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Diffstat (limited to 'chromium/third_party/skia/experimental/Intersection/LineQuadraticIntersection.cpp')
-rw-r--r-- | chromium/third_party/skia/experimental/Intersection/LineQuadraticIntersection.cpp | 369 |
1 files changed, 369 insertions, 0 deletions
diff --git a/chromium/third_party/skia/experimental/Intersection/LineQuadraticIntersection.cpp b/chromium/third_party/skia/experimental/Intersection/LineQuadraticIntersection.cpp new file mode 100644 index 00000000000..f855d97e4c4 --- /dev/null +++ b/chromium/third_party/skia/experimental/Intersection/LineQuadraticIntersection.cpp @@ -0,0 +1,369 @@ +/* + * 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 "CurveIntersection.h" +#include "Intersections.h" +#include "LineUtilities.h" +#include "QuadraticUtilities.h" + +/* +Find the interection of a line and quadratic by solving for valid t values. + +From http://stackoverflow.com/questions/1853637/how-to-find-the-mathematical-function-defining-a-bezier-curve + +"A Bezier curve is a parametric function. A quadratic Bezier curve (i.e. three +control points) can be expressed as: F(t) = A(1 - t)^2 + B(1 - t)t + Ct^2 where +A, B and C are points and t goes from zero to one. + +This will give you two equations: + + x = a(1 - t)^2 + b(1 - t)t + ct^2 + y = d(1 - t)^2 + e(1 - t)t + ft^2 + +If you add for instance the line equation (y = kx + m) to that, you'll end up +with three equations and three unknowns (x, y and t)." + +Similar to above, the quadratic is represented as + x = a(1-t)^2 + 2b(1-t)t + ct^2 + y = d(1-t)^2 + 2e(1-t)t + ft^2 +and the line as + y = g*x + h + +Using Mathematica, solve for the values of t where the quadratic intersects the +line: + + (in) t1 = Resultant[a*(1 - t)^2 + 2*b*(1 - t)*t + c*t^2 - x, + d*(1 - t)^2 + 2*e*(1 - t)*t + f*t^2 - g*x - h, x] + (out) -d + h + 2 d t - 2 e t - d t^2 + 2 e t^2 - f t^2 + + g (a - 2 a t + 2 b t + a t^2 - 2 b t^2 + c t^2) + (in) Solve[t1 == 0, t] + (out) { + {t -> (-2 d + 2 e + 2 a g - 2 b g - + Sqrt[(2 d - 2 e - 2 a g + 2 b g)^2 - + 4 (-d + 2 e - f + a g - 2 b g + c g) (-d + a g + h)]) / + (2 (-d + 2 e - f + a g - 2 b g + c g)) + }, + {t -> (-2 d + 2 e + 2 a g - 2 b g + + Sqrt[(2 d - 2 e - 2 a g + 2 b g)^2 - + 4 (-d + 2 e - f + a g - 2 b g + c g) (-d + a g + h)]) / + (2 (-d + 2 e - f + a g - 2 b g + c g)) + } + } + +Using the results above (when the line tends towards horizontal) + A = (-(d - 2*e + f) + g*(a - 2*b + c) ) + B = 2*( (d - e ) - g*(a - b ) ) + C = (-(d ) + g*(a ) + h ) + +If g goes to infinity, we can rewrite the line in terms of x. + x = g'*y + h' + +And solve accordingly in Mathematica: + + (in) t2 = Resultant[a*(1 - t)^2 + 2*b*(1 - t)*t + c*t^2 - g'*y - h', + d*(1 - t)^2 + 2*e*(1 - t)*t + f*t^2 - y, y] + (out) a - h' - 2 a t + 2 b t + a t^2 - 2 b t^2 + c t^2 - + g' (d - 2 d t + 2 e t + d t^2 - 2 e t^2 + f t^2) + (in) Solve[t2 == 0, t] + (out) { + {t -> (2 a - 2 b - 2 d g' + 2 e g' - + Sqrt[(-2 a + 2 b + 2 d g' - 2 e g')^2 - + 4 (a - 2 b + c - d g' + 2 e g' - f g') (a - d g' - h')]) / + (2 (a - 2 b + c - d g' + 2 e g' - f g')) + }, + {t -> (2 a - 2 b - 2 d g' + 2 e g' + + Sqrt[(-2 a + 2 b + 2 d g' - 2 e g')^2 - + 4 (a - 2 b + c - d g' + 2 e g' - f g') (a - d g' - h')])/ + (2 (a - 2 b + c - d g' + 2 e g' - f g')) + } + } + +Thus, if the slope of the line tends towards vertical, we use: + A = ( (a - 2*b + c) - g'*(d - 2*e + f) ) + B = 2*(-(a - b ) + g'*(d - e ) ) + C = ( (a ) - g'*(d ) - h' ) + */ + + +class LineQuadraticIntersections { +public: + +LineQuadraticIntersections(const Quadratic& q, const _Line& l, Intersections& i) + : quad(q) + , line(l) + , intersections(i) { +} + +int intersectRay(double roots[2]) { +/* + solve by rotating line+quad so line is horizontal, then finding the roots + set up matrix to rotate quad to x-axis + |cos(a) -sin(a)| + |sin(a) cos(a)| + note that cos(a) = A(djacent) / Hypoteneuse + sin(a) = O(pposite) / Hypoteneuse + since we are computing Ts, we can ignore hypoteneuse, the scale factor: + | A -O | + | O A | + A = line[1].x - line[0].x (adjacent side of the right triangle) + O = line[1].y - line[0].y (opposite side of the right triangle) + for each of the three points (e.g. n = 0 to 2) + quad[n].y' = (quad[n].y - line[0].y) * A - (quad[n].x - line[0].x) * O +*/ + double adj = line[1].x - line[0].x; + double opp = line[1].y - line[0].y; + double r[3]; + for (int n = 0; n < 3; ++n) { + r[n] = (quad[n].y - line[0].y) * adj - (quad[n].x - line[0].x) * opp; + } + double A = r[2]; + double B = r[1]; + double C = r[0]; + A += C - 2 * B; // A = a - 2*b + c + B -= C; // B = -(b - c) + return quadraticRootsValidT(A, 2 * B, C, roots); +} + +int intersect() { + addEndPoints(); + double rootVals[2]; + int roots = intersectRay(rootVals); + for (int index = 0; index < roots; ++index) { + double quadT = rootVals[index]; + double lineT = findLineT(quadT); + if (pinTs(quadT, lineT)) { + _Point pt; + xy_at_t(line, lineT, pt.x, pt.y); + intersections.insert(quadT, lineT, pt); + } + } + return intersections.fUsed; +} + +int horizontalIntersect(double axisIntercept, double roots[2]) { + double D = quad[2].y; // f + double E = quad[1].y; // e + double F = quad[0].y; // d + D += F - 2 * E; // D = d - 2*e + f + E -= F; // E = -(d - e) + F -= axisIntercept; + return quadraticRootsValidT(D, 2 * E, F, roots); +} + +int horizontalIntersect(double axisIntercept, double left, double right, bool flipped) { + addHorizontalEndPoints(left, right, axisIntercept); + double rootVals[2]; + int roots = horizontalIntersect(axisIntercept, rootVals); + for (int index = 0; index < roots; ++index) { + _Point pt; + double quadT = rootVals[index]; + xy_at_t(quad, quadT, pt.x, pt.y); + double lineT = (pt.x - left) / (right - left); + if (pinTs(quadT, lineT)) { + intersections.insert(quadT, lineT, pt); + } + } + if (flipped) { + flip(); + } + return intersections.fUsed; +} + +int verticalIntersect(double axisIntercept, double roots[2]) { + double D = quad[2].x; // f + double E = quad[1].x; // e + double F = quad[0].x; // d + D += F - 2 * E; // D = d - 2*e + f + E -= F; // E = -(d - e) + F -= axisIntercept; + return quadraticRootsValidT(D, 2 * E, F, roots); +} + +int verticalIntersect(double axisIntercept, double top, double bottom, bool flipped) { + addVerticalEndPoints(top, bottom, axisIntercept); + double rootVals[2]; + int roots = verticalIntersect(axisIntercept, rootVals); + for (int index = 0; index < roots; ++index) { + _Point pt; + double quadT = rootVals[index]; + xy_at_t(quad, quadT, pt.x, pt.y); + double lineT = (pt.y - top) / (bottom - top); + if (pinTs(quadT, lineT)) { + intersections.insert(quadT, lineT, pt); + } + } + if (flipped) { + flip(); + } + return intersections.fUsed; +} + +protected: + +// add endpoints first to get zero and one t values exactly +void addEndPoints() +{ + for (int qIndex = 0; qIndex < 3; qIndex += 2) { + for (int lIndex = 0; lIndex < 2; lIndex++) { + if (quad[qIndex] == line[lIndex]) { + intersections.insert(qIndex >> 1, lIndex, line[lIndex]); + } + } + } +} + +void addHorizontalEndPoints(double left, double right, double y) +{ + for (int qIndex = 0; qIndex < 3; qIndex += 2) { + if (quad[qIndex].y != y) { + continue; + } + if (quad[qIndex].x == left) { + intersections.insert(qIndex >> 1, 0, quad[qIndex]); + } + if (quad[qIndex].x == right) { + intersections.insert(qIndex >> 1, 1, quad[qIndex]); + } + } +} + +void addVerticalEndPoints(double top, double bottom, double x) +{ + for (int qIndex = 0; qIndex < 3; qIndex += 2) { + if (quad[qIndex].x != x) { + continue; + } + if (quad[qIndex].y == top) { + intersections.insert(qIndex >> 1, 0, quad[qIndex]); + } + if (quad[qIndex].y == bottom) { + intersections.insert(qIndex >> 1, 1, quad[qIndex]); + } + } +} + +double findLineT(double t) { + double x, y; + xy_at_t(quad, t, x, y); + double dx = line[1].x - line[0].x; + double dy = line[1].y - line[0].y; + if (fabs(dx) > fabs(dy)) { + return (x - line[0].x) / dx; + } + return (y - line[0].y) / dy; +} + +void flip() { + // OPTIMIZATION: instead of swapping, pass original line, use [1].y - [0].y + int roots = intersections.fUsed; + for (int index = 0; index < roots; ++index) { + intersections.fT[1][index] = 1 - intersections.fT[1][index]; + } +} + +static bool pinTs(double& quadT, double& lineT) { + if (!approximately_one_or_less(lineT)) { + return false; + } + if (!approximately_zero_or_more(lineT)) { + return false; + } + if (precisely_less_than_zero(quadT)) { + quadT = 0; + } else if (precisely_greater_than_one(quadT)) { + quadT = 1; + } + if (precisely_less_than_zero(lineT)) { + lineT = 0; + } else if (precisely_greater_than_one(lineT)) { + lineT = 1; + } + return true; +} + +private: + +const Quadratic& quad; +const _Line& line; +Intersections& intersections; +}; + +// utility for pairs of coincident quads +static double horizontalIntersect(const Quadratic& quad, const _Point& pt) { + LineQuadraticIntersections q(quad, *((_Line*) 0), *((Intersections*) 0)); + double rootVals[2]; + int roots = q.horizontalIntersect(pt.y, rootVals); + for (int index = 0; index < roots; ++index) { + double x; + double t = rootVals[index]; + xy_at_t(quad, t, x, *(double*) 0); + if (AlmostEqualUlps(x, pt.x)) { + return t; + } + } + return -1; +} + +static double verticalIntersect(const Quadratic& quad, const _Point& pt) { + LineQuadraticIntersections q(quad, *((_Line*) 0), *((Intersections*) 0)); + double rootVals[2]; + int roots = q.verticalIntersect(pt.x, rootVals); + for (int index = 0; index < roots; ++index) { + double y; + double t = rootVals[index]; + xy_at_t(quad, t, *(double*) 0, y); + if (AlmostEqualUlps(y, pt.y)) { + return t; + } + } + return -1; +} + +double axialIntersect(const Quadratic& q1, const _Point& p, bool vertical) { + if (vertical) { + return verticalIntersect(q1, p); + } + return horizontalIntersect(q1, p); +} + +int horizontalIntersect(const Quadratic& quad, double left, double right, + double y, double tRange[2]) { + LineQuadraticIntersections q(quad, *((_Line*) 0), *((Intersections*) 0)); + double rootVals[2]; + int result = q.horizontalIntersect(y, rootVals); + int tCount = 0; + for (int index = 0; index < result; ++index) { + double x, y; + xy_at_t(quad, rootVals[index], x, y); + if (x < left || x > right) { + continue; + } + tRange[tCount++] = rootVals[index]; + } + return tCount; +} + +int horizontalIntersect(const Quadratic& quad, double left, double right, double y, + bool flipped, Intersections& intersections) { + LineQuadraticIntersections q(quad, *((_Line*) 0), intersections); + return q.horizontalIntersect(y, left, right, flipped); +} + +int verticalIntersect(const Quadratic& quad, double top, double bottom, double x, + bool flipped, Intersections& intersections) { + LineQuadraticIntersections q(quad, *((_Line*) 0), intersections); + return q.verticalIntersect(x, top, bottom, flipped); +} + +int intersect(const Quadratic& quad, const _Line& line, Intersections& i) { + LineQuadraticIntersections q(quad, line, i); + return q.intersect(); +} + +int intersectRay(const Quadratic& quad, const _Line& line, Intersections& i) { + LineQuadraticIntersections q(quad, line, i); + return q.intersectRay(i.fT[0]); +} |