On Funk's parabolas

TL;DR

This paper explores four types of parabolas in the unit disk with Funk metric, revealing two classical conics and two novel quartic-based parabolas with applications to physics and navigation problems.

Contribution

It introduces and characterizes four types of Funk metric parabolas, including two known conics and two new quartic-based parabolas, expanding the understanding of geometric structures in non-reversible metrics.

Findings

Two of the parabolas are classical conics.

The other two are characterized by irreducible quartic equations.

Explicit examples of all four parabolas are provided.

Abstract

We study parabolas in the two dimensional unit disk equipped with a Funk metric. Four types of parabolas are obtained, due to the non-reversibility of the Funk metric, each one with applications to physics in the Zermelo navigation problem. We show that two of the four parabolas obtained are well known conics, and the remaining two are characterized by irreducible quartics. Explicit examples are given.