New Representations of Spherical Indicatricies of Bertrand Curves in Minkowski 3-Space

Fırat Yerli̇kaya ,İsmai̇l Aydemi̇r

doi:10.1155/2015/509058

Abstract

We obtained a new representation for timelike Bertrand curves and their Bertrand mate in 3-dimensional Minkowski space. By using this representation, we expressed new representations of spherical indicatricies of Bertrand curves and computed their curvatures and torsions. Furthermore in case the indicatricies of a Bertrand curve are slant helices, we investigated some new characteristic features of these curves.

Highlights

We obtained a new representation for timelike Bertrand curves and their Bertrand mate in 3-dimensional Minkowski space
In the early 1900s, Bertrand had worked on special curves, which were referred to by his name
Bertrand curve is defined as a particular curve, which shares its principal normal vector with another special curve that called Bertrand pair

Summary

Introduction

In the early 1900s, Bertrand had worked on special curves, which were referred to by his name. A Slant helix is a curve whose principal normal vector makes a constant angle with a fixed direction [6]. We obtained a new representation for timelike Bertrand curves and their Bertrand mate in 3-dimensional Minkowski space By using this representation, we expressed new representations of spherical indicatricies of Bertrand curves and computed their curvatures and torsions. Let α be a unit speed timelike space curve with curvature κ, torsion τ and Frenet vector fields of α be {T, N, B} where T is timelike and N, B are spacelike vector fields. This unique nonnegative real number θ is called The Lorentzian timelike angle between x and y. 󵄨󵄨󵄨󵄨⟨x, y⟩󵄨󵄨󵄨󵄨 = ‖x‖ 󵄩󵄩󵄩󵄩y󵄩󵄩󵄩󵄩 sinh θ This unique nonnegative real number θ is called The Lorentzian timelike angle between x and y (see [7]). Is constant everywhere τ2 − κ2 does not vanish [8]

New Representations of Timelike Bertrand Curves

New Representations of Spherical Indicatrix of Timelike Bertrand Curves