High precision ray tracing in cylindrically symmetric electrostatics

With the recent availability of a high order FDM solution to the curved boundary value problem, it is now possible to determine potentials in such geometries with considerably greater accuracy than had been available with the FDM method. In order for the algorithms used in the accurate potential calculations to be useful in ray tracing, an integration of those algorithms needs to be placed into the ray trace process itself. The object of this paper is to incorporate these algorithms into a solution of the equations of motion of the ray and, having done this, to demonstrate its efficacy. The algorithm incorporation has been accomplished by using power series techniques and the solution constructed has been tested by tracing the medial ray through concentric sphere geometries. The testing has indicated that precisions of ray calculations of 10−20 are now possible. This solution offers a considerable extension to the ray tracing accuracy over the current state of art.