Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 48, No. 2, pp. 493-520 (2007)

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Extremal properties of logarithmic spirals

Michael Bolt

Department of Mathematics and Statistics, Calvin College, 3201 Burton St. SE, Grand Rapids, Michigan 49546 USA, e-mail:

Abstract: Loxodromic arcs are shown to be the maximizers of inversive arclength, which is invariant under Möbius transformations. Previously, these arcs were known to be extremals. The first result says that at any loxodromic arc, the inversive arclength functional is concave with respect to a non-trivial perturbation that fixes the circle elements at the endpoints. The second result says that among curves with monotone curvature that connect fixed circle elements, the loxodromic arcs uniquely maximize inversive arclength. These results prove a conjecture made by Liebmann in 1923.

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Electronic version published on: 7 Sep 2007. This page was last modified: 28 Jun 2010.

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