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 /*

  * Copyright 2012 Google Inc.

  *

  * Use of this source code is governed by a BSD-style license that can be

  * found in the LICENSE file.

  */

 #include "SkDQuadImplicit.h"

 /* from http://tom.cs.byu.edu/~tom/papers/cvgip84.pdf 4.1

  *

  * This paper proves that Syvester's method can compute the implicit form of

  * the quadratic from the parameterized form.

  *

  * Given x = a*t*t + b*t + c  (the parameterized form)

  *       y = d*t*t + e*t + f

  *

  * we want to find an equation of the implicit form:

  *

  * A*x*x + B*x*y + C*y*y + D*x + E*y + F = 0

  *

  * The implicit form can be expressed as a 4x4 determinant, as shown.

  *

  * The resultant obtained by Syvester's method is

  *

  * |   a   b   (c - x)     0     |

  * |   0   a      b     (c - x)  |

  * |   d   e   (f - y)     0     |

  * |   0   d      e     (f - y)  |

  *

  * which expands to

  *

  * d*d*x*x + -2*a*d*x*y + a*a*y*y

  *         + (-2*c*d*d + b*e*d - a*e*e + 2*a*f*d)*x

  *         + (-2*f*a*a + e*b*a - d*b*b + 2*d*c*a)*y

  *         +

  * |   a   b   c   0   |

  * |   0   a   b   c   | == 0.

  * |   d   e   f   0   |

  * |   0   d   e   f   |

  *

  * Expanding the constant determinant results in

  *

  *   | a b c |     | b c 0 |

  * a*| e f 0 | + d*| a b c | ==

  *   | d e f |     | d e f |

  *

  * 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)

  *

  */

 // use the tricky arithmetic path, but leave the original to compare just in case

 static bool straight_forward = false;

 SkDQuadImplicit::SkDQuadImplicit(const SkDQuad& q) {

     double a, b, c;

     SkDQuad::SetABC(&q[0].fX, &a, &b, &c);

     double d, e, f;

     SkDQuad::SetABC(&q[0].fY, &d, &e, &f);

     // compute the implicit coefficients

     if (straight_forward) {  // 42 muls, 13 adds

         fP[kXx_Coeff] = d * d;

         fP[kXy_Coeff] = -2 * a * d;

         fP[kYy_Coeff] = a * a;

         fP[kX_Coeff] = -2*c*d*d + b*e*d - a*e*e + 2*a*f*d;

         fP[kY_Coeff] = -2*f*a*a + e*b*a - d*b*b + 2*d*c*a;

         fP[kC_Coeff] = 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);

     } else {  // 26 muls, 11 adds

         double aa = a * a;

         double ad = a * d;

         double dd = d * d;

         fP[kXx_Coeff] = dd;

         fP[kXy_Coeff] = -2 * ad;

         fP[kYy_Coeff] = aa;

         double be = b * e;

         double bde = be * d;

         double cdd = c * dd;

         double ee = e * e;

         fP[kX_Coeff] =  -2*cdd + bde - a*ee + 2*ad*f;

         double aaf = aa * f;

         double abe = a * be;

         double ac = a * c;

         double bb_2ac = b*b - 2*ac;

         fP[kY_Coeff] = -2*aaf + abe - d*bb_2ac;

         fP[kC_Coeff] = aaf*f + ac*ee + d*f*bb_2ac - abe*f + c*cdd - c*bde;

     }

 }

  /* Given a pair of quadratics, determine their parametric coefficients.

   * If the scaled coefficients are nearly equal, then the part of the quadratics

   * may be coincident.

   * OPTIMIZATION -- since comparison short-circuits on no match,

   * lazily compute the coefficients, comparing the easiest to compute first.

   * xx and yy first; then xy; and so on.

   */

 bool SkDQuadImplicit::match(const SkDQuadImplicit& p2) const {

     int first = 0;

     for (int index = 0; index <= kC_Coeff; ++index) {

         if (approximately_zero(fP[index]) && approximately_zero(p2.fP[index])) {

             first += first == index;

             continue;

         }

         if (first == index) {

             continue;

         }

         if (!AlmostDequalUlps(fP[index] * p2.fP[first], fP[first] * p2.fP[index])) {

             return false;

         }

     }

     return true;

 }

 bool SkDQuadImplicit::Match(const SkDQuad& quad1, const SkDQuad& quad2) {

     SkDQuadImplicit i1(quad1);  // a'xx , b'xy , c'yy , d'x , e'y , f

     SkDQuadImplicit i2(quad2);

     return i1.match(i2);

 }