//TODO: this would work only for C1 pw<d2<sb>> input. Split the input at corners to work with pwd2sb...
//TODO: for more general purpose, return a path...
void toPoly(D2<SBasis> const &f, std::list<Point> &p, double tol, bool include_first=true){
D2<SBasis> df = derivative(f);
D2<SBasis> d2f = derivative(df);
double t=0;
if ( include_first ){ p.push_back( f.at0() );}
while (t<1){
Point v = unit_vector(df.valueAt(t));
//OptInterval bounds = bounds_local(df[X]*v[Y]-df[Y]*v[X], Interval(t,1));
OptInterval bounds = bounds_local(d2f[X]*v[Y]-d2f[Y]*v[X], Interval(t,1));
if (bounds) {
double bds_max = (-bounds->min()>bounds->max() ? -bounds->min() : bounds->max());
double dt;
//if (bds_max<tol) dt = 1;
//else dt = tol/bds_max;
if (bds_max<tol/4) dt = 1;
else dt = 2*std::sqrt( tol / bds_max );
t+=dt*5;
if (t>1) t = 1;
}else{
t = 1;
}
p.push_back( f.valueAt(t) );
}
return;
}
OptInterval bounds_local(const SBasisOf<double> &sb, const OptInterval &i, int order) {
#else
OptInterval bounds_local(const SBasis &sb, const OptInterval &i, int order) {
#endif
double t0=i->min(), t1=i->max(), lo=0., hi=0.;
for(int j = sb.size()-1; j>=order; j--) {
double a=sb[j][0];
double b=sb[j][1];
double t = 0;
if (lo<0) t = ((b-a)/lo+1)*0.5;
if (lo>=0 || t<t0 || t>t1) {
lo = std::min(a*(1-t0)+b*t0+lo*t0*(1-t0),a*(1-t1)+b*t1+lo*t1*(1-t1));
}else{
lo = lerp(t, a+lo*t, b);
}
if (hi>0) t = ((b-a)/hi+1)*0.5;
if (hi<=0 || t<t0 || t>t1) {
hi = std::max(a*(1-t0)+b*t0+hi*t0*(1-t0),a*(1-t1)+b*t1+hi*t1*(1-t1));
}else{
hi = lerp(t, a+hi*t, b);
}
}
Interval res = Interval(lo,hi);
if (order>0) res*=pow(.25,order);
return res;
}
void subdiv_sbasis(SBasis const & s,
std::vector<double> & roots,
double left, double right) {
OptInterval bs = bounds_fast(s);
if(!bs || bs->min() > 0 || bs->max() < 0)
return; // no roots here
if(s.tailError(1) < 1e-7) {
double t = s[0][0] / (s[0][0] - s[0][1]);
roots.push_back(left*(1-t) + t*right);
return;
}
double middle = (left + right)/2;
subdiv_sbasis(compose(s, Linear(0, 0.5)), roots, left, middle);
subdiv_sbasis(compose(s, Linear(0.5, 1.)), roots, middle, right);
}
static
std::vector<double>
solve_lambda0(double a0,double a1,double c0,double c1,
int insist_on_speeds_signs){
SBasis p(3, Linear());
p[0] = Linear( a1*c0*c0+c1, a1*a0*(a0+ 2*c0) +a1*c0*c0 +c1 -1 );
p[1] = Linear( -a1*a0*(a0+2*c0), -a1*a0*(3*a0+2*c0) );
p[2] = Linear( a1*a0*a0 );
OptInterval domain = find_bounds_for_lambda0(a0,a1,c0,c1,insist_on_speeds_signs);
if ( !domain )
return std::vector<double>();
p = compose(p,Linear(domain->min(),domain->max()));
std::vector<double>rts = roots(p);
for (unsigned i=0; i<rts.size(); i++){
rts[i] = domain->min() + rts[i] * domain->extent();
}
return rts;
}