Solitons of the midpoint mapping and affine curvature

TL;DR

This paper characterizes polygons whose midpoint polygons are affine images of the original, showing they form orbits under affine group actions and relate to curves with constant affine curvature, including conic sections.

Contribution

It introduces a new class of polygons called solitons of the midpoint mapping and characterizes their geometric properties and relation to affine curvature and differential equations.

Findings

Polygons with midpoint images under affine maps form orbits of affine group actions.

Smooth curves with these properties have constant generalized-affine curvature.

Special cases include parabolas, ellipses, and hyperbolas.

Abstract

For a polygon $x=(x_j)_{j\in \mathbb{Z}}$ in $\mathbb{R}^n$ we consider the midpoints polygon $(M(x))_j=\left(x_j+x_{j+1}\right)/2\,.$ We call a polygon a soliton of the midpoints mapping $M$ if its midpoints polygon is the image of the polygon under an invertible affine map. We show that a large class of these polygons lie on an orbit of a one-parameter subgroup of the affine group acting on $\mathbb{R}^n.$ These smooth curves are also characterized as solutions of the differential equation $\dot{c}(t)=Bc (t)+d$ for a matrix $B$ and a vector $d.$ For $n=2$ these curves are curves of constant generalized-affine curvature $k_{ga}=k_{ga}(B)$ depending on $B$ parametrized by generalized-affine arc length unless they are parametrizations of a parabola, an ellipse, or a hyperbola.