# Distance between two points on 3D Triangular Mesh Manifold

I'm working on my bachelor thesis (on Computer Science) and right now I'm having a problem about finding shortest path between two points on 3D triangular mesh that is manifold. I already read about MMP, but which computes distance function $$d(x)$$ between given point and vertex $$x$$ on mesh.

I got to know that the problem I'm solving is named Geodesics but What I really couldn't find is some good algorithm which uses A* for finding shortest path between two given points on two given vertices.

I 'invented' also algorithm which uses A* by using Euclidian Distance Heuristics and correction after finding new Point on any Edge.. I also have edges saved in half-edge structure.

So my main idea is this:

1. We will find closest edge by A* algorithm and find on this edge point with minimalizing function $$f(x) + g(x)$$ where $$f$$ is our current distance and $$g$$ is heuristics(euclidean distance)
2. Everytime we find new edge, we will unfold current mesh and find closest path to our starting point

So now my questions: