A Jacobian Group Structure on a Hyperbolic Pencil of circles and its Applications
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866913152806944768 |
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| author | Abi-Khuzam, Faruk F. |
| author_facet | Abi-Khuzam, Faruk F. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_22515 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Jacobian Group Structure on a Hyperbolic Pencil of circles and its Applications Abi-Khuzam, Faruk F. Complex Variables 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. |
| title | A Jacobian Group Structure on a Hyperbolic Pencil of circles and its Applications |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2605.22515 |