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Billiards in ellipses revisited

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Abstract

We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods.

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Acknowledgements

This paper would not be written if not for Dan Reznik’s curiosity and persistence; we are very grateful to him. We also thank R. Garcia and J. Koiller for interesting discussions. It is a pleasure to thank the Mathematical Institute of the University of Heidelberg for its stimulating atmosphere. ST thanks Misha Bialy for interesting discussions and the Tel Aviv University for its invariable hospitality. We are grateful to the referee for useful suggestions.

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Authors and Affiliations

  1. Institute of Science and Technology, 3400, Klosterneuburg, Austria

    Arseniy Akopyan

  2. Department of Mathematics, Brown University, Providence, RI, 02912, USA

    Richard Schwartz

  3. Department of Mathematics, Penn State University, University Park, PA, 16802, USA

    Serge Tabachnikov

Authors
  1. Arseniy Akopyan
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  2. Richard Schwartz
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  3. Serge Tabachnikov
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Correspondence to Serge Tabachnikov.

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AA was supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha). RS is supported by NSF Grant DMS-1807320. ST was supported by NSF Grants DMS-1510055, DMS-2005444, and SFB/TRR 191.

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Akopyan, A., Schwartz, R. & Tabachnikov, S. Billiards in ellipses revisited. European Journal of Mathematics 8, 1313–1327 (2022). https://doi.org/10.1007/s40879-020-00426-9

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  • DOI: https://doi.org/10.1007/s40879-020-00426-9

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