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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 }
    118