/* 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);
}
}
static ON_BOOL32 NurbsCurveArc ( const ON_Arc& arc, int dim, ON_NurbsCurve& nurb )
{
if ( !arc.IsValid() )
return false;
// makes a quadratic nurbs arc
const ON_3dPoint center = arc.Center();
double angle = arc.AngleRadians();
ON_Interval dom = arc.DomainRadians();
const double angle0 = dom[0];
const double angle1 = dom[1];
ON_3dPoint start_point = arc.StartPoint();
//ON_3dPoint mid_point = arc.PointAt(angle0 + 0.5*angle);
ON_3dPoint end_point = arc.IsCircle() ? start_point : arc.EndPoint();
ON_4dPoint CV[9];
double knot[10];
double a, b, c, w, winv;
double *cv;
int j, span_count, cv_count;
a = (0.5 + ON_SQRT_EPSILON)*ON_PI;
if (angle <= a)
span_count = 1;
else if (angle <= 2.0*a)
span_count = 2;
else if (angle <= 3.0*a)
span_count = 4; // TODO - make a 3 span case
else
span_count = 4;
cv_count = 2*span_count + 1;
switch(span_count) {
case 1:
CV[0] = start_point;
CV[1] = arc.PointAt(angle0 + 0.50*angle);
CV[2] = end_point;
break;
case 2:
CV[0] = start_point;
CV[1] = arc.PointAt(angle0 + 0.25*angle);
CV[2] = arc.PointAt(angle0 + 0.50*angle);
CV[3] = arc.PointAt(angle0 + 0.75*angle);
CV[4] = end_point;
angle *= 0.5;
break;
default: // 4 spans
CV[0] = start_point;
CV[1] = arc.PointAt(angle0 + 0.125*angle);
CV[2] = arc.PointAt(angle0 + 0.250*angle);
CV[3] = arc.PointAt(angle0 + 0.375*angle);
CV[4] = arc.PointAt(angle0 + 0.500*angle);
CV[5] = arc.PointAt(angle0 + 0.625*angle);
CV[6] = arc.PointAt(angle0 + 0.750*angle);
CV[7] = arc.PointAt(angle0 + 0.875*angle);
CV[8] = end_point;
angle *= 0.25;
break;
}
a = cos(0.5*angle);
b = a - 1.0;
//c = (radius > 0.0) ? radius*angle : angle;
c = angle;
span_count *= 2;
knot[0] = knot[1] = angle0; //0.0;
for (j = 1; j < span_count; j += 2) {
CV[j].x += b * center.x;
CV[j].y += b * center.y;
CV[j].z += b * center.z;
CV[j].w = a;
CV[j+1].w = 1.0;
knot[j+1] = knot[j+2] = knot[j-1] + c;
}
knot[cv_count-1] = knot[cv_count] = angle1;
for ( j = 1; j < span_count; j += 2 ) {
w = CV[j].w;
winv = 1.0/w;
a = CV[j].x*winv;
b = ArcDeFuzz(a);
if ( a != b ) {
CV[j].x = b*w;
}
a = CV[j].y*winv;
b = ArcDeFuzz(a);
if ( a != b ) {
CV[j].y = b*w;
}
a = CV[j].z*winv;
b = ArcDeFuzz(a);
if ( a != b ) {
CV[j].z = b*w;
}
}
nurb.m_dim = (dim==2) ? 2 : 3;
nurb.m_is_rat = 1;
nurb.m_order = 3;
nurb.m_cv_count = cv_count;
nurb.m_cv_stride = (dim==2 ? 3 : 4);
nurb.ReserveCVCapacity( nurb.m_cv_stride*cv_count );
nurb.ReserveKnotCapacity( cv_count+1 );
for ( j = 0; j < cv_count; j++ ) {
cv = nurb.CV(j);
cv[0] = CV[j].x;
cv[1] = CV[j].y;
if ( dim == 2 ) {
cv[2] = CV[j].w;
}
else {
cv[2] = CV[j].z;
cv[3] = CV[j].w;
}
nurb.m_knot[j] = knot[j];
}
nurb.m_knot[cv_count] = knot[cv_count];
return true;
}