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 } 118
