# Corrigendum to “Smarandache Ruled Surfaces according to Frenet-Serret Frame of a Regular Curve inE3”

Authors: Soukaina Ouarab
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

In this paper, we introduce original definitions of Smarandache ruled surfaces according to Frenet-Serret frame of a curve inE3. 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.


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## Body

## 1. Introduction

In differential geometry of curves and surfaces [1–3], a ruled surface defines the set of a family of straight lines depending on a parameter. The straight lines mentioned are the rulings of the ruled surface. The general parametric representation of a ruled surface isΨs,v=cs+vX⟵s, where cs is a curve through which all rulings pass; it is called the base curve of the surface; the vector X⟵s defines the ruling direction.It is well known that ruled surfaces are of great interest to many applications and have contributed in several areas, such as mathematical physics, kinematics and Computer Aided Geometric Design (CAGD).Ruled surfaces have been studied differently by an important number of researchers. In [4], the authors constructed the ruled surface whose rulings are constant linear combinations of alternative moving frame vectors of its base curve; they studied the ruled surface properties, characterize it, and presented examples with illustrations in the case of general helices [5] and slant helices [6] as base curves. In [7], the authors 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 inE3. Their point of interest was to make a comparative study between the two surfaces (the regular surface of reference and the new constructed ruled surface) along their common curve. Also, they investigated properties of the constructed ruled surface. Moreover, they gave examples with illustrations.The notions of developability and minimalist are two of the most important properties of surfaces.The ruled surfaces with vanishing Gaussian curvature, which can be transformed into the plane without any deformation and distortion, are called developable surfaces; they form a relatively small subset that contains cylinders, cones, and the tangent surfaces [8–10].A minimal surface is a surface that locally minimizes its area. It is referred to the fixed boundary curve of a surface area that is minimal with respect to other surfaces with the same boundary. This is equivalent to having zero mean curvature [11–13].In curve theory,Smarandache curves are one of the special innovated curves that were introduced at the first time in Minkowski space time by the authors in [14]. It is about curves whose position vectors are composed by Frenet-Serret frame vectors on another regular curve. In [15, 16, 18], we can find several research works about Smarandache curves according to different frames such as Frenet-Serret frame, Bishop frame, and Darboux frame in Euclidean and Minkowski spaces.The motivation of this work is inspired by ruled surface and Smarandache curve. We are eager to introduce new definitions that combine those two important notions and study their properties. We are also opening up opportunities to perceive future works that are relative to applications in differential geometry, physics, and medical science.In this paper, we are interested in ruled surfaces generated by Smarandache curves according to Frenet-Serret frame. Indeed, we construct and introduce original definitions of three special ruled surfaces generated by TN-Smarandache curve, TB-Smarandache curve, and NB-Smarandache curve according to Frenet-Serret frame of an arbitrary regular curve inE3. We investigate theorems that give us necessary and sufficient conditions for those three ruled surfaces to be developable and minimal. Finally, we give examples with illustrations.


## 2. Preliminaries

In the Euclidean 3-spaceE3, we consider the usual metric given by
(1),=dx1+dx2+dx3,where x1,x2,x3 is a rectangular coordinate system of E3.A curvec:s∈I⊂ℝ⟶E3 is said to be of unit speed (or parameterized by the arc-length) if c′s=c′s,c′s=1 for any s∈I. For such curves, there is a frame T⟵,N⟵,B⟵ which is called the Frenet-Serret frame, where T⟵=c′,N⟵=c′′/c′′, and B⟵=T⟵×N⟵ are the unit tangent, the principal normal, and the binormal vector of the curve, respectively. The Frenet-Serret formulae of cs is given by
(2)T⟵′N⟵′B⟵′=0κ0−κ0τ0−τ0T⟵N⟵B⟵,where κ=κs and τ=τs are the curvature and the torsion functions of cs and they are given by κ=c′′ and τ=detc′,c′′,c′′′/c′′2, respectively [2].Definition 1 (see [17]).
TN-Smarandache curves according to Frenet-Serret frame of the curvec=cs are given by
(3)βs∗s=12T⟵s+N⟵s.Definition 2 (see [17]).
TB-Smarandache curves according to Frenet-Serret frame of the curvec=cs are given by
(4)γs∗s=12T⟵s+B⟵s.Definition 3 (see [17]).
NB-Smarandache curves according to Frenet-Serret frame of the curvec=cs are given by
(5)δs∗s=12N⟵s+B⟵s.LetΨ:s,v↦cs+vX⟵s be a ruled surface in E3.Let it denoted byn⟵=n⟵s,v, the unit normal on ruled surface Ψ at a regular point Ψs,v, we have
(6)n⟵=Ψs∧ΨvΨs∧Ψv=c′+vX⟵′×X⟵c′+vX⟵′×X⟵,where Ψs=∂Ψs,v/∂s and Ψv=∂Ψs,v/∂v.Definition 4 (see [19]).
The distribution parameterλ=λs of ruled surface Ψ is given by
(7)λ=detc′,X⟵,X⟵′X⟵′2.Definition 5 (see [19]).
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 Ψs,v are defined, respectively, by
(8)IΨsds+Ψvdv=Eds2+2Fdsdv+Gdv2,IIΨsds+Ψvdv=eds2+2fdsdv+gdv2,where
(9)E=Ψs2,F=Ψs,Ψv,G=Ψv2,e=Ψss,n⟵,f=Ψsv,n⟵,g=Ψvv,n⟵=0.Definition 6.
The Gaussian curvatureK and the mean curvature H of ruled surface Ψ at a regular point Ψs,v are given, respectively, by:
(10)K=−f2EG−F2,H=Ge−2Ff2EG−F2.Proposition 7 (see [19]).
A ruled surface is developable if and only if its Gaussian curvature vanishes.Proposition 8 (see [19]).
A regular surface is minimal if and only if its mean curvature vanishes.


## 3. Smarandache Ruled Surfaces according to Frenet-Serret Frame of a Regular Curve inE3

In the first step, we start our section by giving the following new definitions of Smarandache ruled surfaces according to Frenet-Serret frame of a curve inE3.Definition 9.
Let bec=cs the C2-class differentiable unit speed curve whose Frenet-Serret apparatus is T⟵s,N⟵s,B⟵s,κs,τs in E3. The ruled surfaces generated by Smarandache curves according to Frenet-Serret of c=cs are as follows:
(11)1Ψs,v=12T⟵s+N⟵s+vB⟵s,(12)2Ψs,v=12T⟵s+B⟵s+vN⟵s,(13)3Ψs,v=12N⟵s+B⟵s+vT⟵s.These ruled surfaces are called TN-Smarandache ruled surface, TB-Smarandache ruled surface, and NB-Smarandache ruled surface, according to Frenet-Serret frame of the curvec=cs, respectively.In the following of this section, we investigate theorems that give necessary and sufficient conditions for Smarandache ruled surfaces (11), (12), and (13) to be developable and minimal. Also, we present example with illustration for each Smarandache ruled surface.Theorem 10.
TN-Smarandache ruled surface (11) is developable if and only if c=cs is a plane curve.Theorem 11.
TN-Smarandache ruled surface (11) is minimal if and only if the curvature κ and the torsion τ of the curve c=cs satisfy the equation κτ2−κ′−2κ2+v2κ′τ+2κ2τ−v22κτ2=0.Proof.
Differentiating (11) with respect to s and v, respectively, we get
(14)1Ψs=−κ2T⟵+κ2−vτN⟵+τ2B⟵,1Ψv=B⟵.
The crossproduct of these two vectors gives the normal vector on TN-Smarandache ruled surface (11):
(15)1Ψs×1Ψv=κ2−vτT⟵+κ2N⟵,
So, under regularity condition, the unit normal takes the form:(16)1Ψs×1Ψv1Ψs×1Ψv=1κ2−2κτv+τ2v2κ2−vτT⟵+κ2N⟵.Making the norms for (14), we get the components of the first fundamental form of TN-Smarandache ruled surface (11), at regular points, as follows:
(17)1E=2κ2+τ22−2κτv+τ2v2,1F=τ2,1G=1.Differentiating (14) with respect to s and v, respectively, we get
(18)1Ψss=−κ′2+κκ2−vτT⟵+−κ2−τ2+κ′2−vτ′N⟵+τ′2+τκ2−vτB⟵,1Ψvs=−τN⟵,1Ψvv=0.Hence, from (16) to (18), we get the components of the second fundamental form of TN-Smarandache ruled surface (11), at regular points, as follows:
(19)1e=−κ/2−vτκ′+κ2/2−vκτ+κκ2+τ2/2κ2−2κτv+τ2v2,1f=−κτ2κ2−2κτv+τ2v2,1g=0.From (17) to (19), we get the Gaussian curvature and the mean curvature of TN-Smarandache ruled surface (11), at regular points, as follows:
(20)1K=−κτ2κ2−2κτv+τ2v22,1H=−κκ′−2κ3+κτ2+v2κ′τ+2κ2τ−v22κτ24κ2−2κτv+τ2v23/2,which replies to both above theorems.Example 1.
Let us consider the regular plane curve1cs=3s−s3,3s2,0. TN-Smarandache ruled surface according to Frenet-Serret frame of the curve 1cs is given by
(21)1Ψs,v=121−s21−s22+4s2−s1+s22s1−s22+4s2+11+s20+v001−s+2s21−s22+4s21+s2.It is developable but not minimal at its regular points. The following Figure1 is the illustration of (21) drawn for s,v∈−2π,2π×−4,4.Theorem 12.
TB-Smarandache ruled surface (12) is developable.Theorem 13.
TB-Smarandache ruled surface (12) is minimal if and only if the curve c=cs is a general helix.Proof.
Differentiating (12) with respect to s and v, respectively, we get
(22)2Ψs=−vκT⟵+κ−τ2N⟵+vτB⟵,2Ψv=N⟵,which implies that the normal vector on TB-Smarandache ruled surface (12) is
(23)2Ψs×2Ψv=−vτT⟵−vκB⟵,
So, under regularity condition, the unit normal is given by(24)2Ψs×2Ψv2Ψs×2Ψv=∓τT⟵+κB⟵κ2+τ2.
Making the norms for (22), we get the components of the first fundamental form of TB-Smarandache ruled surface (12), at regular points
(25)2E=κ−τ22+v2κ2+τ2,2F=κ−τ2,2G=1.
Differentiating (22) with respect to s and v, respectively, we get
(26)2Ψss=−κκ−τ2+vκ′T⟵+κ′−τ′2−vκ2+τ2N⟵+τκ−τ2+vτ′B⟵,2Ψvs=−κT⟵+τB⟵,2Ψvv=0.
Hence, from (24) to (26), we get the components of the second fundamental form of TB-Smarandache ruled surface (12) at regular points.
(27)2e=∓vκ2τ/κ′κ2+τ2,2f=0,2g=0.
From (25) to (27), we get the Gaussian curvature and the mean curvature of TB-Smarandache ruled surface (12), at regular points, as follows
(28)2K=0,2H=∓κ2τ/κ′2vκ2+τ23/2,which replies to both above theorems.Corollary 14.
Ifcs is a general helix, TB-Smarandache ruled surface (12) is developable and minimal.Example 2.
Here, we chose the regular curve2cs=1/2sin2s/2,−cos2s/2,s as a general helix. Then, TB-Smarandache ruled surface according to Frenet-Serret frame of 2cs is:
(29)2Ψs,v=12cos2s−sin2scos2s+sin2s1+v−cos2s−sin2s1.Figure 1
The TN-Smarandache ruled surface (Equation (21)).It is developable and minimal at its regular points. The following Figure2 is the illustration of (29) drawn for s,v∈−2π,2π×−4,4.Theorem 15.
NB-Smarandache ruled surface (13) is developable if and only if c=cs is a plane curve.Theorem 16.
NB-Smarandache ruled surface (13) is minimal if and only if the curvature κ and the torsion τ of the curve c=cs satisfy the equation τ2κ2τ−2τ2−κ2/2+vκ′τ+2κτ2−κ2τ′−2v2κ2τ=0.Proof.
Differentiating (13) with respect to s and v, respectively, we get
(30)3Ψs=−κ2T⟵+−τ2+vκN⟵+τ2B⟵,3Ψv=T⟵,
Realizing the crossproduct of those two vectors, we get the normal vector on NB-Smarandache ruled surface (13):
(31)3Ψs×3Ψv=τ2N⟵+τ2−vκB⟵,
So, under regularity condition, the unit normal takes the form:(32)3Ψs×3Ψv3Ψs×3Ψv=1τ2/2+τ/2−vκ2τ2N⟵+τ2−vκB⟵.
Applying the norms and the scalar product for both vectors (30), we get the components of the first fundamental form of NB-Smarandache ruled surface (13), at regular points, as follows:
(33)3E=κ2+τ22+−τ2+vκ2,3F=−κ2,3G=1.
Differentiating (30) with respect to s and v, respectively, we get
(34)3Ψss=−κ′−κτ2+vκ2T⟵+−κ2+τ2+τ′2+vκ′N⟵+τ′−τ22+vκτB⟵,3Ψvs=κN⟵,3Ψvv=0.
Hence, from (32) to (34), we get the components of the second fundamental form of NB-Smarandache ruled surface (13), at regular points, as follows:
(35)3e=τ/2−κ2+τ2+τ′/2+vκ′+τ/2−vκτ′−τ2/2+vκττ2/2+τ/2−vκ2,3f=κττ2+τ−2vκ2,3g=0.
From (33) to (35), we get the Gaussian curvature and the mean curvature of NB-Smarandache ruled surface (13), at regular points, as follows:
(36)3K=−2κττ2+τ−2vκ22,1c1c1c3H=2τ2κ2τ−2τ2−κ2/2+vκ′τ+2κτ2−κ2τ′−2v2κ2ττ2+τ−2vκ23/2,which replies to both above theorems.Corollary 17.
If NB-Smarandache ruled surface (13) is developable, it is minimal.Example 3.
Let us consider the plane curve3cs=s2,s3,0. Thus, NB-Smarandache ruled surface according to Frenet-Serret frame of 3cs is parametrized by
(37)Ψs,v=124+36s226s6s24s2+9s4+v4s2+9s42s3s20.Figure 2
The TB-Smarandache ruled surface (Equation (29)).It is developable and minimal. The following Figure3 is the illustration of (37) which is drawn for s,v∈−2π,2π×−4,4.Figure 3
The NB-Smarandache ruled surface (Equation (37)).


## 4. Conclusion

The main results of the present work assure that:TN-Smarandache ruled surface according to Frenet-Serret frame is developable (resp., minimal) ifcs is a plane curve (resp. κ and τ satisfy special equation)TB-Smarandache ruled surface according to Frenet-Serret frame is developable. However, it is minimal ifcs is a general helixNB-Smarandache ruled surface according to Frenet-Serret frame is developable (resp., minimal) ifcs is a plane curve (resp. κ and τ satisfy special equation)The investigated theorems prove that developability and minimality conditions of the introducedSmarandache ruled surfaces according to Frenet-Serret frame are related to differential properties of the reference curve. Therefore, we can easily constat that to construct a developable Smarandache ruled surface or a minimal Smarandache ruled surface according to Frenet-Serret frame, we just need to make the right choice for the reference curve c=cs.References[1] D. J. Struik,Lectures on Classical Differential Geometry, 2nd Ed., Addison-Wesley, Dover, 1988.[2] M.P. Do Carmo,Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ, 1976.[3] S. Yilmaz and M. Turgut, "On the differential geometry of the curves in Minkowski spacetime",International Journal of Contemporary Mathematical Sciences, vol. 27, no. 3, pp. 1343-1349, 2008.[4] S. Ouarab, A. O. Chahdi and M. Izid, "Ruled surfaces with alternative moving frame in Euclidean 3-space",International Journal of Mathematical Sciences and Engineering Applications, vol. 12, no. 2, pp. 43-58, 2018.[5] A. T. Ali, "Position vectors of general helices in Euclidean 3-space",Bulletin of Mathematical Analysis and Applications, vol. 3, no. 2, pp. 198-205, 2011.[6] A. T. Ali, "Position vectors of slant helice in Euclidean spaceE3",Journal of the Egyptian Mathematical Society, vol. 20, no. 1, pp. 1-6, 2012.[7] S. Ouarab, A. O. Chahdi, M. Izid, "Ruled surface generated by a curve lying on a regular surface and its characterizations",Journal for Geometry and Graphics, vol. 24, no. 2, pp. 257-267, 2020.[8] G. Hu, H. Cao, J. Wu and G. Wei, "Construction of developable surfaces using generalized C-Bézier bases with shape parameters",Computational and Applied Mathematics, vol. 39, no. 3, 2020.[9] T. G. Nelson, T. Zimmerman, S. P. Magleby, R. J. Lang and L. L. Howell, "Developable mechanisms on developable surfaces",Science Robotics, vol. 4, no. 27, 2019.[10] P. Alegre, K. Arslan, A. Carriazo, C. Murathan and G. Öztürk, "Some special types of developable ruled surface",Hacettepe Journal of Mathematics and Statistics, vol. 39, no. 3, pp. 319-325, 2010.[11] W. Y. Lam, "Minimal surfaces from infinitesimal deformations of circle packings",Advances in Mathematics, vol. 362, pp. 106-939, 2020.[12] T. H. Colding and C. De Lellis, "The min-max construction of minimal surfaces",Surveys in Differential Geometry, vol. 8, pp. 75-107, 2003.[13] J. L. M. Barbosa and A. G. Colares,Minimal Surfaces in3, Springer Verlag, Berlin Heidelberg, 1986.[14] O. Bektas and S. Yüce, "Special Smarandache curves according to Darboux frame in Euclidean 3- space",Romanian Journal of Mathematics and Computer Science, vol. 3, no. 1, pp. 48-59, 2013.[15] E. M. Solouma and W. M. Mahmoud, "On spacelike equiform-Bishop Smarandache curves onS12,Journal of the Egyptian Mathematical Society, vol. 27, no. 7, 2019.[16] M. Cetin, Y. Tunçer and M. K. Karacan, "Smarandache curves according to Bishop frame in Euclidean 3-space",General Mathematics Notes, vol. 20, no. 2, pp. 50-66, 2014.[17] A. T. Ali, "Special Smarandache curves in the Euclidean space",International Journal of Mathematical Combinatorics, vol. 2, pp. 30-36, 2010.[18] M. Turgut and S. Yilmaz, "Smarandache curves in Minkowski space-time",International Journal of Mathematical Combinatorics, vol. 3, pp. 51-55, 2008.[19] B. O’Neill,Sem-Riemannian Geometry, Academic press, New York, 1983.



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