New Type Direction Curves in 3-Dimensional Compact Lie Group

Ali Çakmak

doi:10.3390/sym11030387

Abstract

In this paper, new types of associated curves, which are defined as rectifying-direction, osculating-direction, and normal-direction, in a three-dimensional Lie group G are achieved by using the general definition of the associated curve, and some characterizations for these curves are obtained. Additionally, connections between the new types of associated curves and the curves, such as helices, general helices, Bertrand, and Mannheim, are given.

Highlights

Many authors have made significant contributions to the theory of curves from past to present. Some of these studies indicated that the relationships between the curvatures of the space curves are quite remarkable, and the new special curves are defined via these relations [1,2,3,4,5]
Within the framework of the definition of associated curves, we introduce new types of direction curves in a three-dimensional Lie group G, and we characterize these curves
We introduce the concepts of osculating-direction, normal-direction, and rectifying-direction curves in

Summary

Introduction

Many authors have made significant contributions to the theory of curves from past to present Some of these studies indicated that the relationships between the curvatures of the space curves are quite remarkable, and the new special curves are defined via these relations [1,2,3,4,5]. The curve pairs are obtained by using the Frenet vectors or curvatures In this respect, involute-evolute, Bertrand, and Mannheim curves are well-known examples of curve pairs, and many studies have been performed on this topic [10,11,12,13,14,15,16]. In [23], the authors explained the notions of both the principal (binormal)-direction curve and principal (binormal)-donor curve of a Frenet curve in E3 They characterized some special curves in E3 by using the relationships between the curves. We determine the relationships between the new types of direction curves (rectifying-direction, osculating-direction, and normal-direction curve curves) and the curves (Bertrand curve, involute-evolute, rectifying curve, etc.)

Preliminaries

Main Results

Osculating-Direction Curves

Normal-Direction Curves

Rectifying-Direction Curves

Conclusions