Abstract: The computation of exact geodesics on triangle meshes is a widely used operation in computer-aided design and computer graphics. Practical algorithms for computing such exact geodesics have been recently proposed by Surazhsky et al. [5]. By applying these geometric algorithms to real-world data, degenerate cases frequently appear. In this paper we classify and enumerate all the degenerate cases in a systematic way. Based on the classification, we present solutions to handle all the degenerate cases consistently and correctly. The common users may find the present techniques useful when they implement a robust code of computing exact geodesic paths on meshes.