Conformal Parametrisation of Loxodromes by Triples of Circles

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.