Frenet Formulation of Three-Dimensional Ray Tracing

An approach to three-dimensional ray tracing, based on the ray equations and Frenet formulas, is developed. It is applied to two families of constant sound-speed surfaces for which Snell's laws are not known: coaxial circular cylinders and concentric ellipsoids. The classically known results for the cases of parallel planes and concentric spheres are rederived in terms of the present formulation. It is shown that the ray-tracing equations for concentric spheres can be transformed exactly into those for parallel planes. For the coaxial cylinder case, pseudo-Snell's laws can be derived even though the ray paths are not plane curves. For the family of ellipsoids, no Snell's law was found, but ray paths are plane curves. The examples considered do not exhaust the applicability of this ray-tracing approach, but are meant only to illustrate the technique.