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#include <2geom/bezier.h>
#include <2geom/point.h>
#include <2geom/solver.h>
#include <algorithm>
namespace Geom {
/*** Find the zeros of the parametric function in 2d defined by two beziers X(t), Y(t). The code subdivides until it happy with the linearity of the bezier. This requires an n^2 subdivision for each step, even when there is only one solution.
*
* Perhaps it would be better to subdivide particularly around nodes with changing sign, rather than simply cutting in half.
*/
#define SGN(a) (((a)<0) ? -1 : 1)
/*
* Forward declarations
*/
unsigned
crossing_count(Geom::Point const *V, unsigned degree);
static unsigned
control_poly_flat_enough(Geom::Point const *V, unsigned degree);
static double
compute_x_intercept(Geom::Point const *V, unsigned degree);
const unsigned MAXDEPTH = 64; /* Maximum depth for recursion */
const double BEPSILON = ldexp(1.0,-MAXDEPTH-1); /*Flatness control value */
unsigned total_steps, total_subs;
/*
* find_bezier_roots : Given an equation in Bernstein-Bezier form, find all
* of the roots in the interval [0, 1]. Return the number of roots found.
*/
void
find_parametric_bezier_roots(Geom::Point 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 */
{
total_steps++;
const unsigned max_crossings = crossing_count(w, degree);
switch (max_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((w[0][Geom::X] + w[degree][Geom::X]) / 2.0);
return;
}
// I thought secant method would be faster here, but it'aint. -- njh
if (control_poly_flat_enough(w, degree)) {
solutions.push_back(compute_x_intercept(w, degree));
return;
}
break;
}
/* Otherwise, solve recursively after subdividing control polygon */
//Geom::Point Left[degree+1], /* New left and right */
// Right[degree+1]; /* control polygons */
std::vector<Geom::Point> Left( degree+1 ), Right(degree+1);
casteljau_subdivision(0.5, w, Left.data(), Right.data(), degree);
total_subs ++;
find_parametric_bezier_roots(Left.data(), degree, solutions, depth+1);
find_parametric_bezier_roots(Right.data(), degree, solutions, depth+1);
}
/*
* crossing_count:
* Count the number of times a Bezier control polygon
* crosses the 0-axis. This number is >= the number of roots.
*
*/
unsigned
crossing_count(Geom::Point const *V, /* Control pts of Bezier curve */
unsigned degree) /* Degree of Bezier curve */
{
unsigned n_crossings = 0; /* Number of zero-crossings */
int old_sign = SGN(V[0][Geom::Y]);
for (unsigned i = 1; i <= degree; i++) {
int sign = SGN(V[i][Geom::Y]);
if (sign != old_sign)
n_crossings++;
old_sign = sign;
}
return n_crossings;
}
/*
* control_poly_flat_enough :
* Check if the control polygon of a Bezier curve is flat enough
* for recursive subdivision to bottom out.
*
*/
static unsigned
control_poly_flat_enough(Geom::Point const *V, /* Control points */
unsigned degree) /* 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][Geom::Y] - V[degree][Geom::Y];
const double b = V[degree][Geom::X] - V[0][Geom::X];
const double c = V[0][Geom::X] * V[degree][Geom::Y] - V[degree][Geom::X] * V[0][Geom::Y];
const double abSquared = (a * a) + (b * b);
//double distance[degree]; /* Distances from pts to line */
std::vector<double> distance(degree); /* Distances from pts to line */
for (unsigned i = 1; i < degree; i++) {
/* Compute distance from each of the points to that line */
double & dist(distance[i-1]);
const double d = a * V[i][Geom::X] + b * V[i][Geom::Y] + c;
dist = d*d / abSquared;
if (d < 0.0)
dist = -dist;
}
// Find the largest distance
double max_distance_above = 0.0;
double max_distance_below = 0.0;
for (unsigned i = 0; i < degree-1; i++) {
const double d = distance[i];
if (d < 0.0)
max_distance_below = std::min(max_distance_below, d);
if (d > 0.0)
max_distance_above = std::max(max_distance_above, d);
}
const double intercept_1 = (c + max_distance_above) / -a;
const double intercept_2 = (c + max_distance_below) / -a;
/* 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)
return 1;
return 0;
}
/*
* compute_x_intercept :
* Compute intersection of chord from first control point to last
* with 0-axis.
*
*/
static double
compute_x_intercept(Geom::Point const *V, /* Control points */
unsigned degree) /* Degree of curve */
{
const Geom::Point A = V[degree] - V[0];
return (A[Geom::X]*V[0][Geom::Y] - A[Geom::Y]*V[0][Geom::X]) / -A[Geom::Y];
}
};
/*
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:fileencoding=utf-8:textwidth=99 :
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