article published yes Arseniy Akopyan author 430D2C90-F248-11E8-B48F-1D18A9856A870000-0002-2548-617X Richard Schwartz author Serge Tabachnikov author HeEd department Alpha Shape Theory Extended project 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. Springer Nature2022 eng 2199-675X 2199-6768 2001.0293410.1007/s40879-020-00426-9 841313-1327 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>. Akopyan A, Schwartz R, Tabachnikov S. 2022. Billiards in ellipses revisited. European Journal of Mathematics. 8(4), 1313–1327. 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> 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>. 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> A. Akopyan, R. Schwartz, S. Tabachnikov, European Journal of Mathematics 8 (2022) 1313–1327. 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. 85382020-09-20T22:01:38Z2025-04-14T07:48:34Z