Solitons of discrete curve shortening

TL;DR

This paper studies discrete curve shortening solitons, describing their properties via differential equations and providing examples of spiral solitons that rotate and shrink under the transformation.

Contribution

It characterizes a broad class of solitons for the discrete curve shortening transformation using linear differential equations.

Findings

Identifies a large class of solitons as solutions to linear differential systems.

Constructs examples of spiral solitons that rotate and shrink.

Provides a framework connecting discrete transformations with continuous differential equations.

Abstract

For a polygon in Euclidean space we consider a transformation T which is obtained by applying the midpoints polygon construction twice and using an index shift. For a closed polygon this is a curve shortening process. A polygon is called (affine) soliton of the transformation T if its image under T is an affine image of the polygon. We describe a large class of solitons by considering smooth curves which are solutions of a linear system of differential equations of second order with constant coefficients. As examples we obtain solitons lying on spiral curves which under the transformation T rotate and shrink.