For a project I'm working on I need to find the minimum distance between two three dimensional catmull rom splines (cubic b-spline defined by at least four control points). At the moment I'm using a very naive algorithm that just recursively finds the closest points on the two curves then refines a new curve (set of points) using the adjacent points as new control points. A few recursions gives the true closest point to enough accuracy but of course if there are multiple closest points it won't find them all.
I'm hoping one of the math experts here can give me some advice on a way to solve this more efficiently.
Frustratingly, I was able to find a reference to a paper which supposedly describes an implementation of a multidimensional Newton's method that can solve this problem but the paper itself is nowhere to be found and I don't understand how Newton's method can be applied here.
Thanks a lot for any help!