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Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(online) ISSN 0273-0979(print)

 

Bicentennial of the Great Poncelet Theorem (1813-2013): Current advances


Authors: Vladimir Dragović and Milena Radnović
Journal: Bull. Amer. Math. Soc. 51 (2014), 373-445
MSC (2010): Primary 37J35, 14H70, 37A05
Published electronically: April 1, 2014
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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.


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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

DOI: http://dx.doi.org/10.1090/S0273-0979-2014-01437-5
PII: S 0273-0979(2014)01437-5
Keywords: Poncelet Theorem, periodic billiard trajectories, pencils of quadrics, relativistic quadrics, integrable line congruences, double reflection nets, pseudo-integrable billiards, interval exchange transformations
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).
Article copyright: © Copyright 2014 American Mathematical Society