Asian Journal of Mathematics

Laguerre Arc Length from Distance Functions

David E. Barrett and Michael Bolt

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For the Laguerre geometry in the dual plane, invariant arc length is shown to arise naturally through the use of a pair of distance functions. These distances are useful for identifying equivalence classes of curves, within which the extremal curves are proved to be strict maximizers of Laguerre arc length among three-times differentiable curves of constant signature in a prescribed isotopy class. For smoother curves, it is shown that Laguerre curvature determines the distortion of the distance functions. These results extend existing work for the Möbius geometry in the complex plane.

Article information

Asian J. Math., Volume 14, Number 2 (2010), 213-234.

First available in Project Euclid: 11 January 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 51B15: Laguerre geometries
Secondary: 53A35: Non-Euclidean differential geometry 58E35: Variational inequalities (global problems)

Distance function dual number Laguerre arc length Laguerre geometry


Barrett, David E.; Bolt, Michael. Laguerre Arc Length from Distance Functions. Asian J. Math. 14 (2010), no. 2, 213--234.

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