A unified framework for computing geodesic distances on nonorientable manifold polyhedral surfaces

Abstract

Nonorientable manifold surfaces not only play an important role in topology, but also have numerous applications in many topics such as visualization and computation of minimal surfaces. From the topological point of view, a 2-manifold surface is locally homeomorphic to an open disk. This property is independent of the global orientability. However, as far as the discrete representation is concerned, orientable manifold surfaces are usually discretized with halfedge data structure, while nonorientable surfaces are discretized into polygon soups, which is inconvenient for digital geometry processing that often takes orientable meshes as input. In this paper, we propose a uni ed framework for transforming geodesic distance problems de ned on nonorientable 2-manifold meshes to the counter-parts on orientable surfaces, and thereby bridging up nonorientable 2-manifold meshes and conventional geometric algorithms. In order to illustrate the universal adaptability, we apply this new approach to study three problems on nonorientable meshes, including computing exact geodesic paths, discrete exponential mapping and farthest point sampling.