Geometry of a Navigation problem: The $\lambda-$Funk Finsler Metrics

TL;DR

This paper explores a geometric approach to a navigation problem involving wind-influenced travel times, deriving formulas for distances using a Randers metric that generalizes Euclidean and Funk metrics.

Contribution

It introduces a novel Randers metric model for navigation in a wind-perturbed environment and derives explicit formulas for various travel time scenarios.

Findings

Formulas for point-to-point travel times

Formulas for point-to-line travel times

Generalization of Euclidean and Funk metrics

Abstract

We investigate the travel time in a navigation problem from a geometric perspective. The setting involves an open subset of the Euclidean plane, representing a lake perturbed by a symmetric wind flow proportional to the distance from the origin. The Randers metric derived from this physical problem generalizes the well-known Euclidean metric on the Cartesian plane and the Funk metric on the unit disk. We obtain formulas for distances, or travel times, from point to point, from point to line, and vice-versa