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arXiv · 2605.22515

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

Faruk F. Abi-Khuzam
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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.

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Faruk F. Abi-Khuzam. 2026-05-21. A Jacobian Group Structure on a Hyperbolic Pencil of circles and its Applications. https://arxiv.org/abs/2605.22515

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