caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 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" |
commit-bot@chromium.org | 2db7fe7 | 2014-05-07 15:31:40 +0000 | [diff] [blame] | 12 | #include "SkTSort.h" |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 13 | |
| 14 | const int SkDCubic::gPrecisionUnit = 256; // FIXME: test different values in test framework |
| 15 | |
commit-bot@chromium.org | 2db7fe7 | 2014-05-07 15:31:40 +0000 | [diff] [blame] | 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 | |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 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 | |
commit-bot@chromium.org | 2db7fe7 | 2014-05-07 15:31:40 +0000 | [diff] [blame] | 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 | |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 166 | bool SkDCubic::serpentine() const { |
commit-bot@chromium.org | 8cb1daa | 2014-04-25 12:59:11 +0000 | [diff] [blame] | 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 |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 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); |
caryclark@google.com | 570863f | 2013-09-16 15:55:01 +0000 | [diff] [blame] | 207 | SkPathOpsDebug::MathematicaIze(str, sizeof(str)); |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 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) { |
caryclark@google.com | 7eaa53d | 2013-10-02 14:49:34 +0000 | [diff] [blame] | 233 | if (AlmostDequalUlps(s[i], 1)) { |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 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; |
caryclark@google.com | 7eaa53d | 2013-10-02 14:49:34 +0000 | [diff] [blame] | 264 | if (!AlmostDequalUlps(s[0], r)) { |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 265 | *roots++ = r; |
| 266 | } |
| 267 | r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3; |
caryclark@google.com | 7eaa53d | 2013-10-02 14:49:34 +0000 | [diff] [blame] | 268 | if (!AlmostDequalUlps(s[0], r) && (roots - s == 1 || !AlmostDequalUlps(s[1], r))) { |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 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; |
commit-bot@chromium.org | 2db7fe7 | 2014-05-07 15:31:40 +0000 | [diff] [blame] | 283 | if (AlmostDequalUlps((double) R2, (double) Q3)) { |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 284 | r = -A / 2 - adiv3; |
caryclark@google.com | 7eaa53d | 2013-10-02 14:49:34 +0000 | [diff] [blame] | 285 | if (!AlmostDequalUlps(s[0], r)) { |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 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]; |
caryclark@google.com | 4fdbb22 | 2013-07-23 15:27:41 +0000 | [diff] [blame] | 382 | SkDPoint mid = ptAtT(t); |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 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 | |
caryclark@google.com | 4fdbb22 | 2013-07-23 15:27:41 +0000 | [diff] [blame] | 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 | } |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 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 { |
caryclark@google.com | d892bd8 | 2013-06-17 14:10:36 +0000 | [diff] [blame] | 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; |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 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; |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 492 | // FIXME: call align() ? |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 493 | return dst; |
| 494 | } |
| 495 | |
skia.committer@gmail.com | 8f6ef40 | 2013-06-05 07:01:06 +0000 | [diff] [blame] | 496 | void SkDCubic::align(int endIndex, int ctrlIndex, SkDPoint* dstPt) const { |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 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 | |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 505 | void SkDCubic::subDivide(const SkDPoint& a, const SkDPoint& d, |
| 506 | double t1, double t2, SkDPoint dst[2]) const { |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 507 | SkASSERT(t1 != t2); |
| 508 | #if 0 |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 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; |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 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 | } |
commit-bot@chromium.org | 4431e77 | 2014-04-14 17:08:59 +0000 | [diff] [blame] | 533 | if (AlmostBequalUlps(dst[0].fX, a.fX)) { |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 534 | dst[0].fX = a.fX; |
| 535 | } |
commit-bot@chromium.org | 4431e77 | 2014-04-14 17:08:59 +0000 | [diff] [blame] | 536 | if (AlmostBequalUlps(dst[0].fY, a.fY)) { |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 537 | dst[0].fY = a.fY; |
| 538 | } |
commit-bot@chromium.org | 4431e77 | 2014-04-14 17:08:59 +0000 | [diff] [blame] | 539 | if (AlmostBequalUlps(dst[1].fX, d.fX)) { |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 540 | dst[1].fX = d.fX; |
| 541 | } |
commit-bot@chromium.org | 4431e77 | 2014-04-14 17:08:59 +0000 | [diff] [blame] | 542 | if (AlmostBequalUlps(dst[1].fY, d.fY)) { |
caryclark@google.com | cffbcc3 | 2013-06-04 17:59:42 +0000 | [diff] [blame] | 543 | dst[1].fY = d.fY; |
| 544 | } |
caryclark@google.com | 07393ca | 2013-04-08 11:47:37 +0000 | [diff] [blame] | 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 | } |