Billiard transformations of parallel flows: A periscope theorem

Alexander Plakhov, Serge Tabachnikov, Dmitry Treschev

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


We consider the following problem: given two parallel and identically oriented bundles of light rays in Rn+1and 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 R2is locally the composition of two gradient diffeomorphisms, and therefore can be realized by 4 mirror reflections of light rays in R3, 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 R3can be realized by 6 or 7 reflections.

Original languageEnglish (US)
Pages (from-to)157-166
Number of pages10
JournalJournal of Geometry and Physics
StatePublished - May 1 2017

All Science Journal Classification (ASJC) codes

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Geometry and Topology


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