Relatively normal-slant helices lying on a surface and their characterizations

TL;DR

This paper introduces and characterizes relatively normal-slant helices on surfaces in Euclidean 3-space, providing new geometric insights, relations to other curves, and methods for generating such helices on given surfaces.

Contribution

It defines relatively normal-slant helices using the Darboux frame, characterizes their properties, and offers a method to generate them on surfaces.

Findings

Characterizations and axis of relatively normal-slant helices.

Relations between these helices and other special curves.

Method for generating such helices on surfaces.

Abstract

In this paper, we consider a regular curve on an oriented surface in Euclidean 3-space with the Darboux frame $\{\mathsf{T},\mathsf{V},\mathsf{U}\}$ along the curve, where $\mathsf{T}$ is the unit tangent vector field of the curve, $\mathsf{U}$ is the surface normal restricted to the curve and $\mathsf{V}=\mathsf{U}\times \mathsf{T}$. We define a new curve on a surface by using the Darboux frame. This new curve whose vector field $\mathsf{V}$ makes a constant angle with a fixed direction is called as relatively normal-slant helix. We give some characterizations for such curves and obtain their axis. Besides we give some relations between some special curves (general helices, integral curves, etc.) and relatively normal-slant helices. Moreover, when a regular surface is given by its implicit or parametric equation, we introduce the method for generating the relatively normal-slant helix with the chosen direction and constant angle on the given surface.