A note on time-optimal paths on perturbed spheroid

TL;DR

This paper studies the problem of finding time-optimal paths on a spheroid under perturbations using Finsler geometry, providing solutions for an oblate ellipsoid with rotational effects.

Contribution

It introduces a novel approach to the Zermelo navigation problem on spheroids with perturbations using Randers-type Finsler metrics, including explicit solutions for specific cases.

Findings

Derived explicit solutions for time-optimal paths on an oblate ellipsoid.

Extended the Zermelo navigation problem to perturbed spheroid geometries.

Applied Finsler geometry to practical navigation scenarios.

Abstract

We consider the Zermelo navigation problem on the ellipsoid of revolution (spheroid) in the presence of a perturbation $W$ determined by a mild velocity vector field, $|W|<1$, with application of Finsler metric of Randers type in the context of the corresponding optimal control represented by a time-efficient ship's heading $\varphi(t)$ (steering direction). As the example we present the solutions to the problem on an oblate ellipsoid with acting infinitesimal rotation.