AuroraMiddleware/skia2
1
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity
27accef223
skia2/experimental/Intersection/CubicParameterization.cpp
caryclark@google.com 27accef223 Intersection work in progress 
Review URL: https://codereview.appspot.com/5576043

git-svn-id: http://skia.googlecode.com/svn/trunk@3087 2bbb7eff-a529-9590-31e7-b0007b416f81
2012-01-25 18:57:23 +00:00

353 lines
11 KiB
C++
Raw Blame History

#include "CubicIntersection.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 parameterzied form.
*
* Given x = a*t*t*t + b*t*t + c*t + d (the parameterized form)
* y = e*t*t*t + f*t*t + g*t + h
*
* we want to find an equation of the implicit form:
*
* A*x^3 + B*x*x*y + C*x*y*y + D*y^3 + E*x*x + F*x*y + G*y*y + H*x + I*y + J = 0
*
* The implicit form can be expressed as a 6x6 determinant, as shown.
*
* The resultant obtained by Syvester's method is
*
* | a b c (d - x) 0 0 |
* | 0 a b c (d - x) 0 |
* | 0 0 a b c (d - x) |
* | e f g (h - y) 0 0 |
* | 0 e f g (h - y) 0 |
* | 0 0 e f g (h - y) |
*
* which, according to Mathematica, expands as shown below.
*
* Resultant[a*t^3 + b*t^2 + c*t + d - x, e*t^3 + f*t^2 + g*t + h - y, t]
*
* -d^3 e^3 + c d^2 e^2 f - b d^2 e f^2 + a d^2 f^3 - c^2 d e^2 g +
* 2 b d^2 e^2 g + b c d e f g - 3 a d^2 e f g - a c d f^2 g -
* b^2 d e g^2 + 2 a c d e g^2 + a b d f g^2 - a^2 d g^3 + c^3 e^2 h -
* 3 b c d e^2 h + 3 a d^2 e^2 h - b c^2 e f h + 2 b^2 d e f h +
* a c d e f h + a c^2 f^2 h - 2 a b d f^2 h + b^2 c e g h -
* 2 a c^2 e g h - a b d e g h - a b c f g h + 3 a^2 d f g h +
* a^2 c g^2 h - b^3 e h^2 + 3 a b c e h^2 - 3 a^2 d e h^2 +
* a b^2 f h^2 - 2 a^2 c f h^2 - a^2 b g h^2 + a^3 h^3 + 3 d^2 e^3 x -
* 2 c d e^2 f x + 2 b d e f^2 x - 2 a d f^3 x + c^2 e^2 g x -
* 4 b d e^2 g x - b c e f g x + 6 a d e f g x + a c f^2 g x +
* b^2 e g^2 x - 2 a c e g^2 x - a b f g^2 x + a^2 g^3 x +
* 3 b c e^2 h x - 6 a d e^2 h x - 2 b^2 e f h x - a c e f h x +
* 2 a b f^2 h x + a b e g h x - 3 a^2 f g h x + 3 a^2 e h^2 x -
* 3 d e^3 x^2 + c e^2 f x^2 - b e f^2 x^2 + a f^3 x^2 +
* 2 b e^2 g x^2 - 3 a e f g x^2 + 3 a e^2 h x^2 + e^3 x^3 -
* c^3 e^2 y + 3 b c d e^2 y - 3 a d^2 e^2 y + b c^2 e f y -
* 2 b^2 d e f y - a c d e f y - a c^2 f^2 y + 2 a b d f^2 y -
* b^2 c e g y + 2 a c^2 e g y + a b d e g y + a b c f g y -
* 3 a^2 d f g y - a^2 c g^2 y + 2 b^3 e h y - 6 a b c e h y +
* 6 a^2 d e h y - 2 a b^2 f h y + 4 a^2 c f h y + 2 a^2 b g h y -
* 3 a^3 h^2 y - 3 b c e^2 x y + 6 a d e^2 x y + 2 b^2 e f x y +
* a c e f x y - 2 a b f^2 x y - a b e g x y + 3 a^2 f g x y -
* 6 a^2 e h x y - 3 a e^2 x^2 y - b^3 e y^2 + 3 a b c e y^2 -
* 3 a^2 d e y^2 + a b^2 f y^2 - 2 a^2 c f y^2 - a^2 b g y^2 +
* 3 a^3 h y^2 + 3 a^2 e x y^2 - a^3 y^3
*/
enum {
xxx_coeff,
xxy_coeff,
xyy_coeff,
yyy_coeff,
xx_coeff,
xy_coeff,
yy_coeff,
x_coeff,
y_coeff,
c_coeff,
coeff_count
};
// FIXME: factoring version unwritten
// static bool straight_forward = true;
/* from CubicParameterizationCode.cpp output:
* double A = e * e * e;
* double B = -3 * a * e * e;
* double C = 3 * a * a * e;
* double D = -a * a * a;
*/
static void calc_ABCD(double a, double e, double p[coeff_count]) {
double ee = e * e;
p[xxx_coeff] = e * ee;
p[xxy_coeff] = -3 * a * ee;
double aa = a * a;
p[xyy_coeff] = 3 * aa * e;
p[yyy_coeff] = -aa * a;
}
/* CubicParameterizationCode.cpp turns Mathematica output into C.
* Rather than edit the lines below, please edit the code there instead.
*/
// start of generated code
static double calc_E(double a, double b, double c, double d,
double e, double f, double g, double h) {
return
-3 * d * e * e * e
+ c * e * e * f
- b * e * f * f
+ a * f * f * f
+ 2 * b * e * e * g
- 3 * a * e * f * g
+ 3 * a * e * e * h;
}
static double calc_F(double a, double b, double c, double d,
double e, double f, double g, double h) {
return
-3 * b * c * e * e
+ 6 * a * d * e * e
+ 2 * b * b * e * f
+ a * c * e * f
- 2 * a * b * f * f
- a * b * e * g
+ 3 * a * a * f * g
- 6 * a * a * e * h;
}
static double calc_G(double a, double b, double c, double d,
double e, double f, double g, double h) {
return
-b * b * b * e
+ 3 * a * b * c * e
- 3 * a * a * d * e
+ a * b * b * f
- 2 * a * a * c * f
- a * a * b * g
+ 3 * a * a * a * h;
}
static double calc_H(double a, double b, double c, double d,
double e, double f, double g, double h) {
return
3 * d * d * e * e * e
- 2 * c * d * e * e * f
+ 2 * b * d * e * f * f
- 2 * a * d * f * f * f
+ c * c * e * e * g
- 4 * b * d * e * e * g
- b * c * e * f * g
+ 6 * a * d * e * f * g
+ a * c * f * f * g
+ b * b * e * g * g
- 2 * a * c * e * g * g
- a * b * f * g * g
+ a * a * g * g * g
+ 3 * b * c * e * e * h
- 6 * a * d * e * e * h
- 2 * b * b * e * f * h
- a * c * e * f * h
+ 2 * a * b * f * f * h
+ a * b * e * g * h
- 3 * a * a * f * g * h
+ 3 * a * a * e * h * h;
}
static double calc_I(double a, double b, double c, double d,
double e, double f, double g, double h) {
return
-c * c * c * e * e
+ 3 * b * c * d * e * e
- 3 * a * d * d * e * e
+ b * c * c * e * f
- 2 * b * b * d * e * f
- a * c * d * e * f
- a * c * c * f * f
+ 2 * a * b * d * f * f
- b * b * c * e * g
+ 2 * a * c * c * e * g
+ a * b * d * e * g
+ a * b * c * f * g
- 3 * a * a * d * f * g
- a * a * c * g * g
+ 2 * b * b * b * e * h
- 6 * a * b * c * e * h
+ 6 * a * a * d * e * h
- 2 * a * b * b * f * h
+ 4 * a * a * c * f * h
+ 2 * a * a * b * g * h
- 3 * a * a * a * h * h;
}
static double calc_J(double a, double b, double c, double d,
double e, double f, double g, double h) {
return
-d * d * d * e * e * e
+ c * d * d * e * e * f
- b * d * d * e * f * f
+ a * d * d * f * f * f
- c * c * d * e * e * g
+ 2 * b * d * d * e * e * g
+ b * c * d * e * f * g
- 3 * a * d * d * e * f * g
- a * c * d * f * f * g
- b * b * d * e * g * g
+ 2 * a * c * d * e * g * g
+ a * b * d * f * g * g
- a * a * d * g * g * g
+ c * c * c * e * e * h
- 3 * b * c * d * e * e * h
+ 3 * a * d * d * e * e * h
- b * c * c * e * f * h
+ 2 * b * b * d * e * f * h
+ a * c * d * e * f * h
+ a * c * c * f * f * h
- 2 * a * b * d * f * f * h
+ b * b * c * e * g * h
- 2 * a * c * c * e * g * h
- a * b * d * e * g * h
- a * b * c * f * g * h
+ 3 * a * a * d * f * g * h
+ a * a * c * g * g * h
- b * b * b * e * h * h
+ 3 * a * b * c * e * h * h
- 3 * a * a * d * e * h * h
+ a * b * b * f * h * h
- 2 * a * a * c * f * h * h
- a * a * b * g * h * h
+ a * a * a * h * h * h;
}
// end of generated code
static double (*calc_proc[])(double a, double b, double c, double d,
double e, double f, double g, double h) = {
calc_E, calc_F, calc_G, calc_H, calc_I, calc_J
};
/* Control points to parametric coefficients
s = 1 - t
Attt + 3Btts + 3Ctss + Dsss ==
Attt + 3B(1 - t)tt + 3C(1 - t)(t - tt) + D(1 - t)(1 - 2t + tt) ==
Attt + 3B(tt - ttt) + 3C(t - tt - tt + ttt) + D(1-2t+tt-t+2tt-ttt) ==
Attt + 3Btt - 3Bttt + 3Ct - 6Ctt + 3Cttt + D - 3Dt + 3Dtt - Dttt ==
D + (3C - 3D)t + (3B - 6C + 3D)tt + (A - 3B + 3C - D)ttt
a = A - 3*B + 3*C - D
b = 3*B - 6*C + 3*D
c = 3*C - 3*D
d = D
*/
/* http://www.algorithmist.net/bezier3.html
p = 3 * A
q = 3 * B
r = 3 * C
a = A
b = q - p
c = p - 2 * q + r
d = D - A + q - r
B(t) = a + t * (b + t * (c + t * d))
so
B(t) = a + t*b + t*t*(c + t*d)
= a + t*b + t*t*c + t*t*t*d
*/
static void set_abcd(const double* cubic, double& a, double& b, double& c,
double& d) {
a = cubic[0]; // a = A
b = 3 * cubic[2]; // b = 3*B (compute rest of b lazily)
c = 3 * cubic[4]; // c = 3*C (compute rest of c lazily)
d = cubic[6]; // d = D
a += -b + c - d; // a = A - 3*B + 3*C - D
}
static void calc_bc(const double d, double& b, double& c) {
b -= 3 * c; // b = 3*B - 3*C
c -= 3 * d; // c = 3*C - 3*D
b -= c; // b = 3*B - 6*C + 3*D
}
static void alt_set_abcd(const double* cubic, double& a, double& b, double& c,
double& d) {
a = cubic[0];
double p = 3 * a;
double q = 3 * cubic[2];
double r = 3 * cubic[4];
b = q - p;
c = p - 2 * q + r;
d = cubic[6] - a + q - r;
}
const bool try_alt = true;
bool implicit_matches(const Cubic& one, const Cubic& two) {
double p1[coeff_count]; // a'xxx , b'xxy , c'xyy , d'xx , e'xy , f'yy, etc.
double p2[coeff_count];
double a1, b1, c1, d1;
if (try_alt)
alt_set_abcd(&one[0].x, a1, b1, c1, d1);
else
set_abcd(&one[0].x, a1, b1, c1, d1);
double e1, f1, g1, h1;
if (try_alt)
alt_set_abcd(&one[0].y, e1, f1, g1, h1);
else
set_abcd(&one[0].y, e1, f1, g1, h1);
calc_ABCD(a1, e1, p1);
double a2, b2, c2, d2;
if (try_alt)
alt_set_abcd(&two[0].x, a2, b2, c2, d2);
else
set_abcd(&two[0].x, a2, b2, c2, d2);
double e2, f2, g2, h2;
if (try_alt)
alt_set_abcd(&two[0].y, e2, f2, g2, h2);
else
set_abcd(&two[0].y, e2, f2, g2, h2);
calc_ABCD(a2, e2, p2);
int first = 0;
for (int index = 0; index < coeff_count; ++index) {
if (!try_alt && index == xx_coeff) {
calc_bc(d1, b1, c1);
calc_bc(h1, f1, g1);
calc_bc(d2, b2, c2);
calc_bc(h2, f2, g2);
}
if (index >= xx_coeff) {
int procIndex = index - xx_coeff;
p1[index] = (*calc_proc[procIndex])(a1, b1, c1, d1, e1, f1, g1, h1);
p2[index] = (*calc_proc[procIndex])(a2, b2, c2, d2, e2, f2, g2, h2);
}
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* cubic, double t) {
double a, b, c, d;
set_abcd(cubic, a, b, c, d);
calc_bc(d, b, c);
return 3 * a * t * t + 2 * b * t + c;
}
void tangent(const Cubic& cubic, double t, _Point& result) {
result.x = tangent(&cubic[0].x, t);
result.y = tangent(&cubic[0].y, t);
}
// unit test to return and validate parametric coefficients
#include "CubicParameterization_TestUtility.cpp"
Reference in New Issue View Git Blame Copy Permalink