New Representations of Bertrand Pairs in Euclidean 3-Space

TL;DR

This paper explores the properties and new representations of spherical indicatrices of Bertrand curves and their mates in Euclidean 3-space, focusing on special cases like slant helices and their geometric relations.

Contribution

It introduces novel representations of spherical indicatrices and examines their properties in relation to Bertrand, Mannheim, and involute-evolute pairs in Euclidean 3-space.

Findings

Characterized spherical indicatrices of Bertrand and mate curves.

Identified conditions for spherical indicatrices to be slant helices.

Explored new curve pairs formed by spherical indicatrices.

Abstract

In this work, we studied the properties of the spherical indicatrices of a Bertrand curve and its mate curve and presented some characteristic properties in the cases that Bertrand curve and its mate curve are slant helices, spherical indicatrices are slant helices and we also researched that whether the spherical indicatrices made new curve pairs in the means of Mannheim, involte-evolute and Bertrand pairs. Further more, we investigated the relations between the spherical images and introduced new representations of spherical indicatrices.