Configuration Spaces of Plane Polygons and a sub-Riemannian Approach to the Equitangent Problem

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 relationship 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. The 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.