Abstract
In this paper, we define and investigate a special kind of ruled surfaces called type-2 Smarandache ruled surfaces related to the type-2 Bishop frame in . From this point and depending on the type-2 Bishop curvature, we provide the necessary and sufficient conditions that allow these surfaces to be developable in a minimal amount of time. Furthermore, an example is given to clear the results.
1. 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 where is the base curve of and 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 , we introduce the definitions of type-2 Smarandache ruled surfaces using the type-2 Bishop frame, namely, , , and 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
Let be a 3-dimensional Euclidean space provided with the metric where is the rectangular coordinate system of .
Representing the moving Frenet frame along its regular curve by in conjunction with curvature functions and in , the Frenet formula is given as follows [1]:
where and .
For any arbitrary curve with in , the type-2 Bishop frame of is given as follows [21]: where and are the type-2 Bishop curvatures and satisfying
where and
Definition 1. [21]. type-2 Smarandache curves of the curve via are given as
Definition 2. [21]. type-2 Smarandache curves of the curve via are given as
Definition 3. [21]. type-2 Smarandache curves of the curve via are given as
A ruled surface in can be reparametrized as
where is really the base curve and is its unit which defines a space curve that characterizes the straight line’s direction [22].
’s unit normal vector is given as follows [23]:
where and . The Gaussian curvature and the mean curvature are given as follows [23]:
where , , , , , and . The normal curvature, geodesic curvature, and geodesic torsion that connects the curve on are computed as follows:
Definition 4. A ruled surface is developable if and only if and minimal if and only if .
3. Main Results
In this part, we define the type-2 Smarandache ruled surfaces within Euclidean 3-space referring to the frame . Furthermore, we evaluate the sufficient and necessary conditions that enable these surfaces to be developable and minimal.
3.1. Type-2 Smarandache Ruled Surface
Definition 5. For a regular curve in related to the frame , the type-2 Smarandache ruled surface is given as
Theorem 6. Let be the type-2 Smarandache ruled surface in defined by (13). Then, we have (1) is a developable surface with asymptotic base curve (2) is a minimal surface if and only if the type-2 Bishop curvatures satisfy the following equation where is real constant.
Proof. Considering that the type-2 Smarandache ruled surface given by (13), then, the velocity vectors of are given as follows: From equation (15), we can obtain that the ’s quantities of fundamental forms are Consequently, from the above data, we obtain and of the 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 on as the following: So, the proof ended.
3.2. Type-2 Smarandache Ruled Surface
Definition 7. For a regular curve in related to the frame , the type-2 Smarandache ruled surface is given as
Theorem 8. Let be the type-2 Smarandache ruled surface in defined by (19). Then, we have (1)If , then, is a developable surface with the geodesic base curve(2) 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 type-2 Smarandache ruled surface given by (19), then, the velocity vectors of are given as follows: From equation (21), the ’s quantities of fundamental forms are
Then, and of the type-2 Smarandache ruled surface is given as follows:
Furthermore, from (12), we have which replies to the above theorem.
3.3. Type-2 Smarandache Ruled Surface
Definition 9. For a regular curve in related to the frame , the type-2 Smarandache ruled surface is given as
Theorem 10. Let be the type-2 Smarandache ruled surface in defined by (25). Then, we have (1)If , then, 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 type-2 Smarandache ruled surface given by (25), then, the velocity vectors of are given as follows: From equation (27), the ’s quantities of fundamental forms are The and of the type-2 Smarandache ruled surface given as follows: So, the proof ended.
Also, from (12), we have
Then, equations (29) and (30) complete the proof.
3.4. Example
Let be a circular helix parameterized as (see Figure 1). Then, we have
Then, and . From (4), we get , . Also, we have
The type-2 Smarandache ruled surface is (see Figure 2)
The type-2 Smarandache ruled surface is (see Figure 3)
The type-2 Smarandache ruled surface is (see Figure 4)
4. 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.
Authors’ Contributions
All authors have equal contributions and finalized the study.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University for funding this work through Research Group no. RG-21-09-04.