# Smarandache Ruled Surfaces according to Darboux Frame inE3

Authors: Soukaina Ouarab
Journal: Journal of Mathematics
 (2021)

---

## Abstract

This paper presents a new approach of constructing special ruled surfaces and aims to study their developability and minimalist conditions. Our concept opens opportunities for application in engineering, surface modeling, and architectural design. The principle of our study is to introduce the original definitions of Smarandache ruled surfaces according to Darboux frame of a curve lying on an arbitrary regular surface inE3. It concerns Tg-Smarandache ruled surface, Tn-smarandache ruled surface, and gn-Smarandache ruled surface. New theorems giving necessary and sufficient conditions for those surfaces to be developable and minimal are investigated. Finally, an example with illustrations is presented.


---

## Body

## 1. Introduction

In the classical differential geometry [1, 2], a ruled surface is generated by a straight line moving along a curve cs. The various positions of the generating line are called the rulings. Such a surface has the following parametric representation:(1)ω:s,v∈I×ℝ⟼cs+vX⟶s,where I is an open interval of ℝ, cs is a curve through which all rulings pass (it is called the base curve), and X⟶s are the rulings directors of the ruled surface.Developability and minimalist notions are two of the most important properties that can be studied for ruled surfaces.Developable ruled surfaces can be transformed into the plane without any deformation and distortion, and they form relatively small subsets that contain cylinders, cones, and the tangent surfaces. They are characterized with vanishing Gaussian curvature [3–5].Minimal surfaces are surfaces that locally minimize their area. They refer to the fixed boundary curve of a surface area that is minimal with respect to other surfaces with the same boundary. They are characterized with vanishing mean curvature [6–8].In light of the existing studies, several researchers were interested in the study of ruled surfaces with Darboux frame. Indeed, in [9], we constructed the ruled surface whose rulings are constant linear combinations of Darboux frame vectors of a regular curve lying on a regular surface of reference in E3. We made a comparative study between the two surfaces (the regular surface of reference and the new constructed ruled surface) along their common curve. We investigated properties of the constructed ruled surface and gave examples with illustrations. Furthermore, in [10], the authors showed that the rulings directors of any ruled surface can be expressed with the Darboux frame vectors in E3. Besides, they studied characteristic properties of that ruled surface and gave the relationship between Darboux frame and Frenet frame. Moreover, in [11], the authors defined the evolute offsets of ruled surface with Darboux frame and studied its characteristic properties in E3.In curve theory, Smarandache curves were introduced for the first time in Minkowski space time [12]. It defines curves whose position vectors are composed by Frenet–Serret vectors on another regular curve. In [13], Smarandache curves according to Darboux frame were introduced in E3. It is a paper that represents an inspiration source for us to make the present work of ruled surface with Smarandache curves according to Darboux frame.This paper introduces three ruled surfaces generated byTg-Smarandache curve, Tn-Smarandache curve, and gn-Smarandache curve, according to Darboux frame in E3. Theorems giving necessary and sufficient conditions for those surfaces to be developable and minimal are investigated. Finally, an illustrative example is given.The main contribution of our study is to introduce original notions of ruled surfaces that combine both notions of Smarandache curves and Darboux frame. Our main objective is to study developability and minimalist conditions of such surfaces. Our approach brings together an important significant value in theory and practical application. Indeed, the constructed form of those special ruled surfaces can be useful in applications to medical science, engineering, surface modeling, computer-aided geometric design, computer-aided manufacturing, and computer graphics. In addition, the developability and minimalist properties are the most important properties that are greatly needed and useful in contemporary architecture and architectural design. Hence, the concept of our study opens new avenues for software engineers and architects to some applications in their research fields. Here are some possibilities:(i)
Developing language of geometrical communication for architecture and engineering.(ii)
Constructing algorithms for control point of approximating surfaces.(iii)
Facilitating reconstruction of special developable and minimal surfaces.(iv)
Constructing algorithms for constructing special developable ruled surfaces.The outline of the paper is structured in the following manner. Some basic notions of ruled surface properties, Darboux frame, and Smarandache curves according to Darboux frame are presented in Section2. In Section 3, the notions of new three Smarandache ruled surfaces according to Darboux frame are introduced, characterized in terms of developability and minimalist properties, and illustrated in an example. Finally, a summarized conclusion is given in Section 4.


## 2. Preliminaries

In the Euclidean 3-spaceE3, we consider the usual metric given by(2),=dx1+dx2+dx3,where x1,x2,x3 is a rectangular coordinate system of E3.LetΨ:s,v∈I×ℝ⟼cs+vX⟶s be a ruled surface in E3. It is said to be non-cylindrical, if its rulings directors are not constant, i.e., X⟶′≠0.Let us denote bym⟶=m⟶s,v the unit normal on the ruled surface Ψ at a regular point, and we have(3)m⟶=Ψs∧ΨvΨs∧Ψv=c′+vX⟶′×X⟶c′+vX⟶′×X⟶,where Ψs=∂Ψs,v/∂s and Ψv=∂Ψs,v/∂v.Definition 1.
(see [14]). The distribution parameter λ=λs of a non-cylindrical ruled surface is given by(4)λ=detc′,X⟶,X⟶′X⟶′2.Definition 2.
(see [14]). A ruled surface is developable if its distribution parameter vanishes.The first, I, and the second, II, fundamental forms of ruled surfaceΨ at a regular point are defined, respectively, by(5)IΨsds+Ψvdv=Eds2+2Fdsdv+Gdv2,IIΨsds+Ψvdv=eds2+2fdsdv+gdv2,where(6)E=Ψs2,F=Ψs,Ψv,G=Ψv2,e=Ψss,m⟶,f=Ψsv,m⟶,g=Ψvv,m⟶=0.Definition 3.
The Gaussian curvatureK and the mean curvature H of the ruled surface Ψ at a regular point are given, respectively, by(7)K=−f2EG−F2,H=Ge−2Ff2EG−F2.Proposition 1 (see [14]).
A ruled surface is developable if and only if its Gaussian curvature vanishes.Proposition 2 (see [14]).
A regular surface is minimal if and only if its mean curvature vanishes.Due to a unit speed curveα=αs that lies on a regular surface ϕ=ϕu,r, there exists the Darboux frame and it is denoted by T⟶s,g⟶s,n⟶s, where T⟶s=α′s is the unit tangent vector of the curve α=αs, n⟶s=ϕu×ϕr/ϕu×ϕrus,rs is the unit normal vector of the surface ϕ=ϕu,r along α=αs, and g⟶s is the unit vector which is defined by g⟶s=n⟶s×T⟶s.The derivative formula of the Darboux frame is given as follows:(8)T⟶′g⟶′n⟶′=0ρgρn−ρg0θg−ρn−θg0T⟶g⟶n⟶,where ρg is the geodesic curvature, ρn is the normal curvature, and θg is the geodesic torsion of the curve α=αs on the surface ϕ=ϕu,r.(i)
αs is a geodesic curve on ϕ if its geodesic curvature ρg vanishes.(ii)
αs is an asymptotic line on ϕ if its normal curvature ρn vanishes.(iii)
αs is a principal line on ϕ if its geodesic torsion θg vanishes.Definition 4.
(see [13]). Tg-Smarandache curves according to Darboux frame of the curve α=αs on the surface ϕ are given by(9)α1s∗s=12T⟶s+g⟶s.Definition 5.
(see [13]). Tn-Smarandache curves according to Darboux frame of the curve α=αs on the surface ϕ are given by(10)α2s∗s=12T⟶s+n⟶s.Definition 6.
(see [13]). gn-Smarandache curves according to Darboux frame of the curve α=αs on the surface ϕ are given by(11)α3s∗s=12g⟶s+n⟶s.


## 3. Smarandache Ruled Surfaces according to Darboux Frame inE3

We start our section by giving original definitions of Smarandache ruled surfaces according to Darboux frame of a regular curve lying on a regular surface inE3.Definition 7.
Letα=αs be the C2-class differentiable unit speed curve lying on the regular surface ϕ in E3. Let us denote by T⟶s,g⟶s,n⟶s the Darboux frame of α=αs on ϕ. The ruled surfaces generated by Smarandache curves according to Darboux frame of α=αs on ϕ are as follows:(12)ω1s,v=12T⟶s+g⟶s+vn⟶s,ω2s,v=12T⟶s+n⟶s+vg⟶s,ω3s,v=12g⟶s+n⟶s+vT⟶s.
These ruled surfaces are calledTg-Smarandache ruled surface, Tn-Smarandache ruled surface, and gn−Smarandache ruled surface, according to Darboux frame of the curve α=αs on the surface ϕ, respectively.In the following, we investigate theorems that give necessary and sufficient conditions for Smarandache ruled surfaces (12) to be developable and minimal.Theorem 1.
Tg-Smarandache ruled surface of (12) is developable if and only if αs is a geodesic curve on the surface ϕ or ρn+θg=0.Theorem 2.
Tg-Smarandache ruled surface of (12) is minimal if and only if ρg−v2θgρg′+ρg2+ρnρn+θg+v2ρn′−ρgθg−ρg+v2ρnρg′−ρg2−θgρn+θg−v2θg′+ρgρn−2ρgρn+θg2=0.Proof.
Differentiating the first line of (12) with respect to s and v, respectively, and using (8), we get(13)1ωs=−ρg2+vρnT⟶+ρg2−vθgg⟶+ρn+θg2n⟶,1ωs=n⟶,
The cross product of these two vectors gives the normal vector onTg-Smarandache ruled surface of (12):(14)ωs1×ωv1=ρg2−vθgT⟶+ρg2+vρng⟶.
So, under regularity condition, the unit normal takes the following form:(15)ωs1×ωv1ωs1×ωv1=1ρg−v2θg2+ρg+v2ρn2ρg−v2θgT⟶+ρg+v2ρng⟶.
Applying the norms and the scalar product for (13), we get the components of the first fundamental form of Tg-Smarandache ruled surface of (12), as follows:(16)1E=ρn+θg2+ρg+v2ρn2+ρg−v2θg22,1F=ρn+θg2,1G=1.
Differentiating (13) with respect to s and v, respectively, and using (8), we get(17)1ωss=−ρg′2+vρn′+ρgρg2−vθg+ρnρn+θg2T⟶+ρg′2−vθg′−ρgρg2+vρn−θgρn+θg2g⟶+ρn′+θg′2−ρnρg2+vρn+θgρg2−vθgn⟶,1ωvs=−ρnT⟶−θgg⟶,1ωvv=0.
Hence, from (15) and (17), we get the components of the second fundamental form of Tg−Smarandache ruled surface of (12), as follows:(18)e1=−ρg−v2θgρg′+ρg2+ρnρn+θg/2+vρn′−ρgθgρg−v2θg2+ρg+v2ρn2+ρg+v2ρnρg′−ρg2−θgρn+θg/2−vθg′+ρgρnρg−v2θg2+ρg+v2ρn2,f1=−ρgρn+θgρg−v2θg2+ρg+v2ρn2,g1=0.
Consequently, from (16) and (18), we obtain the Gaussian curvature and the mean curvature of Tg-Smarandache ruled surface of (12):(19)K1=−2ρg2ρn+θg2ρg−v2θg2+ρg+v2ρn22,H1=−ρg−v2θgρg′+ρg2+ρnρn+θg+v2ρn′−ρgθg2ρg−v2θg2+ρg+v2ρn23/2+ρg+v2ρnρg′−ρg2−θgρn+θg−v2θg′+ρgρn2ρg−v2θg2+ρg+v2ρn23/2+2ρgρn+θg22ρg−v2θg2+ρg+v2ρn23/2,which replies to both above theorems.Corollary 1.
Ifαs is a geodesic curve on the surface ϕ, then(i)
Tg-Smarandache ruled surface is developable.(ii)
Tg-Smarandache ruled surface is minimal if and only if ρn′θg−ρnθg′=0.Theorem 3.
Tn-Smarandache ruled surface of (12) is developable if and only if cs is an asymptotic line on the surface ϕ orθg=ρg.Theorem 4.
Tn-Smarandache ruled surface of (12) is minimal if and only if ρn+v2θgρn′+ρn2+ρgρg−θg+v2ρg′+ρnθg−ρn+v2ρgρn′−ρn2+θgρg−θg+v2θg′−ρnρg−2ρnθg−ρg2=0.Proof.
Differentiating the second line of (12) with respect to s and v, respectively, and using (8), we get(20)1ωs=−ρn2+vρgT⟶+ρg−θg2g⟶+ρn2+vθgn⟶,1ωv=g⟶,which implies that the normal vector on Tn−Smarandache ruled surface of (12) is(21)ωs2×ωv2=−ρn2+vθgT⟶−ρn2+vρgn⟶.
So, under regularity condition, the unit normal is given by(22)ωs2×ωv2ωs2×ωv2=−1ρn+v2θg2+ρn+v2ρg2ρn+v2θgT⟶+ρn+v2ρgn⟶.
Applying the norms and the scalar product for (20), we get the components of the first fundamental form of Tn-Smarandache ruled surface of (12):(23)2E=ρg−θg2+ρn+v2ρg2+ρn+v2θg22,2F=ρg−θg2,2G=1.
Differentiating (20) with respect to s and v, respectively, and using (8), we get(24)2ωss=−ρn′2+vρg′+ρgρg−θg2+ρnρn2+vθgT⟶+ρg′−θg′2−ρgρn2+vρg−θgρn2+vθgg⟶+ρn′2+vθg′−ρnρn2+vρg+θgρg−θg2n⟶,2ωvs=−ρgT⟶+θgn⟶,2ωvv=0.
Hence, from (22) and (24), we get the components of the second fundamental form of Tn-Smarandache ruled surface of (12):(25)e2=ρn+v2θgρn′+ρn2+ρgρg−θg/2+vρg′+ρnθgρn+v2θg2+ρn+v2ρg2−ρn+v2ρgρn′−ρn2+θgρg−θg/2+vθg′−ρnρgρn+v2θg2+ρn+v2ρg2,f2=−ρnθg−ρgρn+v2θg2+ρn+v2ρg2,g2=0.
Thus, from (23) and (25), we obtain the Gaussian curvature and the mean curvature of Tn-Smarandache ruled surface of (12):(26)K2=−2ρn2θg−ρg2ρn+v2θg2+ρn+v2ρg23/2,H2=ρn+v2θgρn′+ρn2+ρgρg−θg+v2ρg′+ρnθg2ρn+v2θg2+ρn+v2ρg23/2−ρn+v2ρgρn′−ρn2+θgρg−θg+v2θg′−ρnρg2ρn+v2θg2+ρn+v2ρg23/2−2ρnθg−ρg22ρn+v2θg2+ρn+v2ρg23/2,which replies to both above theorems.Corollary 2.
Ifαs is an asymptotic line on the surface ϕ, then(i)
Tn-Smarandache ruled surface is developable.(ii)
Tn-Smarandache ruled surface is minimal if and only if ρg′θg−ρgθg′=0.Theorem 5.
gn-Smarandache ruled surface of (12) is developable if and only if αs is a principal line on the surface ϕ or ρg+ρn=0.Theorem 6.
gn-Smarandache ruled surface of (12) is minimal if and only if θg+v2ρn−θg′−ρgρg+ρn−θg2+v2ρg′−ρnθg−−θg+v2ρgθg′−ρnρg+ρn−θg2+v2ρn′+ρgθg+2θgρg+ρn2=0.Proof.
Differentiating the third line of (12) with respect to s and v, respectively, and using (8), we get(27)3ωs=−ρg+ρn2T⟶+−θg2+vρgg⟶+θg2+vρnn⟶,3ωv=T⟶,
By realizing the cross product of both vectors of (27), we get the normal vector on gn-Smarandache ruled surface of (12):(28)ωs3×ωv3=θg2+vρng⟶−−θg2+vρgn⟶.
So, under regularity condition, the unit normal takes the following form:(29)ωs3×ωv3ωs3×ωv3=1θg+v2ρn2+−θg+v2ρg2θg+v2ρng⟶−−θg+v2ρgn⟶.
Applying the norms and the scalar product for (27), we get the components of the first fundamental form of gn-Smarandache ruled surface of (12), as follows:(30)3E=ρg+ρn2+−θg+v2ρg2+θg+v2ρn22,3F=−ρg+ρn2,3G=1.
Differentiating (27) with respect to s and v, respectively, and using (8), we get(31)3ωss=−ρg′+ρn′2+ρg−θg2+vρg+ρnθg2+vρnT⟶+−θg′2+vρg′−ρgρg+ρn2−θgθg2+vρng⟶+θg′2+vρn′−ρnρg+ρn2+θg−θg2+vρgn⟶,3ωvs=ρgg⟶+ρnn⟶,3ωvv=0.
Hence, from (29) and (31), we get the components of the second fundamental form of gn-Smarandache ruled surface of (12), as follows:(32)3e=θg+v2ρn−θg′/2+vρg′−ρgρg+ρn/2−θgθg/2+vρnθg+v2ρn2+−θg+v2ρg2−−θg+v2ρgθg′/2+vρn′−ρnρg+ρn/2+θg−θg/2+vρgθg+v2ρn2+−θg+v2ρg2,3f=θgρg+ρnθg+v2ρn2+−θg+v2ρg2,3g=0.
Consequently, from (30) and (32), we obtain the Gaussian curvature and the mean curvature of gn-Smarandache ruled surface of (12):(33)K3=−2θg2ρg+ρn2θg+v2ρn2+−θg+v2ρg22,H3=θg+v2ρn−θg′−ρgρg+ρn−θg2+v2ρg′−ρnθgθg+v2ρn2+−θg+v2ρg23/2−−θg+v2ρgθg′−ρnρg+ρn−θg2+v2ρn′+ρgθgθg+v2ρn2+−θg+v2ρg23/2+2θgρg+ρn22θg+v2ρn2+−θg+v2ρg3/2,which replies to both above theorems.Corollary 3.
Ifαs is a principal line on the surface ϕ, then(i)
gn-Smarandache ruled surface is developable.(ii)
gn-Smarandache ruled surface is minimal if and only if ρn′ρg−ρnρg′=0.Now, we will give an example for the whole study and present corresponding illustrations.Example 1.
Let us consider the unit speed curveαe=αes and the regular surface ϕe=ϕeu,r defined, respectively, by(34)αes=2coss2,2sins2,0,φeu,r=2cosu2−r2sinu2,2sinu2−r2cosu2,r2.
It is clear thatαe=αes lies on eϕ=eϕu,r; indeed, αes=φes,0. The Darboux frame of αe=αes on ϕe=ϕeu,r is given by(35)T⟶s=−sins2coss20,g⟶s=11+sin2s−sinscoss2−sinssins21,n⟶s=11+sin2scoss2sins2sins,ρns=−121+sin2s,ρgs=sins21+sin2s,θgs=−coss1+sin2s.
Consequently,Tg-Smarandache ruled surface, Tn-Smarandache ruled surface, and gn-Smarandache ruled surface, according to Darboux frame of αe=αes on ϕe=ϕeu,r, are given with their illustrations (Figures 1–3), respectively, as follows:(36)ω1s,v=12−sins2−sinscoss/21+sin2scoss2−sinssins/21+sin2s11+sin2s+v1+sin2scoss2sins2sins,(37)ω2s,v=12−sins2+coss/21+sin2scoss2+sins/21+sin2ssins1+sin2s+v1+sin2s−sinscoss2−sinssins21,(38)ω3s,v=121+sin2s−sinscoss2+coss2−sinssins2+sins21+sins+v−sins2coss20.Figure 1
Tg-Smarandache ruled surface (36).Figure 2
Tn-Smarandache ruled surface (37).Figure 3
gn-Smarandache ruled surface (38).


## 4. Conclusion

In this study, we proposed a new approach of constructing three special ruled surfaces inspired by Smarandache curves according to Darboux frame. It was aboutTg-Smarandache ruled surface, Tn-Smarandache ruled surface, and gn-Smarandache ruled surface according to Darboux frame of a unit speed curve αs lying on an arbitrary regular surface ϕ. We reached our objective by investigating necessary and sufficient conditions for those surfaces to be developable and minimal. We found that the results depend on the properties of the geodesic curvature, the normal curvature, and the geodesic torsion of the curve αs on the reference surface ϕ. Hence, we can anticipate the developability and minimality after making the right choice for the curve αs and the surface ϕ. An example was given with illustrations to see the geometric shapes of such surfaces.On the other hand, the approach of our study opened up opportunities for scientists in several areas such as surface modeling, engineering, contemporary architecture, and architectural design, to use our results as tools for their future applications.



---

*Source: 9912624-2021-05-26.xml*