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

TL;DR

This paper reviews recent advances related to the Poncelet Theorem, exploring its extensions in pseudo-Euclidean spaces, connections to discrete differential geometry, and introducing new pseudo-integrable billiard systems with complex dynamical properties.

Contribution

It presents novel concepts such as relativistic quadrics, the link between billiard algebra and discrete geometry, and a new class of pseudo-integrable billiards with unique dynamical behaviors.

Findings

Introduction of relativistic quadrics for billiard trajectories

Connection between billiard algebra and discrete differential geometry

Discovery of pseudo-integrable billiards with nonconvex confocal arcs

Abstract

The paper gives a review of 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 the general mathematical audience the dramatic historic circumstances which led to its discovery, a glimpse of its intrinsic appeal, and importance of its relationship to the dynamics of billiards within confocal conics. We focus on the three main issues: A) The case of Pseudo-Euclidean spaces, presenting a recent notion of relativistic quadrics, and applying it to the description of periodic trajectories of billiards within quadrics. B) The relationship between so-called billiard algebra and foundations of modern discrete differential geometry which leads to the Double-reflection nets. C) We introduce a new class of dynamical systems -- pseudo-integrable billiards generated by the boundary composed of several arcs of confocal conics having nonconvex angles. The dynamics of such billiards has several extraordinary properties. They are related to the interval exchange transformations and generate families of flows which are minimal but not uniquely ergodic. This type of dynamics provides a novel type of the Poncelet porisms -- the local ones.