A Jacobian Group Structure on a Hyperbolic Pencil of circles and its Applications

TL;DR

The paper introduces a new group structure on a hyperbolic pencil of coaxal circles using Jacobian elliptic functions, leading to novel proofs of classical theorems like Poncelet's closure theorem.

Contribution

It presents a novel parametrization revealing a group structure on coaxal circles, enabling new proofs of Poncelet's theorems and related geometric properties.

Findings

Established a group structure on a hyperbolic pencil of circles.

Provided a new proof of Poncelet's closure theorem.

Characterized interscribed polygons via group elements.

Abstract

Using Jacobian Elliptic functions, we introduce a novel parametrization of a hyperbolic pencil of coaxal circles which reveals a remarkable group structure on the pencil. The geometric properties of the group elements lead to a new proof of of the general Poncelet theorems, which in turn leads to a proof of the so called closure theorem. In particular we prove: if $T$ and $% D $ are members of the pencil, then an interscribed $n$-gon to $T$ and $D$ exists, if and only if $D$, the inside circle, is an element of order $n$ in the group.