 54 int temp = SkScalarCeil(SkScalarSqrt(SkScalarDiv(d, tol))); 55 int pow2 = GrNextPow2(temp); 56 // Because of NaNs & INFs we can wind up with a degenerate temp 57 // such that pow2 comes out negative. Also, our point generator 58 // will always output at least one pt. 59 if (pow2 < 1) { 60 pow2 = 1; 61 } 62 return GrMin(pow2, MAX_POINTS_PER_CURVE); 63 } 64} 65 66uint32_t GrPathUtils::generateQuadraticPoints(const GrPoint& p0, 67 const GrPoint& p1, 68 const GrPoint& p2, 69 SkScalar tolSqd, 70 GrPoint** points, 71 uint32_t pointsLeft) { 72 if (pointsLeft < 2 || 73 (p1.distanceToLineSegmentBetweenSqd(p0, p2)) < tolSqd) { 74 (*points)[0] = p2; 75 *points += 1; 76 return 1; 77 } 78 79 GrPoint q[] = { 80 { SkScalarAve(p0.fX, p1.fX), SkScalarAve(p0.fY, p1.fY) }, 81 { SkScalarAve(p1.fX, p2.fX), SkScalarAve(p1.fY, p2.fY) }, 82 }; 83 GrPoint r = { SkScalarAve(q[0].fX, q[1].fX), SkScalarAve(q[0].fY, q[1].fY) }; 84 85 pointsLeft >>= 1; 86 uint32_t a = generateQuadraticPoints(p0, q[0], r, tolSqd, points, pointsLeft); 87 uint32_t b = generateQuadraticPoints(r, q[1], p2, tolSqd, points, pointsLeft); 88 return a + b; 89} 90 91uint32_t GrPathUtils::cubicPointCount(const GrPoint points[], 92 SkScalar tol) { 93 if (tol < gMinCurveTol) { 94 tol = gMinCurveTol; 95 } 96 SkASSERT(tol > 0); 97 98 SkScalar d = GrMax( 99 points[1].distanceToLineSegmentBetweenSqd(points[0], points[3]), 100 points[2].distanceToLineSegmentBetweenSqd(points[0], points[3])); 101 d = SkScalarSqrt(d); 102 if (d <= tol) { 103 return 1; 104 } else { 105 int temp = SkScalarCeil(SkScalarSqrt(SkScalarDiv(d, tol))); 106 int pow2 = GrNextPow2(temp); 107 // Because of NaNs & INFs we can wind up with a degenerate temp 108 // such that pow2 comes out negative. Also, our point generator 109 // will always output at least one pt. 110 if (pow2 < 1) { 111 pow2 = 1; 112 } 113 return GrMin(pow2, MAX_POINTS_PER_CURVE); 114 } 115} 116 117uint32_t GrPathUtils::generateCubicPoints(const GrPoint& p0, 118 const GrPoint& p1, 119 const GrPoint& p2, 120 const GrPoint& p3, 121 SkScalar tolSqd, 122 GrPoint** points, 123 uint32_t pointsLeft) { 124 if (pointsLeft < 2 || 125 (p1.distanceToLineSegmentBetweenSqd(p0, p3) < tolSqd && 126 p2.distanceToLineSegmentBetweenSqd(p0, p3) < tolSqd)) { 127 (*points)[0] = p3; 128 *points += 1; 129 return 1; 130 } 131 GrPoint q[] = { 132 { SkScalarAve(p0.fX, p1.fX), SkScalarAve(p0.fY, p1.fY) }, 133 { SkScalarAve(p1.fX, p2.fX), SkScalarAve(p1.fY, p2.fY) }, 134 { SkScalarAve(p2.fX, p3.fX), SkScalarAve(p2.fY, p3.fY) } 135 }; 136 GrPoint r[] = { 137 { SkScalarAve(q[0].fX, q[1].fX), SkScalarAve(q[0].fY, q[1].fY) }, 138 { SkScalarAve(q[1].fX, q[2].fX), SkScalarAve(q[1].fY, q[2].fY) } 139 }; 140 GrPoint s = { SkScalarAve(r[0].fX, r[1].fX), SkScalarAve(r[0].fY, r[1].fY) }; 141 pointsLeft >>= 1; 142 uint32_t a = generateCubicPoints(p0, q[0], r[0], s, tolSqd, points, pointsLeft); 143 uint32_t b = generateCubicPoints(s, r[1], q[2], p3, tolSqd, points, pointsLeft); 144 return a + b; 145} 146 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 GrPoint pts[4]; 163 while ((verb = iter.next(pts)) != SkPath::kDone_Verb) { 164 165 switch (verb) { 166 case SkPath::kLine_Verb: 167 pointCount += 1; 168 break; 169 case SkPath::kQuad_Verb: 170 pointCount += quadraticPointCount(pts, tol); 171 break; 172 case SkPath::kCubic_Verb: 173 pointCount += cubicPointCount(pts, tol); 174 break; 175 case SkPath::kMove_Verb: 176 pointCount += 1; 177 if (!first) { 178 ++(*subpaths); 179 } 180 break; 181 default: 182 break; 183 } 184 first = false; 185 } 186 return pointCount; 187} 188 189void GrPathUtils::QuadUVMatrix::set(const GrPoint qPts[3]) { 190 // can't make this static, no cons :( 191 SkMatrix UVpts; 192#ifndef SK_SCALAR_IS_FLOAT 193 GrCrash("Expected scalar is float."); 194#endif 195 SkMatrix m; 196 // We want M such that M * xy_pt = uv_pt 197 // We know M * control_pts = [0 1/2 1] 198 // [0 0 1] 199 // [1 1 1] 200 // We invert the control pt matrix and post concat to both sides to get M. 201 UVpts.setAll(0, SK_ScalarHalf, SK_Scalar1, 202 0, 0, SK_Scalar1, 203 SkScalarToPersp(SK_Scalar1), 204 SkScalarToPersp(SK_Scalar1), 205 SkScalarToPersp(SK_Scalar1)); 206 m.setAll(qPts[0].fX, qPts[1].fX, qPts[2].fX, 207 qPts[0].fY, qPts[1].fY, qPts[2].fY, 208 SkScalarToPersp(SK_Scalar1), 209 SkScalarToPersp(SK_Scalar1), 210 SkScalarToPersp(SK_Scalar1)); 211 if (!m.invert(&m)) { 212 // The quad is degenerate. Hopefully this is rare. Find the pts that are 213 // farthest apart to compute a line (unless it is really a pt). 214 SkScalar maxD = qPts[0].distanceToSqd(qPts[1]); 215 int maxEdge = 0; 216 SkScalar d = qPts[1].distanceToSqd(qPts[2]); 217 if (d > maxD) { 218 maxD = d; 219 maxEdge = 1; 220 } 221 d = qPts[2].distanceToSqd(qPts[0]); 222 if (d > maxD) { 223 maxD = d; 224 maxEdge = 2; 225 } 226 // We could have a tolerance here, not sure if it would improve anything 227 if (maxD > 0) { 228 // Set the matrix to give (u = 0, v = distance_to_line) 229 GrVec lineVec = qPts[(maxEdge + 1)%3] - qPts[maxEdge]; 230 // when looking from the point 0 down the line we want positive 231 // distances to be to the left. This matches the non-degenerate 232 // case. 233 lineVec.setOrthog(lineVec, GrPoint::kLeft_Side); 234 lineVec.dot(qPts[0]); 235 // first row 236 fM[0] = 0; 237 fM[1] = 0; 238 fM[2] = 0; 239 // second row 240 fM[3] = lineVec.fX; 241 fM[4] = lineVec.fY; 242 fM[5] = -lineVec.dot(qPts[maxEdge]); 243 } else { 244 // It's a point. It should cover zero area. Just set the matrix such 245 // that (u, v) will always be far away from the quad. 246 fM[0] = 0; fM[1] = 0; fM[2] = 100.f; 247 fM[3] = 0; fM[4] = 0; fM[5] = 100.f; 248 } 249 } else { 250 m.postConcat(UVpts); 251 252 // The matrix should not have perspective. 253 SkDEBUGCODE(static const SkScalar gTOL = SkFloatToScalar(1.f / 100.f)); 254 SkASSERT(SkScalarAbs(m.get(SkMatrix::kMPersp0)) < gTOL); 255 SkASSERT(SkScalarAbs(m.get(SkMatrix::kMPersp1)) < gTOL); 256 257 // It may not be normalized to have 1.0 in the bottom right 258 float m33 = m.get(SkMatrix::kMPersp2); 259 if (1.f != m33) { 260 m33 = 1.f / m33; 261 fM[0] = m33 * m.get(SkMatrix::kMScaleX); 262 fM[1] = m33 * m.get(SkMatrix::kMSkewX); 263 fM[2] = m33 * m.get(SkMatrix::kMTransX); 264 fM[3] = m33 * m.get(SkMatrix::kMSkewY); 265 fM[4] = m33 * m.get(SkMatrix::kMScaleY); 266 fM[5] = m33 * m.get(SkMatrix::kMTransY); 267 } else { 268 fM[0] = m.get(SkMatrix::kMScaleX); 269 fM[1] = m.get(SkMatrix::kMSkewX); 270 fM[2] = m.get(SkMatrix::kMTransX); 271 fM[3] = m.get(SkMatrix::kMSkewY); 272 fM[4] = m.get(SkMatrix::kMScaleY); 273 fM[5] = m.get(SkMatrix::kMTransY); 274 } 275 } 276} 277 278namespace { 279 280// a is the first control point of the cubic. 281// ab is the vector from a to the second control point. 282// dc is the vector from the fourth to the third control point. 283// d is the fourth control point. 284// p is the candidate quadratic control point. 285// this assumes that the cubic doesn't inflect and is simple 286bool is_point_within_cubic_tangents(const SkPoint& a, 287 const SkVector& ab, 288 const SkVector& dc, 289 const SkPoint& d, 290 SkPath::Direction dir, 291 const SkPoint p) { 292 SkVector ap = p - a; 293 SkScalar apXab = ap.cross(ab); 294 if (SkPath::kCW_Direction == dir) { 295 if (apXab > 0) { 296 return false; 297 } 298 } else { 299 SkASSERT(SkPath::kCCW_Direction == dir); 300 if (apXab < 0) { 301 return false; 302 } 303 } 304 305 SkVector dp = p - d; 306 SkScalar dpXdc = dp.cross(dc); 307 if (SkPath::kCW_Direction == dir) { 308 if (dpXdc < 0) { 309 return false; 310 } 311 } else { 312 SkASSERT(SkPath::kCCW_Direction == dir); 313 if (dpXdc > 0) { 314 return false; 315 } 316 } 317 return true; 318} 319 320void convert_noninflect_cubic_to_quads(const SkPoint p[4], 321 SkScalar toleranceSqd, 322 bool constrainWithinTangents, 323 SkPath::Direction dir, 324 SkTArray<SkPoint, true>* quads, 325 int sublevel = 0) { 326 327 // Notation: Point a is always p[0]. Point b is p[1] unless p[1] == p[0], in which case it is 328 // 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]. 329 330 SkVector ab = p[1] - p[0]; 331 SkVector dc = p[2] - p[3]; 332 333 if (ab.isZero()) { 334 if (dc.isZero()) { 335 SkPoint* degQuad = quads->push_back_n(3); 336 degQuad[0] = p[0]; 337 degQuad[1] = p[0]; 338 degQuad[2] = p[3]; 339 return; 340 } 341 ab = p[2] - p[0]; 342 } 343 if (dc.isZero()) { 344 dc = p[1] - p[3]; 345 } 346 347 // When the ab and cd tangents are nearly parallel with vector from d to a the constraint that 348 // the quad point falls between the tangents becomes hard to enforce and we are likely to hit 349 // the max subdivision count. However, in this case the cubic is approaching a line and the 350 // accuracy of the quad point isn't so important. We check if the two middle cubic control 351 // points are very close to the baseline vector. If so then we just pick quadratic points on the 352 // control polygon. 353 354 if (constrainWithinTangents) { 355 SkVector da = p[0] - p[3]; 356 SkScalar invDALengthSqd = da.lengthSqd(); 357 if (invDALengthSqd > SK_ScalarNearlyZero) { 358 invDALengthSqd = SkScalarInvert(invDALengthSqd); 359 // cross(ab, da)^2/length(da)^2 == sqd distance from b to line from d to a. 360 // same goed for point c using vector cd. 361 SkScalar detABSqd = ab.cross(da); 362 detABSqd = SkScalarSquare(detABSqd); 363 SkScalar detDCSqd = dc.cross(da); 364 detDCSqd = SkScalarSquare(detDCSqd); 365 if (SkScalarMul(detABSqd, invDALengthSqd) < toleranceSqd && 366 SkScalarMul(detDCSqd, invDALengthSqd) < toleranceSqd) { 367 SkPoint b = p[0] + ab; 368 SkPoint c = p[3] + dc; 369 SkPoint mid = b + c; 370 mid.scale(SK_ScalarHalf); 371 // Insert two quadratics to cover the case when ab points away from d and/or dc 372 // points away from a. 373 if (SkVector::DotProduct(da, dc) < 0 || SkVector::DotProduct(ab,da) > 0) { 374 SkPoint* qpts = quads->push_back_n(6); 375 qpts[0] = p[0]; 376 qpts[1] = b; 377 qpts[2] = mid; 378 qpts[3] = mid; 379 qpts[4] = c; 380 qpts[5] = p[3]; 381 } else { 382 SkPoint* qpts = quads->push_back_n(3); 383 qpts[0] = p[0]; 384 qpts[1] = mid; 385 qpts[2] = p[3]; 386 } 387 return; 388 } 389 } 390 } 391 392 static const SkScalar kLengthScale = 3 * SK_Scalar1 / 2; 393 static const int kMaxSubdivs = 10; 394 395 ab.scale(kLengthScale); 396 dc.scale(kLengthScale); 397 398 // e0 and e1 are extrapolations along vectors ab and dc. 399 SkVector c0 = p[0]; 400 c0 += ab; 401 SkVector c1 = p[3]; 402 c1 += dc; 403 404 SkScalar dSqd = sublevel > kMaxSubdivs ? 0 : c0.distanceToSqd(c1); 405 if (dSqd < toleranceSqd) { 406 SkPoint cAvg = c0; 407 cAvg += c1; 408 cAvg.scale(SK_ScalarHalf); 409 410 bool subdivide = false; 411 412 if (constrainWithinTangents && 413 !is_point_within_cubic_tangents(p[0], ab, dc, p[3], dir, cAvg)) { 414 // choose a new cAvg that is the intersection of the two tangent lines. 415 ab.setOrthog(ab); 416 SkScalar z0 = -ab.dot(p[0]); 417 dc.setOrthog(dc); 418 SkScalar z1 = -dc.dot(p[3]); 419 cAvg.fX = SkScalarMul(ab.fY, z1) - SkScalarMul(z0, dc.fY); 420 cAvg.fY = SkScalarMul(z0, dc.fX) - SkScalarMul(ab.fX, z1); 421 SkScalar z = SkScalarMul(ab.fX, dc.fY) - SkScalarMul(ab.fY, dc.fX); 422 z = SkScalarInvert(z); 423 cAvg.fX *= z; 424 cAvg.fY *= z; 425 if (sublevel <= kMaxSubdivs) { 426 SkScalar d0Sqd = c0.distanceToSqd(cAvg); 427 SkScalar d1Sqd = c1.distanceToSqd(cAvg); 428 // We need to subdivide if d0 + d1 > tolerance but we have the sqd values. We know 429 // the distances and tolerance can't be negative. 430 // (d0 + d1)^2 > toleranceSqd 431 // d0Sqd + 2*d0*d1 + d1Sqd > toleranceSqd 432 SkScalar d0d1 = SkScalarSqrt(SkScalarMul(d0Sqd, d1Sqd)); 433 subdivide = 2 * d0d1 + d0Sqd + d1Sqd > toleranceSqd; 434 } 435 } 436 if (!subdivide) { 437 SkPoint* pts = quads->push_back_n(3); 438 pts[0] = p[0]; 439 pts[1] = cAvg; 440 pts[2] = p[3]; 441 return; 442 } 443 } 444 SkPoint choppedPts[7]; 445 SkChopCubicAtHalf(p, choppedPts); 446 convert_noninflect_cubic_to_quads(choppedPts + 0, 447 toleranceSqd, 448 constrainWithinTangents, 449 dir, 450 quads, 451 sublevel + 1); 452 convert_noninflect_cubic_to_quads(choppedPts + 3, 453 toleranceSqd, 454 constrainWithinTangents, 455 dir, 456 quads, 457 sublevel + 1); 458} 459} 460 461void GrPathUtils::convertCubicToQuads(const GrPoint p[4], 462 SkScalar tolScale, 463 bool constrainWithinTangents, 464 SkPath::Direction dir, 465 SkTArray<SkPoint, true>* quads) { 466 SkPoint chopped[10]; 467 int count = SkChopCubicAtInflections(p, chopped); 468 469 // base tolerance is 1 pixel. 470 static const SkScalar kTolerance = SK_Scalar1; 471 const SkScalar tolSqd = SkScalarSquare(SkScalarMul(tolScale, kTolerance)); 472 473 for (int i = 0; i < count; ++i) { 474 SkPoint* cubic = chopped + 3*i; 475 convert_noninflect_cubic_to_quads(cubic, tolSqd, constrainWithinTangents, dir, quads); 476 } 477 478} 479 480//////////////////////////////////////////////////////////////////////////////// 481 482enum CubicType { 483 kSerpentine_CubicType, 484 kCusp_CubicType, 485 kLoop_CubicType, 486 kQuadratic_CubicType, 487 kLine_CubicType, 488 kPoint_CubicType 489}; 490 491// discr(I) = d0^2 * (3*d1^2 - 4*d0*d2) 492// Classification: 493// discr(I) > 0 Serpentine 494// discr(I) = 0 Cusp 495// discr(I) < 0 Loop 496// d0 = d1 = 0 Quadratic 497// d0 = d1 = d2 = 0 Line 498// p0 = p1 = p2 = p3 Point 499static CubicType classify_cubic(const SkPoint p[4], const SkScalar d[3]) { 500 if (p[0] == p[1] && p[0] == p[2] && p[0] == p[3]) { 501 return kPoint_CubicType; 502 } 503 const SkScalar discr = d[0] * d[0] * (3.f * d[1] * d[1] - 4.f * d[0] * d[2]); 504 if (discr > SK_ScalarNearlyZero) { 505 return kSerpentine_CubicType; 506 } else if (discr < -SK_ScalarNearlyZero) { 507 return kLoop_CubicType; 508 } else { 509 if (0.f == d[0] && 0.f == d[1]) { 510 return (0.f == d[2] ? kLine_CubicType : kQuadratic_CubicType); 511 } else { 512 return kCusp_CubicType; 513 } 514 } 515} 516 517// Assumes the third component of points is 1. 518// Calcs p0 . (p1 x p2) 519static SkScalar calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) { 520 const SkScalar xComp = p0.fX * (p1.fY - p2.fY); 521 const SkScalar yComp = p0.fY * (p2.fX - p1.fX); 522 const SkScalar wComp = p1.fX * p2.fY - p1.fY * p2.fX; 523 return (xComp + yComp + wComp); 524} 525 526// Solves linear system to extract klm 527// P.K = k (similarly for l, m) 528// Where P is matrix of control points 529// K is coefficients for the line K 530// k is vector of values of K evaluated at the control points 531// Solving for K, thus K = P^(-1) . k 532static void calc_cubic_klm(const SkPoint p[4], const SkScalar controlK[4], 533 const SkScalar controlL[4], const SkScalar controlM[4], 534 SkScalar k[3], SkScalar l[3], SkScalar m[3]) { 535 SkMatrix matrix; 536 matrix.setAll(p[0].fX, p[0].fY, 1.f, 537 p[1].fX, p[1].fY, 1.f, 538 p[2].fX, p[2].fY, 1.f); 539 SkMatrix inverse; 540 if (matrix.invert(&inverse)) { 541 inverse.mapHomogeneousPoints(k, controlK, 1); 542 inverse.mapHomogeneousPoints(l, controlL, 1); 543 inverse.mapHomogeneousPoints(m, controlM, 1); 544 } 545 546} 547 548static void set_serp_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { 549 SkScalar tempSqrt = SkScalarSqrt(9.f * d[1] * d[1] - 12.f * d[0] * d[2]); 550 SkScalar ls = 3.f * d[1] - tempSqrt; 551 SkScalar lt = 6.f * d[0]; 552 SkScalar ms = 3.f * d[1] + tempSqrt; 553 SkScalar mt = 6.f * d[0]; 554 555 k[0] = ls * ms; 556 k[1] = (3.f * ls * ms - ls * mt - lt * ms) / 3.f; 557 k[2] = (lt * (mt - 2.f * ms) + ls * (3.f * ms - 2.f * mt)) / 3.f; 558 k[3] = (lt - ls) * (mt - ms); 559 560 l[0] = ls * ls * ls; 561 const SkScalar lt_ls = lt - ls; 562 l[1] = ls * ls * lt_ls * -1.f; 563 l[2] = lt_ls * lt_ls * ls; 564 l[3] = -1.f * lt_ls * lt_ls * lt_ls; 565 566 m[0] = ms * ms * ms; 567 const SkScalar mt_ms = mt - ms; 568 m[1] = ms * ms * mt_ms * -1.f; 569 m[2] = mt_ms * mt_ms * ms; 570 m[3] = -1.f * mt_ms * mt_ms * mt_ms; 571 572 // If d0 < 0 we need to flip the orientation of our curve 573 // This is done by negating the k and l values 574 // We want negative distance values to be on the inside 575 if ( d[0] > 0) { 576 for (int i = 0; i < 4; ++i) { 577 k[i] = -k[i]; 578 l[i] = -l[i]; 579 } 580 } 581} 582 583static void set_loop_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { 584 SkScalar tempSqrt = SkScalarSqrt(4.f * d[0] * d[2] - 3.f * d[1] * d[1]); 585 SkScalar ls = d[1] - tempSqrt; 586 SkScalar lt = 2.f * d[0]; 587 SkScalar ms = d[1] + tempSqrt; 588 SkScalar mt = 2.f * d[0]; 589 590 k[0] = ls * ms; 591 k[1] = (3.f * ls*ms - ls * mt - lt * ms) / 3.f; 592 k[2] = (lt * (mt - 2.f * ms) + ls * (3.f * ms - 2.f * mt)) / 3.f; 593 k[3] = (lt - ls) * (mt - ms); 594 595 l[0] = ls * ls * ms; 596 l[1] = (ls * (ls * (mt - 3.f * ms) + 2.f * lt * ms))/-3.f; 597 l[2] = ((lt - ls) * (ls * (2.f * mt - 3.f * ms) + lt * ms))/3.f; 598 l[3] = -1.f * (lt - ls) * (lt - ls) * (mt - ms); 599 600 m[0] = ls * ms * ms; 601 m[1] = (ms * (ls * (2.f * mt - 3.f * ms) + lt * ms))/-3.f; 602 m[2] = ((mt - ms) * (ls * (mt - 3.f * ms) + 2.f * lt * ms))/3.f; 603 m[3] = -1.f * (lt - ls) * (mt - ms) * (mt - ms); 604 605 606 // If (d0 < 0 && sign(k1) > 0) || (d0 > 0 && sign(k1) < 0), 607 // we need to flip the orientation of our curve. 608 // This is done by negating the k and l values 609 if ( (d[0] < 0 && k[1] < 0) || (d[0] > 0 && k[1] > 0)) { 610 for (int i = 0; i < 4; ++i) { 611 k[i] = -k[i]; 612 l[i] = -l[i]; 613 } 614 } 615} 616 617static void set_cusp_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { 618 const SkScalar ls = d[2]; 619 const SkScalar lt = 3.f * d[1]; 620 621 k[0] = ls; 622 k[1] = ls - lt / 3.f; 623 k[2] = ls - 2.f * lt / 3.f; 624 k[3] = ls - lt; 625 626 l[0] = ls * ls * ls; 627 const SkScalar ls_lt = ls - lt; 628 l[1] = ls * ls * ls_lt; 629 l[2] = ls_lt * ls_lt * ls; 630 l[3] = ls_lt * ls_lt * ls_lt; 631 632 m[0] = 1.f; 633 m[1] = 1.f; 634 m[2] = 1.f; 635 m[3] = 1.f; 636} 637 638// For the case when a cubic is actually a quadratic 639// M = 640// 0 0 0 641// 1/3 0 1/3 642// 2/3 1/3 2/3 643// 1 1 1 644static void set_quadratic_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { 645 k[0] = 0.f; 646 k[1] = 1.f/3.f; 647 k[2] = 2.f/3.f; 648 k[3] = 1.f; 649 650 l[0] = 0.f; 651 l[1] = 0.f; 652 l[2] = 1.f/3.f; 653 l[3] = 1.f; 654 655 m[0] = 0.f; 656 m[1] = 1.f/3.f; 657 m[2] = 2.f/3.f; 658 m[3] = 1.f; 659 660 // If d2 < 0 we need to flip the orientation of our curve 661 // This is done by negating the k and l values 662 if ( d[2] > 0) { 663 for (int i = 0; i < 4; ++i) { 664 k[i] = -k[i]; 665 l[i] = -l[i]; 666 } 667 } 668} 669 670// Calc coefficients of I(s,t) where roots of I are inflection points of curve 671// I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2) 672// d0 = a1 - 2*a2+3*a3 673// d1 = -a2 + 3*a3 674// d2 = 3*a3 675// a1 = p0 . (p3 x p2) 676// a2 = p1 . (p0 x p3) 677// a3 = p2 . (p1 x p0) 678// Places the values of d1, d2, d3 in array d passed in 679static void calc_cubic_inflection_func(const SkPoint p[4], SkScalar d[3]) { 680 SkScalar a1 = calc_dot_cross_cubic(p[0], p[3], p[2]); 681 SkScalar a2 = calc_dot_cross_cubic(p[1], p[0], p[3]); 682 SkScalar a3 = calc_dot_cross_cubic(p[2], p[1], p[0]); 683 684 // need to scale a's or values in later calculations will grow to high 685 SkScalar max = SkScalarAbs(a1); 686 max = SkMaxScalar(max, SkScalarAbs(a2)); 687 max = SkMaxScalar(max, SkScalarAbs(a3)); 688 max = 1.f/max; 689 a1 = a1 * max; 690 a2 = a2 * max; 691 a3 = a3 * max; 692 693 d[2] = 3.f * a3; 694 d[1] = d[2] - a2; 695 d[0] = d[1] - a2 + a1; 696} 697 698int GrPathUtils::chopCubicAtLoopIntersection(const SkPoint src[4], SkPoint dst[10], SkScalar klm[9], 699 SkScalar klm_rev[3]) { 700 // Variable to store the two parametric values at the loop double point 701 SkScalar smallS = 0.f; 702 SkScalar largeS = 0.f; 703 704 SkScalar d[3]; 705 calc_cubic_inflection_func(src, d); 706 707 CubicType cType = classify_cubic(src, d); 708 709 int chop_count = 0; 710 if (kLoop_CubicType == cType) { 711 SkScalar tempSqrt = SkScalarSqrt(4.f * d[0] * d[2] - 3.f * d[1] * d[1]); 712 SkScalar ls = d[1] - tempSqrt; 713 SkScalar lt = 2.f * d[0]; 714 SkScalar ms = d[1] + tempSqrt; 715 SkScalar mt = 2.f * d[0]; 716 ls = ls / lt; 717 ms = ms / mt; 718 // need to have t values sorted since this is what is expected by SkChopCubicAt 719 if (ls <= ms) { 720 smallS = ls; 721 largeS = ms; 722 } else { 723 smallS = ms; 724 largeS = ls; 725 } 726 727 SkScalar chop_ts[2]; 728 if (smallS > 0.f && smallS < 1.f) { 729 chop_ts[chop_count++] = smallS; 730 } 731 if (largeS > 0.f && largeS < 1.f) { 732 chop_ts[chop_count++] = largeS; 733 } 734 if(dst) { 735 SkChopCubicAt(src, dst, chop_ts, chop_count); 736 } 737 } else { 738 if (dst) { 739 memcpy(dst, src, sizeof(SkPoint) * 4); 740 } 741 } 742 743 if (klm && klm_rev) { 744 // Set klm_rev to to match the sub_section of cubic that needs to have its orientation 745 // flipped. This will always be the section that is the "loop" 746 if (2 == chop_count) { 747 klm_rev[0] = 1.f; 748 klm_rev[1] = -1.f; 749 klm_rev[2] = 1.f; 750 } else if (1 == chop_count) { 751 if (smallS < 0.f) { 752 klm_rev[0] = -1.f; 753 klm_rev[1] = 1.f; 754 } else { 755 klm_rev[0] = 1.f; 756 klm_rev[1] = -1.f; 757 } 758 } else { 759 if (smallS < 0.f && largeS > 1.f) { 760 klm_rev[0] = -1.f; 761 } else { 762 klm_rev[0] = 1.f; 763 } 764 } 765 SkScalar controlK[4]; 766 SkScalar controlL[4]; 767 SkScalar controlM[4]; 768 769 if (kSerpentine_CubicType == cType || (kCusp_CubicType == cType && 0.f != d[0])) { 770 set_serp_klm(d, controlK, controlL, controlM); 771 } else if (kLoop_CubicType == cType) { 772 set_loop_klm(d, controlK, controlL, controlM); 773 } else if (kCusp_CubicType == cType) { 774 SkASSERT(0.f == d[0]); 775 set_cusp_klm(d, controlK, controlL, controlM); 776 } else if (kQuadratic_CubicType == cType) { 777 set_quadratic_klm(d, controlK, controlL, controlM); 778 } 779 780 calc_cubic_klm(src, controlK, controlL, controlM, klm, &klm[3], &klm[6]); 781 } 782 return chop_count + 1; 783} 784 785void GrPathUtils::getCubicKLM(const SkPoint p[4], SkScalar klm[9]) { 786 SkScalar d[3]; 787 calc_cubic_inflection_func(p, d); 788 789 CubicType cType = classify_cubic(p, d); 790 791 SkScalar controlK[4]; 792 SkScalar controlL[4]; 793 SkScalar controlM[4]; 794 795 if (kSerpentine_CubicType == cType || (kCusp_CubicType == cType && 0.f != d[0])) { 796 set_serp_klm(d, controlK, controlL, controlM); 797 } else if (kLoop_CubicType == cType) { 798 set_loop_klm(d, controlK, controlL, controlM); 799 } else if (kCusp_CubicType == cType) { 800 SkASSERT(0.f == d[0]); 801 set_cusp_klm(d, controlK, controlL, controlM); 802 } else if (kQuadratic_CubicType == cType) { 803 set_quadratic_klm(d, controlK, controlL, controlM); 804 } 805 806 calc_cubic_klm(p, controlK, controlL, controlM, klm, &klm[3], &klm[6]); 807} 808