Rotational Surfaces Generated by Planar Curves in E3 with Density

H. Bayram Karadağ ,Ahmet Kazan ,Mustafa Altın

doi:10.28924/2291-8639-17-2019-311

Abstract

In this paper, we obtain the parametric expressions of curves which have zero weighted curvature in a plane with density $e^{ax+by}$ and create the Smarandache curves of the obtaining curves. Also, we construct the rotational surfaces which are generated by planar curves with vanishing weighted curvature and give some characterizations for them.

Highlights

Differential geometers have studied the curves in a plane for a long time and the differential geometry of curves has been interested widely in different spaces such as Euclidean, Minkowski, Galilean, pseudo GalileanReceived 2019-01-12; accepted 2019-02-21; published 2019-05-01. 2010 Mathematics Subject Classification. 53A04, 53C21
We obtain the parametric expressions of curves which have zero weighted curvature in a plane with density eax+by and create the Smarandache curves of the obtaining curves
Work In the present paper, we’ve obtained the weighted curvature of a curve in a plane with density eax+by, where a, b ∈ R not all zero constants and investigated the curves with vanishing weighted curvature according to the cases a = 0, b = 0 and a = 0, b = 0

Summary

Introduction

Differential geometers have studied the curves in a plane for a long time and the differential geometry of curves has been interested widely in different spaces such as Euclidean, Minkowski, Galilean, pseudo Galilean. Plane with density; smarandache curve; weighted curvature; rotational surfaces. From (2.1), the weighted curvature κφ of the curve α(u) = (x(u), y(u), 0) in a plane with density eax is obtained as. Let α(u) be a curve with vanishing weighted curvature in a plane with density eax. We obtain the parametric representations of rotational surfaces which are generated by curves with vanishing weighted curvature in a plane with density eax+by and give their mean and Gaussian curvatures.

The Gaussian and mean curvatures of this surface are obtained by

The Gaussian and mean curvatures of this surface are

Conclusion and Future Work