1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/gfx/skia/trunk/src/pathops/SkDQuadImplicit.cpp Wed Dec 31 06:09:35 2014 +0100
1.3 @@ -0,0 +1,117 @@
1.4 +/*
1.5 + * Copyright 2012 Google Inc.
1.6 + *
1.7 + * Use of this source code is governed by a BSD-style license that can be
1.8 + * found in the LICENSE file.
1.9 + */
1.10 +#include "SkDQuadImplicit.h"
1.11 +
1.12 +/* from http://tom.cs.byu.edu/~tom/papers/cvgip84.pdf 4.1
1.13 + *
1.14 + * This paper proves that Syvester's method can compute the implicit form of
1.15 + * the quadratic from the parameterized form.
1.16 + *
1.17 + * Given x = a*t*t + b*t + c (the parameterized form)
1.18 + * y = d*t*t + e*t + f
1.19 + *
1.20 + * we want to find an equation of the implicit form:
1.21 + *
1.22 + * A*x*x + B*x*y + C*y*y + D*x + E*y + F = 0
1.23 + *
1.24 + * The implicit form can be expressed as a 4x4 determinant, as shown.
1.25 + *
1.26 + * The resultant obtained by Syvester's method is
1.27 + *
1.28 + * | a b (c - x) 0 |
1.29 + * | 0 a b (c - x) |
1.30 + * | d e (f - y) 0 |
1.31 + * | 0 d e (f - y) |
1.32 + *
1.33 + * which expands to
1.34 + *
1.35 + * d*d*x*x + -2*a*d*x*y + a*a*y*y
1.36 + * + (-2*c*d*d + b*e*d - a*e*e + 2*a*f*d)*x
1.37 + * + (-2*f*a*a + e*b*a - d*b*b + 2*d*c*a)*y
1.38 + * +
1.39 + * | a b c 0 |
1.40 + * | 0 a b c | == 0.
1.41 + * | d e f 0 |
1.42 + * | 0 d e f |
1.43 + *
1.44 + * Expanding the constant determinant results in
1.45 + *
1.46 + * | a b c | | b c 0 |
1.47 + * a*| e f 0 | + d*| a b c | ==
1.48 + * | d e f | | d e f |
1.49 + *
1.50 + * 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)
1.51 + *
1.52 + */
1.53 +
1.54 +// use the tricky arithmetic path, but leave the original to compare just in case
1.55 +static bool straight_forward = false;
1.56 +
1.57 +SkDQuadImplicit::SkDQuadImplicit(const SkDQuad& q) {
1.58 + double a, b, c;
1.59 + SkDQuad::SetABC(&q[0].fX, &a, &b, &c);
1.60 + double d, e, f;
1.61 + SkDQuad::SetABC(&q[0].fY, &d, &e, &f);
1.62 + // compute the implicit coefficients
1.63 + if (straight_forward) { // 42 muls, 13 adds
1.64 + fP[kXx_Coeff] = d * d;
1.65 + fP[kXy_Coeff] = -2 * a * d;
1.66 + fP[kYy_Coeff] = a * a;
1.67 + fP[kX_Coeff] = -2*c*d*d + b*e*d - a*e*e + 2*a*f*d;
1.68 + fP[kY_Coeff] = -2*f*a*a + e*b*a - d*b*b + 2*d*c*a;
1.69 + fP[kC_Coeff] = a*(a*f*f + c*e*e - c*f*d - b*e*f)
1.70 + + d*(b*b*f + c*c*d - c*a*f - c*e*b);
1.71 + } else { // 26 muls, 11 adds
1.72 + double aa = a * a;
1.73 + double ad = a * d;
1.74 + double dd = d * d;
1.75 + fP[kXx_Coeff] = dd;
1.76 + fP[kXy_Coeff] = -2 * ad;
1.77 + fP[kYy_Coeff] = aa;
1.78 + double be = b * e;
1.79 + double bde = be * d;
1.80 + double cdd = c * dd;
1.81 + double ee = e * e;
1.82 + fP[kX_Coeff] = -2*cdd + bde - a*ee + 2*ad*f;
1.83 + double aaf = aa * f;
1.84 + double abe = a * be;
1.85 + double ac = a * c;
1.86 + double bb_2ac = b*b - 2*ac;
1.87 + fP[kY_Coeff] = -2*aaf + abe - d*bb_2ac;
1.88 + fP[kC_Coeff] = aaf*f + ac*ee + d*f*bb_2ac - abe*f + c*cdd - c*bde;
1.89 + }
1.90 +}
1.91 +
1.92 + /* Given a pair of quadratics, determine their parametric coefficients.
1.93 + * If the scaled coefficients are nearly equal, then the part of the quadratics
1.94 + * may be coincident.
1.95 + * OPTIMIZATION -- since comparison short-circuits on no match,
1.96 + * lazily compute the coefficients, comparing the easiest to compute first.
1.97 + * xx and yy first; then xy; and so on.
1.98 + */
1.99 +bool SkDQuadImplicit::match(const SkDQuadImplicit& p2) const {
1.100 + int first = 0;
1.101 + for (int index = 0; index <= kC_Coeff; ++index) {
1.102 + if (approximately_zero(fP[index]) && approximately_zero(p2.fP[index])) {
1.103 + first += first == index;
1.104 + continue;
1.105 + }
1.106 + if (first == index) {
1.107 + continue;
1.108 + }
1.109 + if (!AlmostDequalUlps(fP[index] * p2.fP[first], fP[first] * p2.fP[index])) {
1.110 + return false;
1.111 + }
1.112 + }
1.113 + return true;
1.114 +}
1.115 +
1.116 +bool SkDQuadImplicit::Match(const SkDQuad& quad1, const SkDQuad& quad2) {
1.117 + SkDQuadImplicit i1(quad1); // a'xx , b'xy , c'yy , d'x , e'y , f
1.118 + SkDQuadImplicit i2(quad2);
1.119 + return i1.match(i2);
1.120 +}
