---
_id: '8538'
abstract:
- lang: eng
  text: 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.
acknowledgement: " 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 M. Bialy
  for interesting discussions and the Tel Aviv\r\nUniversity for its invariable hospitality.
  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 grant DMS-1510055 and SFB/TRR
  191."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Arseniy
  full_name: Akopyan, Arseniy
  id: 430D2C90-F248-11E8-B48F-1D18A9856A87
  last_name: Akopyan
  orcid: 0000-0002-2548-617X
- first_name: Richard
  full_name: Schwartz, Richard
  last_name: Schwartz
- first_name: Serge
  full_name: Tabachnikov, Serge
  last_name: Tabachnikov
citation:
  ama: Akopyan A, Schwartz R, Tabachnikov S. Billiards in ellipses revisited. <i>European
    Journal of Mathematics</i>. 2022;8(4):1313-1327. doi:<a href="https://doi.org/10.1007/s40879-020-00426-9">10.1007/s40879-020-00426-9</a>
  apa: Akopyan, A., Schwartz, R., &#38; Tabachnikov, S. (2022). Billiards in ellipses
    revisited. <i>European Journal of Mathematics</i>. Springer Nature. <a href="https://doi.org/10.1007/s40879-020-00426-9">https://doi.org/10.1007/s40879-020-00426-9</a>
  chicago: Akopyan, Arseniy, Richard Schwartz, and Serge Tabachnikov. “Billiards in
    Ellipses Revisited.” <i>European Journal of Mathematics</i>. Springer Nature,
    2022. <a href="https://doi.org/10.1007/s40879-020-00426-9">https://doi.org/10.1007/s40879-020-00426-9</a>.
  ieee: A. Akopyan, R. Schwartz, and S. Tabachnikov, “Billiards in ellipses revisited,”
    <i>European Journal of Mathematics</i>, vol. 8, no. 4. Springer Nature, pp. 1313–1327,
    2022.
  ista: Akopyan A, Schwartz R, Tabachnikov S. 2022. Billiards in ellipses revisited.
    European Journal of Mathematics. 8(4), 1313–1327.
  mla: Akopyan, Arseniy, et al. “Billiards in Ellipses Revisited.” <i>European Journal
    of Mathematics</i>, vol. 8, no. 4, Springer Nature, 2022, pp. 1313–27, doi:<a
    href="https://doi.org/10.1007/s40879-020-00426-9">10.1007/s40879-020-00426-9</a>.
  short: A. Akopyan, R. Schwartz, S. Tabachnikov, European Journal of Mathematics
    8 (2022) 1313–1327.
date_created: 2020-09-20T22:01:38Z
date_published: 2022-12-01T00:00:00Z
date_updated: 2025-04-14T07:48:34Z
day: '01'
department:
- _id: HeEd
doi: 10.1007/s40879-020-00426-9
ec_funded: 1
external_id:
  arxiv:
  - '2001.02934'
intvolume: '         8'
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2001.02934
month: '12'
oa: 1
oa_version: Preprint
page: 1313-1327
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '788183'
  name: Alpha Shape Theory Extended
publication: European Journal of Mathematics
publication_identifier:
  eissn:
  - 2199-6768
  issn:
  - 2199-675X
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Billiards in ellipses revisited
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 8
year: '2022'
...