On spacelike equiform-Bishop Smarandache curves on S_{1}^{2}

E M Solouma,W M Mahmoud

doi:10.1186/s42787-019-0009-x

Abstract

In this paper, we introduce the equiform-Bishop frame of a spacelike curve r lying fully on S_{1}^{2} in Minkowski 3-space mathbb {R}^{3}_{1}. By using this frame, we investigate the equiform-Bishop Frenet invariants of special spacelike equiform-Bishop Smarandache curves of a spacelike base curve in mathbb {R}^{3}_{1}. Furthermore, we study the geometric properties of these curves when the spacelike base curve r is specially contained in a plane. Finally, we give a computational example to illustrate these curves.

Highlights

In the theory of curves in the Euclidean and Minkowski spaces, a regular curve whose position vector is composed by Frenet frame vectors on another regular curve is called a Smarandache curve [1]
Special Smarandache curves have been studied by some authors [7,8,9,10,11]
We introduce the equiform-Bishop frame of a spacelike curve r lying fully on S12 in Minkowski 3-space R31

Summary

Introduction

In the theory of curves in the Euclidean and Minkowski spaces, a regular curve whose position vector is composed by Frenet frame vectors on another regular curve is called a Smarandache curve [1]. We introduce the equiform-Bishop frame of a spacelike curve r lying fully on S12 in Minkowski 3-space R31. B1B2, and TB1B2-Smarandache curves in terms of the equiform-Bishop curvature functions K1(σ ), and K2(σ ) of the spacelike curve r in R31. Definition 1 A regular curve in Minkowski 3-space, whose position vector is composed by Frenet frame vectors on another curve, is called a Smarandache curve [15]. Let us consider the Bishop frame {t, b1, b2} of the spacelike curve r(s) with a spacelike or timelike normal b1(ε = 1 or ε = −1).

Main results

Tφ Nφ Bφ

Tφ Nφ

It easy to show that