c682590538
git-svn-id: http://skia.googlecode.com/svn/trunk@3141 2bbb7eff-a529-9590-31e7-b0007b416f81
135 lines
3.9 KiB
C++
135 lines
3.9 KiB
C++
#include "CurveIntersection.h"
|
|
#include "QuadraticUtilities.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)
|
|
*
|
|
*/
|
|
|
|
enum {
|
|
xx_coeff,
|
|
xy_coeff,
|
|
yy_coeff,
|
|
x_coeff,
|
|
y_coeff,
|
|
c_coeff,
|
|
coeff_count
|
|
};
|
|
|
|
static bool straight_forward = true;
|
|
|
|
static void implicit_coefficients(const Quadratic& q, double p[coeff_count]) {
|
|
double a, b, c;
|
|
set_abc(&q[0].x, a, b, c);
|
|
double d, e, f;
|
|
set_abc(&q[0].y, d, e, f);
|
|
// compute the implicit coefficients
|
|
if (straight_forward) { // 42 muls, 13 adds
|
|
p[xx_coeff] = d * d;
|
|
p[xy_coeff] = -2 * a * d;
|
|
p[yy_coeff] = a * a;
|
|
p[x_coeff] = -2*c*d*d + b*e*d - a*e*e + 2*a*f*d;
|
|
p[y_coeff] = -2*f*a*a + e*b*a - d*b*b + 2*d*c*a;
|
|
p[c_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;
|
|
p[xx_coeff] = dd;
|
|
p[xy_coeff] = -2 * ad;
|
|
p[yy_coeff] = aa;
|
|
double be = b * e;
|
|
double bde = be * d;
|
|
double cdd = c * dd;
|
|
double ee = e * e;
|
|
p[x_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;
|
|
p[y_coeff] = -2*aaf + abe - d*bb_2ac;
|
|
p[c_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.
|
|
* FIXME: 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 implicit_matches(const Quadratic& one, const Quadratic& two) {
|
|
double p1[coeff_count]; // a'xx , b'xy , c'yy , d'x , e'y , f
|
|
double p2[coeff_count];
|
|
implicit_coefficients(one, p1);
|
|
implicit_coefficients(two, p2);
|
|
int first = 0;
|
|
for (int index = 0; index < coeff_count; ++index) {
|
|
if (approximately_zero(p1[index]) || approximately_zero(p2[index])) {
|
|
first += first == index;
|
|
continue;
|
|
}
|
|
if (first == index) {
|
|
continue;
|
|
}
|
|
if (!approximately_equal(p1[index] * p2[first],
|
|
p1[first] * p2[index])) {
|
|
return false;
|
|
}
|
|
}
|
|
return true;
|
|
}
|
|
|
|
static double tangent(const double* quadratic, double t) {
|
|
double a, b, c;
|
|
set_abc(quadratic, a, b, c);
|
|
return 2 * a * t + b;
|
|
}
|
|
|
|
void tangent(const Quadratic& quadratic, double t, _Point& result) {
|
|
result.x = tangent(&quadratic[0].x, t);
|
|
result.y = tangent(&quadratic[0].y, t);
|
|
}
|
|
|
|
// unit test to return and validate parametric coefficients
|
|
#include "QuadraticParameterization_TestUtility.cpp"
|