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#include <2geom/solver.h>
#include <2geom/point.h>
#include <algorithm>
#include <valarray>
/*** 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)); }
const unsigned MAXDEPTH = 23; // Maximum depth for recursion. Using floats means 23 bits precision max
const double BEPSILON = ldexp(1.0,(-MAXDEPTH-1)); /*Flatness control value */
const double SECANT_EPSILON = 1e-13; // secant method converges much faster, get a bit more precision
/**
* This function is called _a lot_. We have included various manual memory management stuff to reduce the amount of mallocing that goes on. In the future it is possible that this will hurt performance.
**/
class Bernsteins{
public:
double *Vtemp;
unsigned N,degree;
std::vector<double> &solutions;
bool use_secant;
Bernsteins(int degr, std::vector<double> &so) : N(degr+1), degree(degr),solutions(so), use_secant(false) {
Vtemp = new double[N*2];
}
~Bernsteins() {
delete[] Vtemp;
}
void subdivide(double const *V,
double t,
double *Left,
double *Right);
double horner(const double *b, double t);
unsigned
control_poly_flat_enough(double const *V);
void
find_bernstein_roots(double const *w, /* The control points */
unsigned depth, /* The depth of the recursion */
double left_t, double right_t);
};
/*
* 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, bool use_secant)
{
Bernsteins B(degree, solutions);
B.use_secant = use_secant;
B.find_bernstein_roots(w, depth, left_t, right_t);
}
void
Bernsteins::find_bernstein_roots(double const *w, /* The control points */
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 < N; i++) {
int sign = SGN(w[i]);
if (sign) {
if (sign != old_sign && old_sign) {
n_crossings++;
}
old_sign = sign;
}
}
if (n_crossings == 0) // no solutions here
return;
if (n_crossings == 1) {
/* Unique solution */
/* Stop recursion when the tree is deep enough */
/* if deep enough, return 1 solution at midpoint */
if (depth >= MAXDEPTH) {
//printf("bottom out %d\n", depth);
const double Ax = right_t - left_t;
const double Ay = w[degree] - w[0];
solutions.push_back(left_t - Ax*w[0] / Ay);
return;
solutions.push_back((left_t + right_t) / 2.0);
return;
}
// I thought secant method would be faster here, but it'aint. -- njh
// Actually, it was, I just was using the wrong method for bezier evaluation. Horner's rule results in a very efficient algorithm - 10* faster (20080816)
// Future work: try using brent's method
if(use_secant) { // false position
double s = 0;double t = 1;
double e = 1e-10;
int n,side=0;
double r,fr,fs = w[0],ft = w[degree];
for (n = 1; n <= 100; n++)
{
r = (fs*t - ft*s) / (fs - ft);
if (fabs(t-s) < e*fabs(t+s)) break;
fr = horner(w, r);
if (fr * ft > 0)
{
t = r; ft = fr;
if (side==-1) fs /= 2;
side = -1;
}
else if (fs * fr > 0)
{
s = r; fs = fr;
if (side==+1) ft /= 2;
side = +1;
}
else break;
}
solutions.push_back(r*right_t + (1-r)*left_t);
return;
}
}
/* Otherwise, solve recursively after subdividing control polygon */
std::valarray<double> new_controls(2*N); // New left and right control polygons
const double t = 0.5;
/*
* Bernstein :
* Evaluate a Bernstein function at a particular parameter value
* Fill in control points for resulting sub-curves.
*
*/
for (unsigned i = 0; i < N; i++)
Vtemp[i] = w[i];
/* Triangle computation */
const double omt = (1-t);
new_controls[0] = Vtemp[0];
new_controls[N+degree] = Vtemp[degree];
double *prev_row = Vtemp;
double *row = Vtemp + N;
for (unsigned i = 1; i < N; i++) {
for (unsigned j = 0; j < N - i; j++) {
row[j] = omt*prev_row[j] + t*prev_row[j+1];
}
new_controls[i] = row[0];
new_controls[N+degree-i] = row[degree-i];
std::swap(prev_row, row);
}
double mid_t = left_t*(1-t) + right_t*t;
find_bernstein_roots(&new_controls[0], depth+1, left_t, mid_t);
/* Solution is exactly on the subdivision point. */
if (new_controls[N] == 0)
solutions.push_back(mid_t);
find_bernstein_roots(&new_controls[N], 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.
*
*/
unsigned
Bernsteins::control_poly_flat_enough(double const *V)
{
/* 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];
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 - a;
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 = 1./((a * a) + 1);
const double intercept_1 = (a - max_distance_above * abSquared);
const double intercept_2 = (a - 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);
//printf("error %g %g %g\n", error, a, BEPSILON * a);
return error < BEPSILON * a;
}
// suggested by Sederberg.
double Bernsteins::horner(const double *b, double t) {
int n = degree;
double u, bc, tn, tmp;
int i;
u = 1.0 - t;
bc = 1;
tn = 1;
tmp = b[0]*u;
for(i=1; i<n; i++){
tn = tn*t;
bc = bc*(n-i+1)/i;
tmp = (tmp + tn*bc*b[i])*u;
}
return (tmp + tn*t*b[n]);
}
};
/*
Local Variables:
mode:c++
c-file-style:"stroustrup"
c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
indent-tabs-mode:nil
fill-column:99
End:
*/
// vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:encoding=utf-8:textwidth=99 :
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