Bicycle paths, elasticae and sub-Riemannian geometry

TL;DR

This paper explores the connection between sub-Riemannian geometry on the plane's rigid motions and bicycle path mathematics, revealing that geodesics correspond to elasticae or straight lines, with implications for understanding optimal bike paths.

Contribution

It establishes a novel link between sub-Riemannian geometry and bicycle path curves, identifying geodesics with Euler elasticae and solitons, advancing geometric understanding of bicycle motion.

Findings

Geodesics correspond to Euler elasticae or straight lines.

Infinite geodesics relate to Euler's solitons or straight lines.

Provides a geometric framework connecting bicycle paths and sub-Riemannian geometry.

Abstract

We relate the sub-Riemannian geometry on the group of rigid motions of the plane to `bicycling mathematics'. We show that this geometry's geodesics correspond to bike paths whose front tracks are either non-inflectional Euler elasticae or straight lines, and that its infinite minimizing geodesics (or `metric lines') correspond to bike paths whose front tracks are either straight lines or `Euler's solitons' (also known as Syntractrix or Convicts' curves).