Slant helices and Darboux helices in Myller Configuration

TL;DR

This paper investigates the properties of slant and Darboux helices within Myller configuration, establishing their equivalence and deriving differential equations using different frames.

Contribution

It introduces an alternative frame for curves in Myller configuration and characterizes these helices through new differential equations.

Findings

A curve in M is a slant helix if and only if it is a Darboux helix.

Derived differential equations for curves in M using Frenet and alternative frames.

Established the equivalence of slant and Darboux helices in Myller configuration.

Abstract

In this paper, we study slant helix (or $\overset{\_}{\xi}_{2}$-helix) and Darboux helix in Myller configuration $M$. We show that a curve in $M$ is a slant helix if and only if it is a Darboux helix. We give the alternative frame of a curve in $M$. Furthermore, we obtain the differential equations characterizing the curves in $M$ by means of both Frenet type frame and alternative frame.