Minimum distance between ellipses

[Coelum]

Forum,
I'm addressing the problem of computing the minimum possible distance between two non-interacting bodies on elliptical orbits. From a general point of view, it looks like a minimization problem of a function of two variables, e.g. in the domain [0,2*pi)*[0,2*pi). This problem can be numerically addressed in a standard fashion, e.g. by a conjugate gradient method. But I wonder if an analytical approach exists that can simplify the problem - maybe reducing it to unidimensional - and significantly speed-up the computation.
I have posted the some question on the Celestial Mechanics Forum - no reply so far.

 

[philosophking]

If the path is ellptical, wouldn't it just be a straight line representing the minor axis of the ellipse?

 

[Coelum]

I guess I've not been clear enough: I'm dealing with two ellipses (e.g., representing the paths of two minor planets around the sun) in 3D. Of course, you may assume that they share one of the foci.