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 13 SkScalar 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 36 static const int MAX_POINTS_PER_CURVE = 1 << 10; 37 static const SkScalar gMinCurveTol = 0.0001f; 38 39 uint32_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 66 uint32_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 91 uint32_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 } 116 117 uint32_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 } 146 147 int 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 } 188 189 void GrPathUtils::QuadUVMatrix::set(const GrPoint qPts[3]) { 190 #ifndef SK_SCALAR_IS_FLOAT 191 GrCrash("Expected scalar is float."); 192 #endif 193 SkMatrix m; 194 // We want M such that M * xy_pt = uv_pt 195 // We know M * control_pts = [0 1/2 1] 196 // [0 0 1] 197 // [1 1 1] 198 // And control_pts = [x0 x1 x2] 199 // [y0 y1 y2] 200 // [1 1 1 ] 201 // We invert the control pt matrix and post concat to both sides to get M. 202 // Using the known form of the control point matrix and the result, we can 203 // optimize and improve precision. 204 205 double x0 = qPts[0].fX; 206 double y0 = qPts[0].fY; 207 double x1 = qPts[1].fX; 208 double y1 = qPts[1].fY; 209 double x2 = qPts[2].fX; 210 double y2 = qPts[2].fY; 211 double det = x0*y1 - y0*x1 + x2*y0 - y2*x0 + x1*y2 - y1*x2; 212 213 if (!sk_float_isfinite(det) 214 || SkScalarNearlyZero((float)det, SK_ScalarNearlyZero * SK_ScalarNearlyZero)) { 215 // The quad is degenerate. Hopefully this is rare. Find the pts that are 216 // farthest apart to compute a line (unless it is really a pt). 217 SkScalar maxD = qPts[0].distanceToSqd(qPts[1]); 218 int maxEdge = 0; 219 SkScalar d = qPts[1].distanceToSqd(qPts[2]); 220 if (d > maxD) { 221 maxD = d; 222 maxEdge = 1; 223 } 224 d = qPts[2].distanceToSqd(qPts[0]); 225 if (d > maxD) { 226 maxD = d; 227 maxEdge = 2; 228 } 229 // We could have a tolerance here, not sure if it would improve anything 230 if (maxD > 0) { 231 // Set the matrix to give (u = 0, v = distance_to_line) 232 GrVec lineVec = qPts[(maxEdge + 1)%3] - qPts[maxEdge]; 233 // when looking from the point 0 down the line we want positive 234 // distances to be to the left. This matches the non-degenerate 235 // case. 236 lineVec.setOrthog(lineVec, GrPoint::kLeft_Side); 237 lineVec.dot(qPts[0]); 238 // first row 239 fM[0] = 0; 240 fM[1] = 0; 241 fM[2] = 0; 242 // second row 243 fM[3] = lineVec.fX; 244 fM[4] = lineVec.fY; 245 fM[5] = -lineVec.dot(qPts[maxEdge]); 246 } else { 247 // It's a point. It should cover zero area. Just set the matrix such 248 // that (u, v) will always be far away from the quad. 249 fM[0] = 0; fM[1] = 0; fM[2] = 100.f; 250 fM[3] = 0; fM[4] = 0; fM[5] = 100.f; 251 } 252 } else { 253 double scale = 1.0/det; 254 255 // compute adjugate matrix 256 double a0, a1, a2, a3, a4, a5, a6, a7, a8; 257 a0 = y1-y2; 258 a1 = x2-x1; 259 a2 = x1*y2-x2*y1; 260 261 a3 = y2-y0; 262 a4 = x0-x2; 263 a5 = x2*y0-x0*y2; 264 265 a6 = y0-y1; 266 a7 = x1-x0; 267 a8 = x0*y1-x1*y0; 268 269 // this performs the uv_pts*adjugate(control_pts) multiply, 270 // then does the scale by 1/det afterwards to improve precision 271 m[SkMatrix::kMScaleX] = (float)((0.5*a3 + a6)*scale); 272 m[SkMatrix::kMSkewX] = (float)((0.5*a4 + a7)*scale); 273 m[SkMatrix::kMTransX] = (float)((0.5*a5 + a8)*scale); 274 275 m[SkMatrix::kMSkewY] = (float)(a6*scale); 276 m[SkMatrix::kMScaleY] = (float)(a7*scale); 277 m[SkMatrix::kMTransY] = (float)(a8*scale); 278 279 m[SkMatrix::kMPersp0] = (float)((a0 + a3 + a6)*scale); 280 m[SkMatrix::kMPersp1] = (float)((a1 + a4 + a7)*scale); 281 m[SkMatrix::kMPersp2] = (float)((a2 + a5 + a8)*scale); 282 283 // The matrix should not have perspective. 284 SkDEBUGCODE(static const SkScalar gTOL = 1.f / 100.f); 285 SkASSERT(SkScalarAbs(m.get(SkMatrix::kMPersp0)) < gTOL); 286 SkASSERT(SkScalarAbs(m.get(SkMatrix::kMPersp1)) < gTOL); 287 288 // It may not be normalized to have 1.0 in the bottom right 289 float m33 = m.get(SkMatrix::kMPersp2); 290 if (1.f != m33) { 291 m33 = 1.f / m33; 292 fM[0] = m33 * m.get(SkMatrix::kMScaleX); 293 fM[1] = m33 * m.get(SkMatrix::kMSkewX); 294 fM[2] = m33 * m.get(SkMatrix::kMTransX); 295 fM[3] = m33 * m.get(SkMatrix::kMSkewY); 296 fM[4] = m33 * m.get(SkMatrix::kMScaleY); 297 fM[5] = m33 * m.get(SkMatrix::kMTransY); 298 } else { 299 fM[0] = m.get(SkMatrix::kMScaleX); 300 fM[1] = m.get(SkMatrix::kMSkewX); 301 fM[2] = m.get(SkMatrix::kMTransX); 302 fM[3] = m.get(SkMatrix::kMSkewY); 303 fM[4] = m.get(SkMatrix::kMScaleY); 304 fM[5] = m.get(SkMatrix::kMTransY); 305 } 306 } 307 } 308 309 //////////////////////////////////////////////////////////////////////////////// 310 311 // k = (y2 - y0, x0 - x2, (x2 - x0)*y0 - (y2 - y0)*x0 ) 312 // l = (2*w * (y1 - y0), 2*w * (x0 - x1), 2*w * (x1*y0 - x0*y1)) 313 // m = (2*w * (y2 - y1), 2*w * (x1 - x2), 2*w * (x2*y1 - x1*y2)) 314 void GrPathUtils::getConicKLM(const SkPoint p[3], const SkScalar weight, SkScalar klm[9]) { 315 const SkScalar w2 = 2.f * weight; 316 klm[0] = p[2].fY - p[0].fY; 317 klm[1] = p[0].fX - p[2].fX; 318 klm[2] = (p[2].fX - p[0].fX) * p[0].fY - (p[2].fY - p[0].fY) * p[0].fX; 319 320 klm[3] = w2 * (p[1].fY - p[0].fY); 321 klm[4] = w2 * (p[0].fX - p[1].fX); 322 klm[5] = w2 * (p[1].fX * p[0].fY - p[0].fX * p[1].fY); 323 324 klm[6] = w2 * (p[2].fY - p[1].fY); 325 klm[7] = w2 * (p[1].fX - p[2].fX); 326 klm[8] = w2 * (p[2].fX * p[1].fY - p[1].fX * p[2].fY); 327 328 // scale the max absolute value of coeffs to 10 329 SkScalar scale = 0.f; 330 for (int i = 0; i < 9; ++i) { 331 scale = SkMaxScalar(scale, SkScalarAbs(klm[i])); 332 } 333 SkASSERT(scale > 0.f); 334 scale = 10.f / scale; 335 for (int i = 0; i < 9; ++i) { 336 klm[i] *= scale; 337 } 338 } 339 340 //////////////////////////////////////////////////////////////////////////////// 341 342 namespace { 343 344 // a is the first control point of the cubic. 345 // ab is the vector from a to the second control point. 346 // dc is the vector from the fourth to the third control point. 347 // d is the fourth control point. 348 // p is the candidate quadratic control point. 349 // this assumes that the cubic doesn't inflect and is simple 350 bool is_point_within_cubic_tangents(const SkPoint& a, 351 const SkVector& ab, 352 const SkVector& dc, 353 const SkPoint& d, 354 SkPath::Direction dir, 355 const SkPoint p) { 356 SkVector ap = p - a; 357 SkScalar apXab = ap.cross(ab); 358 if (SkPath::kCW_Direction == dir) { 359 if (apXab > 0) { 360 return false; 361 } 362 } else { 363 SkASSERT(SkPath::kCCW_Direction == dir); 364 if (apXab < 0) { 365 return false; 366 } 367 } 368 369 SkVector dp = p - d; 370 SkScalar dpXdc = dp.cross(dc); 371 if (SkPath::kCW_Direction == dir) { 372 if (dpXdc < 0) { 373 return false; 374 } 375 } else { 376 SkASSERT(SkPath::kCCW_Direction == dir); 377 if (dpXdc > 0) { 378 return false; 379 } 380 } 381 return true; 382 } 383 384 void convert_noninflect_cubic_to_quads(const SkPoint p[4], 385 SkScalar toleranceSqd, 386 bool constrainWithinTangents, 387 SkPath::Direction dir, 388 SkTArray<SkPoint, true>* quads, 389 int sublevel = 0) { 390 391 // Notation: Point a is always p[0]. Point b is p[1] unless p[1] == p[0], in which case it is 392 // 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]. 393 394 SkVector ab = p[1] - p[0]; 395 SkVector dc = p[2] - p[3]; 396 397 if (ab.isZero()) { 398 if (dc.isZero()) { 399 SkPoint* degQuad = quads->push_back_n(3); 400 degQuad[0] = p[0]; 401 degQuad[1] = p[0]; 402 degQuad[2] = p[3]; 403 return; 404 } 405 ab = p[2] - p[0]; 406 } 407 if (dc.isZero()) { 408 dc = p[1] - p[3]; 409 } 410 411 // When the ab and cd tangents are nearly parallel with vector from d to a the constraint that 412 // the quad point falls between the tangents becomes hard to enforce and we are likely to hit 413 // the max subdivision count. However, in this case the cubic is approaching a line and the 414 // accuracy of the quad point isn't so important. We check if the two middle cubic control 415 // points are very close to the baseline vector. If so then we just pick quadratic points on the 416 // control polygon. 417 418 if (constrainWithinTangents) { 419 SkVector da = p[0] - p[3]; 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 goed 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 SkPoint b = p[0] + ab; 432 SkPoint c = p[3] + dc; 433 SkPoint mid = b + c; 434 mid.scale(SK_ScalarHalf); 435 // Insert two quadratics to cover the case when ab points away from d and/or dc 436 // points away from a. 437 if (SkVector::DotProduct(da, dc) < 0 || SkVector::DotProduct(ab,da) > 0) { 438 SkPoint* qpts = quads->push_back_n(6); 439 qpts[0] = p[0]; 440 qpts[1] = b; 441 qpts[2] = mid; 442 qpts[3] = mid; 443 qpts[4] = c; 444 qpts[5] = p[3]; 445 } else { 446 SkPoint* qpts = quads->push_back_n(3); 447 qpts[0] = p[0]; 448 qpts[1] = mid; 449 qpts[2] = p[3]; 450 } 451 return; 452 } 453 } 454 } 455 456 static const SkScalar kLengthScale = 3 * SK_Scalar1 / 2; 457 static const int kMaxSubdivs = 10; 458 459 ab.scale(kLengthScale); 460 dc.scale(kLengthScale); 461 462 // e0 and e1 are extrapolations along vectors ab and dc. 463 SkVector c0 = p[0]; 464 c0 += ab; 465 SkVector c1 = p[3]; 466 c1 += dc; 467 468 SkScalar dSqd = sublevel > kMaxSubdivs ? 0 : c0.distanceToSqd(c1); 469 if (dSqd < toleranceSqd) { 470 SkPoint cAvg = c0; 471 cAvg += c1; 472 cAvg.scale(SK_ScalarHalf); 473 474 bool subdivide = false; 475 476 if (constrainWithinTangents && 477 !is_point_within_cubic_tangents(p[0], ab, dc, p[3], dir, cAvg)) { 478 // choose a new cAvg that is the intersection of the two tangent lines. 479 ab.setOrthog(ab); 480 SkScalar z0 = -ab.dot(p[0]); 481 dc.setOrthog(dc); 482 SkScalar z1 = -dc.dot(p[3]); 483 cAvg.fX = SkScalarMul(ab.fY, z1) - SkScalarMul(z0, dc.fY); 484 cAvg.fY = SkScalarMul(z0, dc.fX) - SkScalarMul(ab.fX, z1); 485 SkScalar z = SkScalarMul(ab.fX, dc.fY) - SkScalarMul(ab.fY, dc.fX); 486 z = SkScalarInvert(z); 487 cAvg.fX *= z; 488 cAvg.fY *= z; 489 if (sublevel <= kMaxSubdivs) { 490 SkScalar d0Sqd = c0.distanceToSqd(cAvg); 491 SkScalar d1Sqd = c1.distanceToSqd(cAvg); 492 // We need to subdivide if d0 + d1 > tolerance but we have the sqd values. We know 493 // the distances and tolerance can't be negative. 494 // (d0 + d1)^2 > toleranceSqd 495 // d0Sqd + 2*d0*d1 + d1Sqd > toleranceSqd 496 SkScalar d0d1 = SkScalarSqrt(SkScalarMul(d0Sqd, d1Sqd)); 497 subdivide = 2 * d0d1 + d0Sqd + d1Sqd > toleranceSqd; 498 } 499 } 500 if (!subdivide) { 501 SkPoint* pts = quads->push_back_n(3); 502 pts[0] = p[0]; 503 pts[1] = cAvg; 504 pts[2] = p[3]; 505 return; 506 } 507 } 508 SkPoint choppedPts[7]; 509 SkChopCubicAtHalf(p, choppedPts); 510 convert_noninflect_cubic_to_quads(choppedPts + 0, 511 toleranceSqd, 512 constrainWithinTangents, 513 dir, 514 quads, 515 sublevel + 1); 516 convert_noninflect_cubic_to_quads(choppedPts + 3, 517 toleranceSqd, 518 constrainWithinTangents, 519 dir, 520 quads, 521 sublevel + 1); 522 } 523 } 524 525 void GrPathUtils::convertCubicToQuads(const GrPoint p[4], 526 SkScalar tolScale, 527 bool constrainWithinTangents, 528 SkPath::Direction dir, 529 SkTArray<SkPoint, true>* quads) { 530 SkPoint chopped[10]; 531 int count = SkChopCubicAtInflections(p, chopped); 532 533 // base tolerance is 1 pixel. 534 static const SkScalar kTolerance = SK_Scalar1; 535 const SkScalar tolSqd = SkScalarSquare(SkScalarMul(tolScale, kTolerance)); 536 537 for (int i = 0; i < count; ++i) { 538 SkPoint* cubic = chopped + 3*i; 539 convert_noninflect_cubic_to_quads(cubic, tolSqd, constrainWithinTangents, dir, quads); 540 } 541 542 } 543 544 //////////////////////////////////////////////////////////////////////////////// 545 546 enum CubicType { 547 kSerpentine_CubicType, 548 kCusp_CubicType, 549 kLoop_CubicType, 550 kQuadratic_CubicType, 551 kLine_CubicType, 552 kPoint_CubicType 553 }; 554 555 // discr(I) = d0^2 * (3*d1^2 - 4*d0*d2) 556 // Classification: 557 // discr(I) > 0 Serpentine 558 // discr(I) = 0 Cusp 559 // discr(I) < 0 Loop 560 // d0 = d1 = 0 Quadratic 561 // d0 = d1 = d2 = 0 Line 562 // p0 = p1 = p2 = p3 Point 563 static CubicType classify_cubic(const SkPoint p[4], const SkScalar d[3]) { 564 if (p[0] == p[1] && p[0] == p[2] && p[0] == p[3]) { 565 return kPoint_CubicType; 566 } 567 const SkScalar discr = d[0] * d[0] * (3.f * d[1] * d[1] - 4.f * d[0] * d[2]); 568 if (discr > SK_ScalarNearlyZero) { 569 return kSerpentine_CubicType; 570 } else if (discr < -SK_ScalarNearlyZero) { 571 return kLoop_CubicType; 572 } else { 573 if (0.f == d[0] && 0.f == d[1]) { 574 return (0.f == d[2] ? kLine_CubicType : kQuadratic_CubicType); 575 } else { 576 return kCusp_CubicType; 577 } 578 } 579 } 580 581 // Assumes the third component of points is 1. 582 // Calcs p0 . (p1 x p2) 583 static SkScalar calc_dot_cross_cubic(const SkPoint& p0, const SkPoint& p1, const SkPoint& p2) { 584 const SkScalar xComp = p0.fX * (p1.fY - p2.fY); 585 const SkScalar yComp = p0.fY * (p2.fX - p1.fX); 586 const SkScalar wComp = p1.fX * p2.fY - p1.fY * p2.fX; 587 return (xComp + yComp + wComp); 588 } 589 590 // Solves linear system to extract klm 591 // P.K = k (similarly for l, m) 592 // Where P is matrix of control points 593 // K is coefficients for the line K 594 // k is vector of values of K evaluated at the control points 595 // Solving for K, thus K = P^(-1) . k 596 static void calc_cubic_klm(const SkPoint p[4], const SkScalar controlK[4], 597 const SkScalar controlL[4], const SkScalar controlM[4], 598 SkScalar k[3], SkScalar l[3], SkScalar m[3]) { 599 SkMatrix matrix; 600 matrix.setAll(p[0].fX, p[0].fY, 1.f, 601 p[1].fX, p[1].fY, 1.f, 602 p[2].fX, p[2].fY, 1.f); 603 SkMatrix inverse; 604 if (matrix.invert(&inverse)) { 605 inverse.mapHomogeneousPoints(k, controlK, 1); 606 inverse.mapHomogeneousPoints(l, controlL, 1); 607 inverse.mapHomogeneousPoints(m, controlM, 1); 608 } 609 610 } 611 612 static void set_serp_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { 613 SkScalar tempSqrt = SkScalarSqrt(9.f * d[1] * d[1] - 12.f * d[0] * d[2]); 614 SkScalar ls = 3.f * d[1] - tempSqrt; 615 SkScalar lt = 6.f * d[0]; 616 SkScalar ms = 3.f * d[1] + tempSqrt; 617 SkScalar mt = 6.f * d[0]; 618 619 k[0] = ls * ms; 620 k[1] = (3.f * ls * ms - ls * mt - lt * ms) / 3.f; 621 k[2] = (lt * (mt - 2.f * ms) + ls * (3.f * ms - 2.f * mt)) / 3.f; 622 k[3] = (lt - ls) * (mt - ms); 623 624 l[0] = ls * ls * ls; 625 const SkScalar lt_ls = lt - ls; 626 l[1] = ls * ls * lt_ls * -1.f; 627 l[2] = lt_ls * lt_ls * ls; 628 l[3] = -1.f * lt_ls * lt_ls * lt_ls; 629 630 m[0] = ms * ms * ms; 631 const SkScalar mt_ms = mt - ms; 632 m[1] = ms * ms * mt_ms * -1.f; 633 m[2] = mt_ms * mt_ms * ms; 634 m[3] = -1.f * mt_ms * mt_ms * mt_ms; 635 636 // If d0 < 0 we need to flip the orientation of our curve 637 // This is done by negating the k and l values 638 // We want negative distance values to be on the inside 639 if ( d[0] > 0) { 640 for (int i = 0; i < 4; ++i) { 641 k[i] = -k[i]; 642 l[i] = -l[i]; 643 } 644 } 645 } 646 647 static void set_loop_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { 648 SkScalar tempSqrt = SkScalarSqrt(4.f * d[0] * d[2] - 3.f * d[1] * d[1]); 649 SkScalar ls = d[1] - tempSqrt; 650 SkScalar lt = 2.f * d[0]; 651 SkScalar ms = d[1] + tempSqrt; 652 SkScalar mt = 2.f * d[0]; 653 654 k[0] = ls * ms; 655 k[1] = (3.f * ls*ms - ls * mt - lt * ms) / 3.f; 656 k[2] = (lt * (mt - 2.f * ms) + ls * (3.f * ms - 2.f * mt)) / 3.f; 657 k[3] = (lt - ls) * (mt - ms); 658 659 l[0] = ls * ls * ms; 660 l[1] = (ls * (ls * (mt - 3.f * ms) + 2.f * lt * ms))/-3.f; 661 l[2] = ((lt - ls) * (ls * (2.f * mt - 3.f * ms) + lt * ms))/3.f; 662 l[3] = -1.f * (lt - ls) * (lt - ls) * (mt - ms); 663 664 m[0] = ls * ms * ms; 665 m[1] = (ms * (ls * (2.f * mt - 3.f * ms) + lt * ms))/-3.f; 666 m[2] = ((mt - ms) * (ls * (mt - 3.f * ms) + 2.f * lt * ms))/3.f; 667 m[3] = -1.f * (lt - ls) * (mt - ms) * (mt - ms); 668 669 670 // If (d0 < 0 && sign(k1) > 0) || (d0 > 0 && sign(k1) < 0), 671 // we need to flip the orientation of our curve. 672 // This is done by negating the k and l values 673 if ( (d[0] < 0 && k[1] > 0) || (d[0] > 0 && k[1] < 0)) { 674 for (int i = 0; i < 4; ++i) { 675 k[i] = -k[i]; 676 l[i] = -l[i]; 677 } 678 } 679 } 680 681 static void set_cusp_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { 682 const SkScalar ls = d[2]; 683 const SkScalar lt = 3.f * d[1]; 684 685 k[0] = ls; 686 k[1] = ls - lt / 3.f; 687 k[2] = ls - 2.f * lt / 3.f; 688 k[3] = ls - lt; 689 690 l[0] = ls * ls * ls; 691 const SkScalar ls_lt = ls - lt; 692 l[1] = ls * ls * ls_lt; 693 l[2] = ls_lt * ls_lt * ls; 694 l[3] = ls_lt * ls_lt * ls_lt; 695 696 m[0] = 1.f; 697 m[1] = 1.f; 698 m[2] = 1.f; 699 m[3] = 1.f; 700 } 701 702 // For the case when a cubic is actually a quadratic 703 // M = 704 // 0 0 0 705 // 1/3 0 1/3 706 // 2/3 1/3 2/3 707 // 1 1 1 708 static void set_quadratic_klm(const SkScalar d[3], SkScalar k[4], SkScalar l[4], SkScalar m[4]) { 709 k[0] = 0.f; 710 k[1] = 1.f/3.f; 711 k[2] = 2.f/3.f; 712 k[3] = 1.f; 713 714 l[0] = 0.f; 715 l[1] = 0.f; 716 l[2] = 1.f/3.f; 717 l[3] = 1.f; 718 719 m[0] = 0.f; 720 m[1] = 1.f/3.f; 721 m[2] = 2.f/3.f; 722 m[3] = 1.f; 723 724 // If d2 < 0 we need to flip the orientation of our curve 725 // This is done by negating the k and l values 726 if ( d[2] > 0) { 727 for (int i = 0; i < 4; ++i) { 728 k[i] = -k[i]; 729 l[i] = -l[i]; 730 } 731 } 732 } 733 734 // Calc coefficients of I(s,t) where roots of I are inflection points of curve 735 // I(s,t) = t*(3*d0*s^2 - 3*d1*s*t + d2*t^2) 736 // d0 = a1 - 2*a2+3*a3 737 // d1 = -a2 + 3*a3 738 // d2 = 3*a3 739 // a1 = p0 . (p3 x p2) 740 // a2 = p1 . (p0 x p3) 741 // a3 = p2 . (p1 x p0) 742 // Places the values of d1, d2, d3 in array d passed in 743 static void calc_cubic_inflection_func(const SkPoint p[4], SkScalar d[3]) { 744 SkScalar a1 = calc_dot_cross_cubic(p[0], p[3], p[2]); 745 SkScalar a2 = calc_dot_cross_cubic(p[1], p[0], p[3]); 746 SkScalar a3 = calc_dot_cross_cubic(p[2], p[1], p[0]); 747 748 // need to scale a's or values in later calculations will grow to high 749 SkScalar max = SkScalarAbs(a1); 750 max = SkMaxScalar(max, SkScalarAbs(a2)); 751 max = SkMaxScalar(max, SkScalarAbs(a3)); 752 max = 1.f/max; 753 a1 = a1 * max; 754 a2 = a2 * max; 755 a3 = a3 * max; 756 757 d[2] = 3.f * a3; 758 d[1] = d[2] - a2; 759 d[0] = d[1] - a2 + a1; 760 } 761 762 int GrPathUtils::chopCubicAtLoopIntersection(const SkPoint src[4], SkPoint dst[10], SkScalar klm[9], 763 SkScalar klm_rev[3]) { 764 // Variable to store the two parametric values at the loop double point 765 SkScalar smallS = 0.f; 766 SkScalar largeS = 0.f; 767 768 SkScalar d[3]; 769 calc_cubic_inflection_func(src, d); 770 771 CubicType cType = classify_cubic(src, d); 772 773 int chop_count = 0; 774 if (kLoop_CubicType == cType) { 775 SkScalar tempSqrt = SkScalarSqrt(4.f * d[0] * d[2] - 3.f * d[1] * d[1]); 776 SkScalar ls = d[1] - tempSqrt; 777 SkScalar lt = 2.f * d[0]; 778 SkScalar ms = d[1] + tempSqrt; 779 SkScalar mt = 2.f * d[0]; 780 ls = ls / lt; 781 ms = ms / mt; 782 // need to have t values sorted since this is what is expected by SkChopCubicAt 783 if (ls <= ms) { 784 smallS = ls; 785 largeS = ms; 786 } else { 787 smallS = ms; 788 largeS = ls; 789 } 790 791 SkScalar chop_ts[2]; 792 if (smallS > 0.f && smallS < 1.f) { 793 chop_ts[chop_count++] = smallS; 794 } 795 if (largeS > 0.f && largeS < 1.f) { 796 chop_ts[chop_count++] = largeS; 797 } 798 if(dst) { 799 SkChopCubicAt(src, dst, chop_ts, chop_count); 800 } 801 } else { 802 if (dst) { 803 memcpy(dst, src, sizeof(SkPoint) * 4); 804 } 805 } 806 807 if (klm && klm_rev) { 808 // Set klm_rev to to match the sub_section of cubic that needs to have its orientation 809 // flipped. This will always be the section that is the "loop" 810 if (2 == chop_count) { 811 klm_rev[0] = 1.f; 812 klm_rev[1] = -1.f; 813 klm_rev[2] = 1.f; 814 } else if (1 == chop_count) { 815 if (smallS < 0.f) { 816 klm_rev[0] = -1.f; 817 klm_rev[1] = 1.f; 818 } else { 819 klm_rev[0] = 1.f; 820 klm_rev[1] = -1.f; 821 } 822 } else { 823 if (smallS < 0.f && largeS > 1.f) { 824 klm_rev[0] = -1.f; 825 } else { 826 klm_rev[0] = 1.f; 827 } 828 } 829 SkScalar controlK[4]; 830 SkScalar controlL[4]; 831 SkScalar controlM[4]; 832 833 if (kSerpentine_CubicType == cType || (kCusp_CubicType == cType && 0.f != d[0])) { 834 set_serp_klm(d, controlK, controlL, controlM); 835 } else if (kLoop_CubicType == cType) { 836 set_loop_klm(d, controlK, controlL, controlM); 837 } else if (kCusp_CubicType == cType) { 838 SkASSERT(0.f == d[0]); 839 set_cusp_klm(d, controlK, controlL, controlM); 840 } else if (kQuadratic_CubicType == cType) { 841 set_quadratic_klm(d, controlK, controlL, controlM); 842 } 843 844 calc_cubic_klm(src, controlK, controlL, controlM, klm, &klm[3], &klm[6]); 845 } 846 return chop_count + 1; 847 } 848 849 void GrPathUtils::getCubicKLM(const SkPoint p[4], SkScalar klm[9]) { 850 SkScalar d[3]; 851 calc_cubic_inflection_func(p, d); 852 853 CubicType cType = classify_cubic(p, d); 854 855 SkScalar controlK[4]; 856 SkScalar controlL[4]; 857 SkScalar controlM[4]; 858 859 if (kSerpentine_CubicType == cType || (kCusp_CubicType == cType && 0.f != d[0])) { 860 set_serp_klm(d, controlK, controlL, controlM); 861 } else if (kLoop_CubicType == cType) { 862 set_loop_klm(d, controlK, controlL, controlM); 863 } else if (kCusp_CubicType == cType) { 864 SkASSERT(0.f == d[0]); 865 set_cusp_klm(d, controlK, controlL, controlM); 866 } else if (kQuadratic_CubicType == cType) { 867 set_quadratic_klm(d, controlK, controlL, controlM); 868 } 869 870 calc_cubic_klm(p, controlK, controlL, controlM, klm, &klm[3], &klm[6]); 871 } 872
