Billiard transformations of parallel flows: A periscope theorem

JOURNAL OF GEOMETRY AND PHYSICS

卷:115

Article

Plakhov, Alexander^1,2  Tabachnikov, Serge³  Treschev, Dmitry^4,5 

[1] Univ Aveiro, Ctr R&D Math & Applicat, Dept Math, Aveiro, Portugal

[2] Inst Informat Transmiss Problems, Moscow, Russia

[3] Penn State Univ, Dept Math, University Pk, PA 16802 USA

[4] Russian Acad Sci, Steklov Math Inst, Moscow, Russia

[5] Lomonosov Moscow State Univ, Moscow, Russia

关键词: Billiards;    Freeform surfaces;    Geometrical optics;    Imaging;   

DOI  :  10.1016/j.geomphys.2016.04.006

来源: Elsevier

【 摘 要 】

We consider the following problem: given two parallel and identically oriented bundles of light rays in Rn+1 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 R-2 is locally the composition of two gradient diffeomorphisms, and therefore can be realized by 4 mirror reflections of light rays in IR3, 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 R-3 can be realized by 6 or 7 reflections. (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

Free