Billiard transformations of parallel flows: a periscope theorem

TL;DR

This paper investigates the conditions under which diffeomorphisms of light ray bundles can be realized through mirror reflections, establishing precise bounds on the number of reflections needed based on the nature of the diffeomorphism.

Contribution

It provides a characterization of when certain light ray transformations can be achieved with a limited number of mirror reflections, linking geometric diffeomorphisms to physical mirror arrangements.

Findings

2-mirror realization iff the diffeomorphism is a gradient

Orientation reversing diffeomorphisms can be realized with 4 reflections

Orientation preserving diffeomorphisms require 6 reflections

Abstract

We consider the following problem: given two parallel and identically oriented bundles of light rays in n-dimensional Euclidean space and given a diffeomorphism between the rays of the former bundle and the rays of the latter one, is it possible to realize this diffeomorphism by means of several mirror reflections? We prove that a 2-mirror realization is possible if and only if the diffeomorphism is the gradient of a function. We further prove that any orientation reversing diffeomorphism of domains in the plane is locally the composition of two gradient diffeomorphisms, and therefore can be realized by 4 mirror reflections of light rays in 3-space, while an orientation preserving diffeomorphism can be realized by 6 reflections. In general, we prove that an (orientation reversing or preserving) diffeomorphism of wave fronts of two normal families of light rays in 3-space can be realized by 6 or 7 reflections.