6d0032a8ec
git-svn-id: http://skia.googlecode.com/svn/trunk@7031 2bbb7eff-a529-9590-31e7-b0007b416f81
136 lines
4.0 KiB
C++
136 lines
4.0 KiB
C++
/*
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* Copyright 2012 Google Inc.
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*
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* Use of this source code is governed by a BSD-style license that can be
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* found in the LICENSE file.
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*/
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#include "CurveIntersection.h"
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#include "QuadraticParameterization.h"
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#include "QuadraticUtilities.h"
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/* from http://tom.cs.byu.edu/~tom/papers/cvgip84.pdf 4.1
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*
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* This paper proves that Syvester's method can compute the implicit form of
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* the quadratic from the parameterized form.
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*
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* Given x = a*t*t + b*t + c (the parameterized form)
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* y = d*t*t + e*t + f
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*
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* we want to find an equation of the implicit form:
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*
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* A*x*x + B*x*y + C*y*y + D*x + E*y + F = 0
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*
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* The implicit form can be expressed as a 4x4 determinant, as shown.
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*
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* The resultant obtained by Syvester's method is
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*
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* | a b (c - x) 0 |
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* | 0 a b (c - x) |
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* | d e (f - y) 0 |
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* | 0 d e (f - y) |
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*
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* which expands to
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*
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* d*d*x*x + -2*a*d*x*y + a*a*y*y
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* + (-2*c*d*d + b*e*d - a*e*e + 2*a*f*d)*x
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* + (-2*f*a*a + e*b*a - d*b*b + 2*d*c*a)*y
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* +
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* | a b c 0 |
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* | 0 a b c | == 0.
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* | d e f 0 |
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* | 0 d e f |
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*
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* Expanding the constant determinant results in
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*
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* | a b c | | b c 0 |
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* a*| e f 0 | + d*| a b c | ==
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* | d e f | | d e f |
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*
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* 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)
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*
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*/
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static bool straight_forward = true;
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QuadImplicitForm::QuadImplicitForm(const Quadratic& q) {
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double a, b, c;
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set_abc(&q[0].x, a, b, c);
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double d, e, f;
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set_abc(&q[0].y, d, e, f);
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// compute the implicit coefficients
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if (straight_forward) { // 42 muls, 13 adds
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p[xx_coeff] = d * d;
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p[xy_coeff] = -2 * a * d;
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p[yy_coeff] = a * a;
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p[x_coeff] = -2*c*d*d + b*e*d - a*e*e + 2*a*f*d;
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p[y_coeff] = -2*f*a*a + e*b*a - d*b*b + 2*d*c*a;
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p[c_coeff] = a*(a*f*f + c*e*e - c*f*d - b*e*f)
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+ d*(b*b*f + c*c*d - c*a*f - c*e*b);
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} else { // 26 muls, 11 adds
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double aa = a * a;
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double ad = a * d;
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double dd = d * d;
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p[xx_coeff] = dd;
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p[xy_coeff] = -2 * ad;
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p[yy_coeff] = aa;
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double be = b * e;
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double bde = be * d;
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double cdd = c * dd;
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double ee = e * e;
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p[x_coeff] = -2*cdd + bde - a*ee + 2*ad*f;
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double aaf = aa * f;
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double abe = a * be;
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double ac = a * c;
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double bb_2ac = b*b - 2*ac;
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p[y_coeff] = -2*aaf + abe - d*bb_2ac;
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p[c_coeff] = aaf*f + ac*ee + d*f*bb_2ac - abe*f + c*cdd - c*bde;
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}
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}
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/* Given a pair of quadratics, determine their parametric coefficients.
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* If the scaled coefficients are nearly equal, then the part of the quadratics
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* may be coincident.
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* FIXME: optimization -- since comparison short-circuits on no match,
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* lazily compute the coefficients, comparing the easiest to compute first.
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* xx and yy first; then xy; and so on.
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*/
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bool QuadImplicitForm::implicit_match(const QuadImplicitForm& p2) const {
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int first = 0;
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for (int index = 0; index < coeff_count; ++index) {
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if (approximately_zero(p[index]) && approximately_zero(p2.p[index])) {
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first += first == index;
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continue;
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}
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if (first == index) {
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continue;
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}
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if (!AlmostEqualUlps(p[index] * p2.p[first], p[first] * p2.p[index])) {
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return false;
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}
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}
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return true;
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}
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bool implicit_matches(const Quadratic& quad1, const Quadratic& quad2) {
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QuadImplicitForm i1(quad1); // a'xx , b'xy , c'yy , d'x , e'y , f
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QuadImplicitForm i2(quad2);
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return i1.implicit_match(i2);
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}
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static double tangent(const double* quadratic, double t) {
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double a, b, c;
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set_abc(quadratic, a, b, c);
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return 2 * a * t + b;
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}
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void tangent(const Quadratic& quadratic, double t, _Point& result) {
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result.x = tangent(&quadratic[0].x, t);
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result.y = tangent(&quadratic[0].y, t);
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}
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// unit test to return and validate parametric coefficients
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#include "QuadraticParameterization_TestUtility.cpp"
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