Logarithmic spirals on surfaces of constant Gaussian curvature

Blacker, Casey; Tsyganenko, Pavel

doi:10.2140/involve.2024.17.689

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Logarithmic spirals on surfaces of constant Gaussian curvature

Casey Blacker and Pavel Tsyganenko

Vol. 17 (2024), No. 4, 689–708

DOI: 10.2140/involve.2024.17.689

Abstract

We compute the geodesic curvature of logarithmic spirals on surfaces of constant Gaussian curvature. In addition, we show that the asymptotic behavior of the geodesic curvature is independent of the curvature of the ambient surface. We also show that, at a fixed distance from the center of the spiral, the geodesic curvature is continuously differentiable as a function of the Gaussian curvature.

Keywords

geometry of curves and surfaces, logarithmic spirals, geodesic curvature

Mathematical Subject Classification

Primary: 53A04

Secondary: 53A05, 53B20

Milestones

Received: 23 February 2023

Revised: 21 May 2023

Accepted: 30 May 2023

Published: 2 October 2024

Communicated by Michael Dorff

Authors

Casey Blacker
Department of Mathematical Sciences George Mason University Fairfax, VA United States

Pavel Tsyganenko
Department of Mathematics and Computer Science Saint Petersburg State University Saint Petersburg Russia

Vol. 17, No. 4, 2024