TY - JOUR

T1 - Billiard transformations of parallel flows

T2 - A periscope theorem

AU - Plakhov, Alexander

AU - Tabachnikov, Serge

AU - Treschev, Dmitry

N1 - Funding Information:
The first author was supported by Portuguese funds through CIDMA ? Center for Research and Development in Mathematics and Applications, and FCT ? Portuguese Foundation for Science and Technology, within the project UID/MAT/04106/2013, as well as by the FCT research project PTDC/MAT/113470/2009. The second author was supported by the NSF grants DMS-1105442 and DMS-1510055. The third author was supported by the grant RFBR ? 15-01-03747.
Publisher Copyright:
© 2016 Elsevier B.V.

PY - 2017/5/1

Y1 - 2017/5/1

N2 - 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.

AB - 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.

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U2 - 10.1016/j.geomphys.2016.04.006

DO - 10.1016/j.geomphys.2016.04.006

M3 - Article

AN - SCOPUS:85028236472

VL - 115

SP - 157

EP - 166

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

ER -