Conformal Parametrisation of Loxodromes by Triples of Circles

TL;DR

This paper introduces a conformally invariant parametrisation of loxodromes using triples of circles, extending Lie sphere geometry to include these spiraling curves and highlighting their geometric properties.

Contribution

It presents a novel covariant parametrisation of loxodromes with triples of circles, expanding the geometric framework to include these curves in Lie sphere geometry.

Findings

Parametrisation is covariant under fractional linear transformations.

The approach encodes conformal properties of loxodromes.

Extends Lie sphere geometry to include loxodromes.

Abstract

We provide a parametrisation of a loxodrome by three specially arranged cycles. The parametrisation is covariant under fractional linear transformations of the complex plane and naturally encodes conformal properties of loxodromes. Selected geometrical examples illustrate the usage of parametrisation. Our work extends the set of objects in Lie sphere geometry---circle, lines and points---to the natural maximal conformally-invariant family, which also includes loxodromes.