Sympletic ray tracing: a new frontier in non-linear ray tracing

Abstract

This paper describes a method of symplectic ray tracing for visualizing non-linear dynamical systems. Symplectic ray tracing is simply an extended version of the ray tracing techniques commonly used to generate computer graphics. However, high performance in analyzing non-linear dynamical systems is achieved by applying Hamiltonian dynamics, symplectic numerical integration, and automatic differentiation. First, symplectic ray tracing calculates a path of light rays from the Hamiltonian. Since the Hamiltonian is a scalar function, the calculation of symplectic ray tracing has no relation to dimensions. Secondly, symplectic numerical integration is suitable for tracing light rays in long term. The example of long-term calculation is a visualization of black hole in the universe. Moreover, since symplectic integration can preserve theoretical invariants of Hamiltonian systems, backward error analysis is possible. Thirdly, Hamilton's canonical equations are constructed automatically because automatic differentiation calculates partial differentials of the Hamiltonian without truncation error. Differentiation by hand or by mathematical software is not required. This paper also demonstrates some visualization results for non-linear optical phenomena such as gravitational lens effects and mirages.