107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com/* 207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com * Copyright 2012 Google Inc. 307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com * 407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com * Use of this source code is governed by a BSD-style license that can be 507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com * found in the LICENSE file. 607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com */ 707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com#include "SkLineParameters.h" 807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com#include "SkPathOpsCubic.h" 907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com#include "SkPathOpsLine.h" 1007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com#include "SkPathOpsQuad.h" 1107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com#include "SkPathOpsRect.h" 122db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org#include "SkTSort.h" 1307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 1407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comconst int SkDCubic::gPrecisionUnit = 256; // FIXME: test different values in test framework 1507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 162db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org// give up when changing t no longer moves point 172db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org// also, copy point rather than recompute it when it does change 182db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.orgdouble SkDCubic::binarySearch(double min, double max, double axisIntercept, 192db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org SearchAxis xAxis) const { 202db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org double t = (min + max) / 2; 212db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org double step = (t - min) / 2; 222db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org SkDPoint cubicAtT = ptAtT(t); 232db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org double calcPos = (&cubicAtT.fX)[xAxis]; 242db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org double calcDist = calcPos - axisIntercept; 252db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org do { 262db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org double priorT = t - step; 272db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org SkASSERT(priorT >= min); 282db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org SkDPoint lessPt = ptAtT(priorT); 292db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org if (approximately_equal(lessPt.fX, cubicAtT.fX) 302db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org && approximately_equal(lessPt.fY, cubicAtT.fY)) { 312db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org return -1; // binary search found no point at this axis intercept 322db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org } 332db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org double lessDist = (&lessPt.fX)[xAxis] - axisIntercept; 342db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org#if DEBUG_CUBIC_BINARY_SEARCH 352db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org SkDebugf("t=%1.9g calc=%1.9g dist=%1.9g step=%1.9g less=%1.9g\n", t, calcPos, calcDist, 362db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org step, lessDist); 372db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org#endif 382db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org double lastStep = step; 392db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org step /= 2; 402db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org if (calcDist > 0 ? calcDist > lessDist : calcDist < lessDist) { 412db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org t = priorT; 422db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org } else { 432db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org double nextT = t + lastStep; 442db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org SkASSERT(nextT <= max); 452db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org SkDPoint morePt = ptAtT(nextT); 462db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org if (approximately_equal(morePt.fX, cubicAtT.fX) 472db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org && approximately_equal(morePt.fY, cubicAtT.fY)) { 482db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org return -1; // binary search found no point at this axis intercept 492db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org } 502db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org double moreDist = (&morePt.fX)[xAxis] - axisIntercept; 512db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org if (calcDist > 0 ? calcDist <= moreDist : calcDist >= moreDist) { 522db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org continue; 532db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org } 542db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org t = nextT; 552db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org } 562db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org SkDPoint testAtT = ptAtT(t); 572db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org cubicAtT = testAtT; 582db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org calcPos = (&cubicAtT.fX)[xAxis]; 592db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org calcDist = calcPos - axisIntercept; 602db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org } while (!approximately_equal(calcPos, axisIntercept)); 612db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org return t; 622db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org} 632db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org 6407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// FIXME: cache keep the bounds and/or precision with the caller? 6507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comdouble SkDCubic::calcPrecision() const { 6607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDRect dRect; 6707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dRect.setBounds(*this); // OPTIMIZATION: just use setRawBounds ? 6807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double width = dRect.fRight - dRect.fLeft; 6907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double height = dRect.fBottom - dRect.fTop; 7007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return (width > height ? width : height) / gPrecisionUnit; 7107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 7207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 7307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.combool SkDCubic::clockwise() const { 7407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double sum = (fPts[0].fX - fPts[3].fX) * (fPts[0].fY + fPts[3].fY); 7507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com for (int idx = 0; idx < 3; ++idx) { 7607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com sum += (fPts[idx + 1].fX - fPts[idx].fX) * (fPts[idx + 1].fY + fPts[idx].fY); 7707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 7807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return sum <= 0; 7907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 8007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 8107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comvoid SkDCubic::Coefficients(const double* src, double* A, double* B, double* C, double* D) { 8207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *A = src[6]; // d 8307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *B = src[4] * 3; // 3*c 8407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *C = src[2] * 3; // 3*b 8507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *D = src[0]; // a 8607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *A -= *D - *C + *B; // A = -a + 3*b - 3*c + d 8707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *B += 3 * *D - 2 * *C; // B = 3*a - 6*b + 3*c 8807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *C -= 3 * *D; // C = -3*a + 3*b 8907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 9007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 9107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.combool SkDCubic::controlsContainedByEnds() const { 9207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDVector startTan = fPts[1] - fPts[0]; 9307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (startTan.fX == 0 && startTan.fY == 0) { 9407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com startTan = fPts[2] - fPts[0]; 9507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 9607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDVector endTan = fPts[2] - fPts[3]; 9707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (endTan.fX == 0 && endTan.fY == 0) { 9807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com endTan = fPts[1] - fPts[3]; 9907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 10007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (startTan.dot(endTan) >= 0) { 10107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return false; 10207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 10307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDLine startEdge = {{fPts[0], fPts[0]}}; 10407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com startEdge[1].fX -= startTan.fY; 10507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com startEdge[1].fY += startTan.fX; 10607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDLine endEdge = {{fPts[3], fPts[3]}}; 10707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com endEdge[1].fX -= endTan.fY; 10807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com endEdge[1].fY += endTan.fX; 10907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double leftStart1 = startEdge.isLeft(fPts[1]); 11007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (leftStart1 * startEdge.isLeft(fPts[2]) < 0) { 11107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return false; 11207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 11307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double leftEnd1 = endEdge.isLeft(fPts[1]); 11407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (leftEnd1 * endEdge.isLeft(fPts[2]) < 0) { 11507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return false; 11607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 11707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return leftStart1 * leftEnd1 >= 0; 11807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 11907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 12007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.combool SkDCubic::endsAreExtremaInXOrY() const { 12107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return (between(fPts[0].fX, fPts[1].fX, fPts[3].fX) 12207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com && between(fPts[0].fX, fPts[2].fX, fPts[3].fX)) 12307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com || (between(fPts[0].fY, fPts[1].fY, fPts[3].fY) 12407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com && between(fPts[0].fY, fPts[2].fY, fPts[3].fY)); 12507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 12607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 12707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.combool SkDCubic::isLinear(int startIndex, int endIndex) const { 12807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkLineParameters lineParameters; 12907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com lineParameters.cubicEndPoints(*this, startIndex, endIndex); 13007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com // FIXME: maybe it's possible to avoid this and compare non-normalized 13107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com lineParameters.normalize(); 13207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double distance = lineParameters.controlPtDistance(*this, 1); 13307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (!approximately_zero(distance)) { 13407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return false; 13507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 13607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com distance = lineParameters.controlPtDistance(*this, 2); 13707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return approximately_zero(distance); 13807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 13907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 14007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.combool SkDCubic::monotonicInY() const { 14107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return between(fPts[0].fY, fPts[1].fY, fPts[3].fY) 14207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com && between(fPts[0].fY, fPts[2].fY, fPts[3].fY); 14307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 14407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 1452db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.orgint SkDCubic::searchRoots(double extremeTs[6], int extrema, double axisIntercept, 1462db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org SearchAxis xAxis, double* validRoots) const { 1472db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org extrema += findInflections(&extremeTs[extrema]); 1482db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org extremeTs[extrema++] = 0; 1492db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org extremeTs[extrema] = 1; 1502db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org SkTQSort(extremeTs, extremeTs + extrema); 1512db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org int validCount = 0; 1522db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org for (int index = 0; index < extrema; ) { 1532db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org double min = extremeTs[index]; 1542db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org double max = extremeTs[++index]; 1552db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org if (min == max) { 1562db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org continue; 1572db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org } 1582db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org double newT = binarySearch(min, max, axisIntercept, xAxis); 1592db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org if (newT >= 0) { 1602db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org validRoots[validCount++] = newT; 1612db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org } 1622db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org } 1632db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org return validCount; 1642db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org} 1652db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org 16607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.combool SkDCubic::serpentine() const { 1678cb1daaa1e4343eb60a7c4f21c12e33de30dad64commit-bot@chromium.org#if 0 // FIXME: enabling this fixes cubicOp114 but breaks cubicOp58d and cubicOp53d 1688cb1daaa1e4343eb60a7c4f21c12e33de30dad64commit-bot@chromium.org double tValues[2]; 1698cb1daaa1e4343eb60a7c4f21c12e33de30dad64commit-bot@chromium.org // OPTIMIZATION : another case where caching the present of cubic inflections would be useful 1708cb1daaa1e4343eb60a7c4f21c12e33de30dad64commit-bot@chromium.org return findInflections(tValues) > 1; 1718cb1daaa1e4343eb60a7c4f21c12e33de30dad64commit-bot@chromium.org#endif 17207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (!controlsContainedByEnds()) { 17307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return false; 17407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 17507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double wiggle = (fPts[0].fX - fPts[2].fX) * (fPts[0].fY + fPts[2].fY); 17607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com for (int idx = 0; idx < 2; ++idx) { 17707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com wiggle += (fPts[idx + 1].fX - fPts[idx].fX) * (fPts[idx + 1].fY + fPts[idx].fY); 17807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 17907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double waggle = (fPts[1].fX - fPts[3].fX) * (fPts[1].fY + fPts[3].fY); 18007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com for (int idx = 1; idx < 3; ++idx) { 18107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com waggle += (fPts[idx + 1].fX - fPts[idx].fX) * (fPts[idx + 1].fY + fPts[idx].fY); 18207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 18307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return wiggle * waggle < 0; 18407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 18507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 18607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// cubic roots 18707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 18807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comstatic const double PI = 3.141592653589793; 18907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 19007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// from SkGeometry.cpp (and Numeric Solutions, 5.6) 19107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comint SkDCubic::RootsValidT(double A, double B, double C, double D, double t[3]) { 19207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double s[3]; 19307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com int realRoots = RootsReal(A, B, C, D, s); 19407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com int foundRoots = SkDQuad::AddValidTs(s, realRoots, t); 19507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return foundRoots; 19607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 19707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 19807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comint SkDCubic::RootsReal(double A, double B, double C, double D, double s[3]) { 19907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com#ifdef SK_DEBUG 20007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com // create a string mathematica understands 20107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com // GDB set print repe 15 # if repeated digits is a bother 20207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com // set print elements 400 # if line doesn't fit 20307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com char str[1024]; 20407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com sk_bzero(str, sizeof(str)); 20507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SK_SNPRINTF(str, sizeof(str), "Solve[%1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]", 20607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com A, B, C, D); 207570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com SkPathOpsDebug::MathematicaIze(str, sizeof(str)); 20807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com#if ONE_OFF_DEBUG && ONE_OFF_DEBUG_MATHEMATICA 20907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDebugf("%s\n", str); 21007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com#endif 21107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com#endif 21207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (approximately_zero(A) 21307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com && approximately_zero_when_compared_to(A, B) 21407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com && approximately_zero_when_compared_to(A, C) 21507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com && approximately_zero_when_compared_to(A, D)) { // we're just a quadratic 21607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return SkDQuad::RootsReal(B, C, D, s); 21707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 21807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (approximately_zero_when_compared_to(D, A) 21907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com && approximately_zero_when_compared_to(D, B) 22007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com && approximately_zero_when_compared_to(D, C)) { // 0 is one root 22107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com int num = SkDQuad::RootsReal(A, B, C, s); 22207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com for (int i = 0; i < num; ++i) { 22307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (approximately_zero(s[i])) { 22407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return num; 22507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 22607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 22707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com s[num++] = 0; 22807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return num; 22907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 23007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (approximately_zero(A + B + C + D)) { // 1 is one root 23107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com int num = SkDQuad::RootsReal(A, A + B, -D, s); 23207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com for (int i = 0; i < num; ++i) { 2337eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com if (AlmostDequalUlps(s[i], 1)) { 23407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return num; 23507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 23607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 23707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com s[num++] = 1; 23807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return num; 23907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 24007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double a, b, c; 24107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com { 24207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double invA = 1 / A; 24307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com a = B * invA; 24407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com b = C * invA; 24507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com c = D * invA; 24607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 24707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double a2 = a * a; 24807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double Q = (a2 - b * 3) / 9; 24907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54; 25007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double R2 = R * R; 25107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double Q3 = Q * Q * Q; 25207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double R2MinusQ3 = R2 - Q3; 25307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double adiv3 = a / 3; 25407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double r; 25507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double* roots = s; 25607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (R2MinusQ3 < 0) { // we have 3 real roots 25707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double theta = acos(R / sqrt(Q3)); 25807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double neg2RootQ = -2 * sqrt(Q); 25907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 26007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com r = neg2RootQ * cos(theta / 3) - adiv3; 26107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *roots++ = r; 26207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 26307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3; 2647eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com if (!AlmostDequalUlps(s[0], r)) { 26507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *roots++ = r; 26607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 26707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3; 2687eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com if (!AlmostDequalUlps(s[0], r) && (roots - s == 1 || !AlmostDequalUlps(s[1], r))) { 26907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *roots++ = r; 27007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 27107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } else { // we have 1 real root 27207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double sqrtR2MinusQ3 = sqrt(R2MinusQ3); 27307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double A = fabs(R) + sqrtR2MinusQ3; 27407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com A = SkDCubeRoot(A); 27507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (R > 0) { 27607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com A = -A; 27707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 27807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (A != 0) { 27907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com A += Q / A; 28007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 28107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com r = A - adiv3; 28207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *roots++ = r; 2832db7fe7d3b7ee875e1099a22f0af17520696f5d7commit-bot@chromium.org if (AlmostDequalUlps((double) R2, (double) Q3)) { 28407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com r = -A / 2 - adiv3; 2857eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com if (!AlmostDequalUlps(s[0], r)) { 28607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *roots++ = r; 28707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 28807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 28907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 29007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return static_cast<int>(roots - s); 29107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 29207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 29307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// from http://www.cs.sunysb.edu/~qin/courses/geometry/4.pdf 29407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// c(t) = a(1-t)^3 + 3bt(1-t)^2 + 3c(1-t)t^2 + dt^3 29507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// c'(t) = -3a(1-t)^2 + 3b((1-t)^2 - 2t(1-t)) + 3c(2t(1-t) - t^2) + 3dt^2 29607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// = 3(b-a)(1-t)^2 + 6(c-b)t(1-t) + 3(d-c)t^2 29707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comstatic double derivative_at_t(const double* src, double t) { 29807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double one_t = 1 - t; 29907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double a = src[0]; 30007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double b = src[2]; 30107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double c = src[4]; 30207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double d = src[6]; 30307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return 3 * ((b - a) * one_t * one_t + 2 * (c - b) * t * one_t + (d - c) * t * t); 30407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 30507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 30607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// OPTIMIZE? compute t^2, t(1-t), and (1-t)^2 and pass them to another version of derivative at t? 30707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comSkDVector SkDCubic::dxdyAtT(double t) const { 30807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDVector result = { derivative_at_t(&fPts[0].fX, t), derivative_at_t(&fPts[0].fY, t) }; 30907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return result; 31007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 31107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 31207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// OPTIMIZE? share code with formulate_F1DotF2 31307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comint SkDCubic::findInflections(double tValues[]) const { 31407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double Ax = fPts[1].fX - fPts[0].fX; 31507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double Ay = fPts[1].fY - fPts[0].fY; 31607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double Bx = fPts[2].fX - 2 * fPts[1].fX + fPts[0].fX; 31707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double By = fPts[2].fY - 2 * fPts[1].fY + fPts[0].fY; 31807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double Cx = fPts[3].fX + 3 * (fPts[1].fX - fPts[2].fX) - fPts[0].fX; 31907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double Cy = fPts[3].fY + 3 * (fPts[1].fY - fPts[2].fY) - fPts[0].fY; 32007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return SkDQuad::RootsValidT(Bx * Cy - By * Cx, Ax * Cy - Ay * Cx, Ax * By - Ay * Bx, tValues); 32107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 32207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 32307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comstatic void formulate_F1DotF2(const double src[], double coeff[4]) { 32407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double a = src[2] - src[0]; 32507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double b = src[4] - 2 * src[2] + src[0]; 32607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double c = src[6] + 3 * (src[2] - src[4]) - src[0]; 32707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com coeff[0] = c * c; 32807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com coeff[1] = 3 * b * c; 32907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com coeff[2] = 2 * b * b + c * a; 33007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com coeff[3] = a * b; 33107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 33207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 33307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com/** SkDCubic'(t) = At^2 + Bt + C, where 33407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com A = 3(-a + 3(b - c) + d) 33507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com B = 6(a - 2b + c) 33607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com C = 3(b - a) 33707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com Solve for t, keeping only those that fit between 0 < t < 1 33807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com*/ 33907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comint SkDCubic::FindExtrema(double a, double b, double c, double d, double tValues[2]) { 34007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com // we divide A,B,C by 3 to simplify 34107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double A = d - a + 3*(b - c); 34207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double B = 2*(a - b - b + c); 34307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double C = b - a; 34407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 34507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return SkDQuad::RootsValidT(A, B, C, tValues); 34607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 34707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 34807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com/* from SkGeometry.cpp 34907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com Looking for F' dot F'' == 0 35007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 35107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com A = b - a 35207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com B = c - 2b + a 35307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com C = d - 3c + 3b - a 35407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 35507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com F' = 3Ct^2 + 6Bt + 3A 35607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com F'' = 6Ct + 6B 35707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 35807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 35907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com*/ 36007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comint SkDCubic::findMaxCurvature(double tValues[]) const { 36107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double coeffX[4], coeffY[4]; 36207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com int i; 36307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com formulate_F1DotF2(&fPts[0].fX, coeffX); 36407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com formulate_F1DotF2(&fPts[0].fY, coeffY); 36507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com for (i = 0; i < 4; i++) { 36607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com coeffX[i] = coeffX[i] + coeffY[i]; 36707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 36807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return RootsValidT(coeffX[0], coeffX[1], coeffX[2], coeffX[3], tValues); 36907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 37007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 37107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comSkDPoint SkDCubic::top(double startT, double endT) const { 37207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDCubic sub = subDivide(startT, endT); 37307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDPoint topPt = sub[0]; 37407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (topPt.fY > sub[3].fY || (topPt.fY == sub[3].fY && topPt.fX > sub[3].fX)) { 37507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com topPt = sub[3]; 37607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 37707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double extremeTs[2]; 37807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (!sub.monotonicInY()) { 37907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com int roots = FindExtrema(sub[0].fY, sub[1].fY, sub[2].fY, sub[3].fY, extremeTs); 38007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com for (int index = 0; index < roots; ++index) { 38107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double t = startT + (endT - startT) * extremeTs[index]; 3824fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com SkDPoint mid = ptAtT(t); 38307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (topPt.fY > mid.fY || (topPt.fY == mid.fY && topPt.fX > mid.fX)) { 38407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com topPt = mid; 38507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 38607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 38707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 38807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return topPt; 38907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 39007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 3914fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.comSkDPoint SkDCubic::ptAtT(double t) const { 3924fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com if (0 == t) { 3934fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com return fPts[0]; 3944fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com } 3954fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com if (1 == t) { 3964fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com return fPts[3]; 3974fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com } 39807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double one_t = 1 - t; 39907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double one_t2 = one_t * one_t; 40007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double a = one_t2 * one_t; 40107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double b = 3 * one_t2 * t; 40207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double t2 = t * t; 40307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double c = 3 * one_t * t2; 40407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double d = t2 * t; 40507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDPoint result = {a * fPts[0].fX + b * fPts[1].fX + c * fPts[2].fX + d * fPts[3].fX, 40607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com a * fPts[0].fY + b * fPts[1].fY + c * fPts[2].fY + d * fPts[3].fY}; 40707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return result; 40807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 40907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 41007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com/* 41107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com Given a cubic c, t1, and t2, find a small cubic segment. 41207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 41307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com The new cubic is defined as points A, B, C, and D, where 41407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com s1 = 1 - t1 41507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com s2 = 1 - t2 41607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com A = c[0]*s1*s1*s1 + 3*c[1]*s1*s1*t1 + 3*c[2]*s1*t1*t1 + c[3]*t1*t1*t1 41707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com D = c[0]*s2*s2*s2 + 3*c[1]*s2*s2*t2 + 3*c[2]*s2*t2*t2 + c[3]*t2*t2*t2 41807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 41907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com We don't have B or C. So We define two equations to isolate them. 42007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com First, compute two reference T values 1/3 and 2/3 from t1 to t2: 42107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 42207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com c(at (2*t1 + t2)/3) == E 42307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com c(at (t1 + 2*t2)/3) == F 42407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 42507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com Next, compute where those values must be if we know the values of B and C: 42607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 42707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com _12 = A*2/3 + B*1/3 42807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 12_ = A*1/3 + B*2/3 42907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com _23 = B*2/3 + C*1/3 43007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 23_ = B*1/3 + C*2/3 43107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com _34 = C*2/3 + D*1/3 43207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 34_ = C*1/3 + D*2/3 43307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com _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 43407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 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 43507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com _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 43607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 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 43707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com _1234 = (A*4/9 + B*4/9 + C*1/9)*2/3 + (B*4/9 + C*4/9 + D*1/9)*1/3 43807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com = A*8/27 + B*12/27 + C*6/27 + D*1/27 43907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com = E 44007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 1234_ = (A*1/9 + B*4/9 + C*4/9)*1/3 + (B*1/9 + C*4/9 + D*4/9)*2/3 44107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com = A*1/27 + B*6/27 + C*12/27 + D*8/27 44207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com = F 44307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com E*27 = A*8 + B*12 + C*6 + D 44407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com F*27 = A + B*6 + C*12 + D*8 44507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 44607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comGroup the known values on one side: 44707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 44807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com M = E*27 - A*8 - D = B*12 + C* 6 44907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com N = F*27 - A - D*8 = B* 6 + C*12 45007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com M*2 - N = B*18 45107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com N*2 - M = C*18 45207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com B = (M*2 - N)/18 45307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com C = (N*2 - M)/18 45407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com */ 45507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 45607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comstatic double interp_cubic_coords(const double* src, double t) { 45707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ab = SkDInterp(src[0], src[2], t); 45807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double bc = SkDInterp(src[2], src[4], t); 45907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double cd = SkDInterp(src[4], src[6], t); 46007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double abc = SkDInterp(ab, bc, t); 46107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double bcd = SkDInterp(bc, cd, t); 46207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double abcd = SkDInterp(abc, bcd, t); 46307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return abcd; 46407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 46507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 46607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comSkDCubic SkDCubic::subDivide(double t1, double t2) const { 467d892bd8ba676d34d4ce4a73ac7aad88e102fad70caryclark@google.com if (t1 == 0 || t2 == 1) { 468d892bd8ba676d34d4ce4a73ac7aad88e102fad70caryclark@google.com if (t1 == 0 && t2 == 1) { 469d892bd8ba676d34d4ce4a73ac7aad88e102fad70caryclark@google.com return *this; 470d892bd8ba676d34d4ce4a73ac7aad88e102fad70caryclark@google.com } 471d892bd8ba676d34d4ce4a73ac7aad88e102fad70caryclark@google.com SkDCubicPair pair = chopAt(t1 == 0 ? t2 : t1); 472d892bd8ba676d34d4ce4a73ac7aad88e102fad70caryclark@google.com SkDCubic dst = t1 == 0 ? pair.first() : pair.second(); 473d892bd8ba676d34d4ce4a73ac7aad88e102fad70caryclark@google.com return dst; 47407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 47507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDCubic dst; 47607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ax = dst[0].fX = interp_cubic_coords(&fPts[0].fX, t1); 47707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ay = dst[0].fY = interp_cubic_coords(&fPts[0].fY, t1); 47807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ex = interp_cubic_coords(&fPts[0].fX, (t1*2+t2)/3); 47907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ey = interp_cubic_coords(&fPts[0].fY, (t1*2+t2)/3); 48007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double fx = interp_cubic_coords(&fPts[0].fX, (t1+t2*2)/3); 48107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double fy = interp_cubic_coords(&fPts[0].fY, (t1+t2*2)/3); 48207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double dx = dst[3].fX = interp_cubic_coords(&fPts[0].fX, t2); 48307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double dy = dst[3].fY = interp_cubic_coords(&fPts[0].fY, t2); 48407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double mx = ex * 27 - ax * 8 - dx; 48507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double my = ey * 27 - ay * 8 - dy; 48607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double nx = fx * 27 - ax - dx * 8; 48707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ny = fy * 27 - ay - dy * 8; 48807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com /* bx = */ dst[1].fX = (mx * 2 - nx) / 18; 48907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com /* by = */ dst[1].fY = (my * 2 - ny) / 18; 49007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com /* cx = */ dst[2].fX = (nx * 2 - mx) / 18; 49107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com /* cy = */ dst[2].fY = (ny * 2 - my) / 18; 492cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com // FIXME: call align() ? 49307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return dst; 49407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 49507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 4968f6ef4010f6835c5ce9ede180e50a6a58512a81eskia.committer@gmail.comvoid SkDCubic::align(int endIndex, int ctrlIndex, SkDPoint* dstPt) const { 497cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com if (fPts[endIndex].fX == fPts[ctrlIndex].fX) { 498cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com dstPt->fX = fPts[endIndex].fX; 499cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com } 500cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com if (fPts[endIndex].fY == fPts[ctrlIndex].fY) { 501cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com dstPt->fY = fPts[endIndex].fY; 502cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com } 503cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com} 504cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com 50507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comvoid SkDCubic::subDivide(const SkDPoint& a, const SkDPoint& d, 50607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double t1, double t2, SkDPoint dst[2]) const { 507cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com SkASSERT(t1 != t2); 508cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com#if 0 50907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ex = interp_cubic_coords(&fPts[0].fX, (t1 * 2 + t2) / 3); 51007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ey = interp_cubic_coords(&fPts[0].fY, (t1 * 2 + t2) / 3); 51107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double fx = interp_cubic_coords(&fPts[0].fX, (t1 + t2 * 2) / 3); 51207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double fy = interp_cubic_coords(&fPts[0].fY, (t1 + t2 * 2) / 3); 51307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double mx = ex * 27 - a.fX * 8 - d.fX; 51407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double my = ey * 27 - a.fY * 8 - d.fY; 51507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double nx = fx * 27 - a.fX - d.fX * 8; 51607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ny = fy * 27 - a.fY - d.fY * 8; 51707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com /* bx = */ dst[0].fX = (mx * 2 - nx) / 18; 51807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com /* by = */ dst[0].fY = (my * 2 - ny) / 18; 51907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com /* cx = */ dst[1].fX = (nx * 2 - mx) / 18; 52007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com /* cy = */ dst[1].fY = (ny * 2 - my) / 18; 521cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com#else 522cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com // this approach assumes that the control points computed directly are accurate enough 523cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com SkDCubic sub = subDivide(t1, t2); 524cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com dst[0] = sub[1] + (a - sub[0]); 525cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com dst[1] = sub[2] + (d - sub[3]); 526cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com#endif 527cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com if (t1 == 0 || t2 == 0) { 528cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com align(0, 1, t1 == 0 ? &dst[0] : &dst[1]); 529cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com } 530cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com if (t1 == 1 || t2 == 1) { 531cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com align(3, 2, t1 == 1 ? &dst[0] : &dst[1]); 532cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com } 5334431e7757cfcb8cfa99535eed0e9f156dabf95c2commit-bot@chromium.org if (AlmostBequalUlps(dst[0].fX, a.fX)) { 534cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com dst[0].fX = a.fX; 535cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com } 5364431e7757cfcb8cfa99535eed0e9f156dabf95c2commit-bot@chromium.org if (AlmostBequalUlps(dst[0].fY, a.fY)) { 537cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com dst[0].fY = a.fY; 538cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com } 5394431e7757cfcb8cfa99535eed0e9f156dabf95c2commit-bot@chromium.org if (AlmostBequalUlps(dst[1].fX, d.fX)) { 540cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com dst[1].fX = d.fX; 541cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com } 5424431e7757cfcb8cfa99535eed0e9f156dabf95c2commit-bot@chromium.org if (AlmostBequalUlps(dst[1].fY, d.fY)) { 543cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com dst[1].fY = d.fY; 544cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com } 54507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 54607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 54707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com/* classic one t subdivision */ 54807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comstatic void interp_cubic_coords(const double* src, double* dst, double t) { 54907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ab = SkDInterp(src[0], src[2], t); 55007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double bc = SkDInterp(src[2], src[4], t); 55107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double cd = SkDInterp(src[4], src[6], t); 55207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double abc = SkDInterp(ab, bc, t); 55307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double bcd = SkDInterp(bc, cd, t); 55407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double abcd = SkDInterp(abc, bcd, t); 55507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 55607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst[0] = src[0]; 55707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst[2] = ab; 55807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst[4] = abc; 55907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst[6] = abcd; 56007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst[8] = bcd; 56107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst[10] = cd; 56207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst[12] = src[6]; 56307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 56407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 56507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comSkDCubicPair SkDCubic::chopAt(double t) const { 56607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDCubicPair dst; 56707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (t == 0.5) { 56807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[0] = fPts[0]; 56907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[1].fX = (fPts[0].fX + fPts[1].fX) / 2; 57007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[1].fY = (fPts[0].fY + fPts[1].fY) / 2; 57107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[2].fX = (fPts[0].fX + 2 * fPts[1].fX + fPts[2].fX) / 4; 57207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[2].fY = (fPts[0].fY + 2 * fPts[1].fY + fPts[2].fY) / 4; 57307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[3].fX = (fPts[0].fX + 3 * (fPts[1].fX + fPts[2].fX) + fPts[3].fX) / 8; 57407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[3].fY = (fPts[0].fY + 3 * (fPts[1].fY + fPts[2].fY) + fPts[3].fY) / 8; 57507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[4].fX = (fPts[1].fX + 2 * fPts[2].fX + fPts[3].fX) / 4; 57607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[4].fY = (fPts[1].fY + 2 * fPts[2].fY + fPts[3].fY) / 4; 57707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[5].fX = (fPts[2].fX + fPts[3].fX) / 2; 57807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[5].fY = (fPts[2].fY + fPts[3].fY) / 2; 57907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[6] = fPts[3]; 58007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return dst; 58107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 58207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com interp_cubic_coords(&fPts[0].fX, &dst.pts[0].fX, t); 58307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com interp_cubic_coords(&fPts[0].fY, &dst.pts[0].fY, t); 58407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return dst; 58507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 586