Homogeneous sub-Riemannian geodesics on the group of motions of the plane

TL;DR

This paper investigates the properties of homogeneous sub-Riemannian geodesics on the SE(2) group, revealing that the structure is not geodesically orbital but has invariant cut time, contributing to geometric control theory.

Contribution

It provides a detailed description of homogeneous sub-Riemannian geodesics on SE(2) and analyzes their orbital properties and invariance of cut time.

Findings

The structure is not geodesically orbital.

Cut time is invariant under initial point shifts.

Provides explicit characterization of geodesics on SE(2).

Abstract

We describe homogeneous sub-Riemannian geodesics for the standard sub-Riemannian structure on the group of proper motions of the plane SE(2). We show that this structure is not geodesically orbital, although the cut time is invariant w.r.t. shift of the initial point along geodesic.