Add intersections for path ops

src/pathops/SkDQuadImplicit.cpp

+/*
+ * 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 "SkDQuadImplicit.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 parameterized 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)
+ *
+ */
+
+// OPTIMIZATION: test, verify tricky arithmetic
+static bool straight_forward = true;
+
+SkDQuadImplicit::SkDQuadImplicit(const SkDQuad& q) {
+    double a, b, c;
+    SkDQuad::SetABC(&q[0].fX, &a, &b, &c);
+    double d, e, f;
+    SkDQuad::SetABC(&q[0].fY, &d, &e, &f);
+    // compute the implicit coefficients
+    if (straight_forward) {  // 42 muls, 13 adds
+        fP[kXx_Coeff] = d * d;
+        fP[kXy_Coeff] = -2 * a * d;
+        fP[kYy_Coeff] = a * a;
+        fP[kX_Coeff] = -2*c*d*d + b*e*d - a*e*e + 2*a*f*d;
+        fP[kY_Coeff] = -2*f*a*a + e*b*a - d*b*b + 2*d*c*a;
+        fP[kC_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;
+        fP[kXx_Coeff] = dd;
+        fP[kXy_Coeff] = -2 * ad;
+        fP[kYy_Coeff] = aa;
+        double be = b * e;
+        double bde = be * d;
+        double cdd = c * dd;
+        double ee = e * e;
+        fP[kX_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;
+        fP[kY_Coeff] = -2*aaf + abe - d*bb_2ac;
+        fP[kC_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 SkDQuadImplicit::match(const SkDQuadImplicit& p2) const {
+    int first = 0;
+    for (int index = 0; index <= kC_Coeff; ++index) {
+        if (approximately_zero(fP[index]) && approximately_zero(p2.fP[index])) {
+            first += first == index;
+            continue;
+        }
+        if (first == index) {
+            continue;
+        }
+        if (!AlmostEqualUlps(fP[index] * p2.fP[first], fP[first] * p2.fP[index])) {
+            return false;
+        }
+    }
+    return true;
+}
+
+bool SkDQuadImplicit::Match(const SkDQuad& quad1, const SkDQuad& quad2) {
+    SkDQuadImplicit i1(quad1);  // a'xx , b'xy , c'yy , d'x , e'y , f
+    SkDQuadImplicit i2(quad2);
+    return i1.match(i2);
+}