void Bernsteins::find_bernstein_roots(Bezier bz,
unsigned depth,
double left_t,
double right_t)
{
debug(std::cout << left_t << ", " << right_t << std::endl);
size_t n_crossings = 0;
int old_sign = SGN(bz[0]);
//std::cout << "w[0] = " << bz[0] << std::endl;
int sign;
for (size_t i = 1; i < bz.size(); i++)
{
//std::cout << "w[" << i << "] = " << w[i] << std::endl;
sign = SGN(bz[i]);
if (sign != 0)
{
if (sign != old_sign && old_sign != 0)
{
++n_crossings;
}
old_sign = sign;
}
}
//std::cout << "n_crossings = " << n_crossings << std::endl;
if (n_crossings == 0) return; // no solutions here
if (n_crossings == 1) /* Unique solution */
{
//std::cout << "depth = " << depth << std::endl;
/* Stop recursion when the tree is deep enough */
/* if deep enough, return 1 solution at midpoint */
if (depth > MAX_DEPTH)
{
//printf("bottom out %d\n", depth);
const double Ax = right_t - left_t;
const double Ay = bz.at1() - bz.at0();
solutions.push_back(left_t - Ax*bz.at0() / Ay);
return;
}
double r = secant(bz);
solutions.push_back(r*right_t + (1-r)*left_t);
return;
}
/* Otherwise, solve recursively after subdividing control polygon */
Bezier::Order o(bz);
Bezier Left(o), Right = bz;
double split_t = (left_t + right_t) * 0.5;
// If subdivision is working poorly, split around the leftmost root of the derivative
if (depth > 2) {
debug(std::cout << "derivative mode\n");
Bezier dbz = derivative(bz);
debug(std::cout << "initial = " << dbz << std::endl);
std::vector<double> dsolutions = dbz.roots(Interval(left_t, right_t));
debug(std::cout << "dsolutions = " << dsolutions << std::endl);
double dsplit_t = 0.5;
if(!dsolutions.empty()) {
dsplit_t = dsolutions[0];
split_t = left_t + (right_t - left_t)*dsplit_t;
debug(std::cout << "split_value = " << bz(split_t) << std::endl);
debug(std::cout << "spliting around " << dsplit_t << " = "
<< split_t << "\n");
}
std::pair<Bezier, Bezier> LR = bz.subdivide(dsplit_t);
Left = LR.first;
Right = LR.second;
} else {
// split at midpoint, because it is cheap
Left[0] = Right[0];
for (size_t i = 1; i < bz.size(); ++i)
{
for (size_t j = 0; j < bz.size()-i; ++j)
{
Right[j] = (Right[j] + Right[j+1]) * 0.5;
}
Left[i] = Right[0];
}
}
debug(std::cout << "Solution is exactly on the subdivision point.\n");
debug(std::cout << Left << " , " << Right << std::endl);
Left = reverse(Left);
while(Right.order() > 0 and fabs(Right[0]) <= 1e-10) {
debug(std::cout << "deflate\n");
Right = Right.deflate();
Left = Left.deflate();
solutions.push_back(split_t);
}
Left = reverse(Left);
if (Right.order() > 0) {
debug(std::cout << Left << " , " << Right << std::endl);
find_bernstein_roots(Left, depth+1, left_t, split_t);
find_bernstein_roots(Right, depth+1, split_t, 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)
{
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);
}
