double Bernsteins::secant(Bezier bz) {
double s = 0, t = 1;
double e = 1e-14;
int side = 0;
double r, fr, fs = bz.at0(), ft = bz.at1();
for (size_t n = 0; n < 100; ++n)
{
r = (fs*t - ft*s) / (fs - ft);
if (fabs(t-s) < e * fabs(t+s)) {
debug(std::cout << "error small " << fabs(t-s)
<< ", accepting solution " << r
<< "after " << n << "iterations\n");
return r;
}
fr = horner(bz, 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;
}
return r;
}
// suggested by Sederberg.
double Bernsteins::horner(Bezier bz, double t)
{
double u, tn, tmp;
u = 1.0 - t;
tn = 1.0;
tmp = bz.at0() * u;
for(size_t i = 1; i < bz.degree(); ++i)
{
tn *= t;
tmp = (tmp + tn*choose<double>(bz.order(), (unsigned)i)*bz[i]) * u;
}
return (tmp + tn*t*bz.at1());
}
vector<double> find_all_roots(Bezier b) {
vector<double> rts = b.roots();
if(b.at0() == 0) rts.push_back(0);
if(b.at1() == 0) rts.push_back(1);
return rts;
}
if(a.size() != b.size()) return;
for(int i = 0; i < a.size(); i++) {
EXPECT_FLOAT_EQ(a[i], b[i]);
}
}
vector<double> find_all_roots(Bezier b) {
vector<double> rts = b.roots();
if(b.at0() == 0) rts.push_back(0);
if(b.at1() == 0) rts.push_back(1);
return rts;
}
TEST_F(ChainTest, Deflate) {
Bezier b = array_roots(vector_from_array((const double[]){0,0.25,0.5}));
EXPECT_FLOAT_EQ(0, b.at0());
b = b.deflate();
EXPECT_FLOAT_EQ(0, b.valueAt(0.25));
b = b.subdivide(0.25).second;
EXPECT_FLOAT_EQ(0, b.at0());
b = b.deflate();
const double rootposition = (0.5-0.25) / (1-0.25);
EXPECT_FLOAT_EQ(0, b.valueAt(rootposition));
b = b.subdivide(rootposition).second;
EXPECT_FLOAT_EQ(0, b.at0());
}
TEST_F(ChainTest, Roots) {
expect_array((const double[]){0.5}, wiggle.roots());
Bezier bigun(Bezier::Order(30));
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);
}
}
