/** Reparameterise M to have unit speed.
\param M the Element.
\param tol the maximum error allowed.
\param order the maximum degree to use for approximation
\relates Piecewise, D2
*/
Piecewise<D2<SBasis> >
Geom::arc_length_parametrization(D2<SBasis> const &M,
unsigned order,
double tol){
Piecewise<D2<SBasis> > u;
u.push_cut(0);
Piecewise<SBasis> s = arcLengthSb(Piecewise<D2<SBasis> >(M),tol);
for (unsigned i=0; i < s.size();i++){
double t0=s.cuts[i],t1=s.cuts[i+1];
if ( are_near(s(t0),s(t1)) ) {
continue;
}
D2<SBasis> sub_M = compose(M,Linear(t0,t1));
D2<SBasis> sub_u;
for (unsigned dim=0;dim<2;dim++){
SBasis sub_s = s.segs[i];
sub_s-=sub_s.at0();
sub_s/=sub_s.at1();
sub_u[dim]=compose_inverse(sub_M[dim],sub_s, order, tol);
}
u.push(sub_u,s(t1));
}
return u;
}
void draw(cairo_t *cr, std::ostringstream *notify, int width, int height, bool save) {
Geom::Point dir(1,-2);
D2<Piecewise<SBasis> > B = make_cuts_independent(path_a_pw);
cairo_set_source_rgba (cr, 0., 0.125, 0, 1);
if(0) {
D2<Piecewise<SBasis> > tB(cos(B[0]*0.1)*(brush_handle.pos[0]/100) + B[0],
cos(B[1]*0.1)*(brush_handle.pos[1]/100) + B[1]);
cairo_d2_pw_sb(cr, tB);
} else {
Piecewise<SBasis> r2 = (dot(path_a_pw - brush_handle.pos, path_a_pw - brush_handle.pos));
Piecewise<SBasis> rc;
rc.push_cut(0);
rc.push(SBasis(Linear(1, 1)), 2);
rc.push(SBasis(Linear(1, 0)), 4);
rc.push(SBasis(Linear(0, 0)), 30);
rc *= 10;
rc.scaleDomain(1000);
Piecewise<SBasis> swr;
swr.push_cut(0);
swr.push(SBasis(Linear(0, 1)), 2);
swr.push(SBasis(Linear(1, 0)), 4);
swr.push(SBasis(Linear(0, 0)), 30);
swr *= 10;
swr.scaleDomain(1000);
cairo_pw(cr, swr);// + (height - 100));
D2<Piecewise<SBasis> > uB = make_cuts_independent(unitVector(path_a_pw - brush_handle.pos));
D2<Piecewise<SBasis> > tB(compose(rc, (r2))*uB[0] + B[0],
compose(rc, (r2))*uB[1] + B[1]);
cairo_d2_pw_sb(cr, tB);
//path_a_pw = sectionize(tB);
}
cairo_stroke(cr);
*notify << path_a_pw.size();
Toy::draw(cr, notify, width, height, save);
}
Piecewise<SBasis> convole(SBasisOf<double> const &f, Interval dom_f,
SBasisOf<double> const &g, Interval dom_g,
bool f_cst_ends = false){
if ( dom_f.extent() < dom_g.extent() ) return convole(g, dom_g, f, dom_f);
Piecewise<SBasis> result;
SBasisOf<SBasisOf<double> > u,v;
u.push_back(LinearOf<SBasisOf<double> >(SBasisOf<double>(LinearOf<double>(0,1))));
v.push_back(LinearOf<SBasisOf<double> >(SBasisOf<double>(LinearOf<double>(0,0)),
SBasisOf<double>(LinearOf<double>(1,1))));
SBasisOf<SBasisOf<double> > v_u = (v - u)*(dom_f.extent()/dom_g.extent());
v_u += SBasisOf<SBasisOf<double> >(SBasisOf<double>(-dom_g.min()/dom_g.extent()));
SBasisOf<SBasisOf<double> > gg = multi_compose(g,v_u);
SBasisOf<SBasisOf<double> > ff = SBasisOf<SBasisOf<double> >(f);
SBasisOf<SBasisOf<double> > hh = integral(ff*gg,0);
result.cuts.push_back(dom_f.min()+dom_g.min());
//Note: we know dom_f.extent() >= dom_g.extent()!!
//double rho = dom_f.extent()/dom_g.extent();
double t0 = dom_g.min()/dom_f.extent();
double t1 = dom_g.max()/dom_f.extent();
double t2 = t0+1;
double t3 = t1+1;
SBasisOf<double> a,b,t;
SBasis seg;
a = SBasisOf<double>(LinearOf<double>(0,0));
b = SBasisOf<double>(LinearOf<double>(0,t1-t0));
t = SBasisOf<double>(LinearOf<double>(t0,t1));
seg = toSBasis(compose(hh,b,t)-compose(hh,a,t));
result.push(seg,dom_f.min() + dom_g.max());
if (dom_f.extent() > dom_g.extent()){
a = SBasisOf<double>(LinearOf<double>(0,t2-t1));
b = SBasisOf<double>(LinearOf<double>(t1-t0,1));
t = SBasisOf<double>(LinearOf<double>(t1,t2));
seg = toSBasis(compose(hh,b,t)-compose(hh,a,t));
result.push(seg,dom_f.max() + dom_g.min());
}
a = SBasisOf<double>(LinearOf<double>(t2-t1,1.));
b = SBasisOf<double>(LinearOf<double>(1.,1.));
t = SBasisOf<double>(LinearOf<double>(t2,t3));
seg = toSBasis(compose(hh,b,t)-compose(hh,a,t));
result.push(seg,dom_f.max() + dom_g.max());
result*=dom_f.extent();
//--constant ends correction--------------
if (f_cst_ends){
SBasis ig = toSBasis(integraaal(g))*dom_g.extent();
ig -= ig.at0();
Piecewise<SBasis> cor;
cor.cuts.push_back(dom_f.min()+dom_g.min());
cor.push(reverse(ig)*f.at0(),dom_f.min()+dom_g.max());
cor.push(Linear(0),dom_f.max()+dom_g.min());
cor.push(ig*f.at1(),dom_f.max()+dom_g.max());
result+=cor;
}
//----------------------------------------
return result;
}
/**
* \brief Retruns a Piecewise SBasis with prescribed values at prescribed times.
*
* \param times: vector of times at which the values are given. Should be sorted in increasing order.
* \param values: vector of prescribed values. Should have the same size as times and be sorted accordingly.
* \param smoothness: (defaults to 1) regularity class of the result: 0=piecewise linear, 1=continuous derivative, etc...
*/
Piecewise<SBasis> interpolate(std::vector<double> times, std::vector<double> values, unsigned smoothness){
assert ( values.size() == times.size() );
if ( values.size() == 0 ) return Piecewise<SBasis>();
if ( values.size() == 1 ) return Piecewise<SBasis>(values[0]);//what about time??
SBasis sk = shift(Linear(1.),smoothness);
SBasis bump_in = integral(sk);
bump_in -= bump_in.at0();
bump_in /= bump_in.at1();
SBasis bump_out = reverse( bump_in );
Piecewise<SBasis> result;
result.cuts.push_back(times[0]);
for (unsigned i = 0; i<values.size()-1; i++){
result.push(bump_out*values[i]+bump_in*values[i+1],times[i+1]);
}
return result;
}