1 /* 2 * Copyright 2012 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 #include "SkLineParameters.h" 8 #include "SkPathOpsCubic.h" 9 #include "SkPathOpsLine.h" 10 #include "SkPathOpsQuad.h" 11 #include "SkPathOpsRect.h" 12 #include "SkTSort.h" 13 14 const int SkDCubic::gPrecisionUnit = 256; // FIXME: test different values in test framework 15 16 // give up when changing t no longer moves point 17 // also, copy point rather than recompute it when it does change 18 double SkDCubic::binarySearch(double min, double max, double axisIntercept, 19 SearchAxis xAxis) const { 20 double t = (min + max) / 2; 21 double step = (t - min) / 2; 22 SkDPoint cubicAtT = ptAtT(t); 23 double calcPos = (&cubicAtT.fX)[xAxis]; 24 double calcDist = calcPos - axisIntercept; 25 do { 26 double priorT = t - step; 27 SkASSERT(priorT >= min); 28 SkDPoint lessPt = ptAtT(priorT); 29 if (approximately_equal(lessPt.fX, cubicAtT.fX) 30 && approximately_equal(lessPt.fY, cubicAtT.fY)) { 31 return -1; // binary search found no point at this axis intercept 32 } 33 double lessDist = (&lessPt.fX)[xAxis] - axisIntercept; 34 #if DEBUG_CUBIC_BINARY_SEARCH 35 SkDebugf("t=%1.9g calc=%1.9g dist=%1.9g step=%1.9g less=%1.9g\n", t, calcPos, calcDist, 36 step, lessDist); 37 #endif 38 double lastStep = step; 39 step /= 2; 40 if (calcDist > 0 ? calcDist > lessDist : calcDist < lessDist) { 41 t = priorT; 42 } else { 43 double nextT = t + lastStep; 44 SkASSERT(nextT <= max); 45 SkDPoint morePt = ptAtT(nextT); 46 if (approximately_equal(morePt.fX, cubicAtT.fX) 47 && approximately_equal(morePt.fY, cubicAtT.fY)) { 48 return -1; // binary search found no point at this axis intercept 49 } 50 double moreDist = (&morePt.fX)[xAxis] - axisIntercept; 51 if (calcDist > 0 ? calcDist <= moreDist : calcDist >= moreDist) { 52 continue; 53 } 54 t = nextT; 55 } 56 SkDPoint testAtT = ptAtT(t); 57 cubicAtT = testAtT; 58 calcPos = (&cubicAtT.fX)[xAxis]; 59 calcDist = calcPos - axisIntercept; 60 } while (!approximately_equal(calcPos, axisIntercept)); 61 return t; 62 } 63 64 // FIXME: cache keep the bounds and/or precision with the caller? 65 double SkDCubic::calcPrecision() const { 66 SkDRect dRect; 67 dRect.setBounds(*this); // OPTIMIZATION: just use setRawBounds ? 68 double width = dRect.fRight - dRect.fLeft; 69 double height = dRect.fBottom - dRect.fTop; 70 return (width > height ? width : height) / gPrecisionUnit; 71 } 72 73 bool SkDCubic::clockwise() const { 74 double sum = (fPts[0].fX - fPts[3].fX) * (fPts[0].fY + fPts[3].fY); 75 for (int idx = 0; idx < 3; ++idx) { 76 sum += (fPts[idx + 1].fX - fPts[idx].fX) * (fPts[idx + 1].fY + fPts[idx].fY); 77 } 78 return sum <= 0; 79 } 80 81 void SkDCubic::Coefficients(const double* src, double* A, double* B, double* C, double* D) { 82 *A = src[6]; // d 83 *B = src[4] * 3; // 3*c 84 *C = src[2] * 3; // 3*b 85 *D = src[0]; // a 86 *A -= *D - *C + *B; // A = -a + 3*b - 3*c + d 87 *B += 3 * *D - 2 * *C; // B = 3*a - 6*b + 3*c 88 *C -= 3 * *D; // C = -3*a + 3*b 89 } 90 91 bool SkDCubic::controlsContainedByEnds() const { 92 SkDVector startTan = fPts[1] - fPts[0]; 93 if (startTan.fX == 0 && startTan.fY == 0) { 94 startTan = fPts[2] - fPts[0]; 95 } 96 SkDVector endTan = fPts[2] - fPts[3]; 97 if (endTan.fX == 0 && endTan.fY == 0) { 98 endTan = fPts[1] - fPts[3]; 99 } 100 if (startTan.dot(endTan) >= 0) { 101 return false; 102 } 103 SkDLine startEdge = {{fPts[0], fPts[0]}}; 104 startEdge[1].fX -= startTan.fY; 105 startEdge[1].fY += startTan.fX; 106 SkDLine endEdge = {{fPts[3], fPts[3]}}; 107 endEdge[1].fX -= endTan.fY; 108 endEdge[1].fY += endTan.fX; 109 double leftStart1 = startEdge.isLeft(fPts[1]); 110 if (leftStart1 * startEdge.isLeft(fPts[2]) < 0) { 111 return false; 112 } 113 double leftEnd1 = endEdge.isLeft(fPts[1]); 114 if (leftEnd1 * endEdge.isLeft(fPts[2]) < 0) { 115 return false; 116 } 117 return leftStart1 * leftEnd1 >= 0; 118 } 119 120 bool SkDCubic::endsAreExtremaInXOrY() const { 121 return (between(fPts[0].fX, fPts[1].fX, fPts[3].fX) 122 && between(fPts[0].fX, fPts[2].fX, fPts[3].fX)) 123 || (between(fPts[0].fY, fPts[1].fY, fPts[3].fY) 124 && between(fPts[0].fY, fPts[2].fY, fPts[3].fY)); 125 } 126 127 bool SkDCubic::isLinear(int startIndex, int endIndex) const { 128 SkLineParameters lineParameters; 129 lineParameters.cubicEndPoints(*this, startIndex, endIndex); 130 // FIXME: maybe it's possible to avoid this and compare non-normalized 131 lineParameters.normalize(); 132 double distance = lineParameters.controlPtDistance(*this, 1); 133 if (!approximately_zero(distance)) { 134 return false; 135 } 136 distance = lineParameters.controlPtDistance(*this, 2); 137 return approximately_zero(distance); 138 } 139 140 bool SkDCubic::monotonicInY() const { 141 return between(fPts[0].fY, fPts[1].fY, fPts[3].fY) 142 && between(fPts[0].fY, fPts[2].fY, fPts[3].fY); 143 } 144 145 int SkDCubic::searchRoots(double extremeTs[6], int extrema, double axisIntercept, 146 SearchAxis xAxis, double* validRoots) const { 147 extrema += findInflections(&extremeTs[extrema]); 148 extremeTs[extrema++] = 0; 149 extremeTs[extrema] = 1; 150 SkTQSort(extremeTs, extremeTs + extrema); 151 int validCount = 0; 152 for (int index = 0; index < extrema; ) { 153 double min = extremeTs[index]; 154 double max = extremeTs[++index]; 155 if (min == max) { 156 continue; 157 } 158 double newT = binarySearch(min, max, axisIntercept, xAxis); 159 if (newT >= 0) { 160 validRoots[validCount++] = newT; 161 } 162 } 163 return validCount; 164 } 165 166 bool SkDCubic::serpentine() const { 167 #if 0 // FIXME: enabling this fixes cubicOp114 but breaks cubicOp58d and cubicOp53d 168 double tValues[2]; 169 // OPTIMIZATION : another case where caching the present of cubic inflections would be useful 170 return findInflections(tValues) > 1; 171 #endif 172 if (!controlsContainedByEnds()) { 173 return false; 174 } 175 double wiggle = (fPts[0].fX - fPts[2].fX) * (fPts[0].fY + fPts[2].fY); 176 for (int idx = 0; idx < 2; ++idx) { 177 wiggle += (fPts[idx + 1].fX - fPts[idx].fX) * (fPts[idx + 1].fY + fPts[idx].fY); 178 } 179 double waggle = (fPts[1].fX - fPts[3].fX) * (fPts[1].fY + fPts[3].fY); 180 for (int idx = 1; idx < 3; ++idx) { 181 waggle += (fPts[idx + 1].fX - fPts[idx].fX) * (fPts[idx + 1].fY + fPts[idx].fY); 182 } 183 return wiggle * waggle < 0; 184 } 185 186 // cubic roots 187 188 static const double PI = 3.141592653589793; 189 190 // from SkGeometry.cpp (and Numeric Solutions, 5.6) 191 int SkDCubic::RootsValidT(double A, double B, double C, double D, double t[3]) { 192 double s[3]; 193 int realRoots = RootsReal(A, B, C, D, s); 194 int foundRoots = SkDQuad::AddValidTs(s, realRoots, t); 195 return foundRoots; 196 } 197 198 int SkDCubic::RootsReal(double A, double B, double C, double D, double s[3]) { 199 #ifdef SK_DEBUG 200 // create a string mathematica understands 201 // GDB set print repe 15 # if repeated digits is a bother 202 // set print elements 400 # if line doesn't fit 203 char str[1024]; 204 sk_bzero(str, sizeof(str)); 205 SK_SNPRINTF(str, sizeof(str), "Solve[%1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]", 206 A, B, C, D); 207 SkPathOpsDebug::MathematicaIze(str, sizeof(str)); 208 #if ONE_OFF_DEBUG && ONE_OFF_DEBUG_MATHEMATICA 209 SkDebugf("%s\n", str); 210 #endif 211 #endif 212 if (approximately_zero(A) 213 && approximately_zero_when_compared_to(A, B) 214 && approximately_zero_when_compared_to(A, C) 215 && approximately_zero_when_compared_to(A, D)) { // we're just a quadratic 216 return SkDQuad::RootsReal(B, C, D, s); 217 } 218 if (approximately_zero_when_compared_to(D, A) 219 && approximately_zero_when_compared_to(D, B) 220 && approximately_zero_when_compared_to(D, C)) { // 0 is one root 221 int num = SkDQuad::RootsReal(A, B, C, s); 222 for (int i = 0; i < num; ++i) { 223 if (approximately_zero(s[i])) { 224 return num; 225 } 226 } 227 s[num++] = 0; 228 return num; 229 } 230 if (approximately_zero(A + B + C + D)) { // 1 is one root 231 int num = SkDQuad::RootsReal(A, A + B, -D, s); 232 for (int i = 0; i < num; ++i) { 233 if (AlmostDequalUlps(s[i], 1)) { 234 return num; 235 } 236 } 237 s[num++] = 1; 238 return num; 239 } 240 double a, b, c; 241 { 242 double invA = 1 / A; 243 a = B * invA; 244 b = C * invA; 245 c = D * invA; 246 } 247 double a2 = a * a; 248 double Q = (a2 - b * 3) / 9; 249 double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54; 250 double R2 = R * R; 251 double Q3 = Q * Q * Q; 252 double R2MinusQ3 = R2 - Q3; 253 double adiv3 = a / 3; 254 double r; 255 double* roots = s; 256 if (R2MinusQ3 < 0) { // we have 3 real roots 257 double theta = acos(R / sqrt(Q3)); 258 double neg2RootQ = -2 * sqrt(Q); 259 260 r = neg2RootQ * cos(theta / 3) - adiv3; 261 *roots++ = r; 262 263 r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3; 264 if (!AlmostDequalUlps(s[0], r)) { 265 *roots++ = r; 266 } 267 r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3; 268 if (!AlmostDequalUlps(s[0], r) && (roots - s == 1 || !AlmostDequalUlps(s[1], r))) { 269 *roots++ = r; 270 } 271 } else { // we have 1 real root 272 double sqrtR2MinusQ3 = sqrt(R2MinusQ3); 273 double A = fabs(R) + sqrtR2MinusQ3; 274 A = SkDCubeRoot(A); 275 if (R > 0) { 276 A = -A; 277 } 278 if (A != 0) { 279 A += Q / A; 280 } 281 r = A - adiv3; 282 *roots++ = r; 283 if (AlmostDequalUlps((double) R2, (double) Q3)) { 284 r = -A / 2 - adiv3; 285 if (!AlmostDequalUlps(s[0], r)) { 286 *roots++ = r; 287 } 288 } 289 } 290 return static_cast<int>(roots - s); 291 } 292 293 // from http://www.cs.sunysb.edu/~qin/courses/geometry/4.pdf 294 // c(t) = a(1-t)^3 + 3bt(1-t)^2 + 3c(1-t)t^2 + dt^3 295 // c'(t) = -3a(1-t)^2 + 3b((1-t)^2 - 2t(1-t)) + 3c(2t(1-t) - t^2) + 3dt^2 296 // = 3(b-a)(1-t)^2 + 6(c-b)t(1-t) + 3(d-c)t^2 297 static double derivative_at_t(const double* src, double t) { 298 double one_t = 1 - t; 299 double a = src[0]; 300 double b = src[2]; 301 double c = src[4]; 302 double d = src[6]; 303 return 3 * ((b - a) * one_t * one_t + 2 * (c - b) * t * one_t + (d - c) * t * t); 304 } 305 306 // OPTIMIZE? compute t^2, t(1-t), and (1-t)^2 and pass them to another version of derivative at t? 307 SkDVector SkDCubic::dxdyAtT(double t) const { 308 SkDVector result = { derivative_at_t(&fPts[0].fX, t), derivative_at_t(&fPts[0].fY, t) }; 309 return result; 310 } 311 312 // OPTIMIZE? share code with formulate_F1DotF2 313 int SkDCubic::findInflections(double tValues[]) const { 314 double Ax = fPts[1].fX - fPts[0].fX; 315 double Ay = fPts[1].fY - fPts[0].fY; 316 double Bx = fPts[2].fX - 2 * fPts[1].fX + fPts[0].fX; 317 double By = fPts[2].fY - 2 * fPts[1].fY + fPts[0].fY; 318 double Cx = fPts[3].fX + 3 * (fPts[1].fX - fPts[2].fX) - fPts[0].fX; 319 double Cy = fPts[3].fY + 3 * (fPts[1].fY - fPts[2].fY) - fPts[0].fY; 320 return SkDQuad::RootsValidT(Bx * Cy - By * Cx, Ax * Cy - Ay * Cx, Ax * By - Ay * Bx, tValues); 321 } 322 323 static void formulate_F1DotF2(const double src[], double coeff[4]) { 324 double a = src[2] - src[0]; 325 double b = src[4] - 2 * src[2] + src[0]; 326 double c = src[6] + 3 * (src[2] - src[4]) - src[0]; 327 coeff[0] = c * c; 328 coeff[1] = 3 * b * c; 329 coeff[2] = 2 * b * b + c * a; 330 coeff[3] = a * b; 331 } 332 333 /** SkDCubic'(t) = At^2 + Bt + C, where 334 A = 3(-a + 3(b - c) + d) 335 B = 6(a - 2b + c) 336 C = 3(b - a) 337 Solve for t, keeping only those that fit between 0 < t < 1 338 */ 339 int SkDCubic::FindExtrema(double a, double b, double c, double d, double tValues[2]) { 340 // we divide A,B,C by 3 to simplify 341 double A = d - a + 3*(b - c); 342 double B = 2*(a - b - b + c); 343 double C = b - a; 344 345 return SkDQuad::RootsValidT(A, B, C, tValues); 346 } 347 348 /* from SkGeometry.cpp 349 Looking for F' dot F'' == 0 350 351 A = b - a 352 B = c - 2b + a 353 C = d - 3c + 3b - a 354 355 F' = 3Ct^2 + 6Bt + 3A 356 F'' = 6Ct + 6B 357 358 F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 359 */ 360 int SkDCubic::findMaxCurvature(double tValues[]) const { 361 double coeffX[4], coeffY[4]; 362 int i; 363 formulate_F1DotF2(&fPts[0].fX, coeffX); 364 formulate_F1DotF2(&fPts[0].fY, coeffY); 365 for (i = 0; i < 4; i++) { 366 coeffX[i] = coeffX[i] + coeffY[i]; 367 } 368 return RootsValidT(coeffX[0], coeffX[1], coeffX[2], coeffX[3], tValues); 369 } 370 371 SkDPoint SkDCubic::top(double startT, double endT) const { 372 SkDCubic sub = subDivide(startT, endT); 373 SkDPoint topPt = sub[0]; 374 if (topPt.fY > sub[3].fY || (topPt.fY == sub[3].fY && topPt.fX > sub[3].fX)) { 375 topPt = sub[3]; 376 } 377 double extremeTs[2]; 378 if (!sub.monotonicInY()) { 379 int roots = FindExtrema(sub[0].fY, sub[1].fY, sub[2].fY, sub[3].fY, extremeTs); 380 for (int index = 0; index < roots; ++index) { 381 double t = startT + (endT - startT) * extremeTs[index]; 382 SkDPoint mid = ptAtT(t); 383 if (topPt.fY > mid.fY || (topPt.fY == mid.fY && topPt.fX > mid.fX)) { 384 topPt = mid; 385 } 386 } 387 } 388 return topPt; 389 } 390 391 SkDPoint SkDCubic::ptAtT(double t) const { 392 if (0 == t) { 393 return fPts[0]; 394 } 395 if (1 == t) { 396 return fPts[3]; 397 } 398 double one_t = 1 - t; 399 double one_t2 = one_t * one_t; 400 double a = one_t2 * one_t; 401 double b = 3 * one_t2 * t; 402 double t2 = t * t; 403 double c = 3 * one_t * t2; 404 double d = t2 * t; 405 SkDPoint result = {a * fPts[0].fX + b * fPts[1].fX + c * fPts[2].fX + d * fPts[3].fX, 406 a * fPts[0].fY + b * fPts[1].fY + c * fPts[2].fY + d * fPts[3].fY}; 407 return result; 408 } 409 410 /* 411 Given a cubic c, t1, and t2, find a small cubic segment. 412 413 The new cubic is defined as points A, B, C, and D, where 414 s1 = 1 - t1 415 s2 = 1 - t2 416 A = c[0]*s1*s1*s1 + 3*c[1]*s1*s1*t1 + 3*c[2]*s1*t1*t1 + c[3]*t1*t1*t1 417 D = c[0]*s2*s2*s2 + 3*c[1]*s2*s2*t2 + 3*c[2]*s2*t2*t2 + c[3]*t2*t2*t2 418 419 We don't have B or C. So We define two equations to isolate them. 420 First, compute two reference T values 1/3 and 2/3 from t1 to t2: 421 422 c(at (2*t1 + t2)/3) == E 423 c(at (t1 + 2*t2)/3) == F 424 425 Next, compute where those values must be if we know the values of B and C: 426 427 _12 = A*2/3 + B*1/3 428 12_ = A*1/3 + B*2/3 429 _23 = B*2/3 + C*1/3 430 23_ = B*1/3 + C*2/3 431 _34 = C*2/3 + D*1/3 432 34_ = C*1/3 + D*2/3 433 _123 = (A*2/3 + B*1/3)*2/3 + (B*2/3 + C*1/3)*1/3 = A*4/9 + B*4/9 + C*1/9 434 123_ = (A*1/3 + B*2/3)*1/3 + (B*1/3 + C*2/3)*2/3 = A*1/9 + B*4/9 + C*4/9 435 _234 = (B*2/3 + C*1/3)*2/3 + (C*2/3 + D*1/3)*1/3 = B*4/9 + C*4/9 + D*1/9 436 234_ = (B*1/3 + C*2/3)*1/3 + (C*1/3 + D*2/3)*2/3 = B*1/9 + C*4/9 + D*4/9 437 _1234 = (A*4/9 + B*4/9 + C*1/9)*2/3 + (B*4/9 + C*4/9 + D*1/9)*1/3 438 = A*8/27 + B*12/27 + C*6/27 + D*1/27 439 = E 440 1234_ = (A*1/9 + B*4/9 + C*4/9)*1/3 + (B*1/9 + C*4/9 + D*4/9)*2/3 441 = A*1/27 + B*6/27 + C*12/27 + D*8/27 442 = F 443 E*27 = A*8 + B*12 + C*6 + D 444 F*27 = A + B*6 + C*12 + D*8 445 446 Group the known values on one side: 447 448 M = E*27 - A*8 - D = B*12 + C* 6 449 N = F*27 - A - D*8 = B* 6 + C*12 450 M*2 - N = B*18 451 N*2 - M = C*18 452 B = (M*2 - N)/18 453 C = (N*2 - M)/18 454 */ 455 456 static double interp_cubic_coords(const double* src, double t) { 457 double ab = SkDInterp(src[0], src[2], t); 458 double bc = SkDInterp(src[2], src[4], t); 459 double cd = SkDInterp(src[4], src[6], t); 460 double abc = SkDInterp(ab, bc, t); 461 double bcd = SkDInterp(bc, cd, t); 462 double abcd = SkDInterp(abc, bcd, t); 463 return abcd; 464 } 465 466 SkDCubic SkDCubic::subDivide(double t1, double t2) const { 467 if (t1 == 0 || t2 == 1) { 468 if (t1 == 0 && t2 == 1) { 469 return *this; 470 } 471 SkDCubicPair pair = chopAt(t1 == 0 ? t2 : t1); 472 SkDCubic dst = t1 == 0 ? pair.first() : pair.second(); 473 return dst; 474 } 475 SkDCubic dst; 476 double ax = dst[0].fX = interp_cubic_coords(&fPts[0].fX, t1); 477 double ay = dst[0].fY = interp_cubic_coords(&fPts[0].fY, t1); 478 double ex = interp_cubic_coords(&fPts[0].fX, (t1*2+t2)/3); 479 double ey = interp_cubic_coords(&fPts[0].fY, (t1*2+t2)/3); 480 double fx = interp_cubic_coords(&fPts[0].fX, (t1+t2*2)/3); 481 double fy = interp_cubic_coords(&fPts[0].fY, (t1+t2*2)/3); 482 double dx = dst[3].fX = interp_cubic_coords(&fPts[0].fX, t2); 483 double dy = dst[3].fY = interp_cubic_coords(&fPts[0].fY, t2); 484 double mx = ex * 27 - ax * 8 - dx; 485 double my = ey * 27 - ay * 8 - dy; 486 double nx = fx * 27 - ax - dx * 8; 487 double ny = fy * 27 - ay - dy * 8; 488 /* bx = */ dst[1].fX = (mx * 2 - nx) / 18; 489 /* by = */ dst[1].fY = (my * 2 - ny) / 18; 490 /* cx = */ dst[2].fX = (nx * 2 - mx) / 18; 491 /* cy = */ dst[2].fY = (ny * 2 - my) / 18; 492 // FIXME: call align() ? 493 return dst; 494 } 495 496 void SkDCubic::align(int endIndex, int ctrlIndex, SkDPoint* dstPt) const { 497 if (fPts[endIndex].fX == fPts[ctrlIndex].fX) { 498 dstPt->fX = fPts[endIndex].fX; 499 } 500 if (fPts[endIndex].fY == fPts[ctrlIndex].fY) { 501 dstPt->fY = fPts[endIndex].fY; 502 } 503 } 504 505 void SkDCubic::subDivide(const SkDPoint& a, const SkDPoint& d, 506 double t1, double t2, SkDPoint dst[2]) const { 507 SkASSERT(t1 != t2); 508 #if 0 509 double ex = interp_cubic_coords(&fPts[0].fX, (t1 * 2 + t2) / 3); 510 double ey = interp_cubic_coords(&fPts[0].fY, (t1 * 2 + t2) / 3); 511 double fx = interp_cubic_coords(&fPts[0].fX, (t1 + t2 * 2) / 3); 512 double fy = interp_cubic_coords(&fPts[0].fY, (t1 + t2 * 2) / 3); 513 double mx = ex * 27 - a.fX * 8 - d.fX; 514 double my = ey * 27 - a.fY * 8 - d.fY; 515 double nx = fx * 27 - a.fX - d.fX * 8; 516 double ny = fy * 27 - a.fY - d.fY * 8; 517 /* bx = */ dst[0].fX = (mx * 2 - nx) / 18; 518 /* by = */ dst[0].fY = (my * 2 - ny) / 18; 519 /* cx = */ dst[1].fX = (nx * 2 - mx) / 18; 520 /* cy = */ dst[1].fY = (ny * 2 - my) / 18; 521 #else 522 // this approach assumes that the control points computed directly are accurate enough 523 SkDCubic sub = subDivide(t1, t2); 524 dst[0] = sub[1] + (a - sub[0]); 525 dst[1] = sub[2] + (d - sub[3]); 526 #endif 527 if (t1 == 0 || t2 == 0) { 528 align(0, 1, t1 == 0 ? &dst[0] : &dst[1]); 529 } 530 if (t1 == 1 || t2 == 1) { 531 align(3, 2, t1 == 1 ? &dst[0] : &dst[1]); 532 } 533 if (AlmostBequalUlps(dst[0].fX, a.fX)) { 534 dst[0].fX = a.fX; 535 } 536 if (AlmostBequalUlps(dst[0].fY, a.fY)) { 537 dst[0].fY = a.fY; 538 } 539 if (AlmostBequalUlps(dst[1].fX, d.fX)) { 540 dst[1].fX = d.fX; 541 } 542 if (AlmostBequalUlps(dst[1].fY, d.fY)) { 543 dst[1].fY = d.fY; 544 } 545 } 546 547 /* classic one t subdivision */ 548 static void interp_cubic_coords(const double* src, double* dst, double t) { 549 double ab = SkDInterp(src[0], src[2], t); 550 double bc = SkDInterp(src[2], src[4], t); 551 double cd = SkDInterp(src[4], src[6], t); 552 double abc = SkDInterp(ab, bc, t); 553 double bcd = SkDInterp(bc, cd, t); 554 double abcd = SkDInterp(abc, bcd, t); 555 556 dst[0] = src[0]; 557 dst[2] = ab; 558 dst[4] = abc; 559 dst[6] = abcd; 560 dst[8] = bcd; 561 dst[10] = cd; 562 dst[12] = src[6]; 563 } 564 565 SkDCubicPair SkDCubic::chopAt(double t) const { 566 SkDCubicPair dst; 567 if (t == 0.5) { 568 dst.pts[0] = fPts[0]; 569 dst.pts[1].fX = (fPts[0].fX + fPts[1].fX) / 2; 570 dst.pts[1].fY = (fPts[0].fY + fPts[1].fY) / 2; 571 dst.pts[2].fX = (fPts[0].fX + 2 * fPts[1].fX + fPts[2].fX) / 4; 572 dst.pts[2].fY = (fPts[0].fY + 2 * fPts[1].fY + fPts[2].fY) / 4; 573 dst.pts[3].fX = (fPts[0].fX + 3 * (fPts[1].fX + fPts[2].fX) + fPts[3].fX) / 8; 574 dst.pts[3].fY = (fPts[0].fY + 3 * (fPts[1].fY + fPts[2].fY) + fPts[3].fY) / 8; 575 dst.pts[4].fX = (fPts[1].fX + 2 * fPts[2].fX + fPts[3].fX) / 4; 576 dst.pts[4].fY = (fPts[1].fY + 2 * fPts[2].fY + fPts[3].fY) / 4; 577 dst.pts[5].fX = (fPts[2].fX + fPts[3].fX) / 2; 578 dst.pts[5].fY = (fPts[2].fY + fPts[3].fY) / 2; 579 dst.pts[6] = fPts[3]; 580 return dst; 581 } 582 interp_cubic_coords(&fPts[0].fX, &dst.pts[0].fX, t); 583 interp_cubic_coords(&fPts[0].fY, &dst.pts[0].fY, t); 584 return dst; 585 } 586