Polygonal Bicycle Paths and the Darboux Transformation

TL;DR

This paper explores the properties of periodic bicycle $(n,k)$-paths, a variation of equilateral polygons with equal diagonals, and investigates their geometric transformations.

Contribution

It introduces the concept of periodic bicycle $(n,k)$-paths and examines their geometric properties and transformations, extending the theory of bicycle polygons.

Findings

Periodic bicycle $(n,k)$-paths exhibit specific symmetry properties.

The Darboux transformation can be applied to these paths to generate new configurations.

The study reveals structural invariants under the Darboux transformation.

Abstract

A Bicycle $(n,k)$-gon is an equilateral $n$-gon whose $k$ diagonals are of equal length. In this paper we consider periodic bicycle $(n,k)$-paths, which are a natural variation in which the polygon is replaced with a periodic polygonal path.