Symplectic Ray Tracing:
A New Frontier in Nonlinear Ray Tracing
Tetsu R. Satoh
Communications Research Laboratory (CRL)
Information and
Network Systems Division
6190237 Kyoto
Japan
email:tetus@crl.go.jp

http://www2.crl.go.jp/jt/a135/tetus/ 
Keywords: Symplectic Integration, Automatic
Differentiation, Nonlinear Ray Tracing
Abstract
This paper describes a method of symplectic ray tracing for visualizing
nonlinear 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
nonlinear 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 longterm 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 nonlinear optical phenomena such as
gravitational lens effects and mirages.