Poncelet pairs of a circle and parabolas from a confocal family and Painlevé VI equations
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866914378890084352 |
|---|---|
| author | Dragović, Vladimir Murad, Mohammad Hassan |
| author_facet | Dragović, Vladimir Murad, Mohammad Hassan |
| contents | We study pairs of conics $(\mathcal{D},\mathcal{P})$, called \textit{$n$-Poncelet pairs}, such that an $n$-gon, called an \textit{$n$-Poncelet polygon}, can be inscribed into $\mathcal{D}$ and circumscribed about $\mathcal{P}$. Here $\mathcal{D}$ is a circle and $\mathcal{P}$ is a parabola from a confocal pencil $\mathcal{F}$ with the focus $F$. We prove that the circle contains $F$ if and only if every parabola $\mathcal{P}\in\mathcal{F}$ forms a $3$-Poncelet pair with the circle. We prove that the center of $\mathcal{D}$ coincides with $F$ if and only if every parabola $\mathcal{P}\in \mathcal{F}$ forms a $4$-Poncelet pair with the circle. We refer to such property, observed for $n=3$ and $n=4$, as \textit{$n$-isoperiodicity}. We prove that $\mathcal{F}$ is not $n$-isoperiodic with any circle $\mathcal{D}$ for $n$ different from $3$ and $4$. Using isoperiodicity, we construct explicit algebraic solutions to Painlevé VI equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10795 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Poncelet pairs of a circle and parabolas from a confocal family and Painlevé VI equations Dragović, Vladimir Murad, Mohammad Hassan Classical Analysis and ODEs Algebraic Geometry Dynamical Systems Metric Geometry 14H70, 34M55, 37J70, 37A10, 51N20 We study pairs of conics $(\mathcal{D},\mathcal{P})$, called \textit{$n$-Poncelet pairs}, such that an $n$-gon, called an \textit{$n$-Poncelet polygon}, can be inscribed into $\mathcal{D}$ and circumscribed about $\mathcal{P}$. Here $\mathcal{D}$ is a circle and $\mathcal{P}$ is a parabola from a confocal pencil $\mathcal{F}$ with the focus $F$. We prove that the circle contains $F$ if and only if every parabola $\mathcal{P}\in\mathcal{F}$ forms a $3$-Poncelet pair with the circle. We prove that the center of $\mathcal{D}$ coincides with $F$ if and only if every parabola $\mathcal{P}\in \mathcal{F}$ forms a $4$-Poncelet pair with the circle. We refer to such property, observed for $n=3$ and $n=4$, as \textit{$n$-isoperiodicity}. We prove that $\mathcal{F}$ is not $n$-isoperiodic with any circle $\mathcal{D}$ for $n$ different from $3$ and $4$. Using isoperiodicity, we construct explicit algebraic solutions to Painlevé VI equations. |
| title | Poncelet pairs of a circle and parabolas from a confocal family and Painlevé VI equations |
| topic | Classical Analysis and ODEs Algebraic Geometry Dynamical Systems Metric Geometry 14H70, 34M55, 37J70, 37A10, 51N20 |
| url | https://arxiv.org/abs/2501.10795 |