On Helices and Bertrand Curves in Euclidean 3-Space

TL;DR

This paper explores the properties of Bertrand curves associated with spherical images of various indicatrices of space curves in Euclidean 3-space, focusing on helices and their geometric characteristics.

Contribution

It characterizes Bertrand curves linked to spherical images of tangent, binormal, and principal normal indicatrices for general and slant helices in Euclidean 3-space.

Findings

Curves from tangent and binormal indicatrices of general helices are Bertrand and circular helices.

The principal normal indicatrix of a slant helix yields a Bertrand and circular helix.

Provides new geometric insights into the relationship between helices and Bertrand curves.

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.