Characterizations of the Quaternionic Bertrand Curve in Euclidean Space E4

TL;DR

This paper defines and explores quaternionic Bertrand curves in four-dimensional Euclidean space, establishing their properties, existence conditions, and introducing a new class called quaternionic (N - B2) Bertrand curves.

Contribution

It introduces the concept of quaternionic Bertrand curves in E4, analyzes their properties, and defines a new subclass called quaternionic (N - B2) Bertrand curves.

Findings

Quaternionic Bertrand curves exist under certain conditions in E4.

No quaternionic Bertrand curves exist when r - K = 0.

A new class called quaternionic (N - B2) Bertrand curves is characterized.

Abstract

In [18], L. R. Pears proved that Bertrand curves in E-n(n > 3) are degenerate curves. This result restate in [16] by Matsuda and Yorozu. They proved that there is no special Bertrand curves in E-n(n > 3) and they define new kind of Bertrand curves called (1, 3)-type Bertrand curves in 4-dimensional Euclidean space. In this study, we define a quaternionic Bertrand curve ?(4) in Euclidean space E4 and investigate its properties for two cases. In the first case; we consider quaternionic Bertrand curve in the Euclidean space E4 for r-K = 0 where r is the torsion of the spatial quaternionic curve ?; K is the principal curvature of the quaternionic curve ?(4): And then, in the other case, we prove that there is no quaternionic Bertrand curve in the Euclidean space E4 for r - K = 0: So, we give an idea of quaternionic Bertrand curve which we call quaternionic (N - B2) Bertrand curve in the Euclidean space E4 by using the similar method in [16] and we give some characterizations of such curves.