michael@0: /*
michael@0:  * Copyright 2012 Google Inc.
michael@0:  *
michael@0:  * Use of this source code is governed by a BSD-style license that can be
michael@0:  * found in the LICENSE file.
michael@0:  */
michael@0: #include "SkDQuadImplicit.h"
michael@0: 
michael@0: /* from http://tom.cs.byu.edu/~tom/papers/cvgip84.pdf 4.1
michael@0:  *
michael@0:  * This paper proves that Syvester's method can compute the implicit form of
michael@0:  * the quadratic from the parameterized form.
michael@0:  *
michael@0:  * Given x = a*t*t + b*t + c  (the parameterized form)
michael@0:  *       y = d*t*t + e*t + f
michael@0:  *
michael@0:  * we want to find an equation of the implicit form:
michael@0:  *
michael@0:  * A*x*x + B*x*y + C*y*y + D*x + E*y + F = 0
michael@0:  *
michael@0:  * The implicit form can be expressed as a 4x4 determinant, as shown.
michael@0:  *
michael@0:  * The resultant obtained by Syvester's method is
michael@0:  *
michael@0:  * |   a   b   (c - x)     0     |
michael@0:  * |   0   a      b     (c - x)  |
michael@0:  * |   d   e   (f - y)     0     |
michael@0:  * |   0   d      e     (f - y)  |
michael@0:  *
michael@0:  * which expands to
michael@0:  *
michael@0:  * d*d*x*x + -2*a*d*x*y + a*a*y*y
michael@0:  *         + (-2*c*d*d + b*e*d - a*e*e + 2*a*f*d)*x
michael@0:  *         + (-2*f*a*a + e*b*a - d*b*b + 2*d*c*a)*y
michael@0:  *         +
michael@0:  * |   a   b   c   0   |
michael@0:  * |   0   a   b   c   | == 0.
michael@0:  * |   d   e   f   0   |
michael@0:  * |   0   d   e   f   |
michael@0:  *
michael@0:  * Expanding the constant determinant results in
michael@0:  *
michael@0:  *   | a b c |     | b c 0 |
michael@0:  * a*| e f 0 | + d*| a b c | ==
michael@0:  *   | d e f |     | d e f |
michael@0:  *
michael@0:  * 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:  *
michael@0:  */
michael@0: 
michael@0: // use the tricky arithmetic path, but leave the original to compare just in case
michael@0: static bool straight_forward = false;
michael@0: 
michael@0: SkDQuadImplicit::SkDQuadImplicit(const SkDQuad& q) {
michael@0:     double a, b, c;
michael@0:     SkDQuad::SetABC(&q[0].fX, &a, &b, &c);
michael@0:     double d, e, f;
michael@0:     SkDQuad::SetABC(&q[0].fY, &d, &e, &f);
michael@0:     // compute the implicit coefficients
michael@0:     if (straight_forward) {  // 42 muls, 13 adds
michael@0:         fP[kXx_Coeff] = d * d;
michael@0:         fP[kXy_Coeff] = -2 * a * d;
michael@0:         fP[kYy_Coeff] = a * a;
michael@0:         fP[kX_Coeff] = -2*c*d*d + b*e*d - a*e*e + 2*a*f*d;
michael@0:         fP[kY_Coeff] = -2*f*a*a + e*b*a - d*b*b + 2*d*c*a;
michael@0:         fP[kC_Coeff] = a*(a*f*f + c*e*e - c*f*d - b*e*f)
michael@0:                    + d*(b*b*f + c*c*d - c*a*f - c*e*b);
michael@0:     } else {  // 26 muls, 11 adds
michael@0:         double aa = a * a;
michael@0:         double ad = a * d;
michael@0:         double dd = d * d;
michael@0:         fP[kXx_Coeff] = dd;
michael@0:         fP[kXy_Coeff] = -2 * ad;
michael@0:         fP[kYy_Coeff] = aa;
michael@0:         double be = b * e;
michael@0:         double bde = be * d;
michael@0:         double cdd = c * dd;
michael@0:         double ee = e * e;
michael@0:         fP[kX_Coeff] =  -2*cdd + bde - a*ee + 2*ad*f;
michael@0:         double aaf = aa * f;
michael@0:         double abe = a * be;
michael@0:         double ac = a * c;
michael@0:         double bb_2ac = b*b - 2*ac;
michael@0:         fP[kY_Coeff] = -2*aaf + abe - d*bb_2ac;
michael@0:         fP[kC_Coeff] = aaf*f + ac*ee + d*f*bb_2ac - abe*f + c*cdd - c*bde;
michael@0:     }
michael@0: }
michael@0: 
michael@0:  /* Given a pair of quadratics, determine their parametric coefficients.
michael@0:   * If the scaled coefficients are nearly equal, then the part of the quadratics
michael@0:   * may be coincident.
michael@0:   * OPTIMIZATION -- since comparison short-circuits on no match,
michael@0:   * lazily compute the coefficients, comparing the easiest to compute first.
michael@0:   * xx and yy first; then xy; and so on.
michael@0:   */
michael@0: bool SkDQuadImplicit::match(const SkDQuadImplicit& p2) const {
michael@0:     int first = 0;
michael@0:     for (int index = 0; index <= kC_Coeff; ++index) {
michael@0:         if (approximately_zero(fP[index]) && approximately_zero(p2.fP[index])) {
michael@0:             first += first == index;
michael@0:             continue;
michael@0:         }
michael@0:         if (first == index) {
michael@0:             continue;
michael@0:         }
michael@0:         if (!AlmostDequalUlps(fP[index] * p2.fP[first], fP[first] * p2.fP[index])) {
michael@0:             return false;
michael@0:         }
michael@0:     }
michael@0:     return true;
michael@0: }
michael@0: 
michael@0: bool SkDQuadImplicit::Match(const SkDQuad& quad1, const SkDQuad& quad2) {
michael@0:     SkDQuadImplicit i1(quad1);  // a'xx , b'xy , c'yy , d'x , e'y , f
michael@0:     SkDQuadImplicit i2(quad2);
michael@0:     return i1.match(i2);
michael@0: }