A curve on surface is a 2d object, it just happens to lie on a surface. now the curve does not have 3d cordinates, but the surface does.
In this specific case use nurbs they give the aswer directly.
But for a polygonal suface this is indeed very tricky since you need to do the iterpolation yourself by finding th uv then tha face associated with taht uv than interpolate the barycentric cordinate of the facce subtriangle and so on. Not a monday task by the way.
