  We present the foundations of a new approach to the Casimir effect based on
classical ray optics. We show that a very useful approximation to the Casimir
force between arbitrarily shaped smooth conductors can be obtained from
knowledge of the paths of light rays that originate at points between these
bodies and close on themselves. Although an approximation, the optical method
is exact for flat bodies, and is surprisingly accurate and versatile. In this
paper we present a self-contained derivation of our approximation, discuss its
range of validity and possible improvements, and work out three examples in
detail. The results are in excellent agreement with recent precise numerical
analysis for the experimentally interesting configuration of a sphere opposite
an infinite plane.
