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+/*
+ * 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]);
+}