Characteristic Properties of Type-2 Smarandache Ruled Surfaces According to the Type-2 Bishop Frame in E 3

In the classical differential geometry, the theory of ruled surfaces is one of its branches which has been developed by several researchers. A ruled surface is generally defined as the set of a family of straight lines that depend on a parameter that is mentioned as the ruled surface’s rulings. A ruled surface’s parametric representation is Yðσ, υÞ = cðσÞ + υXðσÞ where cðσÞ is the base curve of Yðσ, υÞ and XðσÞ define the ruling directions [1, 2]. Surfaces’ developability and minimalist notions are two of their most important properties. One of the most interesting points is the study of ruled surfaces with different moving frames, as seen in this example [3–7]. The Smarandache curve in Euclidean and Minkowski spaces is the curve whose position vector is made by Frenet frame vectors on another regular curve [8–11]. Several researchers [12–20] have recently studied Smarandache curves in Minkowski and the Euclidean spaces. In this work, in E3, we introduce the definitions of type-2 Smarandache ruled surfaces using the type-2 Bishop frame, namely, μ1μ2, μ1B, and μ2B type-2 Smarandache ruled surfaces. Our main results are presented in theorems that look into the necessary and sufficient conditions for those surfaces to be developable and minimal. Throughout the response, an example with illustrations is created. 2. Preliminaries


Introduction
In the classical differential geometry, the theory of ruled surfaces is one of its branches which has been developed by several researchers. A ruled surface is generally defined as the set of a family of straight lines that depend on a parameter that is mentioned as the ruled surface's rulings. A ruled surface's parametric representation is Yðσ, υÞ = cðσÞ + υXðσÞ where cðσÞ is the base curve of Yðσ, υÞ and XðσÞ define the ruling directions [1,2]. Surfaces' developability and minimalist notions are two of their most important properties. One of the most interesting points is the study of ruled surfaces with different moving frames, as seen in this example [3][4][5][6][7].
In this work, in E 3 , we introduce the definitions of type-2 Smarandache ruled surfaces using the type-2 Bishop frame, namely, μ 1 μ 2 , μ 1 B, and μ 2 B type-2 Smarandache ruled surfaces. Our main results are presented in theorems that look into the necessary and sufficient conditions for those surfaces to be developable and minimal. Throughout the response, an example with illustrations is created.

Preliminaries
Let E 3 be a 3-dimensional Euclidean space provided with the metric where ðu 1 , u 2 , u 3 Þ is the rectangular coordinate system of E 3 . Representing the moving Frenet frame along its regular curve ψ by fT, N, Bg in conjunction with curvature functions κ and τ in E 3 , the Frenet formula is given as follows [1]: where hT, Ti = hN, Ni = hB, Bi = 1 and hT, Ni = hT, Bi = hN, Bi = 0.
For any arbitrary curve ψ with τ ≠ 0 in E 3 , the type-2 Bishop frame of ψ is given as follows [21]: where k 1 and k 2 are the type-2 Bishop curvatures and satisfying where θðσÞ = arctan ðk 2 /k 1 Þ and Definition 1. [21]. μ 1 μ 2 type-2 Smarandache curves of the curve ψðσÞ via fμ 1 , μ 2 , Bg are given as Definition 2. [21]. μ 1 B type-2 Smarandache curves of the curve ψðσÞ via fμ 1 , μ 2 , Bg are given as Definition 3. [21]. μ 2 B type-2 Smarandache curves of the curve ψðσÞ via fμ 1 , μ 2 , Bg are given as A ruled surface υ in E 3 can be reparametrized as where ψðσÞ is really the base curve and χðσÞ is its unit which defines a space curve that characterizes the straight line's direction [22]. υ's unit normal vector N is given as follows [23]: where υ σ = ∂υ/∂σ and υ υ = ∂υ/∂υ. The Gaussian curvature K and the mean curvature H are given as follows [23]: where The normal curvature, geodesic curvature, and geodesic torsion that connects the curve ψðσÞ on Y are computed as follows:

Main Results
In this part, we define the type-2 Smarandache ruled surfaces within Euclidean 3-space E 3 referring to the frame fμ 1 , μ 2 , Bg. Furthermore, we evaluate the sufficient and necessary conditions that enable these surfaces to be developable and minimal.
Proof. Considering that the μ 1 μ 2 type-2 Smarandache ruled surface given by (13), then, the velocity vectors of Ω are given as follows:

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From equation (15), we can obtain that the Ω's quantities of fundamental forms are Consequently, from the above data, we obtain K Ω and H Ω of the μ 1 μ 2 type-2 Smarandache ruled surface given as follows: Also, we use (12) to get the normal curvature, the geodesic curvature, and the geodesic torsion that associate ψðσÞ on Ω as the following: So, the proof ended.

μ 1 B Type-2 Smarandache Ruled Surface
Definition 7. For a regular curve ψ = ψðσÞ in E 3 related to the frame fμ 1 , μ 2 , Bg, the μ 1 B type-2 Smarandache ruled surface is given as Theorem 8. Let Φ = Φðσ, υÞ be the μ 1 B type-2 Smarandache ruled surface in E 3 defined by (19). Then, we have (1) If k 1 k 2 = 0, then, Φ is a developable surface with the geodesic base curve (2) Φ is a minimal surface with the geodesic base curve if and only if the type-2 Bishop curvatures satisfy the following differential equation Proof. Considering the μ 1 B type-2 Smarandache ruled surface given by (19), then, the velocity vectors of Φ are given as follows: From equation (21), the Φ's quantities of fundamental forms are Then, K Φ and H Φ of the μ 1 B type-2 Smarandache ruled surface is given as follows:

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Furthermore, from (12), we have which replies to the above theorem.

μ 2 B Type-2 Smarandache Ruled Surface
Definition 9. For a regular curve ψ = ψðσÞ in E 3 related to the frame fμ 1 , μ 2 , Bg, the μ 2 B type-2 Smarandache ruled surface is given as Theorem 10. Let Ψ = Ψðσ, υÞ be the μ 2 B type-2 Smarandache ruled surface in E 3 defined by (25). Then, we have (1) If k 1 k 2 = 0, then, Ψ is a developable surface with the principal base curve (2) Ψ is a minimal surface if and only if the type-2 Bishop curvatures satisfy the following differential equation Proof. Considering the μ 2 B type-2 Smarandache ruled surface given by (25), then, the velocity vectors of Ψ are given as follows: From equation (27), the Ψ's quantities of fundamental forms are The K Ψ and H Ψ of the μ 2 B type-2 Smarandache ruled surface given as follows: So, the proof ended. − cos σ 9 + υ 3 :

Conclusion
The study of ruled surfaces with different moving frames is one of the most interesting points of this paper. The researchers found that these surfaces could be developed in a minimal amount of time. In this work, we describe and study type-2 Smarandache ruled surfaces, which are a specific form of ruled surfaces. We create the essential and adequate circumstances for these surfaces to be developable in a minimal amount of time.

Data Availability
No data is used in this study.

Conflicts of Interest
The authors declare no competing interest.