Geodesic Computation on Implicit Surfaces

Geodesics have a wide range of applications in CAD, shape design and machine learning. 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 a novel 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 approach provides a solution to two existing fundamental problems, which are (1) to produce a Cartesian grid that can tightly embed the implicit surface concerned, which remains challenging; and (2) to formulate exact geodesics on the original implicit surface itself. Our algorithm consists of two steps, Cartesian grid based geodesic computation and geodesic correction. The later corrects an approximate geodesic path so that it can be on the implicit surface. In addition, in comparison with other existing work, our methods can handle both low dimensional and high dimensional surfaces (hyper-surfaces). HONGCHUAN YU and JIAN J. ZHANG 2