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/* From Sanchez-Reyes 1997
W_{j,k} = W_{n0j, n-k} = choose(n-2k-1, j-k)choose(2k+1,k)/choose(n,j)
for k=0,...,q-1; j = k, ...,n-k-1
W_{q,q} = 1 (n even)
This is wrong, it should read
W_{j,k} = W_{n0j, n-k} = choose(n-2k-1, j-k)/choose(n,j)
for k=0,...,q-1; j = k, ...,n-k-1
W_{q,q} = 1 (n even)
*/
#include "sbasis-to-bezier.h"
#include "choose.h"
#include "svg-path.h"
#include <iostream>
#include "exception.h"
namespace Geom{
double W(unsigned n, unsigned j, unsigned k) {
unsigned q = (n+1)/2;
if((n & 1) == 0 && j == q && k == q)
return 1;
if(k > n-k) return W(n, n-j, n-k);
assert((k <= q));
if(k >= q) return 0;
//assert(!(j >= n-k));
if(j >= n-k) return 0;
//assert(!(j < k));
if(j < k) return 0;
return choose<double>(n-2*k-1, j-k) /
choose<double>(n,j);
}
// this produces a degree 2q bezier from a degree k sbasis
Bezier
sbasis_to_bezier(SBasis const &B, unsigned q) {
if(q == 0) {
q = B.size();
/*if(B.back()[0] == B.back()[1]) {
n--;
}*/
}
unsigned n = q*2;
Bezier result = Bezier(Bezier::Order(n-1));
if(q > B.size())
q = B.size();
n--;
for(unsigned k = 0; k < q; k++) {
for(unsigned j = 0; j <= n-k; j++) {
result[j] += (W(n, j, k)*B[k][0] +
W(n, n-j, k)*B[k][1]);
}
}
return result;
}
double mopi(int i) {
return (i&1)?-1:1;
}
// this produces a degree k sbasis from a degree 2q bezier
SBasis
bezier_to_sbasis(Bezier const &B) {
unsigned n = B.size();
unsigned q = (n+1)/2;
SBasis result;
result.resize(q+1);
for(unsigned k = 0; k < q; k++) {
result[k][0] = result[k][1] = 0;
for(unsigned j = 0; j <= n-k; j++) {
result[k][0] += mopi(int(j)-int(k))*W(n, j, k)*B[j];
result[k][1] += mopi(int(j)-int(k))*W(n, j, k)*B[j];
//W(n, n-j, k)*B[k][1]);
}
}
return result;
}
// this produces a 2q point bezier from a degree q sbasis
std::vector<Geom::Point>
sbasis_to_bezier(D2<SBasis> const &B, unsigned qq) {
std::vector<Geom::Point> result;
if(qq == 0) {
qq = sbasis_size(B);
}
unsigned n = qq * 2;
result.resize(n, Geom::Point(0,0));
n--;
for(unsigned dim = 0; dim < 2; dim++) {
unsigned q = qq;
if(q > B[dim].size())
q = B[dim].size();
for(unsigned k = 0; k < q; k++) {
for(unsigned j = 0; j <= n-k; j++) {
result[j][dim] += (W(n, j, k)*B[dim][k][0] +
W(n, n-j, k)*B[dim][k][1]);
}
}
}
return result;
}
/*
template <unsigned order>
D2<Bezier<order> > sbasis_to_bezier(D2<SBasis> const &B) {
return D2<Bezier<order> >(sbasis_to_bezier<order>(B[0]), sbasis_to_bezier<order>(B[1]));
}
*/
#if 0 // using old path
//std::vector<Geom::Point>
// mutating
void
subpath_from_sbasis(Geom::OldPathSetBuilder &pb, D2<SBasis> const &B, double tol, bool initial) {
assert(B.is_finite());
if(B.tail_error(2) < tol || B.size() == 2) { // nearly cubic enough
if(B.size() == 1) {
if (initial) {
pb.start_subpath(Geom::Point(B[0][0][0], B[1][0][0]));
}
pb.push_line(Geom::Point(B[0][0][1], B[1][0][1]));
} else {
std::vector<Geom::Point> bez = sbasis_to_bezier(B, 2);
if (initial) {
pb.start_subpath(bez[0]);
}
pb.push_cubic(bez[1], bez[2], bez[3]);
}
} else {
subpath_from_sbasis(pb, compose(B, Linear(0, 0.5)), tol, initial);
subpath_from_sbasis(pb, compose(B, Linear(0.5, 1)), tol, false);
}
}
/*
* This version works by inverting a reasonable upper bound on the error term after subdividing the
* curve at $a$. We keep biting off pieces until there is no more curve left.
*
* Derivation: The tail of the power series is $a_ks^k + a_{k+1}s^{k+1} + \ldots = e$. A
* subdivision at $a$ results in a tail error of $e*A^k, A = (1-a)a$. Let this be the desired
* tolerance tol $= e*A^k$ and invert getting $A = e^{1/k}$ and $a = 1/2 - \sqrt{1/4 - A}$
*/
void
subpath_from_sbasis_incremental(Geom::OldPathSetBuilder &pb, D2<SBasis> B, double tol, bool initial) {
const unsigned k = 2; // cubic bezier
double te = B.tail_error(k);
assert(B[0].is_finite());
assert(B[1].is_finite());
//std::cout << "tol = " << tol << std::endl;
while(1) {
double A = std::sqrt(tol/te); // pow(te, 1./k)
double a = A;
if(A < 1) {
A = std::min(A, 0.25);
a = 0.5 - std::sqrt(0.25 - A); // quadratic formula
if(a > 1) a = 1; // clamp to the end of the segment
} else
a = 1;
assert(a > 0);
//std::cout << "te = " << te << std::endl;
//std::cout << "A = " << A << "; a=" << a << std::endl;
D2<SBasis> Bs = compose(B, Linear(0, a));
assert(Bs.tail_error(k));
std::vector<Geom::Point> bez = sbasis_to_bezier(Bs, 2);
reverse(bez.begin(), bez.end());
if (initial) {
pb.start_subpath(bez[0]);
initial = false;
}
pb.push_cubic(bez[1], bez[2], bez[3]);
// move to next piece of curve
if(a >= 1) break;
B = compose(B, Linear(a, 1));
te = B.tail_error(k);
}
}
#endif
void build_from_sbasis(Geom::PathBuilder &pb, D2<SBasis> const &B, double tol) {
if (!B.isFinite()) {
throwException("assertion failed: B.isFinite()");
}
if(tail_error(B, 2) < tol || sbasis_size(B) == 2) { // nearly cubic enough
if(sbasis_size(B) <= 1) {
pb.lineTo(B.at1());
} else {
std::vector<Geom::Point> bez = sbasis_to_bezier(B, 2);
pb.curveTo(bez[1], bez[2], bez[3]);
}
} else {
build_from_sbasis(pb, compose(B, Linear(0, 0.5)), tol);
build_from_sbasis(pb, compose(B, Linear(0.5, 1)), tol);
}
}
Path
path_from_sbasis(D2<SBasis> const &B, double tol) {
PathBuilder pb;
pb.moveTo(B.at0());
build_from_sbasis(pb, B, tol);
pb.finish();
return pb.peek().front();
}
//TODO: some of this logic should be lifted into svg-path
std::vector<Geom::Path>
path_from_piecewise(Geom::Piecewise<Geom::D2<Geom::SBasis> > const &B, double tol) {
Geom::PathBuilder pb;
if(B.size() == 0) return pb.peek();
Geom::Point start = B[0].at0();
pb.moveTo(start);
for(unsigned i = 0; ; i++) {
if(i+1 == B.size() || !are_near(B[i+1].at0(), B[i].at1(), tol)) {
//start of a new path
if(are_near(start, B[i].at1()) && sbasis_size(B[i]) <= 1) {
//last line seg already there
goto no_add;
}
build_from_sbasis(pb, B[i], tol);
if(are_near(start, B[i].at1())) {
//it's closed
pb.closePath();
}
no_add:
if(i+1 >= B.size()) break;
start = B[i+1].at0();
pb.moveTo(start);
} else {
build_from_sbasis(pb, B[i], tol);
}
}
pb.finish();
return pb.peek();
}
};
/*
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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