/* === S Y N F I G ========================================================= */
/*! \file curve.cpp
** \brief Operations with cubic curves
**
** $Id$
**
** \legal
** ......... ... 2019 Ivan Mahonin
**
** This package is free software; you can redistribute it and/or
** modify it under the terms of the GNU General Public License as
** published by the Free Software Foundation; either version 2 of
** the License, or (at your option) any later version.
**
** This package is distributed in the hope that it will be useful,
** but WITHOUT ANY WARRANTY; without even the implied warranty of
** MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
** General Public License for more details.
** \endlegal
*/
/* ========================================================================= */
/* === H E A D E R S ======================================================= */
#ifdef USING_PCH
# include "pch.h"
#else
#ifdef HAVE_CONFIG_H
# include <config.h>
#endif
#include <cmath>
#include "curve.h"
#endif
/* === U S I N G =========================================================== */
using namespace synfig;
/* === M A C R O S ========================================================= */
/* === G L O B A L S ======================================================= */
int
synfig::solve_equation(Real* roots, Real k0, Real k1)
{
if (approximate_zero(k1)) return 0;
if (roots) roots[0] = -k0/k1;
return 1;
}
int
synfig::solve_equation(Real* roots, Real k0, Real k1, Real k2)
{
if (approximate_zero(k2)) return solve_equation(roots, k0, k1);
Real D = k1*k1 - 4*k2*k0;
if (approximate_zero(D)) {
if (roots) roots[0] = -0.5*k1/k2;
return 1;
} else
if (D > 0) {
if (roots) {
Real a = sqrt(D);
Real b = -0.5/k2;
roots[0] = (k1 - a)*b;
roots[1] = (k1 + a)*b;
}
return 2;
}
return 0;
}
int
synfig::solve_equation(Real* roots, Real k0, Real k1, Real k2, Real k3)
{
if (approximate_zero(k3)) return solve_equation(roots, k0, k1, k2);
Real k = 1/k3;
Real a = k2*k;
Real b = k1*k;
Real c = k0*k;
Real Q = (a*a - 3*b)/9;
Real Q3 = Q*Q*Q;
Real R = (2*a*a*a - 9*a*b + 27*c)/54;
Real S = Q3 - R*R;
if (approximate_zero(S)) {
if (roots) {
Real rr = cbrt(R);
Real aa = -a/3;
roots[0] = aa - 2*rr;
roots[1] = aa + rr;
}
return 2;
} else
if (S > 0) {
if (roots) {
Real ph = acos(R/sqrt(Q3))/3;
Real qq = -2*sqrt(Q);
Real aa = -a/3;
roots[0] = qq*cos(ph) + aa;
roots[1] = qq*cos(ph + 2*PI/3) + aa;
roots[2] = qq*cos(ph - 2*PI/3) + aa;
}
return 3;
} else
if (approximate_zero(Q)) {
if (roots) roots[0] = -cbrt(c - a*a*a/27) - a/3;
return 1;
} else
if (Q > 0) {
if (roots) {
Real ph = acosh( fabs(R)/sqrt(Q3) )/3;
Real sign = approximate_zero(R) ? 0 : R < 0 ? -1 : 1;
roots[0] = -2*sign*sqrt(Q)*cosh(ph) - a/3;
}
return 1;
} else {
if (roots) {
Real ph = asinh( fabs(R)/sqrt(-Q3) )/3;
Real sign = approximate_zero(R) ? 0 : R < 0 ? -1 : 1;
roots[0] = -2*sign*sqrt(-Q)*sinh(ph) - a/3;
}
return 1;
}
return 0;
}
/* === M E T H O D S ======================================================= */
// Hermite
int Hermite::inflection(Real *l, Real p0, Real p1, Real t0, Real t1) {
Real root;
if (solve_equation(
&root,
-3*p0 + 3*p1 - 2*t0 - t1,
6*p0 - 6*p1 + 3*t0 + 3*t1 ))
{
if (approximate_less(Real(0), root) && approximate_less(root, Real(1))) {
if (l) *l = root;
return 1;
}
}
return 0;
}
int Hermite::bends(Real *l, Real p0, Real p1, Real t0, Real t1) {
Real roots[2];
int count = solve_equation(
roots,
t0 ,
-6*p0 + 6*p1 - 4*t0 - 2*t1,
6*p0 - 6*p1 + 3*t0 + 3*t1 );
int valid_count = 0;
for(Real *i = roots, *end = i + count; i != end; ++i)
if (approximate_less(Real(0), *i) && approximate_less(*i, Real(1))) {
if (l) l[valid_count] = *i;
++valid_count;
}
return valid_count;
}
Range Hermite::bounds_accurate(Real p0, Real p1, Real t0, Real t1) {
Range range(p0);
range.expand(p1);
Real roots[2];
int count = bends(roots, p0, p1, t0, t1);
for(Real *i = roots, *end = i + count; i != end; ++i)
range.expand( p(*i, p0, p1, t0, t1) );
return range;
}
int Hermite::intersections(Real *l, Real p, Real p0, Real p1, Real t0, Real t1) {
Real roots[3];
int count = solve_equation(
roots,
p0 - p ,
t0 ,
-3*p0 + 3*p1 - 2*t0 - t1,
2*p0 - 2*p1 + t0 + t1 );
int valid_count = 0;
for(Real *i = roots, *end = i + count; i != end; ++i)
if (approximate_less(Real(0), *i) && approximate_less(*i, Real(1))) {
if (l) l[valid_count] = *i;
++valid_count;
}
return valid_count;
}
// Bezier
int Bezier::inflection(Real *l, Real p0, Real p1, Real pp0, Real pp1) {
Real root;
if (solve_equation(
&root,
p0 - 2*pp0 + pp1 ,
-p0 + 3*pp0 - 3*pp1 + p1 ))
{
if (approximate_less(Real(0), root) && approximate_less(root, Real(1))) {
if (l) *l = root;
return 1;
}
}
return 0;
}
int Bezier::bends(Real *l, Real p0, Real p1, Real pp0, Real pp1) {
Real roots[2];
int count = solve_equation(
roots,
-p0 + pp0 ,
2*p0 - 4*pp0 + 2*pp1 ,
-p0 + 3*pp0 - 3*pp1 + p1 );
int valid_count = 0;
for(Real *i = roots, *end = i + count; i != end; ++i)
if (approximate_less(Real(0), *i) && approximate_less(*i, Real(1))) {
if (l) l[valid_count] = *i;
++valid_count;
}
return valid_count;
}
Range Bezier::bounds_accurate(Real p0, Real p1, Real pp0, Real pp1) {
Range range(p0);
range.expand(p1);
Real roots[2];
int count = bends(roots, p0, p1, pp0, pp1);
for(Real *i = roots, *end = i + count; i != end; ++i)
range.expand( p(*i, p0, p1, pp0, pp1) );
return range;
}
int Bezier::intersections(Real *l, Real p, Real p0, Real p1, Real pp0, Real pp1) {
Real roots[3];
int count = solve_equation(
roots,
p0 - p ,
-3*p0 + 3*pp0 ,
3*p0 - 6*pp0 + 3*pp1 ,
-p0 + 3*pp0 - 3*pp1 + p1 );
int valid_count = 0;
for(Real *i = roots, *end = i + count; i != end; ++i)
if (approximate_less(Real(0), *i) && approximate_less(*i, Real(1))) {
if (l) l[valid_count] = *i;
++valid_count;
}
return valid_count;
}