Smarandache Ruled Surfaces according to Frenet-Serret Frame of a Regular Curve in E 3

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.

POINTWISE 1-TYPE GAUSS MAP OF DEVELOPABLE SMARANDACHE RULED SURFACES IN $\mathbb{E}^{3}$

“…In [16], Ouarab introduced the Smarandache ruled surfaces whose directrix are (2.2), (2.3), and (2.4) in E 3 , and defined as: Definition 2.1. Let α = α(s) be a regular unit-speed curve and denote (T (s), N (s), B(s)) as the Frenet frame of α in E 3 .…”

“…In this section, we study the developable T N -Smarandache ruled surface with a harmonic Gauss map and a pointwise 1-type Gauss map of the first kind. The T N -Smarandache ruled surface x(s, v), as given by (2.5) is developable if and only if α(s) is a plane curve, i.e., τ = 0 [16].…”

“…In 2021, Ourab [16] defined a Smarandache ruled surfaces whose directrix are T N , T B, and N B-Smarandache curves derived from Frenet vectors of the curve in E 3 , and studied the geometric properties of such surfaces based on the mean curvature and Gaussian curvature. Recently, in 2022, Senyurt [18] introduced some new special ruled surfaces with a directrix as the T N B-Smarandache curve and studied their geometric properties such as mean curvature and Gaussian curvature in E 3 .…”

Another application of Smarandache based ruled surfaces with the Darboux vector according to Frenet frame in $E^{3}$

“…In addition, studies with Smarandache curves are available in [1, 4, 6, 24, 26-31, 34, 35]. Recently, in [18][19][20], generating the way of new ruled surfaces have been given by exploiting the idea of Smarandache geometry and using the Frenet, the Darboux or the alternative frame. In the light of all these informations in this study, we obtain the ruled surfaces from the direction vector obtained from the Frenet and Darboux vectors of any curve and from the Smarandache base curve obtained in the same way.…”