Distance from 3d point to an ellipse

I want to calculate the distance from a 3d point (x,y,z) to an ellipse (described by xc,yc,zc,a,b,theta,phi,psi). (xc,yc,zc) is the center of the ellipse and the others are the standard variables from the equation of an ellipse.

The 3d point may not be on the same plane as the ellipse.

A lot of previous work on this problem has been in 2 dimensions.

Summary: Question: How do I find distance from 3d point to an ellipse given (x,y,z,xc,yc,zc,a,b,theta,phi,psi)?

Answer 1

[Alessandro Masullo]

A simple and straightforward solution could be discretizing the points of the ellipse, evaluating the distance from all the points, and then looking for the minimum.

[Mohammad Abouali]

Discretization may work fine, but (1) requires more memory, (2) you are as accurate as your discretization, and (3) you end up generally do more calculation.

An extra note on discretization, the finer you make the discretization (in order to have better accuracy) you need more memory and more computation. Since this is 3D problem the increase in amount of memory and computation is scaled by power 3; meaning if you make your resolution half (divide by 2) then you need 8 times (2^3) more work and memory.

[browser]

Thanks - I had considered this but as has been mentioned the amount of computation becomes very large. I have this method implemented currently.