Conformal Loxodromes
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.
Keywords
Cite
@article{arxiv.2302.06775,
title = {Conformal Loxodromes},
author = {Michael Eastwood},
journal= {arXiv preprint arXiv:2302.06775},
year = {2023}
}
Comments
13 pages. Following helpful feedback from Maciej Dunajski, a discussion of parameterised curves is added as Section 5: the updated version is 14 pages. Following yet more helpful feedback from Maciej Dunajski, Josef Silhan, Lenka Zalabova, Vojtech Zadnik, and an anonymous referee, further improvements have been made and some additional references have been added: the update is now 16 pages