On projective evolutes of polygons

TL;DR

This paper introduces a projectively natural discrete analog of the evolute for polygons, analyzing its properties and dynamics, especially for pentagons and hexagons, revealing integrable structures and conjugacy to simple maps.

Contribution

It defines projective perpendicular bisectors for polygon sides and studies the resulting map's dynamics on moduli spaces, uncovering integrability and conjugacy properties.

Findings

Second iteration has an integral with cubic level curves.

The map on these curves is conjugate to a simple map $x\mapsto -4x$ mod 1.

Experimental results for hexagons suggest further structure.

Abstract

The evolute of a curve is the envelope of its normals. In this note we consider a projectively natural discrete analog of this construction: we define projective perpendicular bisectors of the sides of a polygon in the projective plane, and study the map that sends a polygon to the new polygon formed by the projective perpendicular bisectors of its sides. We consider this map acting on the moduli space of projective polygons. We analyze the case of pentagons; the moduli space is 2-dimensional in this case. The second iteration of the map has one integral whose level curves are cubic curves, and the transformation on these level curves is conjugated to the map $x\mapsto -4x$ mod 1. We also present the results of an experimental study in the case of hexagons.