The Geodesics of Rolling Ball Systems

TL;DR

This paper characterizes the sub-Riemannian geodesics of two balls rolling without slipping or twisting in D space, showing they are horizontal curves on intersections with Euclidean 5-planes.

Contribution

It provides a geometric description of geodesics in the rolling ball system, linking them to intersections with Euclidean planes.

Findings

Geodesics are horizontal curves on intersections of balls with Euclidean 5-planes.

The work offers a geometric interpretation of the sub-Riemannian structure.

Clarifies the nature of shortest paths in the rolling ball configuration.

Abstract

A well-known and interesting family of sub-Riemannian space are the systems involving two balls rolling against each other without slipping or twisting. In this note, we show how the sub-Riemannian geodesics of these space, when the two balls are embedded in $\mathbb{R}^3 \times \mathbb{R}^3$, are horizontal curves on the intersections of these balls with Euclidean 5-planes.