In this report, we present an efficient algorithm for finding the shortest path between two points along a sequence of triangles on some polyhedral surface, using the idea of Method of Orienting Curves, here the meaning of "curves" is "geodesics". In the framework of this method, the concepts of "final geodesics'' and "orienting geodesics'' (the special cases of straightest geodesics, modified from the concept straightest geodesics on polyhedral surfaces of Polthier and Schmies) are introduced. Then, the shortest path between two points along the sequence of triangles can be determined without planar unfolding technique. It takes $O(n^2)$ time, where $n$ is the number of edges of the sequence, and it is visualized by JavaView software.
This is a joint work with P. T. An, H. X. Phu and K. Polthier.