Asian Journal of Mathematics

Laguerre Arc Length from Distance Functions

David E. Barrett and Michael Bolt

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 11 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.ajm/1294789786

Mathematical Reviews number (MathSciNet)
MR2746121

Zentralblatt MATH identifier
1225.51002

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

Keywords
Distance function dual number Laguerre arc length Laguerre geometry

Citation

Barrett, David E.; Bolt, Michael. Laguerre Arc Length from Distance Functions. Asian J. Math. 14 (2010), no. 2, 213--234. https://projecteuclid.org/euclid.ajm/1294789786


Export citation