Polygons with prescribed edge slopes: configuration space and extremal points of perimeter

Abstract

We describe the configuration space $$\mathbf {S}$$ of polygons with prescribed edge slopes, and study the perimeter $${\mathcal {P}}$$ as a Morse function on $$\mathbf {S}$$ . We characterize critical points of $${\mathcal {P}}$$ (these are tangential polygons) and compute their Morse indices. This setup is motivated by a number of results about critical points and Morse indices of the oriented area function defined on the configuration space of polygons with prescribed edge lengths (flexible polygons). As a by-product, we present an independent computation of the Morse index of the area function (obtained earlier by Panina and Zhukova).