Osculating direction curves and their applications

TL;DR

This paper introduces osculating direction curves in Euclidean 3-space, characterizes them, and explores their relationships with well-known special curves like helices and rectifying curves.

Contribution

It defines a new class of direction curves, provides their characterizations, and establishes connections with existing special curves in Euclidean space.

Findings

Osculating-direction curves are characterized in Euclidean 3-space.

Relationships between osculating-direction curves and helices, slant helices, rectifying curves are established.

New insights into the geometric properties of these curves are provided.

Abstract

In this study, we define a new type of direction curves in the Euclidean 3-space such as osculating-direction curve. We give the characterizations for these curves. Moreover, we obtain the relationships between osculating direction curves and some special curves such as helix, slant helix or rectifying curves.