Please use this identifier to cite or link to this item:
http://hdl.handle.net/10773/21278
Title: | Billiard transformations of parallel flows: a periscope theorem |
Author: | Plakhov, Alexander Tabachnikov, Sergey Treschev, Dmitry |
Keywords: | Billiards Freeform surfaces Imaging Geometrical optics |
Issue Date: | May-2017 |
Publisher: | Elsevier |
Abstract: | We consider the following problem: given two parallel and identically oriented bundles of light rays in R^{n+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 R^3, 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. |
Peer review: | yes |
URI: | http://hdl.handle.net/10773/21278 |
DOI: | 10.1016/j.geomphys.2016.04.006 |
ISSN: | 0393-0440 |
Appears in Collections: | CIDMA - Artigos OGTCG - Artigos |
Files in This Item:
File | Description | Size | Format | |
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ParFlows5.pdf | documento principal | 162.12 kB | Adobe PDF | View/Open |
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