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#### Mercurial (d8847129d134)

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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);
``````}
``````