Laguerre Arc Length from Distance Functions

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.

Keywords

Distance function, dual number, Laguerre arc length, Laguerre geometry

2010 Mathematics Subject Classification

Primary 51B15. Secondary 53A35, 58E35.

Full Text (PDF format)

Published 1 January 2010