| /* |
| * Copyright 2012 Google Inc. |
| * |
| * Use of this source code is governed by a BSD-style license that can be |
| * found in the LICENSE file. |
| */ |
| #include "SkIntersections.h" |
| #include "SkPathOpsCubic.h" |
| #include "SkPathOpsLine.h" |
| |
| /* |
| Find the interection of a line and cubic by solving for valid t values. |
| |
| Analogous to line-quadratic intersection, solve line-cubic intersection by |
| representing the cubic as: |
| x = a(1-t)^3 + 2b(1-t)^2t + c(1-t)t^2 + dt^3 |
| y = e(1-t)^3 + 2f(1-t)^2t + g(1-t)t^2 + ht^3 |
| and the line as: |
| y = i*x + j (if the line is more horizontal) |
| or: |
| x = i*y + j (if the line is more vertical) |
| |
| Then using Mathematica, solve for the values of t where the cubic intersects the |
| line: |
| |
| (in) Resultant[ |
| a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - x, |
| e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - i*x - j, x] |
| (out) -e + j + |
| 3 e t - 3 f t - |
| 3 e t^2 + 6 f t^2 - 3 g t^2 + |
| e t^3 - 3 f t^3 + 3 g t^3 - h t^3 + |
| i ( a - |
| 3 a t + 3 b t + |
| 3 a t^2 - 6 b t^2 + 3 c t^2 - |
| a t^3 + 3 b t^3 - 3 c t^3 + d t^3 ) |
| |
| if i goes to infinity, we can rewrite the line in terms of x. Mathematica: |
| |
| (in) Resultant[ |
| a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - i*y - j, |
| e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - y, y] |
| (out) a - j - |
| 3 a t + 3 b t + |
| 3 a t^2 - 6 b t^2 + 3 c t^2 - |
| a t^3 + 3 b t^3 - 3 c t^3 + d t^3 - |
| i ( e - |
| 3 e t + 3 f t + |
| 3 e t^2 - 6 f t^2 + 3 g t^2 - |
| e t^3 + 3 f t^3 - 3 g t^3 + h t^3 ) |
| |
| Solving this with Mathematica produces an expression with hundreds of terms; |
| instead, use Numeric Solutions recipe to solve the cubic. |
| |
| The near-horizontal case, in terms of: Ax^3 + Bx^2 + Cx + D == 0 |
| A = (-(-e + 3*f - 3*g + h) + i*(-a + 3*b - 3*c + d) ) |
| B = 3*(-( e - 2*f + g ) + i*( a - 2*b + c ) ) |
| C = 3*(-(-e + f ) + i*(-a + b ) ) |
| D = (-( e ) + i*( a ) + j ) |
| |
| The near-vertical case, in terms of: Ax^3 + Bx^2 + Cx + D == 0 |
| A = ( (-a + 3*b - 3*c + d) - i*(-e + 3*f - 3*g + h) ) |
| B = 3*( ( a - 2*b + c ) - i*( e - 2*f + g ) ) |
| C = 3*( (-a + b ) - i*(-e + f ) ) |
| D = ( ( a ) - i*( e ) - j ) |
| |
| For horizontal lines: |
| (in) Resultant[ |
| a*(1 - t)^3 + 3*b*(1 - t)^2*t + 3*c*(1 - t)*t^2 + d*t^3 - j, |
| e*(1 - t)^3 + 3*f*(1 - t)^2*t + 3*g*(1 - t)*t^2 + h*t^3 - y, y] |
| (out) e - j - |
| 3 e t + 3 f t + |
| 3 e t^2 - 6 f t^2 + 3 g t^2 - |
| e t^3 + 3 f t^3 - 3 g t^3 + h t^3 |
| */ |
| |
| class LineCubicIntersections { |
| public: |
| |
| LineCubicIntersections(const SkDCubic& c, const SkDLine& l, SkIntersections& i) |
| : cubic(c) |
| , line(l) |
| , intersections(i) { |
| } |
| |
| // see parallel routine in line quadratic intersections |
| int intersectRay(double roots[3]) { |
| double adj = line[1].fX - line[0].fX; |
| double opp = line[1].fY - line[0].fY; |
| SkDCubic r; |
| for (int n = 0; n < 4; ++n) { |
| r[n].fX = (cubic[n].fY - line[0].fY) * adj - (cubic[n].fX - line[0].fX) * opp; |
| } |
| double A, B, C, D; |
| SkDCubic::Coefficients(&r[0].fX, &A, &B, &C, &D); |
| return SkDCubic::RootsValidT(A, B, C, D, roots); |
| } |
| |
| int intersect() { |
| addEndPoints(); |
| double rootVals[3]; |
| int roots = intersectRay(rootVals); |
| for (int index = 0; index < roots; ++index) { |
| double cubicT = rootVals[index]; |
| double lineT = findLineT(cubicT); |
| if (pinTs(&cubicT, &lineT)) { |
| SkDPoint pt = line.xyAtT(lineT); |
| intersections.insert(cubicT, lineT, pt); |
| } |
| } |
| return intersections.used(); |
| } |
| |
| int horizontalIntersect(double axisIntercept, double roots[3]) { |
| double A, B, C, D; |
| SkDCubic::Coefficients(&cubic[0].fY, &A, &B, &C, &D); |
| D -= axisIntercept; |
| return SkDCubic::RootsValidT(A, B, C, D, roots); |
| } |
| |
| int horizontalIntersect(double axisIntercept, double left, double right, bool flipped) { |
| addHorizontalEndPoints(left, right, axisIntercept); |
| double rootVals[3]; |
| int roots = horizontalIntersect(axisIntercept, rootVals); |
| for (int index = 0; index < roots; ++index) { |
| double cubicT = rootVals[index]; |
| SkDPoint pt = cubic.xyAtT(cubicT); |
| double lineT = (pt.fX - left) / (right - left); |
| if (pinTs(&cubicT, &lineT)) { |
| intersections.insert(cubicT, lineT, pt); |
| } |
| } |
| if (flipped) { |
| intersections.flip(); |
| } |
| return intersections.used(); |
| } |
| |
| int verticalIntersect(double axisIntercept, double roots[3]) { |
| double A, B, C, D; |
| SkDCubic::Coefficients(&cubic[0].fX, &A, &B, &C, &D); |
| D -= axisIntercept; |
| return SkDCubic::RootsValidT(A, B, C, D, roots); |
| } |
| |
| int verticalIntersect(double axisIntercept, double top, double bottom, bool flipped) { |
| addVerticalEndPoints(top, bottom, axisIntercept); |
| double rootVals[3]; |
| int roots = verticalIntersect(axisIntercept, rootVals); |
| for (int index = 0; index < roots; ++index) { |
| double cubicT = rootVals[index]; |
| SkDPoint pt = cubic.xyAtT(cubicT); |
| double lineT = (pt.fY - top) / (bottom - top); |
| if (pinTs(&cubicT, &lineT)) { |
| intersections.insert(cubicT, lineT, pt); |
| } |
| } |
| if (flipped) { |
| intersections.flip(); |
| } |
| return intersections.used(); |
| } |
| |
| protected: |
| |
| void addEndPoints() { |
| for (int cIndex = 0; cIndex < 4; cIndex += 3) { |
| for (int lIndex = 0; lIndex < 2; lIndex++) { |
| if (cubic[cIndex] == line[lIndex]) { |
| intersections.insert(cIndex >> 1, lIndex, line[lIndex]); |
| } |
| } |
| } |
| } |
| |
| void addHorizontalEndPoints(double left, double right, double y) { |
| for (int cIndex = 0; cIndex < 4; cIndex += 3) { |
| if (cubic[cIndex].fY != y) { |
| continue; |
| } |
| if (cubic[cIndex].fX == left) { |
| intersections.insert(cIndex >> 1, 0, cubic[cIndex]); |
| } |
| if (cubic[cIndex].fX == right) { |
| intersections.insert(cIndex >> 1, 1, cubic[cIndex]); |
| } |
| } |
| } |
| |
| void addVerticalEndPoints(double top, double bottom, double x) { |
| for (int cIndex = 0; cIndex < 4; cIndex += 3) { |
| if (cubic[cIndex].fX != x) { |
| continue; |
| } |
| if (cubic[cIndex].fY == top) { |
| intersections.insert(cIndex >> 1, 0, cubic[cIndex]); |
| } |
| if (cubic[cIndex].fY == bottom) { |
| intersections.insert(cIndex >> 1, 1, cubic[cIndex]); |
| } |
| } |
| } |
| |
| double findLineT(double t) { |
| SkDPoint xy = cubic.xyAtT(t); |
| double dx = line[1].fX - line[0].fX; |
| double dy = line[1].fY - line[0].fY; |
| if (fabs(dx) > fabs(dy)) { |
| return (xy.fX - line[0].fX) / dx; |
| } |
| return (xy.fY - line[0].fY) / dy; |
| } |
| |
| static bool pinTs(double* cubicT, double* lineT) { |
| if (!approximately_one_or_less(*lineT)) { |
| return false; |
| } |
| if (!approximately_zero_or_more(*lineT)) { |
| return false; |
| } |
| if (precisely_less_than_zero(*cubicT)) { |
| *cubicT = 0; |
| } else if (precisely_greater_than_one(*cubicT)) { |
| *cubicT = 1; |
| } |
| if (precisely_less_than_zero(*lineT)) { |
| *lineT = 0; |
| } else if (precisely_greater_than_one(*lineT)) { |
| *lineT = 1; |
| } |
| return true; |
| } |
| |
| private: |
| |
| const SkDCubic& cubic; |
| const SkDLine& line; |
| SkIntersections& intersections; |
| }; |
| |
| int SkIntersections::horizontal(const SkDCubic& cubic, double left, double right, double y, |
| bool flipped) { |
| LineCubicIntersections c(cubic, *(static_cast<SkDLine*>(0)), *this); |
| return c.horizontalIntersect(y, left, right, flipped); |
| } |
| |
| int SkIntersections::vertical(const SkDCubic& cubic, double top, double bottom, double x, |
| bool flipped) { |
| LineCubicIntersections c(cubic, *(static_cast<SkDLine*>(0)), *this); |
| return c.verticalIntersect(x, top, bottom, flipped); |
| } |
| |
| int SkIntersections::intersect(const SkDCubic& cubic, const SkDLine& line) { |
| LineCubicIntersections c(cubic, line, *this); |
| return c.intersect(); |
| } |
| |
| int SkIntersections::intersectRay(const SkDCubic& cubic, const SkDLine& line) { |
| LineCubicIntersections c(cubic, line, *this); |
| fUsed = c.intersectRay(fT[0]); |
| for (int index = 0; index < fUsed; ++index) { |
| fPt[index] = cubic.xyAtT(fT[0][index]); |
| } |
| return fUsed; |
| } |