The clothoid helices obtained via the Lie-Darboux method

TL;DR

This paper derives and studies clothoid helices with linearly proportional curvature and torsion using the Lie-Darboux method, including shifted variants.

Contribution

It introduces a novel application of the Lie-Darboux method to generate and analyze clothoid helices with specific geometric properties.

Findings

Derived explicit forms of clothoid helices with proportional curvature and torsion.

Introduced shifted versions of these helices within the same framework.

Provided detailed geometric analysis of the obtained helices.

Abstract

The clothoid helices that have both curvature and torsion directly proportional to the arclength are obtained via the Lie-Darboux method and studied in some detail. Shifted counterparts are also introduced and presented in the same approach.