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Logarithmic spirals on surfaces of constant Gaussian curvature

Casey Blacker and Pavel Tsyganenko

Vol. 17 (2024), No. 4, 689–708
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