Geometry of a navigation problem: the λ−Funk Finsler metrics

Abstract

We investigate the travel time in a navigation problem from a geometric perspective, with respect to a new class of Finsler metrics. We present the λ−Funk Finsler Metrics. The setting involves an open disk centered at the origin, representing a circular lake perturbed by a symmetric wind flow proportional to the distance from the origin with proportionality factor λ. The Randers metric, which is an important Finsler metric, derived from this physical problem, generalizes the well-known Euclidean metric (λ = 0) on the Cartesian plane and the Funk metric on the unit disk (λ = 1). We obtain the formula for distance, or travel time, from point to point, and the circumference equations. In addition, we obtain the distance formulas from point to line and vice versa.