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