Geodesics have a wide range of applications in computer graphics, geometric modeling and machine learning, to name just a few. Current research on geodesic computation focuses primarily on parametric surfaces and mesh representations. There is little work on implicit surfaces. In this paper, we present an algorithm able to compute the exact geodesics on implicit surfaces. Although the existing Fast Marching Method can generate a geodesic path on a Cartesian grid that envelopes the implicit surface in question, this method, as well as other existing methods, is unable to compute a geodesic on the original surface. The computed geodesic path is actually a polyline offsetting from the surface. Our research presented in this paper provides a solution to this fundamental problem, that is, to formulate exact geodesics on the original implicit surface itself. In addition, in comparison with other existing work, our methods can handle both low dimensional and high dimensional surfaces (hyper-surfaces).

geodesics, implicit surface, geodesic curvature flow.