Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 50, No. 2, pp. 369-387 (2009)

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Conformal width in M{ö}bius geometry

Rémi Langevin and Eberhard Teufel

Université de Bourgogne, Institut de Mathématiques de Bourgogne, CNRS Laboratory $\mbox{N}^{\circ}$ 5584, 21078 Dijon, France. e-mail:; Universit{ä}t Stuttgart, Fakultät 8: Mathematik und Physik, Institut für Geometrie und Topologie, 70550 Stuttgart, Germany

Abstract: In this article we extend the euclidean concept of width to Möbius geometry. For pairs of curves in the plane or in the 2-sphere $S^2$ which are the two folds of an envelope of circles, the conformal width will be defined as the conformal distance between the osculating circles at corresponding points. We mainly study pairs of curves having constant conformal width. The main results characterize constant conformal width in terms of the geodesic curvature of the family of circles enveloping the pair of curves, seen as a curve in the 3-dimensional de Sitter space, and in terms of the conformal arc-lengths of the two folds of the envelope.

Keywords: curves, conformal width, constant width, conformal 2-sphere, Möbius geometry

Classification (MSC2000): 53A30, 53C40, 53A04

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Electronic version published on: 28 Aug 2009. This page was last modified: 28 Jan 2013.

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