work in progress for shape operations
A experimental/Intersection
A experimental/Intersection/Intersections.h
A experimental/Intersection/DataTypes.cpp
A experimental/Intersection/QuadraticReduceOrder.cpp
A experimental/Intersection/IntersectionUtilities.cpp
A experimental/Intersection/CubicIntersection_Tests.h
A experimental/Intersection/LineParameteters_Test.cpp
A experimental/Intersection/ReduceOrder.cpp
A experimental/Intersection/QuadraticIntersection.cpp
A experimental/Intersection/Extrema.h
A experimental/Intersection/CubicIntersection_TestData.h
A experimental/Intersection/QuadraticParameterization_Test.cpp
A experimental/Intersection/TestUtilities.cpp
A experimental/Intersection/CubicRoots.cpp
A experimental/Intersection/QuadraticParameterization.cpp
A experimental/Intersection/QuadraticSubDivide.cpp
A experimental/Intersection/LineIntersection_Test.cpp
A experimental/Intersection/LineIntersection.cpp
A experimental/Intersection/CubicParameterizationCode.cpp
A experimental/Intersection/LineParameters.h
A experimental/Intersection/CubicIntersection.h
A experimental/Intersection/CubeRoot.cpp
A experimental/Intersection/SkAntiEdge.h
A experimental/Intersection/ConvexHull_Test.cpp
A experimental/Intersection/CubicBezierClip_Test.cpp
A experimental/Intersection/CubicIntersection_Tests.cpp
A experimental/Intersection/CubicBezierClip.cpp
A experimental/Intersection/CubicIntersectionT.cpp
A experimental/Intersection/Inline_Tests.cpp
A experimental/Intersection/ReduceOrder_Test.cpp
A experimental/Intersection/QuadraticIntersection_TestData.h
A experimental/Intersection/DataTypes.h
A experimental/Intersection/Extrema.cpp
A experimental/Intersection/EdgeApp.cpp
A experimental/Intersection/CubicIntersection_TestData.cpp
A experimental/Intersection/IntersectionUtilities.h
A experimental/Intersection/CubicReduceOrder.cpp
A experimental/Intersection/CubicCoincidence.cpp
A experimental/Intersection/CubicIntersection_Test.cpp
A experimental/Intersection/CubicIntersection.cpp
A experimental/Intersection/QuadraticUtilities.h
A experimental/Intersection/SkAntiEdge.cpp
A experimental/Intersection/TestUtilities.h
A experimental/Intersection/CubicParameterization_Test.cpp
A experimental/Intersection/LineIntersection.h
A experimental/Intersection/CubicSubDivide.cpp
A experimental/Intersection/CubicParameterization.cpp
A experimental/Intersection/QuadraticBezierClip_Test.cpp
A experimental/Intersection/QuadraticBezierClip.cpp
A experimental/Intersection/BezierClip_Test.cpp
A experimental/Intersection/ConvexHull.cpp
A experimental/Intersection/BezierClip.cpp
A experimental/Intersection/QuadraticIntersection_TestData.cpp
git-svn-id: http://skia.googlecode.com/svn/trunk@3005 2bbb7eff-a529-9590-31e7-b0007b416f81
diff --git a/experimental/Intersection/QuadraticParameterization.cpp b/experimental/Intersection/QuadraticParameterization.cpp
new file mode 100644
index 0000000..a78aa9f
--- /dev/null
+++ b/experimental/Intersection/QuadraticParameterization.cpp
@@ -0,0 +1,132 @@
+#include "CubicIntersection.h"
+#include "QuadraticUtilities.h"
+
+/* from http://tom.cs.byu.edu/~tom/papers/cvgip84.pdf 4.1
+ *
+ * This paper proves that Syvester's method can compute the implicit form of
+ * the quadratic from the parameterzied form.
+ *
+ * Given x = a*t*t + b*t + c (the parameterized form)
+ * y = d*t*t + e*t + f
+ *
+ * we want to find an equation of the implicit form:
+ *
+ * A*x*x + B*x*y + C*y*y + D*x + E*y + F = 0
+ *
+ * The implicit form can be expressed as a 4x4 determinant, as shown.
+ *
+ * The resultant obtained by Syvester's method is
+ *
+ * | a b (c - x) 0 |
+ * | 0 a b (c - x) |
+ * | d e (f - y) 0 |
+ * | 0 d e (f - y) |
+ *
+ * which expands to
+ *
+ * d*d*x*x + -2*a*d*x*y + a*a*y*y
+ * + (-2*c*d*d + b*e*d - a*e*e + 2*a*f*d)*x
+ * + (-2*f*a*a + e*b*a - d*b*b + 2*d*c*a)*y
+ * +
+ * | a b c 0 |
+ * | 0 a b c | == 0.
+ * | d e f 0 |
+ * | 0 d e f |
+ *
+ * Expanding the constant determinant results in
+ *
+ * | a b c | | b c 0 |
+ * a*| e f 0 | + d*| a b c | ==
+ * | d e f | | d e f |
+ *
+ * a*(a*f*f + c*e*e - c*f*d - b*e*f) + d*(b*b*f + c*c*d - c*a*f - c*e*b)
+ *
+ */
+
+enum {
+ xx_coeff,
+ xy_coeff,
+ yy_coeff,
+ x_coeff,
+ y_coeff,
+ c_coeff,
+ coeff_count
+};
+
+static bool straight_forward = true;
+
+static void implicit_coefficients(const Quadratic& q, double p[coeff_count]) {
+ double a, b, c;
+ set_abc(&q[0].x, a, b, c);
+ double d, e, f;
+ set_abc(&q[0].y, d, e, f);
+ // compute the implicit coefficients
+ if (straight_forward) { // 42 muls, 13 adds
+ p[xx_coeff] = d * d;
+ p[xy_coeff] = -2 * a * d;
+ p[yy_coeff] = a * a;
+ p[x_coeff] = -2*c*d*d + b*e*d - a*e*e + 2*a*f*d;
+ p[y_coeff] = -2*f*a*a + e*b*a - d*b*b + 2*d*c*a;
+ p[c_coeff] = a*(a*f*f + c*e*e - c*f*d - b*e*f)
+ + d*(b*b*f + c*c*d - c*a*f - c*e*b);
+ } else { // 26 muls, 11 adds
+ double aa = a * a;
+ double ad = a * d;
+ double dd = d * d;
+ p[xx_coeff] = dd;
+ p[xy_coeff] = -2 * ad;
+ p[yy_coeff] = aa;
+ double be = b * e;
+ double bde = be * d;
+ double cdd = c * dd;
+ double ee = e * e;
+ p[x_coeff] = -2*cdd + bde - a*ee + 2*ad*f;
+ double aaf = aa * f;
+ double abe = a * be;
+ double ac = a * c;
+ double bb_2ac = b*b - 2*ac;
+ p[y_coeff] = -2*aaf + abe - d*bb_2ac;
+ p[c_coeff] = aaf*f + ac*ee + d*f*bb_2ac - abe*f + c*cdd - c*bde;
+ }
+}
+
+ /* Given a pair of quadratics, determine their parametric coefficients.
+ * If the scaled coefficients are nearly equal, then the part of the quadratics
+ * may be coincident.
+ * FIXME: optimization -- since comparison short-circuits on no match,
+ * lazily compute the coefficients, comparing the easiest to compute first.
+ * xx and yy first; then xy; and so on.
+ */
+bool implicit_matches(const Quadratic& one, const Quadratic& two) {
+ double p1[coeff_count]; // a'xx , b'xy , c'yy , d'x , e'y , f
+ double p2[coeff_count];
+ implicit_coefficients(one, p1);
+ implicit_coefficients(two, p2);
+ int first = 0;
+ for (int index = 0; index < coeff_count; ++index) {
+ if (approximately_zero(p1[index]) || approximately_zero(p2[index])) {
+ first += first == index;
+ continue;
+ }
+ if (first == index) {
+ continue;
+ }
+ if (!approximately_equal(p1[index] * p2[first],
+ p1[first] * p2[index])) {
+ return false;
+ }
+ }
+ return true;
+}
+
+static double tangent(const double* quadratic, double t) {
+ double a, b, c;
+ set_abc(quadratic, a, b, c);
+ return 2 * a * t + b;
+}
+
+void tangent(const Quadratic& quadratic, double t, _Point& result) {
+ result.x = tangent(&quadratic[0].x, t);
+ result.y = tangent(&quadratic[0].y, t);
+}
+