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Minimum distance between two catmull rom splines

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.

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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!