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#include "solver.h"
#include "point.h"
#include <algorithm>
/*** Find the zeros of the bernstein function. The code subdivides until it is happy with the
* linearity of the function. This requires an O(degree^2) subdivision for each step, even when
* there is only one solution.
*/
namespace Geom{
template<class t>
static int SGN(t x) { return (x > 0 ? 1 : (x < 0 ? -1 : 0)); }
/*
* Forward declarations
*/
static void
Bernstein(double const *V,
unsigned degree,
double t,
double *Left,
double *Right);
static unsigned
control_poly_flat_enough(double const *V, unsigned degree,
double left_t, double right_t);
const unsigned MAXDEPTH = 64; /* Maximum depth for recursion */
const double BEPSILON = ldexp(1.0,((signed)-1)-MAXDEPTH); /*Flatness control value */
/*
* find_bernstein_roots : Given an equation in Bernstein-Bernstein form, find all
* of the roots in the open interval (0, 1). Return the number of roots found.
*/
void
find_bernstein_roots(double const *w, /* The control points */
unsigned degree, /* The degree of the polynomial */
std::vector<double> &solutions, /* RETURN candidate t-values */
unsigned depth, /* The depth of the recursion */
double left_t, double right_t)
{
unsigned n_crossings = 0; /* Number of zero-crossings */
int old_sign = SGN(w[0]);
for (unsigned i = 1; i <= degree; i++) {
int sign = SGN(w[i]);
if (sign) {
if (sign != old_sign && old_sign) {
n_crossings++;
}
old_sign = sign;
}
}
switch (n_crossings) {
case 0: /* No solutions here */
return;
case 1:
/* Unique solution */
/* Stop recursion when the tree is deep enough */
/* if deep enough, return 1 solution at midpoint */
if (depth >= MAXDEPTH) {
solutions.push_back((left_t + right_t) / 2.0);
return;
}
// I thought secant method would be faster here, but it'aint. -- njh
if (control_poly_flat_enough(w, degree, left_t, right_t)) {
const double Ax = right_t - left_t;
const double Ay = w[degree] - w[0];
solutions.push_back(left_t - Ax*w[0] / Ay);
return;
}
break;
}
/* Otherwise, solve recursively after subdividing control polygon */
std::vector<double> Left(degree+1); /* New left and right */
std::vector<double> Right(degree+1);/* control polygons */
const double split = 0.5;
Bernstein(w, degree, split, &Left[0], &Right[0]);
double mid_t = left_t*(1-split) + right_t*split;
find_bernstein_roots(&Left[0], degree, solutions, depth+1, left_t, mid_t);
/* Solution is exactly on the subdivision point. */
if (Right[0] == 0)
solutions.push_back(mid_t);
find_bernstein_roots(&Right[0], degree, solutions, depth+1, mid_t, right_t);
}
/*
* control_poly_flat_enough :
* Check if the control polygon of a Bernstein curve is flat enough
* for recursive subdivision to bottom out.
*
*/
static unsigned
control_poly_flat_enough(double const *V, /* Control points */
unsigned degree,
double left_t, double right_t) /* Degree of polynomial */
{
/* Find the perpendicular distance from each interior control point to line connecting V[0] and
* V[degree] */
/* Derive the implicit equation for line connecting first */
/* and last control points */
const double a = V[0] - V[degree];
const double b = right_t - left_t;
const double c = left_t * V[degree] - right_t * V[0] + a * left_t;
double max_distance_above = 0.0;
double max_distance_below = 0.0;
double ii = 0, dii = 1./degree;
for (unsigned i = 1; i < degree; i++) {
ii += dii;
/* Compute distance from each of the points to that line */
const double d = (a + V[i]) * ii*b + c;
double dist = d*d;
// Find the largest distance
if (d < 0.0)
max_distance_below = std::min(max_distance_below, -dist);
else
max_distance_above = std::max(max_distance_above, dist);
}
const double abSquared = (a * a) + (b * b);
const double intercept_1 = -(c + max_distance_above / abSquared);
const double intercept_2 = -(c + max_distance_below / abSquared);
/* Compute bounding interval*/
const double left_intercept = std::min(intercept_1, intercept_2);
const double right_intercept = std::max(intercept_1, intercept_2);
const double error = 0.5 * (right_intercept - left_intercept);
if (error < BEPSILON * a)
return 1;
return 0;
}
/*
* Bernstein :
* Evaluate a Bernstein function at a particular parameter value
* Fill in control points for resulting sub-curves.
*
*/
static void
Bernstein(double const *V, /* Control pts */
unsigned degree, /* Degree of bernstein curve */
double t, /* Parameter value */
double *Left, /* RETURN left half ctl pts */
double *Right) /* RETURN right half ctl pts */
{
const unsigned size=degree+1;
std::vector<double> vtemp(V,V+size);
/* Copy control points */
Left[0] = vtemp[0];
Right[degree]= vtemp[degree];
/* Triangle computation */
const double omt = (1-t);
for (unsigned i = 1; i < size; ++i) {
for (unsigned j = 0; j < size - i; ++j) {
vtemp[j] = omt*vtemp[j]+t*vtemp[j+1];
}
Left[i] =vtemp[0];
Right[degree-i]=vtemp[degree-i];
}
}
};
/*
Local Variables:
mode:c++
c-file-style:"stroustrup"
c-file-offsets:((innamespace . 0)(substatement-open . 0))
indent-tabs-mode:nil
c-brace-offset:0
fill-column:99
End:
vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4 :
*/
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