Extremal Area of Polygons sliding along Curves

TL;DR

This paper investigates the extremal area configurations of polygons with vertices sliding along curves, establishing geometric criteria for critical points and introducing a novel inner area billiard system.

Contribution

It provides geometric criteria for critical points of the area function and introduces the concept of inner area billiards, linking Morse theory and polygon configuration topology.

Findings

Derived conditions for extremal polygon areas

Analyzed the Hessian matrix at critical points

Introduced the inner area billiard concept

Abstract

In this paper we study the area function of polygons, where the vertices are sliding along curves. We give geometric criteria for the critical points and determine also the Hesse matrix at those points. This is the starting point for a Morse-theoretic approach, which includes the relation with the topology of the configuration spaces. Moreover the condition for extremal area gives rise to a new type of billiard: the inner area billiard.