Conformal Parametrisation of Loxodromes by Triples of Circles

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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.

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Item Type:	Preprint
Authors/Creators:	Kisil, V.V. https://orcid.org/0000-0002-6593-6147 Reid, J.
Copyright, Publisher and Additional Information:	This is an open access preprint under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits unrestricted use, distribution and reproduction in any medium, provided the original work is properly cited.
Dates:	Published: 6 February 2018
Institution:	The University of Leeds
Academic Units:	The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds)
Depositing User:	Symplectic Publications
Date Deposited:	28 Apr 2025 14:20
Last Modified:	28 Apr 2025 14:20
Published Version:	https://arxiv.org/abs/1802.01864
Identification Number:	10.48550/arxiv.1802.01864
Open Archives Initiative ID (OAI ID):	oai:eprints.whiterose.ac.uk:225846

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Conformal Parametrisation of Loxodromes by Triples of Circles

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