Finding minimum distance between two splines

Question

I have two 3d splines produced by MATLAB:

curve1 = cscvn([0 .5 0; 0 .5 0; 0 .5 1])
curve2 = cscvn([.5 .2 .5; 2 1 0; .1 2 1]);

I want to find the smallest possible distance between any point on curve 1 to any point on curve 2.

Thanks, James

Answer 1

The smallest distance between two parametric curves C1(t) and C2(s) could happen at their interior or at their end points. If it happens at the interior, it will satisfy the following condition:

C1'(t) \dot (C1(t) - C2(s)) = 0
C2'(s) \dot (C1(t) - C2(s)) = 0

This means when the minimum distance happens at t0 and s0, the tangent vector at C1(t0) and the tangent vector at C2(s0) are both perpendicular to the vector defined by C1(t0) and C2(s0). In general, you can use multivariate Newton Raphson method to obtain the parameter t0 and s0. If you do find a solution, you can find the minimum distance.

Please note that above two equations still hold for local maximum. So, you will always need to compare the "minimum distance" obtained to the distances between end points.