GrPathUtils.cpp revision 07e1c3fd5030869c480c15ff30d36bd161718262
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} 277 278//////////////////////////////////////////////////////////////////////////////// 279 280// k = (y2 - y0, x0 - x2, (x2 - x0)*y0 - (y2 - y0)*x0 ) 281// l = (2*w * (y1 - y0), 2*w * (x0 - x1), 2*w * (x1*y0 - x0*y1)) 282// m = (2*w * (y2 - y1), 2*w * (x1 - x2), 2*w * (x2*y1 - x1*y2)) 283void GrPathUtils::getConicKLM(const SkPoint p[3], const SkScalar weight, SkScalar klm[9]) { 284 const SkScalar w2 = 2.f * weight; 285 klm[0] = p[2].fY - p[0].fY; 286 klm[1] = p[0].fX - p[2].fX; 287 klm[2] = (p[2].fX - p[0].fX) * p[0].fY - (p[2].fY - p[0].fY) * p[0].fX; 288 289 klm[3] = w2 * (p[1].fY - p[0].fY); 290 klm[4] = w2 * (p[0].fX - p[1].fX); 291 klm[5] = w2 * (p[1].fX * p[0].fY - p[0].fX * p[1].fY); 292 293 klm[6] = w2 * (p[2].fY - p[1].fY); 294 klm[7] = w2 * (p[1].fX - p[2].fX); 295 klm[8] = w2 * (p[2].fX * p[1].fY - p[1].fX * p[2].fY); 296 297 // scale the max absolute value of coeffs to 10 298 SkScalar scale = 0.f; 299 for (int i = 0; i < 9; ++i) { 300 scale = SkMaxScalar(scale, SkScalarAbs(klm[i])); 301 } 302 SkASSERT(scale > 0.f); 303 scale = 10.f / scale; 304 for (int i = 0; i < 9; ++i) { 305 klm[i] *= scale; 306 } 307} 308 309//////////////////////////////////////////////////////////////////////////////// 310 311namespace { 312 313// a is the first control point of the cubic. 314// ab is the vector from a to the second control point. 315// dc is the vector from the fourth to the third control point. 316// d is the fourth control point. 317// p is the candidate quadratic control point. 318// this assumes that the cubic doesn't inflect and is simple 319bool is_point_within_cubic_tangents(const SkPoint& a, 320 const SkVector& ab, 321 const SkVector& dc, 322 const SkPoint& d, 323 SkPath::Direction dir, 324 const SkPoint p) { 325 SkVector ap = p - a; 326 SkScalar apXab = ap.cross(ab); 327 if (SkPath::kCW_Direction == dir) { 328 if (apXab > 0) { 329 return false; 330 } 331 } else { 332 SkASSERT(SkPath::kCCW_Direction == dir); 333 if (apXab < 0) { 334 return false; 335 } 336 } 337 338 SkVector dp = p - d; 339 SkScalar dpXdc = dp.cross(dc); 340 if (SkPath::kCW_Direction == dir) { 341 if (dpXdc < 0) { 342 return false; 343 } 344 } else { 345 SkASSERT(SkPath::kCCW_Direction == dir); 346 if (dpXdc > 0) { 347 return false; 348 } 349 } 350 return true; 351} 352 353void convert_noninflect_cubic_to_quads(const SkPoint p[4], 354 SkScalar toleranceSqd, 355 bool constrainWithinTangents, 356 SkPath::Direction dir, 357 SkTArray<SkPoint, true>* quads, 358 int sublevel = 0) { 359 360 // Notation: Point a is always p[0]. Point b is p[1] unless p[1] == p[0], in which case it is 361 // 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]. 362 363 SkVector ab = p[1] - p[0]; 364 SkVector dc = p[2] - p[3]; 365 366 if (ab.isZero()) { 367 if (dc.isZero()) { 368 SkPoint* degQuad = quads->push_back_n(3); 369 degQuad[0] = p[0]; 370 degQuad[1] = p[0]; 371 degQuad[2] = p[3]; 372 return; 373 } 374 ab = p[2] - p[0]; 375 } 376 if (dc.isZero()) { 377 dc = p[1] - p[3]; 378 } 379 380 // When the ab and cd tangents are nearly parallel with vector from d to a the constraint that 381 // the quad point falls between the tangents becomes hard to enforce and we are likely to hit 382 // the max subdivision count. However, in this case the cubic is approaching a line and the 383 // accuracy of the quad point isn't so important. We check if the two middle cubic control 384 // points are very close to the baseline vector. If so then we just pick quadratic points on the 385 // control polygon. 386 387 if (constrainWithinTangents) { 388 SkVector da = p[0] - p[3]; 389 SkScalar invDALengthSqd = da.lengthSqd(); 390 if (invDALengthSqd > SK_ScalarNearlyZero) { 391 invDALengthSqd = SkScalarInvert(invDALengthSqd); 392 // cross(ab, da)^2/length(da)^2 == sqd distance from b to line from d to a. 393 // same goed for point c using vector cd. 394 SkScalar detABSqd = ab.cross(da); 395 detABSqd = SkScalarSquare(detABSqd); 396 SkScalar detDCSqd = dc.cross(da); 397 detDCSqd = SkScalarSquare(detDCSqd); 398 if (SkScalarMul(detABSqd, invDALengthSqd) < toleranceSqd && 399 SkScalarMul(detDCSqd, invDALengthSqd) < toleranceSqd) { 400 SkPoint b = p[0] + ab; 401 SkPoint c = p[3] + dc; 402 SkPoint mid = b + c; 403 mid.scale(SK_ScalarHalf); 404 // Insert two quadratics to cover the case when ab points away from d and/or dc 405 // points away from a. 406 if (SkVector::DotProduct(da, dc) < 0 || SkVector::DotProduct(ab,da) > 0) { 407 SkPoint* qpts = quads->push_back_n(6); 408 qpts[0] = p[0]; 409 qpts[1] = b; 410 qpts[2] = mid; 411 qpts[3] = mid; 412 qpts[4] = c; 413 qpts[5] = p[3]; 414 } else { 415 SkPoint* qpts = quads->push_back_n(3); 416 qpts[0] = p[0]; 417 qpts[1] = mid; 418 qpts[2] = p[3]; 419 } 420 return; 421 } 422 } 423 } 424 425 static const SkScalar kLengthScale = 3 * SK_Scalar1 / 2; 426 static const int kMaxSubdivs = 10; 427 428 ab.scale(kLengthScale); 429 dc.scale(kLengthScale); 430 431 // e0 and e1 are extrapolations along vectors ab and dc. 432 SkVector c0 = p[0]; 433 c0 += ab; 434 SkVector c1 = p[3]; 435 c1 += dc; 436 437 SkScalar dSqd = sublevel > kMaxSubdivs ? 0 : c0.distanceToSqd(c1); 438 if (dSqd < toleranceSqd) { 439 SkPoint cAvg = c0; 440 cAvg += c1; 441 cAvg.scale(SK_ScalarHalf); 442 443 bool subdivide = false; 444 445 if (constrainWithinTangents && 446 !is_point_within_cubic_tangents(p[0], ab, dc, p[3], dir, cAvg)) { 447 // choose a new cAvg that is the intersection of the two tangent lines. 448 ab.setOrthog(ab); 449 SkScalar z0 = -ab.dot(p[0]); 450 dc.setOrthog(dc); 451 SkScalar z1 = -dc.dot(p[3]); 452 cAvg.fX = SkScalarMul(ab.fY, z1) - SkScalarMul(z0, dc.fY); 453 cAvg.fY = SkScalarMul(z0, dc.fX) - SkScalarMul(ab.fX, z1); 454 SkScalar z = SkScalarMul(ab.fX, dc.fY) - SkScalarMul(ab.fY, dc.fX); 455 z = SkScalarInvert(z); 456 cAvg.fX *= z; 457 cAvg.fY *= z; 458 if (sublevel <= kMaxSubdivs) { 459 SkScalar d0Sqd = c0.distanceToSqd(cAvg); 460 SkScalar d1Sqd = c1.distanceToSqd(cAvg); 461 // We need to subdivide if d0 + d1 > tolerance but we have the sqd values. We know 462 // the distances and tolerance can't be negative. 463 // (d0 + d1)^2 > toleranceSqd 464 // d0Sqd + 2*d0*d1 + d1Sqd > toleranceSqd 465 SkScalar d0d1 = SkScalarSqrt(SkScalarMul(d0Sqd, d1Sqd)); 466 subdivide = 2 * d0d1 + d0Sqd + d1Sqd > toleranceSqd; 467 } 468 } 469 if (!subdivide) { 470 SkPoint* pts = quads->push_back_n(3); 471 pts[0] = p[0]; 472 pts[1] = cAvg; 473 pts[2] = p[3]; 474 return; 475 } 476 } 477 SkPoint choppedPts[7]; 478 SkChopCubicAtHalf(p, choppedPts); 479 convert_noninflect_cubic_to_quads(choppedPts + 0, 480 toleranceSqd, 481 constrainWithinTangents, 482 dir, 483 quads, 484 sublevel + 1); 485 convert_noninflect_cubic_to_quads(choppedPts + 3, 486 toleranceSqd, 487 constrainWithinTangents, 488 dir, 489 quads, 490 sublevel + 1); 491} 492} 493 494void GrPathUtils::convertCubicToQuads(const GrPoint p[4], 495 SkScalar tolScale, 496 bool constrainWithinTangents, 497 SkPath::Direction dir, 498 SkTArray<SkPoint, true>* quads) { 499 SkPoint chopped[10]; 500 int count = SkChopCubicAtInflections(p, chopped); 501 502 // base tolerance is 1 pixel. 503 static const SkScalar kTolerance = SK_Scalar1; 504 const SkScalar tolSqd = SkScalarSquare(SkScalarMul(tolScale, kTolerance)); 505 506 for (int i = 0; i < count; ++i) { 507 SkPoint* cubic = chopped + 3*i; 508 convert_noninflect_cubic_to_quads(cubic, tolSqd, constrainWithinTangents, dir, quads); 509 } 510 511} 512 513//////////////////////////////////////////////////////////////////////////////// 514 515enum CubicType { 516 kSerpentine_CubicType, 517 kCusp_CubicType, 518 kLoop_CubicType, 519 kQuadratic_CubicType, 520 kLine_CubicType, 521 kPoint_CubicType 522}; 523 524// discr(I) = d0^2 * (3*d1^2 - 4*d0*d2) 525// Classification: 526// discr(I) > 0 Serpentine 527// discr(I) = 0 Cusp 528// discr(I) < 0 Loop 529// d0 = d1 = 0 Quadratic 530// d0 = d1 = d2 = 0 Line 531// p0 = p1 = p2 = p3 Point 532static CubicType classify_cubic(const SkPoint p[4], const SkScalar d[3]) { 533 if (p[0] == p[1] && p[0] == p[2] && p[0] == p[3]) { 534 return kPoint_CubicType; 535 } 536 const SkScalar discr = d[0] * d[0] * (3.f * d[1] * d[1] - 4.f * d[0] * d[2]); 537 if (discr > SK_ScalarNearlyZero) { 538 return kSerpentine_CubicType; 539 } else if (discr < -SK_ScalarNearlyZero) { 540 return kLoop_CubicType; 541 } else { 542 if (0.f == d[0] && 0.f == d[1]) { 543 return (0.f == d[2] ? kLine_CubicType : kQuadratic_CubicType); 544 } else { 545 return kCusp_CubicType; 546 } 547 } 548} 549 550// Assumes the third component of points is 1. 551// Calcs p0 . (p1 x p2) 552static SkScalar calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) { 553 const SkScalar xComp = p0.fX * (p1.fY - p2.fY); 554 const SkScalar yComp = p0.fY * (p2.fX - p1.fX); 555 const SkScalar wComp = p1.fX * p2.fY - p1.fY * p2.fX; 556 return (xComp + yComp + wComp); 557} 558 559// Solves linear system to extract klm 560// P.K = k (similarly for l, m) 561// Where P is matrix of control points 562// K is coefficients for the line K 563// k is vector of values of K evaluated at the control points 564// Solving for K, thus K = P^(-1) . k 565static void calc_cubic_klm(const SkPoint p[4], const SkScalar controlK[4], 566 const SkScalar controlL[4], const SkScalar controlM[4], 567 SkScalar k[3], SkScalar l[3], SkScalar m[3]) { 568 SkMatrix matrix; 569 matrix.setAll(p[0].fX, p[0].fY, 1.f, 570 p[1].fX, p[1].fY, 1.f, 571 p[2].fX, p[2].fY, 1.f); 572 SkMatrix inverse; 573 if (matrix.invert(&inverse)) { 574 inverse.mapHomogeneousPoints(k, controlK, 1); 575 inverse.mapHomogeneousPoints(l, controlL, 1); 576 inverse.mapHomogeneousPoints(m, controlM, 1); 577 } 578 579} 580 581static void set_serp_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { 582 SkScalar tempSqrt = SkScalarSqrt(9.f * d[1] * d[1] - 12.f * d[0] * d[2]); 583 SkScalar ls = 3.f * d[1] - tempSqrt; 584 SkScalar lt = 6.f * d[0]; 585 SkScalar ms = 3.f * d[1] + tempSqrt; 586 SkScalar mt = 6.f * d[0]; 587 588 k[0] = ls * ms; 589 k[1] = (3.f * ls * ms - ls * mt - lt * ms) / 3.f; 590 k[2] = (lt * (mt - 2.f * ms) + ls * (3.f * ms - 2.f * mt)) / 3.f; 591 k[3] = (lt - ls) * (mt - ms); 592 593 l[0] = ls * ls * ls; 594 const SkScalar lt_ls = lt - ls; 595 l[1] = ls * ls * lt_ls * -1.f; 596 l[2] = lt_ls * lt_ls * ls; 597 l[3] = -1.f * lt_ls * lt_ls * lt_ls; 598 599 m[0] = ms * ms * ms; 600 const SkScalar mt_ms = mt - ms; 601 m[1] = ms * ms * mt_ms * -1.f; 602 m[2] = mt_ms * mt_ms * ms; 603 m[3] = -1.f * mt_ms * mt_ms * mt_ms; 604 605 // If d0 < 0 we need to flip the orientation of our curve 606 // This is done by negating the k and l values 607 // We want negative distance values to be on the inside 608 if ( d[0] > 0) { 609 for (int i = 0; i < 4; ++i) { 610 k[i] = -k[i]; 611 l[i] = -l[i]; 612 } 613 } 614} 615 616static void set_loop_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { 617 SkScalar tempSqrt = SkScalarSqrt(4.f * d[0] * d[2] - 3.f * d[1] * d[1]); 618 SkScalar ls = d[1] - tempSqrt; 619 SkScalar lt = 2.f * d[0]; 620 SkScalar ms = d[1] + tempSqrt; 621 SkScalar mt = 2.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 * ms; 629 l[1] = (ls * (ls * (mt - 3.f * ms) + 2.f * lt * ms))/-3.f; 630 l[2] = ((lt - ls) * (ls * (2.f * mt - 3.f * ms) + lt * ms))/3.f; 631 l[3] = -1.f * (lt - ls) * (lt - ls) * (mt - ms); 632 633 m[0] = ls * ms * ms; 634 m[1] = (ms * (ls * (2.f * mt - 3.f * ms) + lt * ms))/-3.f; 635 m[2] = ((mt - ms) * (ls * (mt - 3.f * ms) + 2.f * lt * ms))/3.f; 636 m[3] = -1.f * (lt - ls) * (mt - ms) * (mt - ms); 637 638 639 // If (d0 < 0 && sign(k1) > 0) || (d0 > 0 && sign(k1) < 0), 640 // we need to flip the orientation of our curve. 641 // This is done by negating the k and l values 642 if ( (d[0] < 0 && k[1] > 0) || (d[0] > 0 && k[1] < 0)) { 643 for (int i = 0; i < 4; ++i) { 644 k[i] = -k[i]; 645 l[i] = -l[i]; 646 } 647 } 648} 649 650static void set_cusp_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { 651 const SkScalar ls = d[2]; 652 const SkScalar lt = 3.f * d[1]; 653 654 k[0] = ls; 655 k[1] = ls - lt / 3.f; 656 k[2] = ls - 2.f * lt / 3.f; 657 k[3] = ls - lt; 658 659 l[0] = ls * ls * ls; 660 const SkScalar ls_lt = ls - lt; 661 l[1] = ls * ls * ls_lt; 662 l[2] = ls_lt * ls_lt * ls; 663 l[3] = ls_lt * ls_lt * ls_lt; 664 665 m[0] = 1.f; 666 m[1] = 1.f; 667 m[2] = 1.f; 668 m[3] = 1.f; 669} 670 671// For the case when a cubic is actually a quadratic 672// M = 673// 0 0 0 674// 1/3 0 1/3 675// 2/3 1/3 2/3 676// 1 1 1 677static void set_quadratic_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { 678 k[0] = 0.f; 679 k[1] = 1.f/3.f; 680 k[2] = 2.f/3.f; 681 k[3] = 1.f; 682 683 l[0] = 0.f; 684 l[1] = 0.f; 685 l[2] = 1.f/3.f; 686 l[3] = 1.f; 687 688 m[0] = 0.f; 689 m[1] = 1.f/3.f; 690 m[2] = 2.f/3.f; 691 m[3] = 1.f; 692 693 // If d2 < 0 we need to flip the orientation of our curve 694 // This is done by negating the k and l values 695 if ( d[2] > 0) { 696 for (int i = 0; i < 4; ++i) { 697 k[i] = -k[i]; 698 l[i] = -l[i]; 699 } 700 } 701} 702 703// Calc coefficients of I(s,t) where roots of I are inflection points of curve 704// I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2) 705// d0 = a1 - 2*a2+3*a3 706// d1 = -a2 + 3*a3 707// d2 = 3*a3 708// a1 = p0 . (p3 x p2) 709// a2 = p1 . (p0 x p3) 710// a3 = p2 . (p1 x p0) 711// Places the values of d1, d2, d3 in array d passed in 712static void calc_cubic_inflection_func(const SkPoint p[4], SkScalar d[3]) { 713 SkScalar a1 = calc_dot_cross_cubic(p[0], p[3], p[2]); 714 SkScalar a2 = calc_dot_cross_cubic(p[1], p[0], p[3]); 715 SkScalar a3 = calc_dot_cross_cubic(p[2], p[1], p[0]); 716 717 // need to scale a's or values in later calculations will grow to high 718 SkScalar max = SkScalarAbs(a1); 719 max = SkMaxScalar(max, SkScalarAbs(a2)); 720 max = SkMaxScalar(max, SkScalarAbs(a3)); 721 max = 1.f/max; 722 a1 = a1 * max; 723 a2 = a2 * max; 724 a3 = a3 * max; 725 726 d[2] = 3.f * a3; 727 d[1] = d[2] - a2; 728 d[0] = d[1] - a2 + a1; 729} 730 731int GrPathUtils::chopCubicAtLoopIntersection(const SkPoint src[4], SkPoint dst[10], SkScalar klm[9], 732 SkScalar klm_rev[3]) { 733 // Variable to store the two parametric values at the loop double point 734 SkScalar smallS = 0.f; 735 SkScalar largeS = 0.f; 736 737 SkScalar d[3]; 738 calc_cubic_inflection_func(src, d); 739 740 CubicType cType = classify_cubic(src, d); 741 742 int chop_count = 0; 743 if (kLoop_CubicType == cType) { 744 SkScalar tempSqrt = SkScalarSqrt(4.f * d[0] * d[2] - 3.f * d[1] * d[1]); 745 SkScalar ls = d[1] - tempSqrt; 746 SkScalar lt = 2.f * d[0]; 747 SkScalar ms = d[1] + tempSqrt; 748 SkScalar mt = 2.f * d[0]; 749 ls = ls / lt; 750 ms = ms / mt; 751 // need to have t values sorted since this is what is expected by SkChopCubicAt 752 if (ls <= ms) { 753 smallS = ls; 754 largeS = ms; 755 } else { 756 smallS = ms; 757 largeS = ls; 758 } 759 760 SkScalar chop_ts[2]; 761 if (smallS > 0.f && smallS < 1.f) { 762 chop_ts[chop_count++] = smallS; 763 } 764 if (largeS > 0.f && largeS < 1.f) { 765 chop_ts[chop_count++] = largeS; 766 } 767 if(dst) { 768 SkChopCubicAt(src, dst, chop_ts, chop_count); 769 } 770 } else { 771 if (dst) { 772 memcpy(dst, src, sizeof(SkPoint) * 4); 773 } 774 } 775 776 if (klm && klm_rev) { 777 // Set klm_rev to to match the sub_section of cubic that needs to have its orientation 778 // flipped. This will always be the section that is the "loop" 779 if (2 == chop_count) { 780 klm_rev[0] = 1.f; 781 klm_rev[1] = -1.f; 782 klm_rev[2] = 1.f; 783 } else if (1 == chop_count) { 784 if (smallS < 0.f) { 785 klm_rev[0] = -1.f; 786 klm_rev[1] = 1.f; 787 } else { 788 klm_rev[0] = 1.f; 789 klm_rev[1] = -1.f; 790 } 791 } else { 792 if (smallS < 0.f && largeS > 1.f) { 793 klm_rev[0] = -1.f; 794 } else { 795 klm_rev[0] = 1.f; 796 } 797 } 798 SkScalar controlK[4]; 799 SkScalar controlL[4]; 800 SkScalar controlM[4]; 801 802 if (kSerpentine_CubicType == cType || (kCusp_CubicType == cType && 0.f != d[0])) { 803 set_serp_klm(d, controlK, controlL, controlM); 804 } else if (kLoop_CubicType == cType) { 805 set_loop_klm(d, controlK, controlL, controlM); 806 } else if (kCusp_CubicType == cType) { 807 SkASSERT(0.f == d[0]); 808 set_cusp_klm(d, controlK, controlL, controlM); 809 } else if (kQuadratic_CubicType == cType) { 810 set_quadratic_klm(d, controlK, controlL, controlM); 811 } 812 813 calc_cubic_klm(src, controlK, controlL, controlM, klm, &klm[3], &klm[6]); 814 } 815 return chop_count + 1; 816} 817 818void GrPathUtils::getCubicKLM(const SkPoint p[4], SkScalar klm[9]) { 819 SkScalar d[3]; 820 calc_cubic_inflection_func(p, d); 821 822 CubicType cType = classify_cubic(p, d); 823 824 SkScalar controlK[4]; 825 SkScalar controlL[4]; 826 SkScalar controlM[4]; 827 828 if (kSerpentine_CubicType == cType || (kCusp_CubicType == cType && 0.f != d[0])) { 829 set_serp_klm(d, controlK, controlL, controlM); 830 } else if (kLoop_CubicType == cType) { 831 set_loop_klm(d, controlK, controlL, controlM); 832 } else if (kCusp_CubicType == cType) { 833 SkASSERT(0.f == d[0]); 834 set_cusp_klm(d, controlK, controlL, controlM); 835 } else if (kQuadratic_CubicType == cType) { 836 set_quadratic_klm(d, controlK, controlL, controlM); 837 } 838 839 calc_cubic_klm(p, controlK, controlL, controlM, klm, &klm[3], &klm[6]); 840} 841