/// Uses 2D Newton's method using stereographic parametrization and fancy initial guess
/// Pass:
/// abc - coefficients of the ellipse equation; aka the half-extents
/// p - point in ellipse coordinates
public static Vector3 ClosestPointEllipsoid(
Vector3 abc_, Vector3 p_, int numIters = 30, double threshold = 0)
{
// Doesn't like negative scales & result is the same by symmetry!
vec3 abc = vec3.abs(new vec3(abc_));
// WLOG move to the all-positive octant; move back to original octant when done.
vec3 orig_p = new vec3(p_);
vec3 p = vec3.abs(orig_p);
Stereo uv = Stereo.FromCartesian(InitialGuessClosestPointToEllipsoid(p, abc), abc);
for (int i = 0; i < numIters; ++i)
{
Stereo prevUv = uv;
uv = Iterate_Stereographic(uv, p, abc);
if (Math.Abs(prevUv.x - uv.x) < threshold ||
Math.Abs(prevUv.y - uv.y) < threshold)
{
break;
}
}
// Move back to correct octant
vec3 wrongOctant = uv.ToCartesian(abc);
vec3 rightOctant = wrongOctant * orig_p.sign();
return((Vector3)rightOctant);
}
