Evolutes of plane curves and null curves in Minkowski $3$-space

TL;DR

This paper establishes a geometric correspondence between plane curves and null curves in Minkowski 3-space using Laguerre geometry, providing new insights into their properties and classifications.

Contribution

It introduces a novel method to relate plane curves to null curves in Minkowski space via isotropic projection, enhancing understanding of their geometric relationship.

Findings

Describes null curves in terms of plane curve curvature

Provides an alternative classification of plane curves based on Laguerre congruence

Details the geometry of null curves using Cartan frames and pseudo-torsion

Abstract

We use the isotropic projection of Laguerre geometry in order to establish a correspondence between plane curves and null curves in the Minkowski $3$-space. We describe the geometry of null curves (Cartan frame, pseudo-arc parameter, pseudo-torsion, pairs of associated curves) in terms of the curvature of the corresponding plane curves. This leads to an alternative description of all plane curves which are Laguerre congruent to a given one.