Configuration spaces of plane polygons and a sub-Riemannian approach to the equitangent problem

TL;DR

This paper explores the geometric properties of convex plane curves and their equitangent loci using sub-Riemannian geometry, revealing new insights into the structure of equitangent polygons and their relation to the billiard problem.

Contribution

It introduces a sub-Riemannian geometric framework to analyze equitangent polygons and their families, connecting these to the Birkhoff distribution and billiard dynamics.

Findings

Characterization of equitangent polygons via sub-Riemannian distributions

Identification of conditions for 1-parameter families of equitangent polygons

Relation between equitangent loci and billiard ball problem dynamics

Abstract

The equitangent locus of a convex plane curve consists of the points from which the two tangent segments to the curve have equal length. The equitangent problem concerns the relation between the curve and its equitangent locus. An equitangent n-gon of a convex curve is a circumscribed n-gon whose vertices belong to the equitangent locus. We are interested in curves that admit 1-parameter families of equitangent n-gons. We use methods of sub-Riemannian geometry: we define a distribution on the space of polygons and study its bracket generating properties. 1-parameter families of equitangent polygons correspond to the curves, tangent to this distribution. This distribution is closely related with the Birkhoff distribution on the space of plane polygons with a fixed perimeter length whose study, in the framework of the billiard ball problem, was pioneered by Yu. Baryshnikov, V. Zharnitsky and J. Landsberg