Bicycling geodesics are Kirchhoff rods

TL;DR

This paper characterizes bicycle geodesics as special curves called Kirchhoff rods, showing their geometric properties and relation to elastic curves in higher dimensions.

Contribution

It establishes a connection between bicycle geodesics and Kirchhoff rods, providing variational equations and geometric insights for dimensions n ≥ 3.

Findings

Each bicycle geodesic lies in a 3D affine subspace.

Front tracks of these geodesics are a subfamily of Kirchhoff rods.

The study generalizes planar elastic curves to higher dimensions.

Abstract

A bicycle path is a pair of trajectories in ${\mathbb R}^n$, the `front' and `back' tracks, traced out by the endpoints of a moving line segment of fixed length (the `bicycle frame') and tangent to the back track. Bicycle geodesics are bicycle paths whose front track's length is critical among all bicycle paths connecting two given placements of the line segment. We write down and study the associated variational equations, showing that for $n\geq 3$ each such geodesic is contained in a 3-dimensional affine subspace and that the front tracks of these geodesics form a certain subfamily of Kirchhoff rods, a class of curves introduced in 1859 by G. Kirchhoff, generalizing the planar elastic curves of J. Bernoulli and L. Euler.