New Invariants of Poncelet-Jacobi Bicentric Polygons

TL;DR

This paper discovers new invariants in Poncelet-Jacobi bicentric polygons, showing they have constant sums of internal angle cosines and pedal polygons with invariant perimeter, linking these properties to elliptic billiard N-periodics.

Contribution

It introduces novel invariants for bicentric polygons and connects these to elliptic billiard properties, expanding understanding of geometric invariants in polygonal and billiard systems.

Findings

Invariant sum of internal angle cosines in bicentric polygons

Invariant perimeter of pedal polygons related to the family

Elliptic billiard N-periodics share these invariants

Abstract

The 1d family of Poncelet polygons interscribed between two circles is known as the Bicentric family. Using elliptic functions and Liouville's theorem, we show (i) that this family has invariant sum of internal angle cosines and (ii) that the pedal polygons with respect to the family's limiting points have invariant perimeter. Interestingly, both (i) and (ii) are also properties of elliptic billiard N-periodics. Furthermore, since the pedal polygons in (ii) are identical to inversions of elliptic billiard N-periodics with respect to a focus-centered circle, an important corollary is that (iii) elliptic billiard focus-inversive N-gons have constant perimeter. Interestingly, these also conserve their sum of cosines (except for the N=4 case).