Extremal Area of Polygons, sliding along a Circle

TL;DR

This paper characterizes critical configurations of polygons inscribed in circles or ellipses, analyzing their Morse indices and eigenvalue relations, revealing complex singularities like zigzag trains in even dimensions.

Contribution

It provides a comprehensive classification of critical points for the area function on inscribed polygons, including degenerate cases and their Morse indices.

Findings

Identifies all critical configurations for polygons on circles and ellipses.

Computes Morse indices for isolated critical points.

Describes non-isolated singularities as zigzag trains in even dimensions.

Abstract

We determine all critical configurations for the Area function on polygons with vertices on a circle or an ellipse. For isolated critical points we compute their Morse index, resp index of the gradient vector field. We relate the computation at an isolated degenerate point to an eigenvalue question about combinations. In the even dimensional case non-isolated singularities occur as `zigzag trains'.