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

Abstract

Framed polygons and oriented chains of circlesBy the angle between two vectors u and v we mean the angle through which one needs to rotate u to align it with v (with the usual counter-clock orientation of the plane). We denote the angle by ∠(u, v). By this definition,u). The angles are measured mod 2π. The next definition is motivated by a discussion in Introduction, see Figure 1.Definition 1 A framing of an n-gon B 1 , B 2 , . . . is an assignment of unit vectors u 1 , u 2 , . . . to its respective vertices such thatfor all i = 1, . . . , n (here and elsewhere the index is understood cyclically). If (u i ) is a framing then so is (−u i ), and we consider framing up to this involution, acting simultaneously on all vectors u i .