SkPathOpsCubic.cpp revision 7eaa53d8f7e48fd17d02b5e3bd91f90e9c1899ef
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" 1207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 1307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comconst int SkDCubic::gPrecisionUnit = 256; // FIXME: test different values in test framework 1407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 1507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// FIXME: cache keep the bounds and/or precision with the caller? 1607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comdouble SkDCubic::calcPrecision() const { 1707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDRect dRect; 1807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dRect.setBounds(*this); // OPTIMIZATION: just use setRawBounds ? 1907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double width = dRect.fRight - dRect.fLeft; 2007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double height = dRect.fBottom - dRect.fTop; 2107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return (width > height ? width : height) / gPrecisionUnit; 2207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 2307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 2407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.combool SkDCubic::clockwise() const { 2507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double sum = (fPts[0].fX - fPts[3].fX) * (fPts[0].fY + fPts[3].fY); 2607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com for (int idx = 0; idx < 3; ++idx) { 2707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com sum += (fPts[idx + 1].fX - fPts[idx].fX) * (fPts[idx + 1].fY + fPts[idx].fY); 2807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 2907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return sum <= 0; 3007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 3107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 3207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comvoid SkDCubic::Coefficients(const double* src, double* A, double* B, double* C, double* D) { 3307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *A = src[6]; // d 3407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *B = src[4] * 3; // 3*c 3507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *C = src[2] * 3; // 3*b 3607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *D = src[0]; // a 3707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *A -= *D - *C + *B; // A = -a + 3*b - 3*c + d 3807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *B += 3 * *D - 2 * *C; // B = 3*a - 6*b + 3*c 3907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *C -= 3 * *D; // C = -3*a + 3*b 4007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 4107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 4207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.combool SkDCubic::controlsContainedByEnds() const { 4307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDVector startTan = fPts[1] - fPts[0]; 4407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (startTan.fX == 0 && startTan.fY == 0) { 4507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com startTan = fPts[2] - fPts[0]; 4607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 4707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDVector endTan = fPts[2] - fPts[3]; 4807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (endTan.fX == 0 && endTan.fY == 0) { 4907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com endTan = fPts[1] - fPts[3]; 5007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 5107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (startTan.dot(endTan) >= 0) { 5207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return false; 5307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 5407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDLine startEdge = {{fPts[0], fPts[0]}}; 5507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com startEdge[1].fX -= startTan.fY; 5607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com startEdge[1].fY += startTan.fX; 5707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDLine endEdge = {{fPts[3], fPts[3]}}; 5807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com endEdge[1].fX -= endTan.fY; 5907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com endEdge[1].fY += endTan.fX; 6007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double leftStart1 = startEdge.isLeft(fPts[1]); 6107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (leftStart1 * startEdge.isLeft(fPts[2]) < 0) { 6207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return false; 6307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 6407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double leftEnd1 = endEdge.isLeft(fPts[1]); 6507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (leftEnd1 * endEdge.isLeft(fPts[2]) < 0) { 6607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return false; 6707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 6807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return leftStart1 * leftEnd1 >= 0; 6907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 7007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 7107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.combool SkDCubic::endsAreExtremaInXOrY() const { 7207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return (between(fPts[0].fX, fPts[1].fX, fPts[3].fX) 7307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com && between(fPts[0].fX, fPts[2].fX, fPts[3].fX)) 7407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com || (between(fPts[0].fY, fPts[1].fY, fPts[3].fY) 7507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com && between(fPts[0].fY, fPts[2].fY, fPts[3].fY)); 7607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 7707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 7807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.combool SkDCubic::isLinear(int startIndex, int endIndex) const { 7907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkLineParameters lineParameters; 8007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com lineParameters.cubicEndPoints(*this, startIndex, endIndex); 8107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com // FIXME: maybe it's possible to avoid this and compare non-normalized 8207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com lineParameters.normalize(); 8307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double distance = lineParameters.controlPtDistance(*this, 1); 8407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (!approximately_zero(distance)) { 8507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return false; 8607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 8707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com distance = lineParameters.controlPtDistance(*this, 2); 8807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return approximately_zero(distance); 8907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 9007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 9107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.combool SkDCubic::monotonicInY() const { 9207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return between(fPts[0].fY, fPts[1].fY, fPts[3].fY) 9307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com && between(fPts[0].fY, fPts[2].fY, fPts[3].fY); 9407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 9507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 9607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.combool SkDCubic::serpentine() const { 9707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (!controlsContainedByEnds()) { 9807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return false; 9907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 10007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double wiggle = (fPts[0].fX - fPts[2].fX) * (fPts[0].fY + fPts[2].fY); 10107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com for (int idx = 0; idx < 2; ++idx) { 10207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com wiggle += (fPts[idx + 1].fX - fPts[idx].fX) * (fPts[idx + 1].fY + fPts[idx].fY); 10307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 10407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double waggle = (fPts[1].fX - fPts[3].fX) * (fPts[1].fY + fPts[3].fY); 10507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com for (int idx = 1; idx < 3; ++idx) { 10607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com waggle += (fPts[idx + 1].fX - fPts[idx].fX) * (fPts[idx + 1].fY + fPts[idx].fY); 10707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 10807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return wiggle * waggle < 0; 10907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 11007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 11107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// cubic roots 11207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 11307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comstatic const double PI = 3.141592653589793; 11407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 11507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// from SkGeometry.cpp (and Numeric Solutions, 5.6) 11607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comint SkDCubic::RootsValidT(double A, double B, double C, double D, double t[3]) { 11707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double s[3]; 11807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com int realRoots = RootsReal(A, B, C, D, s); 11907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com int foundRoots = SkDQuad::AddValidTs(s, realRoots, t); 12007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return foundRoots; 12107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 12207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 12307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comint SkDCubic::RootsReal(double A, double B, double C, double D, double s[3]) { 12407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com#ifdef SK_DEBUG 12507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com // create a string mathematica understands 12607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com // GDB set print repe 15 # if repeated digits is a bother 12707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com // set print elements 400 # if line doesn't fit 12807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com char str[1024]; 12907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com sk_bzero(str, sizeof(str)); 13007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SK_SNPRINTF(str, sizeof(str), "Solve[%1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]", 13107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com A, B, C, D); 132570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com SkPathOpsDebug::MathematicaIze(str, sizeof(str)); 13307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com#if ONE_OFF_DEBUG && ONE_OFF_DEBUG_MATHEMATICA 13407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDebugf("%s\n", str); 13507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com#endif 13607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com#endif 13707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (approximately_zero(A) 13807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com && approximately_zero_when_compared_to(A, B) 13907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com && approximately_zero_when_compared_to(A, C) 14007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com && approximately_zero_when_compared_to(A, D)) { // we're just a quadratic 14107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return SkDQuad::RootsReal(B, C, D, s); 14207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 14307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (approximately_zero_when_compared_to(D, A) 14407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com && approximately_zero_when_compared_to(D, B) 14507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com && approximately_zero_when_compared_to(D, C)) { // 0 is one root 14607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com int num = SkDQuad::RootsReal(A, B, C, s); 14707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com for (int i = 0; i < num; ++i) { 14807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (approximately_zero(s[i])) { 14907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return num; 15007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 15107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 15207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com s[num++] = 0; 15307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return num; 15407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 15507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (approximately_zero(A + B + C + D)) { // 1 is one root 15607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com int num = SkDQuad::RootsReal(A, A + B, -D, s); 15707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com for (int i = 0; i < num; ++i) { 1587eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com if (AlmostDequalUlps(s[i], 1)) { 15907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return num; 16007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 16107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 16207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com s[num++] = 1; 16307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return num; 16407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 16507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double a, b, c; 16607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com { 16707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double invA = 1 / A; 16807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com a = B * invA; 16907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com b = C * invA; 17007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com c = D * invA; 17107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 17207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double a2 = a * a; 17307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double Q = (a2 - b * 3) / 9; 17407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54; 17507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double R2 = R * R; 17607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double Q3 = Q * Q * Q; 17707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double R2MinusQ3 = R2 - Q3; 17807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double adiv3 = a / 3; 17907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double r; 18007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double* roots = s; 18107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (R2MinusQ3 < 0) { // we have 3 real roots 18207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double theta = acos(R / sqrt(Q3)); 18307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double neg2RootQ = -2 * sqrt(Q); 18407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 18507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com r = neg2RootQ * cos(theta / 3) - adiv3; 18607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *roots++ = r; 18707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 18807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3; 1897eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com if (!AlmostDequalUlps(s[0], r)) { 19007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *roots++ = r; 19107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 19207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3; 1937eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com if (!AlmostDequalUlps(s[0], r) && (roots - s == 1 || !AlmostDequalUlps(s[1], r))) { 19407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *roots++ = r; 19507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 19607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } else { // we have 1 real root 19707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double sqrtR2MinusQ3 = sqrt(R2MinusQ3); 19807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double A = fabs(R) + sqrtR2MinusQ3; 19907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com A = SkDCubeRoot(A); 20007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (R > 0) { 20107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com A = -A; 20207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 20307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (A != 0) { 20407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com A += Q / A; 20507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 20607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com r = A - adiv3; 20707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *roots++ = r; 2087eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com if (AlmostDequalUlps(R2, Q3)) { 20907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com r = -A / 2 - adiv3; 2107eaa53d8f7e48fd17d02b5e3bd91f90e9c1899efcaryclark@google.com if (!AlmostDequalUlps(s[0], r)) { 21107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com *roots++ = r; 21207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 21307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 21407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 21507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return static_cast<int>(roots - s); 21607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 21707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 21807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// from http://www.cs.sunysb.edu/~qin/courses/geometry/4.pdf 21907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// c(t) = a(1-t)^3 + 3bt(1-t)^2 + 3c(1-t)t^2 + dt^3 22007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// c'(t) = -3a(1-t)^2 + 3b((1-t)^2 - 2t(1-t)) + 3c(2t(1-t) - t^2) + 3dt^2 22107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// = 3(b-a)(1-t)^2 + 6(c-b)t(1-t) + 3(d-c)t^2 22207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comstatic double derivative_at_t(const double* src, double t) { 22307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double one_t = 1 - t; 22407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double a = src[0]; 22507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double b = src[2]; 22607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double c = src[4]; 22707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double d = src[6]; 22807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return 3 * ((b - a) * one_t * one_t + 2 * (c - b) * t * one_t + (d - c) * t * t); 22907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 23007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 23107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// OPTIMIZE? compute t^2, t(1-t), and (1-t)^2 and pass them to another version of derivative at t? 23207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comSkDVector SkDCubic::dxdyAtT(double t) const { 23307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDVector result = { derivative_at_t(&fPts[0].fX, t), derivative_at_t(&fPts[0].fY, t) }; 23407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return result; 23507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 23607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 23707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com// OPTIMIZE? share code with formulate_F1DotF2 23807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comint SkDCubic::findInflections(double tValues[]) const { 23907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double Ax = fPts[1].fX - fPts[0].fX; 24007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double Ay = fPts[1].fY - fPts[0].fY; 24107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double Bx = fPts[2].fX - 2 * fPts[1].fX + fPts[0].fX; 24207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double By = fPts[2].fY - 2 * fPts[1].fY + fPts[0].fY; 24307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double Cx = fPts[3].fX + 3 * (fPts[1].fX - fPts[2].fX) - fPts[0].fX; 24407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double Cy = fPts[3].fY + 3 * (fPts[1].fY - fPts[2].fY) - fPts[0].fY; 24507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return SkDQuad::RootsValidT(Bx * Cy - By * Cx, Ax * Cy - Ay * Cx, Ax * By - Ay * Bx, tValues); 24607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 24707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 24807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comstatic void formulate_F1DotF2(const double src[], double coeff[4]) { 24907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double a = src[2] - src[0]; 25007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double b = src[4] - 2 * src[2] + src[0]; 25107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double c = src[6] + 3 * (src[2] - src[4]) - src[0]; 25207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com coeff[0] = c * c; 25307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com coeff[1] = 3 * b * c; 25407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com coeff[2] = 2 * b * b + c * a; 25507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com coeff[3] = a * b; 25607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 25707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 25807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com/** SkDCubic'(t) = At^2 + Bt + C, where 25907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com A = 3(-a + 3(b - c) + d) 26007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com B = 6(a - 2b + c) 26107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com C = 3(b - a) 26207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com Solve for t, keeping only those that fit between 0 < t < 1 26307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com*/ 26407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comint SkDCubic::FindExtrema(double a, double b, double c, double d, double tValues[2]) { 26507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com // we divide A,B,C by 3 to simplify 26607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double A = d - a + 3*(b - c); 26707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double B = 2*(a - b - b + c); 26807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double C = b - a; 26907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 27007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return SkDQuad::RootsValidT(A, B, C, tValues); 27107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 27207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 27307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com/* from SkGeometry.cpp 27407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com Looking for F' dot F'' == 0 27507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 27607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com A = b - a 27707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com B = c - 2b + a 27807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com C = d - 3c + 3b - a 27907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 28007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com F' = 3Ct^2 + 6Bt + 3A 28107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com F'' = 6Ct + 6B 28207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 28307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB 28407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com*/ 28507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comint SkDCubic::findMaxCurvature(double tValues[]) const { 28607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double coeffX[4], coeffY[4]; 28707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com int i; 28807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com formulate_F1DotF2(&fPts[0].fX, coeffX); 28907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com formulate_F1DotF2(&fPts[0].fY, coeffY); 29007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com for (i = 0; i < 4; i++) { 29107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com coeffX[i] = coeffX[i] + coeffY[i]; 29207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 29307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return RootsValidT(coeffX[0], coeffX[1], coeffX[2], coeffX[3], tValues); 29407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 29507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 29607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comSkDPoint SkDCubic::top(double startT, double endT) const { 29707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDCubic sub = subDivide(startT, endT); 29807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDPoint topPt = sub[0]; 29907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (topPt.fY > sub[3].fY || (topPt.fY == sub[3].fY && topPt.fX > sub[3].fX)) { 30007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com topPt = sub[3]; 30107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 30207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double extremeTs[2]; 30307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (!sub.monotonicInY()) { 30407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com int roots = FindExtrema(sub[0].fY, sub[1].fY, sub[2].fY, sub[3].fY, extremeTs); 30507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com for (int index = 0; index < roots; ++index) { 30607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double t = startT + (endT - startT) * extremeTs[index]; 3074fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com SkDPoint mid = ptAtT(t); 30807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (topPt.fY > mid.fY || (topPt.fY == mid.fY && topPt.fX > mid.fX)) { 30907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com topPt = mid; 31007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 31107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 31207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 31307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return topPt; 31407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 31507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 3164fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.comSkDPoint SkDCubic::ptAtT(double t) const { 3174fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com if (0 == t) { 3184fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com return fPts[0]; 3194fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com } 3204fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com if (1 == t) { 3214fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com return fPts[3]; 3224fdbb229649caf74e5c1b55a1823926df903af34caryclark@google.com } 32307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double one_t = 1 - t; 32407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double one_t2 = one_t * one_t; 32507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double a = one_t2 * one_t; 32607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double b = 3 * one_t2 * t; 32707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double t2 = t * t; 32807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double c = 3 * one_t * t2; 32907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double d = t2 * t; 33007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDPoint result = {a * fPts[0].fX + b * fPts[1].fX + c * fPts[2].fX + d * fPts[3].fX, 33107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com a * fPts[0].fY + b * fPts[1].fY + c * fPts[2].fY + d * fPts[3].fY}; 33207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return result; 33307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 33407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 33507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com/* 33607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com Given a cubic c, t1, and t2, find a small cubic segment. 33707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 33807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com The new cubic is defined as points A, B, C, and D, where 33907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com s1 = 1 - t1 34007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com s2 = 1 - t2 34107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@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 34207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@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 34307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 34407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com We don't have B or C. So We define two equations to isolate them. 34507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com First, compute two reference T values 1/3 and 2/3 from t1 to t2: 34607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 34707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com c(at (2*t1 + t2)/3) == E 34807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com c(at (t1 + 2*t2)/3) == F 34907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 35007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com Next, compute where those values must be if we know the values of B and C: 35107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 35207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com _12 = A*2/3 + B*1/3 35307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 12_ = A*1/3 + B*2/3 35407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com _23 = B*2/3 + C*1/3 35507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 23_ = B*1/3 + C*2/3 35607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com _34 = C*2/3 + D*1/3 35707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 34_ = C*1/3 + D*2/3 35807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@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 35907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@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 36007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@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 36107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@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 36207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@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 36307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com = A*8/27 + B*12/27 + C*6/27 + D*1/27 36407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com = E 36507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@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 36607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com = A*1/27 + B*6/27 + C*12/27 + D*8/27 36707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com = F 36807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com E*27 = A*8 + B*12 + C*6 + D 36907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com F*27 = A + B*6 + C*12 + D*8 37007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 37107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comGroup the known values on one side: 37207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 37307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com M = E*27 - A*8 - D = B*12 + C* 6 37407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com N = F*27 - A - D*8 = B* 6 + C*12 37507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com M*2 - N = B*18 37607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com N*2 - M = C*18 37707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com B = (M*2 - N)/18 37807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com C = (N*2 - M)/18 37907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com */ 38007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 38107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comstatic double interp_cubic_coords(const double* src, double t) { 38207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ab = SkDInterp(src[0], src[2], t); 38307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double bc = SkDInterp(src[2], src[4], t); 38407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double cd = SkDInterp(src[4], src[6], t); 38507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double abc = SkDInterp(ab, bc, t); 38607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double bcd = SkDInterp(bc, cd, t); 38707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double abcd = SkDInterp(abc, bcd, t); 38807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return abcd; 38907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 39007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 39107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comSkDCubic SkDCubic::subDivide(double t1, double t2) const { 392d892bd8ba676d34d4ce4a73ac7aad88e102fad70caryclark@google.com if (t1 == 0 || t2 == 1) { 393d892bd8ba676d34d4ce4a73ac7aad88e102fad70caryclark@google.com if (t1 == 0 && t2 == 1) { 394d892bd8ba676d34d4ce4a73ac7aad88e102fad70caryclark@google.com return *this; 395d892bd8ba676d34d4ce4a73ac7aad88e102fad70caryclark@google.com } 396d892bd8ba676d34d4ce4a73ac7aad88e102fad70caryclark@google.com SkDCubicPair pair = chopAt(t1 == 0 ? t2 : t1); 397d892bd8ba676d34d4ce4a73ac7aad88e102fad70caryclark@google.com SkDCubic dst = t1 == 0 ? pair.first() : pair.second(); 398d892bd8ba676d34d4ce4a73ac7aad88e102fad70caryclark@google.com return dst; 39907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 40007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDCubic dst; 40107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ax = dst[0].fX = interp_cubic_coords(&fPts[0].fX, t1); 40207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ay = dst[0].fY = interp_cubic_coords(&fPts[0].fY, t1); 40307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ex = interp_cubic_coords(&fPts[0].fX, (t1*2+t2)/3); 40407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ey = interp_cubic_coords(&fPts[0].fY, (t1*2+t2)/3); 40507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double fx = interp_cubic_coords(&fPts[0].fX, (t1+t2*2)/3); 40607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double fy = interp_cubic_coords(&fPts[0].fY, (t1+t2*2)/3); 40707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double dx = dst[3].fX = interp_cubic_coords(&fPts[0].fX, t2); 40807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double dy = dst[3].fY = interp_cubic_coords(&fPts[0].fY, t2); 40907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double mx = ex * 27 - ax * 8 - dx; 41007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double my = ey * 27 - ay * 8 - dy; 41107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double nx = fx * 27 - ax - dx * 8; 41207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ny = fy * 27 - ay - dy * 8; 41307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com /* bx = */ dst[1].fX = (mx * 2 - nx) / 18; 41407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com /* by = */ dst[1].fY = (my * 2 - ny) / 18; 41507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com /* cx = */ dst[2].fX = (nx * 2 - mx) / 18; 41607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com /* cy = */ dst[2].fY = (ny * 2 - my) / 18; 417cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com // FIXME: call align() ? 41807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return dst; 41907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 42007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 4218f6ef4010f6835c5ce9ede180e50a6a58512a81eskia.committer@gmail.comvoid SkDCubic::align(int endIndex, int ctrlIndex, SkDPoint* dstPt) const { 422cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com if (fPts[endIndex].fX == fPts[ctrlIndex].fX) { 423cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com dstPt->fX = fPts[endIndex].fX; 424cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com } 425cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com if (fPts[endIndex].fY == fPts[ctrlIndex].fY) { 426cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com dstPt->fY = fPts[endIndex].fY; 427cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com } 428cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com} 429cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com 43007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comvoid SkDCubic::subDivide(const SkDPoint& a, const SkDPoint& d, 43107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double t1, double t2, SkDPoint dst[2]) const { 432cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com SkASSERT(t1 != t2); 433cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com#if 0 43407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ex = interp_cubic_coords(&fPts[0].fX, (t1 * 2 + t2) / 3); 43507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ey = interp_cubic_coords(&fPts[0].fY, (t1 * 2 + t2) / 3); 43607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double fx = interp_cubic_coords(&fPts[0].fX, (t1 + t2 * 2) / 3); 43707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double fy = interp_cubic_coords(&fPts[0].fY, (t1 + t2 * 2) / 3); 43807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double mx = ex * 27 - a.fX * 8 - d.fX; 43907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double my = ey * 27 - a.fY * 8 - d.fY; 44007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double nx = fx * 27 - a.fX - d.fX * 8; 44107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ny = fy * 27 - a.fY - d.fY * 8; 44207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com /* bx = */ dst[0].fX = (mx * 2 - nx) / 18; 44307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com /* by = */ dst[0].fY = (my * 2 - ny) / 18; 44407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com /* cx = */ dst[1].fX = (nx * 2 - mx) / 18; 44507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com /* cy = */ dst[1].fY = (ny * 2 - my) / 18; 446cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com#else 447cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com // this approach assumes that the control points computed directly are accurate enough 448cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com SkDCubic sub = subDivide(t1, t2); 449cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com dst[0] = sub[1] + (a - sub[0]); 450cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com dst[1] = sub[2] + (d - sub[3]); 451cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com#endif 452cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com if (t1 == 0 || t2 == 0) { 453cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com align(0, 1, t1 == 0 ? &dst[0] : &dst[1]); 454cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com } 455cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com if (t1 == 1 || t2 == 1) { 456cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com align(3, 2, t1 == 1 ? &dst[0] : &dst[1]); 457cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com } 458cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com if (precisely_subdivide_equal(dst[0].fX, a.fX)) { 459cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com dst[0].fX = a.fX; 460cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com } 461cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com if (precisely_subdivide_equal(dst[0].fY, a.fY)) { 462cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com dst[0].fY = a.fY; 463cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com } 464cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com if (precisely_subdivide_equal(dst[1].fX, d.fX)) { 465cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com dst[1].fX = d.fX; 466cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com } 467cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com if (precisely_subdivide_equal(dst[1].fY, d.fY)) { 468cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com dst[1].fY = d.fY; 469cffbcc3b9665f2c928544b6fc6b8a0e22a4210fbcaryclark@google.com } 47007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 47107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 47207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com/* classic one t subdivision */ 47307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comstatic void interp_cubic_coords(const double* src, double* dst, double t) { 47407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double ab = SkDInterp(src[0], src[2], t); 47507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double bc = SkDInterp(src[2], src[4], t); 47607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double cd = SkDInterp(src[4], src[6], t); 47707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double abc = SkDInterp(ab, bc, t); 47807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double bcd = SkDInterp(bc, cd, t); 47907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com double abcd = SkDInterp(abc, bcd, t); 48007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 48107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst[0] = src[0]; 48207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst[2] = ab; 48307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst[4] = abc; 48407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst[6] = abcd; 48507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst[8] = bcd; 48607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst[10] = cd; 48707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst[12] = src[6]; 48807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 48907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com 49007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.comSkDCubicPair SkDCubic::chopAt(double t) const { 49107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com SkDCubicPair dst; 49207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com if (t == 0.5) { 49307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[0] = fPts[0]; 49407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[1].fX = (fPts[0].fX + fPts[1].fX) / 2; 49507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[1].fY = (fPts[0].fY + fPts[1].fY) / 2; 49607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[2].fX = (fPts[0].fX + 2 * fPts[1].fX + fPts[2].fX) / 4; 49707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[2].fY = (fPts[0].fY + 2 * fPts[1].fY + fPts[2].fY) / 4; 49807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[3].fX = (fPts[0].fX + 3 * (fPts[1].fX + fPts[2].fX) + fPts[3].fX) / 8; 49907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[3].fY = (fPts[0].fY + 3 * (fPts[1].fY + fPts[2].fY) + fPts[3].fY) / 8; 50007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[4].fX = (fPts[1].fX + 2 * fPts[2].fX + fPts[3].fX) / 4; 50107393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[4].fY = (fPts[1].fY + 2 * fPts[2].fY + fPts[3].fY) / 4; 50207393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[5].fX = (fPts[2].fX + fPts[3].fX) / 2; 50307393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[5].fY = (fPts[2].fY + fPts[3].fY) / 2; 50407393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com dst.pts[6] = fPts[3]; 50507393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return dst; 50607393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com } 50707393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com interp_cubic_coords(&fPts[0].fX, &dst.pts[0].fX, t); 50807393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com interp_cubic_coords(&fPts[0].fY, &dst.pts[0].fY, t); 50907393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com return dst; 51007393cab57ce74a4aae89a31fae9aaa9780fc19dcaryclark@google.com} 511570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com 512570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com#ifdef SK_DEBUG 513570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.comvoid SkDCubic::dump() { 514570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com SkDebugf("{{"); 515570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com int index = 0; 516570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com do { 517570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com fPts[index].dump(); 518570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com SkDebugf(", "); 519570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com } while (++index < 3); 520570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com fPts[index].dump(); 521570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com SkDebugf("}}\n"); 522570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com} 523570863f2e22b8ea7d7c504bd15e4f766af097df2caryclark@google.com#endif 524