Computing geodesic paths on triangulated manifolds

Solving the Eikonal equation on a surface allows establishment of a curvilinear coordinate system which facilitates many types of physics-based simulations and measurements. Existing methods that solve the Eikonal equation on surface triangulations suffer from significant limitations prohibiting their use in many applications. First, the surface may not have any holes. Second, every obtuse triangle in the tessellation must have a neighboring triangle positioned so that partial solution to the equation can be propagated to the obtuse triangle through the temporary addition of a virtual triangle to the mesh. Since these conditions are typically not satisfied in triangulations generated for many real world applications ranging from computer graphics, computer vision and geometric modeling, we provide an optimal time algorithm that removes these requirements. Our algorithm solves the Eikonal equation uniformly for triangulations with or without holes and it places no requirements on obtuse triangles in the mesh. We construct partial solutions in the form of parameter functions along surface elements and extend the Fast Marching Method for the numeric solution of the Eikonal equation. As an application, we apply the Eikonal solution to compute the geodesic distances on such domains and thereby extract the shortest paths on these surfaces Modeling Curves & Surfaces, Modeling Computational Geometry, Modeling 3D Shape Matching