Notes on W-direction curves in Euclidean 3-space

TL;DR

This paper explores the properties of W-direction curves in Euclidean 3-space, focusing on their spherical indicatrices, associated ruled surfaces, and curvature characteristics, providing new insights into their geometric behavior.

Contribution

It introduces the Frenet apparatus for spherical indicatrix curves of W-direction curves and characterizes their helix properties, also analyzing related ruled surfaces and curvature properties.

Findings

Characterization of spherical indicatrix curves as general and slant helices

Properties of normal and binormal ruled surfaces derived from W-direction curves

Expressions for Gaussian, mean curvatures, and other curvature measures

Abstract

In this paper, we study the spherical indicatrices of W-direction curves in three dimensional Euclidean space which were defined by using the unit Darboux vector field W of a Frenet curve, in [11]. We obtain the Frenet apparatus of these spherical indicatrix curves and the characterizations of being general helix and slant helix. Moreover we give some properties between the spherical indicatrix curves and their associated curves. Then we investigate two special ruled surface that are normal and binormal surface by using W-direction curves. We give useful results involving the char- acterizations of these ruled surfaces. From those applications, we make use of such a work to interpret the Gaussian, mean curvatures of these surfaces and geodesic, normal curvature and geodesic torsion of the base curves with respect to these surfaces.