Explicit Constructions for Poncelet Polygons

TL;DR

This paper explores the geometric and algebraic properties of Poncelet polygons, providing explicit constructions and linking them to configurations that were previously considered rigid, thus broadening their potential applications.

Contribution

It introduces explicit constructions of Poncelet n-gons for specific n and connects these to movable (N_4)-configurations, challenging prior assumptions about their rigidity.

Findings

Explicit constructions of Poncelet n-gons for certain n.

Algebraic characterizations using bracket polynomials.

Construction of a large class of movable (N_4)-configurations.

Abstract

We study the geometric structure of Poncelet $n$-gons from a projective point of view. In particular we present explicit constructions of Poncelet $n$-gons for certain $n$ and derive algebraic characterisations in terms of bracket polynomials. Via the connections of Poncelet polygons and $(N_4)$-configurations, the results of this article can be used to construct a large class of specific movable $(N_4)$-configurations, the trivial celestial 4-configurations, which up to this point were all thought to be rigid and to require regular polygons for their construction.