Conformal Loxodromes

TL;DR

This paper explores conformal loxodromes, their geometric properties, and derives a fifth-order invariant differential equation describing these curves in the context of conformal differential geometry.

Contribution

It introduces the concept of conformal loxodromes, providing a curved analogue in Moebius geometry and deriving the associated invariant differential equation.

Findings

Conformal loxodromes are invariant curves under conformal transformations.

A fifth-order invariant ODE characterizes conformal loxodromes.

The paper extends classical loxodrome concepts to conformal geometry.

Abstract

In conformal differential geometry, there are some distinguished curves, often known as 'conformal circles,' since, on the round sphere, they are the round circles (and these are conformally invariant). But on the two-sphere, the curves of constant compass bearing are also conformally invariant. These 'loxodromes' admit a curved analogue in the realm of Moebius geometry. In this article, these curved analogues are explained and the fifth order invariant ODE that they satisfy is derived.