Smarandache Ruled Surfaces according to Frenet-Serret Frame of a Regular Curve in 
                     
                        
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Soukaina Ouarab,Chun Gang Zhu

doi:10.1155/2021/5526536

Abstract

In this paper, we introduce original definitions of Smarandache ruled surfaces according to Frenet-Serret frame of a curve in E 3 . It concerns TN-Smarandache ruled surface, TB-Smarandache ruled surface, and NB-Smarandache ruled surface. We investigate theorems that give necessary and sufficient conditions for those special ruled surfaces to be developable and minimal. Furthermore, we present examples with illustrations.

Highlights

In differential geometry of curves and surfaces [1,2,3], a ruled surface defines the set of a family of straight lines depending on a parameter
In [7], the authors constructed the ruled surface whose rulings are constant linear combinations of Darboux frame vectors of a regular curve lying on a regular surface of reference in E3
The main results of the present work assure the following: (i) The TN-Smarandache ruled surface according to the Frenet-Serret frame is developable if cðsÞ is a plane curve

Summary

Introduction

In differential geometry of curves and surfaces [1,2,3], a ruled surface defines the set of a family of straight lines depending on a parameter. In [4], the authors constructed the ruled surface whose rulings are constant linear combinations of alternative moving frame vectors of its base curve; they studied the ruled surface properties, characterize it, and presented examples with illustrations in the case of general helices [5] and slant helices [6] as base curves. In [7], the authors constructed the ruled surface whose rulings are constant linear combinations of Darboux frame vectors of a regular curve lying on a regular surface of reference in E3. We construct and introduce original definitions of three special ruled surfaces generated by TN-Smarandache curve, TB-Smarandache curve, and NB-Smarandache curve according to the Frenet-Serret frame of an arbitrary regular curve in E3.

Preliminaries

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Conclusion