On Helices and Bertrand Curves in Euclidean 3-Space

Murat Babaarslan,Yusuf Yayli

doi:10.3390/mca18010001

Murat Babaarslan, Yusuf Yayli

Open Access

https://doi.org/10.3390/mca18010001

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Abstract

In this article, we investigate Bertrand curves corresponding to the spherical images of the tangent, binormal, principal normal and Darboux indicatrices of a space curve in Euclidean 3-space. As a result, in case of a space curve is a general helix, we show that the curves corresponding to the spherical images of its the tangent indicatrix and binormal indicatrix are both Bertrand curves and circular helices. Similarly, in case of a space curve is a slant helix, we demonstrate that the curve corresponding to the spherical image of its the principal normal indicatrix is both a Bertrand curve and a circular helix.

Highlights

A curve of constant slope or general helix in Euclidean 3-space is characterized by the property that the tangent lines make a constant angle with a fixed direction
A classical result about helix stated by Lancret in 1802 and first proved by de Saint Venant in 1845 says that: A necessary and sufficient condition that a curve be a general helix is that the ratio / is constant along the curve, where 0
The aim of this paper is to investigate Bertrand curves corresponding to the spherical images of a space curve in Euclidean 3-space

Summary

INTRODUCTION

A curve of constant slope or general helix in Euclidean 3-space is characterized by the property that the tangent lines make a constant angle with a fixed direction (the axis of the general helix). A classical result about helix stated by Lancret in 1802 and first proved by de Saint Venant in 1845 (see [3] for details) says that: A necessary and sufficient condition that a curve be a general helix is that the ratio / is constant along the curve, where 0 If both and are non-zero constants, it is called a circular helix. In case of a space curve is general helix, we show that the curves corresponding to the spherical images of its the tangent indicatrix and binormal indicatrix are both Bertrand curves and circular helices. In case of a space curve is slant helix, we demonstrate that the curve corresponding to the spherical image of its the principal normal indicatrix is both a Bertrand curve and a circular helix

PRELIMINARIES

HELICES AND BERTRAND CURVES IN EUCLIDEAN 3-SPACE