/* find a point of inflection on a bezier curve, if it exists, by finding the
* value of parameter 't' where the signed curvature of the bezier changes
* signs. Returns true if an inflection point is found.
*/
HIDDEN bool
bezier_inflection(const ON_BezierCurve& bezier, fastf_t& inflection_pt)
{
int sign;
fastf_t t, step, crv;
ON_3dVector d1, d2; // first derivative, second derivative
ON_3dPoint tmp; // not used, but needed by Ev2Der
// calculate curvature at t=0
bezier.Ev2Der(0, tmp, d1, d2);
crv = CURVATURE(d1, d2);
// step size decreases as |crv| -> 0
step = GETSTEPSIZE(fabs(crv));
sign = SIGN(crv);
for (t = step; t <= 1.0; t += step) {
bezier.Ev2Der(t, tmp, d1, d2);
crv = CURVATURE(d1, d2);
// if sign changes, t is an inflection point
if (sign != SIGN(crv)) {
inflection_pt = t;
return true;
}
step = GETSTEPSIZE(fabs(crv));
}
return false;
}
/* approximates a bezier curve with a set of circular arcs by dividing where
* the bezier's deviation from its approximating biarc is at a maximum, then
* recursively calling on the subsections until it is approximated to
* tolerance by the biarc
*/
HIDDEN void
approx_bezier(const ON_BezierCurve& bezier, const ON_Arc& biarc, const struct bn_tol *tol, std::vector<ON_Arc>& approx)
{
fastf_t t = 0.0, step = 0.0;
fastf_t crv = 0.0, err = 0.0, max_t = 0.0, max_err = 0.0;
ON_3dPoint test;
ON_3dVector d1, d2;
// walk the bezier curve at interval given by step
for (t = 0; t <= 1.0; t += step) {
bezier.Ev2Der(t, test, d1, d2);
err = fabs((test - biarc.Center()).Length() - biarc.Radius());
// find the maximum point of deviation
if (err > max_err) {
max_t = t;
max_err = err;
}
crv = CURVATURE(d1, d2);
// step size decreases as |crv| -> 1
step = GETSTEPSIZE(1.0 - fabs(crv));
}
if (max_err + VDIVIDE_TOL < tol->dist) {
// max deviation is less than the given tolerance, add the biarc approximation
approx.push_back(biarc);
} else {
ON_BezierCurve head, tail;
// split bezier at point of maximum deviation and recurse on the new subsections
bezier.Split(max_t, head, tail);
approx_bezier(head, make_biarc(head), tol, approx);
approx_bezier(tail, make_biarc(tail), tol, approx);
}
}
