Baking the distance from a point along faces

So I'm looking for a way to create what I would call a distance map. To explain what I mean imagine a plane that is UV mapped 1:1 to fill up the whole uv space. Below are two examples of distance maps where the red dot is where the distance is measured from. The actual maps wouldn't have the red dot. So the map is a white-black measure of distance along the face of the plane from the given point in red.

Making this map is simple do do for a plane, but I need something like this for a complex mesh. I want to specify a point in uv space and then measure the distance from that point for the whole mesh. I don't want the distance through space. I need the distance to be calculated by running along the meshes faces. Any ideas? If my explanation didn't make sense I'll try to clarify more if you let me know. It's sort of hard to explain.