# Characteristic Properties of Type-2 Smarandache Ruled Surfaces According to the Type-2 Bishop Frame inE3

Authors: Ibrahim Al-Dayel; E. M. Solouma
Journal: Advances in Mathematical Physics
 (2021)

---

## 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 inE3. 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.


---

## Body

## 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 isYσ,υ=cσ+υXσ where cσ is the base curve of Yσ,υ and Xσ 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, inE3, 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

LetE3 be a 3-dimensional Euclidean space provided with the metric
(1),=du12+du22+du32,where u1,u2,u3 is the rectangular coordinate system of E3.Representing the moving Frenet frame along its regular curveψ by T,N,B in conjunction with curvature functions κ and τ in E3, the Frenet formula is given as follows [1]:
(2)ddσTσNσBσ=0κσ0−κσ0τσ0−τσ0TσNσBσ,whereT,T=N,N=B,B=1 and T,N=T,B=N,B=0.For any arbitrary curveψ with τ≠0 in E3, the type-2 Bishop frame of ψ is given as follows [21]:
(3)ddσμ1σμ2σBσ=00−k1σ00−k2σk1σk2σ0μ1σμ2σBσ,where k1 and k2 are the type-2 Bishop curvatures and satisfying
(4)TσNσBσ=sinθσ−cosθσ0cosθσsinθσ0001μ1σμ2σBσ,whereθσ=arctank2/k1 and
(5)k1=−τcosθσ,k2=−τsinθσ.Definition 1. [21].
μ1μ2 type-2 Smarandache curves of the curve ψσ via μ1,μ2,B are given as
(6)βσ∗σ=12μ1σ+μ2σ.Definition 2. [21].
μ1B type-2 Smarandache curves of the curve ψσ via μ1,μ2,B are given as
(7)γσ∗σ=12μ1σ+Bσ.Definition 3. [21].
μ2B type-2 Smarandache curves of the curve ψσ via μ1,μ2,B are given as
(8)δσ∗σ=12μ2σ+Bσ.
A ruled surfaceυ in E3 can be reparametrized as
(9)υσ,υ=ψσ+υχσ,where ψσ is really the base curve and χσ 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]:
(10)U=υσ×υυυσ×υυ,where υσ=∂υ/∂σ and υυ=∂υ/∂υ. The Gaussian curvature K and the mean curvature H are given as follows [23]:
(11)K=ℓn−m2EG−F2,H=En+Gℓ−2mF2EG−F2,where E=Yσ2, F=Yσ,Yυ, G=Yυ2, ℓ=Yσσ,U, m=Yσυ,U, and n=Yυυ,U. The normal curvature, geodesic curvature, and geodesic torsion that connects the curve ψσ on Y are computed as follows:
(12)κn=ψ″,U,κg=U×T,T′,τg=U×U′,T′.Definition 4.
A ruled surface is developable if and only ifK=0 and minimal if and only if H=0.


## 3. Main Results

In this part, we define the type-2 Smarandache ruled surfaces within Euclidean 3-spaceE3 referring to the frame μ1,μ2,B. Furthermore, we evaluate the sufficient and necessary conditions that enable these surfaces to be developable and minimal.


### 3.1.μ1μ2 Type-2 Smarandache Ruled Surface

Definition 5.
For a regular curveψ=ψσ in E3 related to the frame μ1,μ2,B, the μ1μ2 type-2 Smarandache ruled surface is given as
(13)Ω=Ωσ,υ=12μ1σ+μ2σ+υBσ.Theorem 6.
LetΩ=Ωσ,υ be the μ1μ2 type-2 Smarandache ruled surface in E3 defined by (13). Then, we have
(1)
Ω is a developable surface with asymptotic base curve ψσ(2)
Ω is a minimal surface if and only if the type-2 Bishop curvatures satisfy the following equation
(14)k1=k2eσ+c,where c is real constant.Proof.
Considering that theμ1μ2 type-2 Smarandache ruled surface given by (13), then, the velocity vectors of Ω are given as follows:
(15)Ωσ=υk1μ1+υk2μ2−k1+k22B,Ωυ=B.
From equation (15), we can obtain that the Ω’s quantities of fundamental forms are
(16)E=υ2τ2+12k1+k22,F=−12k1+k2,G=1,ℓ=k2τυk1′−k1k1+k22−k1τυk2′−k1k1+k22,m=0,n=0.
Consequently, from the above data, we obtainKΩ and HΩ of the μ1μ2 type-2 Smarandache ruled surface given as follows:
(17)KΩ=0,HΩ=k1′k2−k1k2′2υτ3.
Also, we use (12) to get the normal curvature, the geodesic curvature, and the geodesic torsion that associate ψσ on Ω as the following:
(18)κn=0,κg=−k12τ,τg=1τ3τ2k1k2τ′−k1k2τ′.
So, the proof ended.


### 3.2.μ1B Type-2 Smarandache Ruled Surface

Definition 7.
For a regular curveψ=ψσ in E3 related to the frame μ1,μ2,B, the μ1B type-2 Smarandache ruled surface is given as
(19)Φ=Φσ,υ=12μ1σ+Bσ+υμ2σ.Theorem 8.
LetΦ=Φσ,υ be the μ1B type-2 Smarandache ruled surface in E3 defined by (19). Then, we have
(1)
Ifk1k2=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
(20)k1+2νk2k1′−k1k1+2νk2−k1τ2+k1′+2νk2′−2k1k22=0.Proof.
Considering theμ1B type-2 Smarandache ruled surface given by (19), then, the velocity vectors of Φ are given as follows:
(21)Φσ=k12μ1+k22μ2−k1+2υk22B,Φυ=μ2.
From equation (21), the Φ’s quantities of fundamental forms are
(22)E=12τ2+k1+2υk22,F=k22,G=1,ℓ=k1+2υk2k1′−k1k1+2υk2−k1τ2+k1′+2υk2′2k12+k1+2υk22,m=−k1k2k12+k1+2υk22,n=0.Then,KΦ and HΦ of the μ1B type-2 Smarandache ruled surface is given as follows:
(23)KΦ=−2k12k22k12+k1+2υk22τ2−k22+k1+2υk22,HΦ=k1+2υk2k1′−k1k1+2υk2−k1τ2+k1′+2υk2′+2k1k222k12+k1+2υk22.Furthermore, from (12), we have
(24)κn=k1′−k12k1+2υk2−k1τ2+k1′2k12+k1+2υk22,κg=0,τg=−k12k2k1+2υk2k12+k1+2υk22,which replies to the above theorem.


### 3.3.μ2B Type-2 Smarandache Ruled Surface

Definition 9.
For a regular curveψ=ψσ in E3 related to the frame μ1,μ2,B, the μ2B type-2 Smarandache ruled surface is given as
(25)Ψ=Ψσ,υ=12μ2σ+Bσ+υμ1σ.Theorem 10.
LetΨ=Ψσ,υ be the μ2B type-2 Smarandache ruled surface in E3 defined by (25). Then, we have
(1)
Ifk1k2=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
(26)k2τ2+k2′+2υk1′−2k12k2−k2+2υK1k2′−k2k2+2υk1=0.Proof.
Considering theμ2B type-2 Smarandache ruled surface given by (25), then, the velocity vectors of Ψ are given as follows:
(27)Ψσ=k12μ1+k22μ2−k2+2υk22B,Ψυ=μ1.
From equation (27), the Ψ’s quantities of fundamental forms are
(28)E=12τ2+k2+2υk12,F=k12,G=1,ℓ=k2τ2+k2′+2υk1′−k2+2υk1k2′−k2k2+2υk12k22+k2+2υk12,m=k1k2k22+k2+2υk12,n=0.
TheKΨ and HΨ of the μ2B type-2 Smarandache ruled surface given as follows:
(29)KΨ=−2k12k22k22+k2+2υk12τ2−k12+k2+2υk12,HΨ=k2τ2+k2′+2υk1′−2k12k2−k2+2υk1k2′−k2k2+2υk12k22+k2+2υk12.
So, the proof ended.Also, from (12), we have
(30)κn=k2τ2+k2′−k2′−k12k2+2υk12k22+k2+2υk12,κg=k1k2+2υk1k22+k2+2υk12,τg=k12k2k2+2υk1k22+k2+2υk12.Then, equations (29) and (30) complete the proof.


### 3.4. Example

Letψ be a circular helix parameterized as ψσ=cosσ/3,sinσ/3,22σ/3 (see Figure 1). Then, we have
(31)Tσ=−13sinσ3,13cosσ3,223,Nσ=−cosσ3,−sinσ3,0,Bσ=223sinσ3,−223cosσ3,13.Figure 1
Curveψ=ψσ.Then,τ=22/9≠0 and θσ=∫0σ1/9dσ=σ/9. From (4), we get κ1σ=−22/9cosσ/9, κ2σ=−22/9sinσ/9. Also, we have
(32)μ1σ=−cosσ9cosσ3−13sinσ9sinσ3,13cosσ3sinσ9−cosσ9sinσ3,229sinσ9,μ2σ=−cosσ3sinσ9+13cosσ9sinσ3,−13cosσ9cosσ3−sinσ9sinσ3,−229cosσ9.Theμ1μ2 type-2 Smarandache ruled surface Ωσ,υ is (see Figure 2)
(33)Ωσ,υ=1213sinσ3cosσ9−sinσ9−cosσ3sinσ9+cosσ9+23υsinσ3,1213cosσ3sinσ9−cosσ9−sinσ3cosσ9+sinσ9−23υcosσ3,23sinσ9−cosσ9+υ3.Figure 2
μ1μ2 type-2 Smarandache ruled surface Ωσ,υ.Theμ1B type-2 Smarandache ruled surface Φσ,υ is (see Figure 3)
(34)Φσ,υ=13sinσ3υcosσ9−12sinσ9−cosσ3υsinσ9+12cosσ9+23sinσ3,13cosσ312sinσ3−υcosσ3−sinσ3υsinσ3+12cosσ3−23cosσ3,23sinσ9−2υcosσ9+122.Figure 3
μ1B type-2 Smarandache ruled surface Φσ,υ.Theμ2B type-2 Smarandache ruled surface Ψσ,υ is (see Figure 4)
(35)Ψσ,υ=13sinσ312cosσ9−υsinσ9−cosσ312sinσ9+υcosσ9+23sinσ3,13cosσ3υsinσ3−12cosσ3−sinσ312sinσ3+υcosσ3−23cosσ3,232υsinσ9−cosσ9+122.Figure 4
μ2B type-2 Smarandache ruled surface Ψσ,υ.


## 3.1.μ1μ2 Type-2 Smarandache Ruled Surface

Definition 5.
For a regular curveψ=ψσ in E3 related to the frame μ1,μ2,B, the μ1μ2 type-2 Smarandache ruled surface is given as
(13)Ω=Ωσ,υ=12μ1σ+μ2σ+υBσ.Theorem 6.
LetΩ=Ωσ,υ be the μ1μ2 type-2 Smarandache ruled surface in E3 defined by (13). Then, we have
(1)
Ω is a developable surface with asymptotic base curve ψσ(2)
Ω is a minimal surface if and only if the type-2 Bishop curvatures satisfy the following equation
(14)k1=k2eσ+c,where c is real constant.Proof.
Considering that theμ1μ2 type-2 Smarandache ruled surface given by (13), then, the velocity vectors of Ω are given as follows:
(15)Ωσ=υk1μ1+υk2μ2−k1+k22B,Ωυ=B.
From equation (15), we can obtain that the Ω’s quantities of fundamental forms are
(16)E=υ2τ2+12k1+k22,F=−12k1+k2,G=1,ℓ=k2τυk1′−k1k1+k22−k1τυk2′−k1k1+k22,m=0,n=0.
Consequently, from the above data, we obtainKΩ and HΩ of the μ1μ2 type-2 Smarandache ruled surface given as follows:
(17)KΩ=0,HΩ=k1′k2−k1k2′2υτ3.
Also, we use (12) to get the normal curvature, the geodesic curvature, and the geodesic torsion that associate ψσ on Ω as the following:
(18)κn=0,κg=−k12τ,τg=1τ3τ2k1k2τ′−k1k2τ′.
So, the proof ended.


## 3.2.μ1B Type-2 Smarandache Ruled Surface

Definition 7.
For a regular curveψ=ψσ in E3 related to the frame μ1,μ2,B, the μ1B type-2 Smarandache ruled surface is given as
(19)Φ=Φσ,υ=12μ1σ+Bσ+υμ2σ.Theorem 8.
LetΦ=Φσ,υ be the μ1B type-2 Smarandache ruled surface in E3 defined by (19). Then, we have
(1)
Ifk1k2=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
(20)k1+2νk2k1′−k1k1+2νk2−k1τ2+k1′+2νk2′−2k1k22=0.Proof.
Considering theμ1B type-2 Smarandache ruled surface given by (19), then, the velocity vectors of Φ are given as follows:
(21)Φσ=k12μ1+k22μ2−k1+2υk22B,Φυ=μ2.
From equation (21), the Φ’s quantities of fundamental forms are
(22)E=12τ2+k1+2υk22,F=k22,G=1,ℓ=k1+2υk2k1′−k1k1+2υk2−k1τ2+k1′+2υk2′2k12+k1+2υk22,m=−k1k2k12+k1+2υk22,n=0.Then,KΦ and HΦ of the μ1B type-2 Smarandache ruled surface is given as follows:
(23)KΦ=−2k12k22k12+k1+2υk22τ2−k22+k1+2υk22,HΦ=k1+2υk2k1′−k1k1+2υk2−k1τ2+k1′+2υk2′+2k1k222k12+k1+2υk22.Furthermore, from (12), we have
(24)κn=k1′−k12k1+2υk2−k1τ2+k1′2k12+k1+2υk22,κg=0,τg=−k12k2k1+2υk2k12+k1+2υk22,which replies to the above theorem.


## 3.3.μ2B Type-2 Smarandache Ruled Surface

Definition 9.
For a regular curveψ=ψσ in E3 related to the frame μ1,μ2,B, the μ2B type-2 Smarandache ruled surface is given as
(25)Ψ=Ψσ,υ=12μ2σ+Bσ+υμ1σ.Theorem 10.
LetΨ=Ψσ,υ be the μ2B type-2 Smarandache ruled surface in E3 defined by (25). Then, we have
(1)
Ifk1k2=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
(26)k2τ2+k2′+2υk1′−2k12k2−k2+2υK1k2′−k2k2+2υk1=0.Proof.
Considering theμ2B type-2 Smarandache ruled surface given by (25), then, the velocity vectors of Ψ are given as follows:
(27)Ψσ=k12μ1+k22μ2−k2+2υk22B,Ψυ=μ1.
From equation (27), the Ψ’s quantities of fundamental forms are
(28)E=12τ2+k2+2υk12,F=k12,G=1,ℓ=k2τ2+k2′+2υk1′−k2+2υk1k2′−k2k2+2υk12k22+k2+2υk12,m=k1k2k22+k2+2υk12,n=0.
TheKΨ and HΨ of the μ2B type-2 Smarandache ruled surface given as follows:
(29)KΨ=−2k12k22k22+k2+2υk12τ2−k12+k2+2υk12,HΨ=k2τ2+k2′+2υk1′−2k12k2−k2+2υk1k2′−k2k2+2υk12k22+k2+2υk12.
So, the proof ended.Also, from (12), we have
(30)κn=k2τ2+k2′−k2′−k12k2+2υk12k22+k2+2υk12,κg=k1k2+2υk1k22+k2+2υk12,τg=k12k2k2+2υk1k22+k2+2υk12.Then, equations (29) and (30) complete the proof.


## 3.4. Example

Letψ be a circular helix parameterized as ψσ=cosσ/3,sinσ/3,22σ/3 (see Figure 1). Then, we have
(31)Tσ=−13sinσ3,13cosσ3,223,Nσ=−cosσ3,−sinσ3,0,Bσ=223sinσ3,−223cosσ3,13.Figure 1
Curveψ=ψσ.Then,τ=22/9≠0 and θσ=∫0σ1/9dσ=σ/9. From (4), we get κ1σ=−22/9cosσ/9, κ2σ=−22/9sinσ/9. Also, we have
(32)μ1σ=−cosσ9cosσ3−13sinσ9sinσ3,13cosσ3sinσ9−cosσ9sinσ3,229sinσ9,μ2σ=−cosσ3sinσ9+13cosσ9sinσ3,−13cosσ9cosσ3−sinσ9sinσ3,−229cosσ9.Theμ1μ2 type-2 Smarandache ruled surface Ωσ,υ is (see Figure 2)
(33)Ωσ,υ=1213sinσ3cosσ9−sinσ9−cosσ3sinσ9+cosσ9+23υsinσ3,1213cosσ3sinσ9−cosσ9−sinσ3cosσ9+sinσ9−23υcosσ3,23sinσ9−cosσ9+υ3.Figure 2
μ1μ2 type-2 Smarandache ruled surface Ωσ,υ.Theμ1B type-2 Smarandache ruled surface Φσ,υ is (see Figure 3)
(34)Φσ,υ=13sinσ3υcosσ9−12sinσ9−cosσ3υsinσ9+12cosσ9+23sinσ3,13cosσ312sinσ3−υcosσ3−sinσ3υsinσ3+12cosσ3−23cosσ3,23sinσ9−2υcosσ9+122.Figure 3
μ1B type-2 Smarandache ruled surface Φσ,υ.Theμ2B type-2 Smarandache ruled surface Ψσ,υ is (see Figure 4)
(35)Ψσ,υ=13sinσ312cosσ9−υsinσ9−cosσ312sinσ9+υcosσ9+23sinσ3,13cosσ3υsinσ3−12cosσ3−sinσ312sinσ3+υcosσ3−23cosσ3,232υsinσ9−cosσ9+122.Figure 4
μ2B type-2 Smarandache ruled surface Ψσ,υ.


## 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.



---

*Source: 8575443-2021-10-22.xml*