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

Given a strictly convex plane curve γ and a point x in its exterior, there are two tangent segments to γ from x. The equitangent locus of γ is the set of points x for which the two tangent segments have equal lengths. An equitangent n-gon of γ is a circumscribed n-gon such that the tangent segments to γ from the vertices have equal lengths, see Figure 1. For example, if γ is a circle, the equitangent locus is the whole exterior of γ (this property is characteristic of circles [20]), and if γ is an ellipse then its equitangent locus consists of the two axes of symmetry (this implies that an ellipse does not admit equitangent n-gons with n 6= 4). Generically, the equitangent locus of a curve γ is also a curve, say, Γ. Note that Γ is not empty; in fact, every curve γ has pairs of equal tangent segments of an arbitrary length. The equitangent problem is to study the relation between a curve and its equitangent locus. For example, if the equitangent locus contains a line tangent to γ, then γ must be a circle. On the other hand, there exists an infinite-dimensional,