/* 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);
}
}
/* approximates a bezier curve with a set of circular arcs.
* returns approximation in carcs
*/
HIDDEN void
bezier_to_carcs(const ON_BezierCurve& bezier, const struct bn_tol *tol, std::vector<ON_Arc>& carcs)
{
bool skip_while = true, curvature_changed = false;
fastf_t inflection_pt, biarc_angle;
ON_Arc biarc;
ON_BezierCurve current, next;
std::vector<ON_BezierCurve> rest;
// find inflection point, if it exists
if (bezier_inflection(bezier, inflection_pt)) {
curvature_changed = true;
bezier.Split(inflection_pt, current, next);
rest.push_back(next);
} else {
current = bezier;
}
while (skip_while || !rest.empty()) {
if (skip_while) skip_while = false;
biarc = make_biarc(current);
if ((biarc_angle = biarc.AngleRadians()) <= M_PI_2) {
// approximate the current bezier segment and add its biarc
// approximation to carcs
approx_bezier(current, biarc, tol, carcs);
} else if (biarc_angle <= M_PI) {
// divide the current bezier segment in half
current.Split(0.5, current, next);
// approximate first bezier segment
approx_bezier(current, biarc, tol, carcs);
// approximate second bezier segment
approx_bezier(next, biarc, tol, carcs);
} else {
fastf_t t = 1.0;
ON_Arc test_biarc;
ON_BezierCurve test_bezier;
// divide the current bezier such that the first curve segment would
// have an approximating biarc segment <=90 degrees
do {
t *= 0.5;
current.Split(t, test_bezier, next);
test_biarc = make_biarc(test_bezier);
} while(test_biarc.AngleRadians() > M_PI_2);
approx_bezier(test_bezier, test_biarc, tol, carcs);
current = next;
skip_while = true;
continue;
}
if (curvature_changed) {
curvature_changed = false;
current = rest.back();
rest.pop_back();
// continue even if we just popped the last element
skip_while = true;
}
}
}
