Bicentennial of the Great Poncelet Theorem (1813–2013): Current advances
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- by Vladimir Dragović and Milena Radnović PDF
- Bull. Amer. Math. Soc. 51 (2014), 373-445 Request permission
Abstract:
We present very recent results related to the Poncelet Theorem on the occasion of its bicentennial. We are telling the story of one of the most beautiful theorems of geometry, recalling for general mathematical audiences the dramatic historic circumstances which led to its discovery, a glimpse of its intrinsic appeal, and the importance of its relationship to dynamics of billiards within confocal conics. We focus on the three main issues: A) The case of pseudo-Euclidean spaces, for which we present a recent notion of relativistic quadrics and apply it to the description of periodic trajectories of billiards within quadrics. B) The relationship between so-called billiard algebra and the foundations of modern discrete differential geometry which leads to double-reflection nets. C) We present a new class of dynamical systems—pseudo-integrable billiards generated by a boundary composed of several arcs of confocal conics having nonconvex angles. The dynamics of such billiards have several extraordinary properties, which are related to interval exchange transformations and which generate families of flows that are minimal but not uniquely ergodic. This type of dynamics provides a novel type of Poncelet porisms—the local ones.References
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Additional Information
- Vladimir Dragović
- Affiliation: Department of Mathematical Sciences, University of Texas at Dallas, Dallas, Texas; Mathematical Institute SANU, Kneza Mihaila 36, Belgrade, Serbia
- Email: vladimir.dragovic@utdallas.edu
- Milena Radnović
- Affiliation: Mathematical Institute SANU, Kneza Mihaila 36, Belgrade, Serbia
- Email: milena@mi.sanu.ac.rs
- Received by editor(s): November 1, 2012
- Received by editor(s) in revised form: March 25, 2013
- Published electronically: April 1, 2014
- Additional Notes: The research which led to this paper was partially supported by the Serbian Ministry of Education and Science (Project no. 174020: Geometry and Topology of Manifolds and Integrable Dynamical Systems).
- © Copyright 2014 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 51 (2014), 373-445
- MSC (2010): Primary 37J35, 14H70, 37A05
- DOI: https://doi.org/10.1090/S0273-0979-2014-01437-5
- MathSciNet review: 3196793