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

TL;DR

This paper investigates the configuration space of polygons with fixed edge slopes, analyzing the perimeter function as a Morse function, characterizing critical points, and relating findings to known results about area functions in polygon spaces.

Contribution

It introduces a detailed study of the perimeter function on polygons with prescribed edge slopes, including critical point characterization and Morse index computation, extending previous work on area functions.

Findings

Characterization of critical points as tangential polygons

Computation of Morse indices for these critical points

Connection to existing results on area function Morse indices

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 \textit{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 G. Panina and A. Zhukova).