michael@0: /*
michael@0:  * Copyright 2012 Google Inc.
michael@0:  *
michael@0:  * Use of this source code is governed by a BSD-style license that can be
michael@0:  * found in the LICENSE file.
michael@0:  */
michael@0: #include "SkLineParameters.h"
michael@0: #include "SkPathOpsCubic.h"
michael@0: #include "SkPathOpsLine.h"
michael@0: #include "SkPathOpsQuad.h"
michael@0: #include "SkPathOpsRect.h"
michael@0: 
michael@0: const int SkDCubic::gPrecisionUnit = 256;  // FIXME: test different values in test framework
michael@0: 
michael@0: // FIXME: cache keep the bounds and/or precision with the caller?
michael@0: double SkDCubic::calcPrecision() const {
michael@0:     SkDRect dRect;
michael@0:     dRect.setBounds(*this);  // OPTIMIZATION: just use setRawBounds ?
michael@0:     double width = dRect.fRight - dRect.fLeft;
michael@0:     double height = dRect.fBottom - dRect.fTop;
michael@0:     return (width > height ? width : height) / gPrecisionUnit;
michael@0: }
michael@0: 
michael@0: bool SkDCubic::clockwise() const {
michael@0:     double sum = (fPts[0].fX - fPts[3].fX) * (fPts[0].fY + fPts[3].fY);
michael@0:     for (int idx = 0; idx < 3; ++idx) {
michael@0:         sum += (fPts[idx + 1].fX - fPts[idx].fX) * (fPts[idx + 1].fY + fPts[idx].fY);
michael@0:     }
michael@0:     return sum <= 0;
michael@0: }
michael@0: 
michael@0: void SkDCubic::Coefficients(const double* src, double* A, double* B, double* C, double* D) {
michael@0:     *A = src[6];  // d
michael@0:     *B = src[4] * 3;  // 3*c
michael@0:     *C = src[2] * 3;  // 3*b
michael@0:     *D = src[0];  // a
michael@0:     *A -= *D - *C + *B;     // A =   -a + 3*b - 3*c + d
michael@0:     *B += 3 * *D - 2 * *C;  // B =  3*a - 6*b + 3*c
michael@0:     *C -= 3 * *D;           // C = -3*a + 3*b
michael@0: }
michael@0: 
michael@0: bool SkDCubic::controlsContainedByEnds() const {
michael@0:     SkDVector startTan = fPts[1] - fPts[0];
michael@0:     if (startTan.fX == 0 && startTan.fY == 0) {
michael@0:         startTan = fPts[2] - fPts[0];
michael@0:     }
michael@0:     SkDVector endTan = fPts[2] - fPts[3];
michael@0:     if (endTan.fX == 0 && endTan.fY == 0) {
michael@0:         endTan = fPts[1] - fPts[3];
michael@0:     }
michael@0:     if (startTan.dot(endTan) >= 0) {
michael@0:         return false;
michael@0:     }
michael@0:     SkDLine startEdge = {{fPts[0], fPts[0]}};
michael@0:     startEdge[1].fX -= startTan.fY;
michael@0:     startEdge[1].fY += startTan.fX;
michael@0:     SkDLine endEdge = {{fPts[3], fPts[3]}};
michael@0:     endEdge[1].fX -= endTan.fY;
michael@0:     endEdge[1].fY += endTan.fX;
michael@0:     double leftStart1 = startEdge.isLeft(fPts[1]);
michael@0:     if (leftStart1 * startEdge.isLeft(fPts[2]) < 0) {
michael@0:         return false;
michael@0:     }
michael@0:     double leftEnd1 = endEdge.isLeft(fPts[1]);
michael@0:     if (leftEnd1 * endEdge.isLeft(fPts[2]) < 0) {
michael@0:         return false;
michael@0:     }
michael@0:     return leftStart1 * leftEnd1 >= 0;
michael@0: }
michael@0: 
michael@0: bool SkDCubic::endsAreExtremaInXOrY() const {
michael@0:     return (between(fPts[0].fX, fPts[1].fX, fPts[3].fX)
michael@0:             && between(fPts[0].fX, fPts[2].fX, fPts[3].fX))
michael@0:             || (between(fPts[0].fY, fPts[1].fY, fPts[3].fY)
michael@0:             && between(fPts[0].fY, fPts[2].fY, fPts[3].fY));
michael@0: }
michael@0: 
michael@0: bool SkDCubic::isLinear(int startIndex, int endIndex) const {
michael@0:     SkLineParameters lineParameters;
michael@0:     lineParameters.cubicEndPoints(*this, startIndex, endIndex);
michael@0:     // FIXME: maybe it's possible to avoid this and compare non-normalized
michael@0:     lineParameters.normalize();
michael@0:     double distance = lineParameters.controlPtDistance(*this, 1);
michael@0:     if (!approximately_zero(distance)) {
michael@0:         return false;
michael@0:     }
michael@0:     distance = lineParameters.controlPtDistance(*this, 2);
michael@0:     return approximately_zero(distance);
michael@0: }
michael@0: 
michael@0: bool SkDCubic::monotonicInY() const {
michael@0:     return between(fPts[0].fY, fPts[1].fY, fPts[3].fY)
michael@0:             && between(fPts[0].fY, fPts[2].fY, fPts[3].fY);
michael@0: }
michael@0: 
michael@0: bool SkDCubic::serpentine() const {
michael@0:     if (!controlsContainedByEnds()) {
michael@0:         return false;
michael@0:     }
michael@0:     double wiggle = (fPts[0].fX - fPts[2].fX) * (fPts[0].fY + fPts[2].fY);
michael@0:     for (int idx = 0; idx < 2; ++idx) {
michael@0:         wiggle += (fPts[idx + 1].fX - fPts[idx].fX) * (fPts[idx + 1].fY + fPts[idx].fY);
michael@0:     }
michael@0:     double waggle = (fPts[1].fX - fPts[3].fX) * (fPts[1].fY + fPts[3].fY);
michael@0:     for (int idx = 1; idx < 3; ++idx) {
michael@0:         waggle += (fPts[idx + 1].fX - fPts[idx].fX) * (fPts[idx + 1].fY + fPts[idx].fY);
michael@0:     }
michael@0:     return wiggle * waggle < 0;
michael@0: }
michael@0: 
michael@0: // cubic roots
michael@0: 
michael@0: static const double PI = 3.141592653589793;
michael@0: 
michael@0: // from SkGeometry.cpp (and Numeric Solutions, 5.6)
michael@0: int SkDCubic::RootsValidT(double A, double B, double C, double D, double t[3]) {
michael@0:     double s[3];
michael@0:     int realRoots = RootsReal(A, B, C, D, s);
michael@0:     int foundRoots = SkDQuad::AddValidTs(s, realRoots, t);
michael@0:     return foundRoots;
michael@0: }
michael@0: 
michael@0: int SkDCubic::RootsReal(double A, double B, double C, double D, double s[3]) {
michael@0: #ifdef SK_DEBUG
michael@0:     // create a string mathematica understands
michael@0:     // GDB set print repe 15 # if repeated digits is a bother
michael@0:     //     set print elements 400 # if line doesn't fit
michael@0:     char str[1024];
michael@0:     sk_bzero(str, sizeof(str));
michael@0:     SK_SNPRINTF(str, sizeof(str), "Solve[%1.19g x^3 + %1.19g x^2 + %1.19g x + %1.19g == 0, x]",
michael@0:             A, B, C, D);
michael@0:     SkPathOpsDebug::MathematicaIze(str, sizeof(str));
michael@0: #if ONE_OFF_DEBUG && ONE_OFF_DEBUG_MATHEMATICA
michael@0:     SkDebugf("%s\n", str);
michael@0: #endif
michael@0: #endif
michael@0:     if (approximately_zero(A)
michael@0:             && approximately_zero_when_compared_to(A, B)
michael@0:             && approximately_zero_when_compared_to(A, C)
michael@0:             && approximately_zero_when_compared_to(A, D)) {  // we're just a quadratic
michael@0:         return SkDQuad::RootsReal(B, C, D, s);
michael@0:     }
michael@0:     if (approximately_zero_when_compared_to(D, A)
michael@0:             && approximately_zero_when_compared_to(D, B)
michael@0:             && approximately_zero_when_compared_to(D, C)) {  // 0 is one root
michael@0:         int num = SkDQuad::RootsReal(A, B, C, s);
michael@0:         for (int i = 0; i < num; ++i) {
michael@0:             if (approximately_zero(s[i])) {
michael@0:                 return num;
michael@0:             }
michael@0:         }
michael@0:         s[num++] = 0;
michael@0:         return num;
michael@0:     }
michael@0:     if (approximately_zero(A + B + C + D)) {  // 1 is one root
michael@0:         int num = SkDQuad::RootsReal(A, A + B, -D, s);
michael@0:         for (int i = 0; i < num; ++i) {
michael@0:             if (AlmostDequalUlps(s[i], 1)) {
michael@0:                 return num;
michael@0:             }
michael@0:         }
michael@0:         s[num++] = 1;
michael@0:         return num;
michael@0:     }
michael@0:     double a, b, c;
michael@0:     {
michael@0:         double invA = 1 / A;
michael@0:         a = B * invA;
michael@0:         b = C * invA;
michael@0:         c = D * invA;
michael@0:     }
michael@0:     double a2 = a * a;
michael@0:     double Q = (a2 - b * 3) / 9;
michael@0:     double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54;
michael@0:     double R2 = R * R;
michael@0:     double Q3 = Q * Q * Q;
michael@0:     double R2MinusQ3 = R2 - Q3;
michael@0:     double adiv3 = a / 3;
michael@0:     double r;
michael@0:     double* roots = s;
michael@0:     if (R2MinusQ3 < 0) {   // we have 3 real roots
michael@0:         double theta = acos(R / sqrt(Q3));
michael@0:         double neg2RootQ = -2 * sqrt(Q);
michael@0: 
michael@0:         r = neg2RootQ * cos(theta / 3) - adiv3;
michael@0:         *roots++ = r;
michael@0: 
michael@0:         r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3;
michael@0:         if (!AlmostDequalUlps(s[0], r)) {
michael@0:             *roots++ = r;
michael@0:         }
michael@0:         r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3;
michael@0:         if (!AlmostDequalUlps(s[0], r) && (roots - s == 1 || !AlmostDequalUlps(s[1], r))) {
michael@0:             *roots++ = r;
michael@0:         }
michael@0:     } else {  // we have 1 real root
michael@0:         double sqrtR2MinusQ3 = sqrt(R2MinusQ3);
michael@0:         double A = fabs(R) + sqrtR2MinusQ3;
michael@0:         A = SkDCubeRoot(A);
michael@0:         if (R > 0) {
michael@0:             A = -A;
michael@0:         }
michael@0:         if (A != 0) {
michael@0:             A += Q / A;
michael@0:         }
michael@0:         r = A - adiv3;
michael@0:         *roots++ = r;
michael@0:         if (AlmostDequalUlps(R2, Q3)) {
michael@0:             r = -A / 2 - adiv3;
michael@0:             if (!AlmostDequalUlps(s[0], r)) {
michael@0:                 *roots++ = r;
michael@0:             }
michael@0:         }
michael@0:     }
michael@0:     return static_cast<int>(roots - s);
michael@0: }
michael@0: 
michael@0: // from http://www.cs.sunysb.edu/~qin/courses/geometry/4.pdf
michael@0: // c(t)  = a(1-t)^3 + 3bt(1-t)^2 + 3c(1-t)t^2 + dt^3
michael@0: // c'(t) = -3a(1-t)^2 + 3b((1-t)^2 - 2t(1-t)) + 3c(2t(1-t) - t^2) + 3dt^2
michael@0: //       = 3(b-a)(1-t)^2 + 6(c-b)t(1-t) + 3(d-c)t^2
michael@0: static double derivative_at_t(const double* src, double t) {
michael@0:     double one_t = 1 - t;
michael@0:     double a = src[0];
michael@0:     double b = src[2];
michael@0:     double c = src[4];
michael@0:     double d = src[6];
michael@0:     return 3 * ((b - a) * one_t * one_t + 2 * (c - b) * t * one_t + (d - c) * t * t);
michael@0: }
michael@0: 
michael@0: // OPTIMIZE? compute t^2, t(1-t), and (1-t)^2 and pass them to another version of derivative at t?
michael@0: SkDVector SkDCubic::dxdyAtT(double t) const {
michael@0:     SkDVector result = { derivative_at_t(&fPts[0].fX, t), derivative_at_t(&fPts[0].fY, t) };
michael@0:     return result;
michael@0: }
michael@0: 
michael@0: // OPTIMIZE? share code with formulate_F1DotF2
michael@0: int SkDCubic::findInflections(double tValues[]) const {
michael@0:     double Ax = fPts[1].fX - fPts[0].fX;
michael@0:     double Ay = fPts[1].fY - fPts[0].fY;
michael@0:     double Bx = fPts[2].fX - 2 * fPts[1].fX + fPts[0].fX;
michael@0:     double By = fPts[2].fY - 2 * fPts[1].fY + fPts[0].fY;
michael@0:     double Cx = fPts[3].fX + 3 * (fPts[1].fX - fPts[2].fX) - fPts[0].fX;
michael@0:     double Cy = fPts[3].fY + 3 * (fPts[1].fY - fPts[2].fY) - fPts[0].fY;
michael@0:     return SkDQuad::RootsValidT(Bx * Cy - By * Cx, Ax * Cy - Ay * Cx, Ax * By - Ay * Bx, tValues);
michael@0: }
michael@0: 
michael@0: static void formulate_F1DotF2(const double src[], double coeff[4]) {
michael@0:     double a = src[2] - src[0];
michael@0:     double b = src[4] - 2 * src[2] + src[0];
michael@0:     double c = src[6] + 3 * (src[2] - src[4]) - src[0];
michael@0:     coeff[0] = c * c;
michael@0:     coeff[1] = 3 * b * c;
michael@0:     coeff[2] = 2 * b * b + c * a;
michael@0:     coeff[3] = a * b;
michael@0: }
michael@0: 
michael@0: /** SkDCubic'(t) = At^2 + Bt + C, where
michael@0:     A = 3(-a + 3(b - c) + d)
michael@0:     B = 6(a - 2b + c)
michael@0:     C = 3(b - a)
michael@0:     Solve for t, keeping only those that fit between 0 < t < 1
michael@0: */
michael@0: int SkDCubic::FindExtrema(double a, double b, double c, double d, double tValues[2]) {
michael@0:     // we divide A,B,C by 3 to simplify
michael@0:     double A = d - a + 3*(b - c);
michael@0:     double B = 2*(a - b - b + c);
michael@0:     double C = b - a;
michael@0: 
michael@0:     return SkDQuad::RootsValidT(A, B, C, tValues);
michael@0: }
michael@0: 
michael@0: /*  from SkGeometry.cpp
michael@0:     Looking for F' dot F'' == 0
michael@0: 
michael@0:     A = b - a
michael@0:     B = c - 2b + a
michael@0:     C = d - 3c + 3b - a
michael@0: 
michael@0:     F' = 3Ct^2 + 6Bt + 3A
michael@0:     F'' = 6Ct + 6B
michael@0: 
michael@0:     F' dot F'' -> CCt^3 + 3BCt^2 + (2BB + CA)t + AB
michael@0: */
michael@0: int SkDCubic::findMaxCurvature(double tValues[]) const {
michael@0:     double coeffX[4], coeffY[4];
michael@0:     int i;
michael@0:     formulate_F1DotF2(&fPts[0].fX, coeffX);
michael@0:     formulate_F1DotF2(&fPts[0].fY, coeffY);
michael@0:     for (i = 0; i < 4; i++) {
michael@0:         coeffX[i] = coeffX[i] + coeffY[i];
michael@0:     }
michael@0:     return RootsValidT(coeffX[0], coeffX[1], coeffX[2], coeffX[3], tValues);
michael@0: }
michael@0: 
michael@0: SkDPoint SkDCubic::top(double startT, double endT) const {
michael@0:     SkDCubic sub = subDivide(startT, endT);
michael@0:     SkDPoint topPt = sub[0];
michael@0:     if (topPt.fY > sub[3].fY || (topPt.fY == sub[3].fY && topPt.fX > sub[3].fX)) {
michael@0:         topPt = sub[3];
michael@0:     }
michael@0:     double extremeTs[2];
michael@0:     if (!sub.monotonicInY()) {
michael@0:         int roots = FindExtrema(sub[0].fY, sub[1].fY, sub[2].fY, sub[3].fY, extremeTs);
michael@0:         for (int index = 0; index < roots; ++index) {
michael@0:             double t = startT + (endT - startT) * extremeTs[index];
michael@0:             SkDPoint mid = ptAtT(t);
michael@0:             if (topPt.fY > mid.fY || (topPt.fY == mid.fY && topPt.fX > mid.fX)) {
michael@0:                 topPt = mid;
michael@0:             }
michael@0:         }
michael@0:     }
michael@0:     return topPt;
michael@0: }
michael@0: 
michael@0: SkDPoint SkDCubic::ptAtT(double t) const {
michael@0:     if (0 == t) {
michael@0:         return fPts[0];
michael@0:     }
michael@0:     if (1 == t) {
michael@0:         return fPts[3];
michael@0:     }
michael@0:     double one_t = 1 - t;
michael@0:     double one_t2 = one_t * one_t;
michael@0:     double a = one_t2 * one_t;
michael@0:     double b = 3 * one_t2 * t;
michael@0:     double t2 = t * t;
michael@0:     double c = 3 * one_t * t2;
michael@0:     double d = t2 * t;
michael@0:     SkDPoint result = {a * fPts[0].fX + b * fPts[1].fX + c * fPts[2].fX + d * fPts[3].fX,
michael@0:             a * fPts[0].fY + b * fPts[1].fY + c * fPts[2].fY + d * fPts[3].fY};
michael@0:     return result;
michael@0: }
michael@0: 
michael@0: /*
michael@0:  Given a cubic c, t1, and t2, find a small cubic segment.
michael@0: 
michael@0:  The new cubic is defined as points A, B, C, and D, where
michael@0:  s1 = 1 - t1
michael@0:  s2 = 1 - t2
michael@0:  A = c[0]*s1*s1*s1 + 3*c[1]*s1*s1*t1 + 3*c[2]*s1*t1*t1 + c[3]*t1*t1*t1
michael@0:  D = c[0]*s2*s2*s2 + 3*c[1]*s2*s2*t2 + 3*c[2]*s2*t2*t2 + c[3]*t2*t2*t2
michael@0: 
michael@0:  We don't have B or C. So We define two equations to isolate them.
michael@0:  First, compute two reference T values 1/3 and 2/3 from t1 to t2:
michael@0: 
michael@0:  c(at (2*t1 + t2)/3) == E
michael@0:  c(at (t1 + 2*t2)/3) == F
michael@0: 
michael@0:  Next, compute where those values must be if we know the values of B and C:
michael@0: 
michael@0:  _12   =  A*2/3 + B*1/3
michael@0:  12_   =  A*1/3 + B*2/3
michael@0:  _23   =  B*2/3 + C*1/3
michael@0:  23_   =  B*1/3 + C*2/3
michael@0:  _34   =  C*2/3 + D*1/3
michael@0:  34_   =  C*1/3 + D*2/3
michael@0:  _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
michael@0:  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
michael@0:  _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
michael@0:  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
michael@0:  _1234 = (A*4/9 + B*4/9 + C*1/9)*2/3 + (B*4/9 + C*4/9 + D*1/9)*1/3
michael@0:        =  A*8/27 + B*12/27 + C*6/27 + D*1/27
michael@0:        =  E
michael@0:  1234_ = (A*1/9 + B*4/9 + C*4/9)*1/3 + (B*1/9 + C*4/9 + D*4/9)*2/3
michael@0:        =  A*1/27 + B*6/27 + C*12/27 + D*8/27
michael@0:        =  F
michael@0:  E*27  =  A*8    + B*12   + C*6     + D
michael@0:  F*27  =  A      + B*6    + C*12    + D*8
michael@0: 
michael@0: Group the known values on one side:
michael@0: 
michael@0:  M       = E*27 - A*8 - D     = B*12 + C* 6
michael@0:  N       = F*27 - A   - D*8   = B* 6 + C*12
michael@0:  M*2 - N = B*18
michael@0:  N*2 - M = C*18
michael@0:  B       = (M*2 - N)/18
michael@0:  C       = (N*2 - M)/18
michael@0:  */
michael@0: 
michael@0: static double interp_cubic_coords(const double* src, double t) {
michael@0:     double ab = SkDInterp(src[0], src[2], t);
michael@0:     double bc = SkDInterp(src[2], src[4], t);
michael@0:     double cd = SkDInterp(src[4], src[6], t);
michael@0:     double abc = SkDInterp(ab, bc, t);
michael@0:     double bcd = SkDInterp(bc, cd, t);
michael@0:     double abcd = SkDInterp(abc, bcd, t);
michael@0:     return abcd;
michael@0: }
michael@0: 
michael@0: SkDCubic SkDCubic::subDivide(double t1, double t2) const {
michael@0:     if (t1 == 0 || t2 == 1) {
michael@0:         if (t1 == 0 && t2 == 1) {
michael@0:             return *this;
michael@0:         }
michael@0:         SkDCubicPair pair = chopAt(t1 == 0 ? t2 : t1);
michael@0:         SkDCubic dst = t1 == 0 ? pair.first() : pair.second();
michael@0:         return dst;
michael@0:     }
michael@0:     SkDCubic dst;
michael@0:     double ax = dst[0].fX = interp_cubic_coords(&fPts[0].fX, t1);
michael@0:     double ay = dst[0].fY = interp_cubic_coords(&fPts[0].fY, t1);
michael@0:     double ex = interp_cubic_coords(&fPts[0].fX, (t1*2+t2)/3);
michael@0:     double ey = interp_cubic_coords(&fPts[0].fY, (t1*2+t2)/3);
michael@0:     double fx = interp_cubic_coords(&fPts[0].fX, (t1+t2*2)/3);
michael@0:     double fy = interp_cubic_coords(&fPts[0].fY, (t1+t2*2)/3);
michael@0:     double dx = dst[3].fX = interp_cubic_coords(&fPts[0].fX, t2);
michael@0:     double dy = dst[3].fY = interp_cubic_coords(&fPts[0].fY, t2);
michael@0:     double mx = ex * 27 - ax * 8 - dx;
michael@0:     double my = ey * 27 - ay * 8 - dy;
michael@0:     double nx = fx * 27 - ax - dx * 8;
michael@0:     double ny = fy * 27 - ay - dy * 8;
michael@0:     /* bx = */ dst[1].fX = (mx * 2 - nx) / 18;
michael@0:     /* by = */ dst[1].fY = (my * 2 - ny) / 18;
michael@0:     /* cx = */ dst[2].fX = (nx * 2 - mx) / 18;
michael@0:     /* cy = */ dst[2].fY = (ny * 2 - my) / 18;
michael@0:     // FIXME: call align() ?
michael@0:     return dst;
michael@0: }
michael@0: 
michael@0: void SkDCubic::align(int endIndex, int ctrlIndex, SkDPoint* dstPt) const {
michael@0:     if (fPts[endIndex].fX == fPts[ctrlIndex].fX) {
michael@0:         dstPt->fX = fPts[endIndex].fX;
michael@0:     }
michael@0:     if (fPts[endIndex].fY == fPts[ctrlIndex].fY) {
michael@0:         dstPt->fY = fPts[endIndex].fY;
michael@0:     }
michael@0: }
michael@0: 
michael@0: void SkDCubic::subDivide(const SkDPoint& a, const SkDPoint& d,
michael@0:                          double t1, double t2, SkDPoint dst[2]) const {
michael@0:     SkASSERT(t1 != t2);
michael@0: #if 0
michael@0:     double ex = interp_cubic_coords(&fPts[0].fX, (t1 * 2 + t2) / 3);
michael@0:     double ey = interp_cubic_coords(&fPts[0].fY, (t1 * 2 + t2) / 3);
michael@0:     double fx = interp_cubic_coords(&fPts[0].fX, (t1 + t2 * 2) / 3);
michael@0:     double fy = interp_cubic_coords(&fPts[0].fY, (t1 + t2 * 2) / 3);
michael@0:     double mx = ex * 27 - a.fX * 8 - d.fX;
michael@0:     double my = ey * 27 - a.fY * 8 - d.fY;
michael@0:     double nx = fx * 27 - a.fX - d.fX * 8;
michael@0:     double ny = fy * 27 - a.fY - d.fY * 8;
michael@0:     /* bx = */ dst[0].fX = (mx * 2 - nx) / 18;
michael@0:     /* by = */ dst[0].fY = (my * 2 - ny) / 18;
michael@0:     /* cx = */ dst[1].fX = (nx * 2 - mx) / 18;
michael@0:     /* cy = */ dst[1].fY = (ny * 2 - my) / 18;
michael@0: #else
michael@0:     // this approach assumes that the control points computed directly are accurate enough
michael@0:     SkDCubic sub = subDivide(t1, t2);
michael@0:     dst[0] = sub[1] + (a - sub[0]);
michael@0:     dst[1] = sub[2] + (d - sub[3]);
michael@0: #endif
michael@0:     if (t1 == 0 || t2 == 0) {
michael@0:         align(0, 1, t1 == 0 ? &dst[0] : &dst[1]);
michael@0:     }
michael@0:     if (t1 == 1 || t2 == 1) {
michael@0:         align(3, 2, t1 == 1 ? &dst[0] : &dst[1]);
michael@0:     }
michael@0:     if (precisely_subdivide_equal(dst[0].fX, a.fX)) {
michael@0:         dst[0].fX = a.fX;
michael@0:     }
michael@0:     if (precisely_subdivide_equal(dst[0].fY, a.fY)) {
michael@0:         dst[0].fY = a.fY;
michael@0:     }
michael@0:     if (precisely_subdivide_equal(dst[1].fX, d.fX)) {
michael@0:         dst[1].fX = d.fX;
michael@0:     }
michael@0:     if (precisely_subdivide_equal(dst[1].fY, d.fY)) {
michael@0:         dst[1].fY = d.fY;
michael@0:     }
michael@0: }
michael@0: 
michael@0: /* classic one t subdivision */
michael@0: static void interp_cubic_coords(const double* src, double* dst, double t) {
michael@0:     double ab = SkDInterp(src[0], src[2], t);
michael@0:     double bc = SkDInterp(src[2], src[4], t);
michael@0:     double cd = SkDInterp(src[4], src[6], t);
michael@0:     double abc = SkDInterp(ab, bc, t);
michael@0:     double bcd = SkDInterp(bc, cd, t);
michael@0:     double abcd = SkDInterp(abc, bcd, t);
michael@0: 
michael@0:     dst[0] = src[0];
michael@0:     dst[2] = ab;
michael@0:     dst[4] = abc;
michael@0:     dst[6] = abcd;
michael@0:     dst[8] = bcd;
michael@0:     dst[10] = cd;
michael@0:     dst[12] = src[6];
michael@0: }
michael@0: 
michael@0: SkDCubicPair SkDCubic::chopAt(double t) const {
michael@0:     SkDCubicPair dst;
michael@0:     if (t == 0.5) {
michael@0:         dst.pts[0] = fPts[0];
michael@0:         dst.pts[1].fX = (fPts[0].fX + fPts[1].fX) / 2;
michael@0:         dst.pts[1].fY = (fPts[0].fY + fPts[1].fY) / 2;
michael@0:         dst.pts[2].fX = (fPts[0].fX + 2 * fPts[1].fX + fPts[2].fX) / 4;
michael@0:         dst.pts[2].fY = (fPts[0].fY + 2 * fPts[1].fY + fPts[2].fY) / 4;
michael@0:         dst.pts[3].fX = (fPts[0].fX + 3 * (fPts[1].fX + fPts[2].fX) + fPts[3].fX) / 8;
michael@0:         dst.pts[3].fY = (fPts[0].fY + 3 * (fPts[1].fY + fPts[2].fY) + fPts[3].fY) / 8;
michael@0:         dst.pts[4].fX = (fPts[1].fX + 2 * fPts[2].fX + fPts[3].fX) / 4;
michael@0:         dst.pts[4].fY = (fPts[1].fY + 2 * fPts[2].fY + fPts[3].fY) / 4;
michael@0:         dst.pts[5].fX = (fPts[2].fX + fPts[3].fX) / 2;
michael@0:         dst.pts[5].fY = (fPts[2].fY + fPts[3].fY) / 2;
michael@0:         dst.pts[6] = fPts[3];
michael@0:         return dst;
michael@0:     }
michael@0:     interp_cubic_coords(&fPts[0].fX, &dst.pts[0].fX, t);
michael@0:     interp_cubic_coords(&fPts[0].fY, &dst.pts[0].fY, t);
michael@0:     return dst;
michael@0: }
michael@0: 
michael@0: #ifdef SK_DEBUG
michael@0: void SkDCubic::dump() {
michael@0:     SkDebugf("{{");
michael@0:     int index = 0;
michael@0:     do {
michael@0:         fPts[index].dump();
michael@0:         SkDebugf(", ");
michael@0:     } while (++index < 3);
michael@0:     fPts[index].dump();
michael@0:     SkDebugf("}}\n");
michael@0: }
michael@0: #endif