Find shortest cyclic path on a cylinder represented by 3d triangular mesh

Question

Suppose I have a 3d triangular mesh with the topology of a finite cylinder. Let C be a vertex on that mesh.

How can I find the shortest path from C to itself that goes around the cylinder? (I don't know the proper terminology).

I suppose that if the cylinder was transformed to a filled annulus, I would be looking for a closed path through C such that every path from the inner ring to the outer ring must intersect it.

However, I'm looking for a practical approach.

Thank you very much.

edit: I have 3d coordinates for each vertex and I'm looking for the shortest Euclidean distance along the mesh.

Answer 1

I've also asked this question on scicomp.stackexchange. I eventually answered it myself. Here's the answer: https://scicomp.stackexchange.com/a/33559/32988