bool Quadric::intersect(const Point3D &eye, const Point3D &_ray, HitReporter &hr) const
{
Polynomial<2> x, y, z;
const Vector3D ray = _ray - eye;
x[0] = eye[0];
x[1] = ray[0];
y[0] = eye[1];
y[1] = ray[1];
z[0] = eye[2];
z[1] = ray[2];
const Polynomial<2> eqn = A * x * x + B * x * y + C * x * z
+ D * y * y + E * y * z + F * z * z + G * x + H * y + J * z + K;
double ts[2];
auto nhits = eqn.solve(ts);
if(nhits > 0)
{
const Polynomial<2> ddx = 2 * A * x + B * y + C * z + G,
ddy = 2 * D * y + B * x + E * z + H,
ddz = 2 * F * z + C * x + E * y + J;
for(int i = 0; i < nhits; i++)
{
// Figure out the normal.
const double t = ts[i];
const Point3D pt = eye + t * ray;
if(!predicate(pt))
continue;
Vector3D normal(ddx.eval(t), ddy.eval(t), ddz.eval(t));
// Figure out the uv.
Point2D uv;
Vector3D u, v;
get_uv(pt, normal, uv, u, v);
if(!hr.report(ts[i], normal, uv, u, v))
return false;
}
}
return true;
}
int main()
{
Polynomial a ({2,1,0,1,0}) ;
Polynomial b ;
Polynomial c (a);
cout<<a<<endl;
cout<<a.deriv()<<endl;
cout<<endl;
cout<<endl;
cout<<endl;
cout<<a<<endl;
b.setcoefs({1,1,1,1});
cout<<b.getcoef(0)<<endl;
cout<<b<<endl;
cout<< b.eval(2)<<endl;
cout<<endl;
cout<<endl;
cout<<endl;
cout<<b<<endl;
cout<<b*3<<endl;
cout<<endl;
cout<<endl;
cout<<endl;
cout<<b<<endl;
cout<<b+3<<endl;
cout<<endl;
return 0;
}
void isolate_root_with_cluster(double& low, double& hgh, Polynomial *p)
{
dprint(low);
dprint(hgh);
// ensure starting conditions
//
PointVal pv_low(low,p);
// double p_low = eval(low);
if(pv_low.v() < 0.)
{
Polynomial p_minus = -(*p);
p_minus.isolate_root(low,hgh);
return;
}
// initialize algorithm
//
// initialize root bracket.
//
PointVal pv_hgh(hgh,p);
RootBracketClusterState rb(pv_low,pv_hgh);
dprint(rb.low().x());
dprint(rb.low().v());
dprint(rb.hgh().x());
dprint(rb.hgh().v());
//
Polynomial Dp = p->get_derivative();
// requirements of algorithm:
// - second-order convergence in all cases
// - high performance in generic cases
// - handles repeated polynomial roots
// - converges to a root between low and hgh
// - residual of answer is less than machine precision
// conditions maintained by algorithm:
// * low < hgh
// * p(low) > EPSILON
// * p(hgh) < -EPSILON
// using_hgh = fabs(p(hgh)) > fabs(p(low))
// state of algorithm is specified by:
// * low
// * hgh
// * using_hgh
// * num_roots_in_cluster = assumed odd number of roots in cluster
// iteration of algorithm:
// use newton iteration on smaller side to look for root
// - check that guess is between low and hgh
// and that new value has smaller residual
// else switch to secant method
// - if newton does not change the sign
// increment num_roots_in_cluster until you leave interval,
// the sign changes, or the residual increases.
// - if the sign does change
// decrement num_roots_in_cluster until num_roots_in_cluster=1
// or the sign does not change (if residual does not shrink
// sufficiently modify num_roots_in_cluster).
// - do one iteration of bisection if residual did not
// shrink by factor of e.g. .5
// assumed number of roots in root cluster we are converging towards
goto start_newton_iteration;
start_newton_iteration:
// initialize newton iteration
// choose smaller endpoint as starting value.
goto newton_iteration;
done:
dprint(rb.num_evals());
low = rb.low().x();
hgh = rb.hgh().x();
//return rb.x();
return;
// newton's method begins with a point which is one end of a root bracket.
// it determines a displacement based on the value of the function
// and its derivative at the point. To handle root clusters
// it assumes that the function is approximated by a shifted
// and scaled monomial.
newton_iteration:
{
if(rb.done()) goto done;
dprintf("using newton: x0=%24.16e, p(x0)=%24.16e",rb.pv().x(),rb.pv().v());
const PointVal start_pv0(rb.pv());
//
const double Dpx0 = Dp.eval(rb.pv().x());
// bail if the derivative is too small
if(fabs(Dpx0) < 1e-6)
{
dprintf("bad derivative: %f",Dpx0);
goto newton_bailed;
}
double displacement = -rb.pv().v()/Dpx0;
compute_x0:
double x0 = rb.pv().x() + rb.num_roots_in_cluster()*displacement;
// if Newton prediction is outside interval
if(rb.not_in_interval(x0))
{
// reset num_roots_in_cluster and try again.
if(rb.num_roots_in_cluster()!=1)
{
rb.reset_num_roots_in_cluster();
goto compute_x0;
}
dprintf("Newton %24.16e outside interval [%24.16e,%24.16e]",
x0, rb.low().x(), rb.hgh().x());
dprintf("with values [%24.16e,%24.16e]", rb.low().v(), rb.hgh().v());
goto newton_bailed;
}
// evaluate the polynomial at the new guess
if(rb.eval(x0,p)) goto done;
// bail if the new residual is not sufficiently smaller
if(rb.pv().a() > .5*start_pv0.a()) // exp(-1) < .5 < 1
{
dprintf("apx0=%24.16e and start_apx0=%24.16e",rb.pv().a(),start_pv0.a());
dprintf("so Newton didn't help enough, maybe because Dpx0=%24.16e.",Dpx0);
goto newton_bailed;
}
// did the sign change?
bool sign_changed = ((start_pv0.v() < 0. && rb.pv().v() > 0.)
|| (start_pv0.v() > 0. && rb.pv().v() < 0.)) ? true : false;
// does changing number of roots in root cluster help?
const PointVal old_pv0 = rb.pv();
if(sign_changed)
{
// keep decrementing presumed number
// of roots in cluster as long as you can
// until it does not help the residual
if(rb.num_roots_in_cluster()>=3)
{
//double old_x0 = x0;
decrement:
double x0 = old_pv0.x() - 2*displacement;
// displacement stayed in root bracket, so smaller one will too
if(rb.eval(x0,p)) goto done;
if(rb.pv().a() < old_pv0.a())
{
// accept decrement
rb.decrement_num_roots_in_cluster();
goto decrement;
}
else
{
// reject decrement
rb.set_to_using_smaller();
goto newton_iteration;
}
}
// only one root so just iterate
goto newton_iteration;
}
else // !sign_changed
{
// if the residual did not shrink by trigger_ratio then
// we will try incrementing num_roots_in_cluster
//
// compute trigger_ratio
//
const double trigger_ratio
= get_trigger_ratio_for_newton(rb.num_roots_in_cluster());
// if residual shrunk by more than trigger_ratio we're fine
if(rb.pv().a() < trigger_ratio*old_pv0.a())
{
// evidently not near a root cluster
// with a wrong assumption about the number
// of roots, so just iterate
goto newton_iteration;
}
else
{
// keep incrementing presumed number of roots
// in cluster until we are outside the interval
// or until doing so does not improve
// the residual by better than multiplying
// it by a factor of .5
// (can use any number between 1/e and 1).
{
increment:
x0 += 2*displacement;
if(rb.not_in_interval(x0))
{
// reject increment
goto newton_iteration;
}
const PointVal curr_pv0(rb.pv());
if(rb.eval(x0,p)) goto done;
if(rb.pv().a() > .5*curr_pv0.a())
{
// reject increment
//
// the constraint that we always
// tighten the root bracket whenever
// we evaluate the polynomial means
// we cannot completely reject the evaluation
// although we can still refrain from
// incrementing num_roots_in_cluster.
rb.set_to_using_smaller();
goto newton_iteration;
}
else
{
// accept increment
rb.increment_num_roots_in_cluster();
goto increment;
}
}
}
goto newton_iteration;
}
}
newton_bailed:
{
// ideally we should try the secant method
// before reverting to bisection
//
// the secant method should handle repeated roots better
goto secant_bracket_iteration;
}
// the secant rule needs to maintain a list of two
// points that we are "using".
// the root bracket contains the most recently evaluated point.
// The secant method is initialized with a root bracket.
secant_bracket_iteration:
{
dprintf("using secant: low=%24.16e, hgh=%24.16e, p_low=%24.16e, p_hgh=%24.16e",
rb.low().x(),rb.hgh().x(),rb.low().v(),rb.hgh().v());
// rb.set_to_using_smaller();
PointVal pv1(rb.pv_old());
const double start_pva = fmax(pv1.a(),rb.pv().a());
secant_iteration:
// double accepted_num_roots_in_cluster = num_roots_in_cluster;
compute_x0_secant: // for try_num_roots_in_cluster
const double ratio = rb.pv().v()/pv1.v();
compute_secant_rule:
// problem: these numbers can be equal.
//dprint(rb.pv().v());
//dprint(pv1.v());
double x0 = secant_rule(rb.pv().x(),pv1.x(),ratio,
rb.num_roots_in_cluster());
// make sure that x0 is in root bracket
if(x0<rb.low().x() || x0>rb.hgh().x())
{
// different signs would force x0 in root bracket
//assert_gt(ratio, 0.)
// increasing num roots would only make it worse
if(rb.num_roots_in_cluster()>1)
{
rb.decrement_num_roots_in_cluster();
goto compute_secant_rule;
}
//else
//{
// goto bisect;
//}
}
// evaluate at the candidate
dprint(x0);
pv1 = rb.pv(); if(rb.eval(x0,p)) goto done;
if(rb.pv().a() >= pv1.a())
{
dprintf("bisect");
goto bisect;
}
// did residual change signs?
bool sign_changed = ((pv1.v() < 0. && rb.pv().v() > 0.)
|| (pv1.v() > 0. && rb.pv().v() < 0.)) ? true : false;
// if residual changed signs then try decrementing num_roots_in_cluster
// if residual did not change signs then try incrementing.
if(!sign_changed)
{
// if the residual did not shrink by trigger_ratio then
// we will try incrementing num_roots_in_cluster
double secant_trigger = get_trigger_ratio_for_secant(
rb.num_roots_in_cluster());
if(pv1.a() > secant_trigger * rb.pv().a())
{
// try incrementing num_roots_in_cluster
const double ratio = rb.pv().v()/pv1.v();
const double x0 = secant_rule(rb.pv().x(),pv1.x(),ratio,
rb.num_roots_in_cluster()+2);
// make sure root is in interval
if(x0<rb.low().x() || x0>rb.hgh().x())
{
dprintf("bisect");
goto bisect;
}
pv1 = rb.pv(); if(rb.eval(x0,p)) goto done;
if(rb.pv().a() < pv1.a())
{
rb.increment_num_roots_in_cluster();
goto secant_iteration;
}
}
}
else
{
// try decrementing num_roots_in_cluster
if(rb.num_roots_in_cluster()>=3)
{
const double ratio = rb.pv().v()/pv1.v();
const double x0 = secant_rule(rb.pv().x(),pv1.x(),ratio,
rb.num_roots_in_cluster()-2);
// make sure root is in interval
if(x0<rb.low().x() || x0>rb.hgh().x()) goto bisect;
pv1 = rb.pv(); if(rb.eval(x0,p)) goto done;
if(rb.pv().a() < pv1.a())
{
rb.decrement_num_roots_in_cluster();
goto secant_iteration;
}
}
}
secant_iter_done:
{
if(rb.pv().a() > .5*start_pva)
{
dprintf("secant did not help enough: apx=%24.16e, old_apx=%24.16e",
rb.pv().a(), start_pva);
goto bisect;
}
goto secant_bracket_iteration;
}
}
bisect:
{
dprintf("bisecting: x=%24.16e, px=%24.16e",rb.pv().x(),rb.pv().v());
const double x0 = (rb.low().x() + rb.hgh().x())*.5;
if(rb.eval(x0,p)) goto done;
//goto newton_iteration;
goto secant_bracket_iteration;
}
eprintf("should not get here");
return;
}
