gfx/skia/trunk/src/pathops/SkDQuadImplicit.cpp@6474c204b198 (annotated)
gfx/skia/trunk/src/pathops/SkDQuadImplicit.cpp
Wed, 31 Dec 2014 06:09:35 +0100
- author
- Michael Schloh von Bennewitz <michael@schloh.com>
- date
- Wed, 31 Dec 2014 06:09:35 +0100
- changeset 0
- 6474c204b198
- permissions
- -rw-r--r--
Cloned upstream origin tor-browser at tor-browser-31.3.0esr-4.5-1-build1
revision ID fc1c9ff7c1b2defdbc039f12214767608f46423f for hacking purpose.
| michael@0 | 1 | /* |
| michael@0 | 2 | * Copyright 2012 Google Inc. |
| michael@0 | 3 | * |
| michael@0 | 4 | * Use of this source code is governed by a BSD-style license that can be |
| michael@0 | 5 | * found in the LICENSE file. |
| michael@0 | 6 | */ |
| michael@0 | 7 | #include "SkDQuadImplicit.h" |
| michael@0 | 8 | |
| michael@0 | 9 | /* from http://tom.cs.byu.edu/~tom/papers/cvgip84.pdf 4.1 |
| michael@0 | 10 | * |
| michael@0 | 11 | * This paper proves that Syvester's method can compute the implicit form of |
| michael@0 | 12 | * the quadratic from the parameterized form. |
| michael@0 | 13 | * |
| michael@0 | 14 | * Given x = a*t*t + b*t + c (the parameterized form) |
| michael@0 | 15 | * y = d*t*t + e*t + f |
| michael@0 | 16 | * |
| michael@0 | 17 | * we want to find an equation of the implicit form: |
| michael@0 | 18 | * |
| michael@0 | 19 | * A*x*x + B*x*y + C*y*y + D*x + E*y + F = 0 |
| michael@0 | 20 | * |
| michael@0 | 21 | * The implicit form can be expressed as a 4x4 determinant, as shown. |
| michael@0 | 22 | * |
| michael@0 | 23 | * The resultant obtained by Syvester's method is |
| michael@0 | 24 | * |
| michael@0 | 25 | * | a b (c - x) 0 | |
| michael@0 | 26 | * | 0 a b (c - x) | |
| michael@0 | 27 | * | d e (f - y) 0 | |
| michael@0 | 28 | * | 0 d e (f - y) | |
| michael@0 | 29 | * |
| michael@0 | 30 | * which expands to |
| michael@0 | 31 | * |
| michael@0 | 32 | * d*d*x*x + -2*a*d*x*y + a*a*y*y |
| michael@0 | 33 | * + (-2*c*d*d + b*e*d - a*e*e + 2*a*f*d)*x |
| michael@0 | 34 | * + (-2*f*a*a + e*b*a - d*b*b + 2*d*c*a)*y |
| michael@0 | 35 | * + |
| michael@0 | 36 | * | a b c 0 | |
| michael@0 | 37 | * | 0 a b c | == 0. |
| michael@0 | 38 | * | d e f 0 | |
| michael@0 | 39 | * | 0 d e f | |
| michael@0 | 40 | * |
| michael@0 | 41 | * Expanding the constant determinant results in |
| michael@0 | 42 | * |
| michael@0 | 43 | * | a b c | | b c 0 | |
| michael@0 | 44 | * a*| e f 0 | + d*| a b c | == |
| michael@0 | 45 | * | d e f | | d e f | |
| michael@0 | 46 | * |
| michael@0 | 47 | * 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) |
| michael@0 | 48 | * |
| michael@0 | 49 | */ |
| michael@0 | 50 | |
| michael@0 | 51 | // use the tricky arithmetic path, but leave the original to compare just in case |
| michael@0 | 52 | static bool straight_forward = false; |
| michael@0 | 53 | |
| michael@0 | 54 | SkDQuadImplicit::SkDQuadImplicit(const SkDQuad& q) { |
| michael@0 | 55 | double a, b, c; |
| michael@0 | 56 | SkDQuad::SetABC(&q[0].fX, &a, &b, &c); |
| michael@0 | 57 | double d, e, f; |
| michael@0 | 58 | SkDQuad::SetABC(&q[0].fY, &d, &e, &f); |
| michael@0 | 59 | // compute the implicit coefficients |
| michael@0 | 60 | if (straight_forward) { // 42 muls, 13 adds |
| michael@0 | 61 | fP[kXx_Coeff] = d * d; |
| michael@0 | 62 | fP[kXy_Coeff] = -2 * a * d; |
| michael@0 | 63 | fP[kYy_Coeff] = a * a; |
| michael@0 | 64 | fP[kX_Coeff] = -2*c*d*d + b*e*d - a*e*e + 2*a*f*d; |
| michael@0 | 65 | fP[kY_Coeff] = -2*f*a*a + e*b*a - d*b*b + 2*d*c*a; |
| michael@0 | 66 | fP[kC_Coeff] = a*(a*f*f + c*e*e - c*f*d - b*e*f) |
| michael@0 | 67 | + d*(b*b*f + c*c*d - c*a*f - c*e*b); |
| michael@0 | 68 | } else { // 26 muls, 11 adds |
| michael@0 | 69 | double aa = a * a; |
| michael@0 | 70 | double ad = a * d; |
| michael@0 | 71 | double dd = d * d; |
| michael@0 | 72 | fP[kXx_Coeff] = dd; |
| michael@0 | 73 | fP[kXy_Coeff] = -2 * ad; |
| michael@0 | 74 | fP[kYy_Coeff] = aa; |
| michael@0 | 75 | double be = b * e; |
| michael@0 | 76 | double bde = be * d; |
| michael@0 | 77 | double cdd = c * dd; |
| michael@0 | 78 | double ee = e * e; |
| michael@0 | 79 | fP[kX_Coeff] = -2*cdd + bde - a*ee + 2*ad*f; |
| michael@0 | 80 | double aaf = aa * f; |
| michael@0 | 81 | double abe = a * be; |
| michael@0 | 82 | double ac = a * c; |
| michael@0 | 83 | double bb_2ac = b*b - 2*ac; |
| michael@0 | 84 | fP[kY_Coeff] = -2*aaf + abe - d*bb_2ac; |
| michael@0 | 85 | fP[kC_Coeff] = aaf*f + ac*ee + d*f*bb_2ac - abe*f + c*cdd - c*bde; |
| michael@0 | 86 | } |
| michael@0 | 87 | } |
| michael@0 | 88 | |
| michael@0 | 89 | /* Given a pair of quadratics, determine their parametric coefficients. |
| michael@0 | 90 | * If the scaled coefficients are nearly equal, then the part of the quadratics |
| michael@0 | 91 | * may be coincident. |
| michael@0 | 92 | * OPTIMIZATION -- since comparison short-circuits on no match, |
| michael@0 | 93 | * lazily compute the coefficients, comparing the easiest to compute first. |
| michael@0 | 94 | * xx and yy first; then xy; and so on. |
| michael@0 | 95 | */ |
| michael@0 | 96 | bool SkDQuadImplicit::match(const SkDQuadImplicit& p2) const { |
| michael@0 | 97 | int first = 0; |
| michael@0 | 98 | for (int index = 0; index <= kC_Coeff; ++index) { |
| michael@0 | 99 | if (approximately_zero(fP[index]) && approximately_zero(p2.fP[index])) { |
| michael@0 | 100 | first += first == index; |
| michael@0 | 101 | continue; |
| michael@0 | 102 | } |
| michael@0 | 103 | if (first == index) { |
| michael@0 | 104 | continue; |
| michael@0 | 105 | } |
| michael@0 | 106 | if (!AlmostDequalUlps(fP[index] * p2.fP[first], fP[first] * p2.fP[index])) { |
| michael@0 | 107 | return false; |
| michael@0 | 108 | } |
| michael@0 | 109 | } |
| michael@0 | 110 | return true; |
| michael@0 | 111 | } |
| michael@0 | 112 | |
| michael@0 | 113 | bool SkDQuadImplicit::Match(const SkDQuad& quad1, const SkDQuad& quad2) { |
| michael@0 | 114 | SkDQuadImplicit i1(quad1); // a'xx , b'xy , c'yy , d'x , e'y , f |
| michael@0 | 115 | SkDQuadImplicit i2(quad2); |
| michael@0 | 116 | return i1.match(i2); |
| michael@0 | 117 | } |
