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arXiv · 2112.08124

A family of integrable transformations of centroaffine polygons: geometrical aspects

M. Arnold·D. Fuchs·S. Tabachnikov
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Abstract

Two polygons, $(P_1,\ldots,P_n)$ and $(Q_1,\ldots,Q_n)$ in ${\mathbb R}^2$ are $c$-related if $\det(P_i, P_{i+1})=\det(Q_i, Q_{i+1})$ and $\det(P_i, Q_i)=c$ for all $i$. This relation extends to twisted polygons (polygons with monodromy), and it descends to the moduli space of $SL(2,{\mathbb R}^2)$-equivalent polygons. This relation is an equiaffine analog of the discrete bicycle correspondence studied by a number of authors. We study the geometry of this relations, present its integrals, and show that, in an appropriate sense, these relations, considered for different values of the constants $c$, commute. We relate this topic with the dressing chain of Veselov and Shabat. The case of small-gons is investigated in detail.

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M. Arnold, D. Fuchs, S. Tabachnikov. 2021-12-15. A family of integrable transformations of centroaffine polygons: geometrical aspects. https://arxiv.org/abs/2112.08124

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