Smarandache Ruled Surfaces According to Frenet-Serret Frame of a Regular Curve in E3

Hindawi Abstract and Applied Analysis Volume 2021, Article ID 5526536, 8 pages https://doi.org/10.1155/2021/5526536

Research Article Smarandache Ruled Surfaces according to Frenet-Serret Frame of a Regular Curve in E3

Soukaina Ouarab

Department of Mathematics and Computer Science, Hassan II University of Casablanca, Ben M’sik Faculty of Sciences, Morocco

Correspondence should be addressed to Soukaina Ouarab; [email protected]

Received 30 January 2021; Accepted 16 March 2021; Published 1 April 2021

Academic Editor: Chun Gang Zhu

Copyright © 2021 Soukaina Ouarab. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we introduce original deﬁnitions of Smarandache ruled surfaces according to Frenet-Serret frame of a curve in E3.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.

1. Introduction Also, they investigated properties of the constructed ruled surface. Moreover, they gave examples with illustrations. In differential geometry of curves and surfaces [1–3], a ruled The notions of developability and minimalist are two of surface defines the set of a family of straight lines depending the most important properties of surfaces. on a parameter. The straight lines mentioned are the rulings The ruled surfaces with vanishing Gaussian curvature, of the ruled surface. The general parametric representation of which can be transformed into the plane without any defor- ! a ruled surface is Ψðs, vÞ = cðsÞ + vXðsÞ where cðsÞ is a curve mation and distortion, are called developable surfaces; they through which all rulings pass; it is called the base curve of form a relatively small subset that contains cylinders, cones, and the tangent surfaces [8–10]. ! fi the surface; the vector XðsÞ de nes the ruling direction. A minimal surface is a surface that locally minimizes its It is well known that ruled surfaces are of great interest to area. It is referred to the fixed boundary curve of a surface many applications and have contributed in several areas, area that is minimal with respect to other surfaces with the such as mathematical physics, kinematics, and Computer same boundary. This is equivalent to having zero mean cur- Aided Geometric Design (CAGD). vature [11–13]. ff Ruled surfaces have been studied di erently by an impor- In curve theory, Smarandache curves are one of the spe- tant number of researchers. In [4], the authors constructed cial innovated curves that were introduced at the first time the ruled surface whose rulings are constant linear combina- in Minkowski space-time by authors in [14]. It is about tions of alternative moving frame vectors of its base curve; curves whose position vectors are composed by Frenet- they studied the ruled surface properties, characterize it, Serret frame vectors on another regular curve. In [15–17], and presented examples with illustrations in the case of gen- we can find several research works about Smarandache eral helices [5] and slant helices [6] as base curves. In [7], the curves according to different frames such as the Frenet- authors constructed the ruled surface whose rulings are con- Serret frame, Bishop frame, and Darboux frame in Euclidean stant linear combinations of Darboux frame vectors of a reg- and Minkowski spaces. 3 ular curve lying on a regular surface of reference in E . Their The motivation of this work is inspired by ruled surface point of interest was to make a comparative study between and Smarandache curve. We are eager to introduce new def- the two surfaces (the regular surface of reference and the initions that combine those two important notions and study new constructed ruled surface) along their common curve. their properties. We are also opening up opportunities to 2 Abstract and Applied Analysis

! perceive future works that are relative to applications in dif- Let Ψ : ðs, vÞ ↦ cðsÞ + vXðsÞ be a ruled surface in E3. ferential geometry, physics, and medical science. ! ! Let is denote by n = nðs, vÞ the unit normal on ruled sur- In this paper, we are interested in ruled surfaces gener- face Ψ at a regular point Ψðs, vÞ; we have ated by Smarandache curves according to the Frenet-Serret fi frame. Indeed, we construct and introduce original de nitions ! ′ + ′ × ! of three special ruled surfaces generated by TN-Smarandache Ψ ∧ Ψ c vX X curve, TB-Smarandache curve, and NB-Smarandache curve !n = s v = , ð6Þ Ψ ∧ Ψ ! ! according to the Frenet-Serret frame of an arbitrary regular kks v c′ + vX′ × X curve in E3. We investigate theorems that give us necessary and sufficient conditions for those three ruled surfaces to be Ψ = ∂Ψ , /∂ Ψ = ∂Ψ , /∂ developable and minimal. Finally, we give examples with where s ð ðs vÞÞ s and v ð ðs vÞÞ v. illustrations. Definition 4 (see [19]). The distribution parameter λ = λðsÞ of 2. Preliminaries ruled surface Ψ is given by 3 ! In the Euclidean 3-space E , we consider the usual metric det ′, !, ′ given by c X X λ = : 7 2 ð Þ , = + + , 1 ! hi dx1 dx2 dx3 ð Þ X′

3 where (x1, x2, x3) is a rectangular coordinate system of E . 3 Definition 5 (see [19]). A ruled surface is developable if its Acurvec : s ∈ I ⊂ ℝ → E is said to be of unit speed (or qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi distribution parameter vanishes. parameterized by the arc length) if kc′ðsÞk = hc′ðsÞ, c′ðsÞi The first I and the second II fundamental forms of ! ! ! ruled surface Ψ at a regular point Ψðs, vÞ are defined, =1for any s ∈ I. For such curves, there is a frame fT, N, Bg ! ! respectively, by which is called the Frenet-Serret frame, where T = c′, N = c″ 2 2 / ″ ! = ! × ! I ðÞΨ ds + Ψ dv = Eds +2Fdsdv + Gdv , kc k,andB T N are the unit tangent, the principal nor- s v ð8Þ mal, and the binormal vector of the curve, respectively. The Ψ + Ψ = 2 +2 + 2, II ðÞsds vdv eds f dsdv gdv Frenet-Serret formulae of cðsÞ is given by 2 3 = Ψ 2, = Ψ , Ψ , = Ψ 2, = Ψ , ! , = ! 2 32 3 where E k sk F h s vi G k vk e h ss ni f ′ 0 κ 0 ! ! ! 6 T 7 T hΨ , ni, and g = hΨ , ni =0. 6 7 6 76 7 sv vv ! = 6 −κ 0 τ 76 ! 7, 2 6 N′ 7 4 54 N 5 ð Þ 4 5 Definition 6. The Gaussian curvature K and the mean curva- ! 0 −τ 0 ! Ψ Ψ , B′ B ture H of ruled surface at a regular point ðs vÞ are given, respectively, by where κ = κðsÞ are the curvature and the torsion functions of 2 f κ = ″ τ =det ′, ″, ‴ / = − , cðsÞ and they are given by kc k and ðc c c Þ K − 2 ″ 2 EG F 9 kc k , respectively [2]. − 2 ð Þ = ÀÁGe Ff : H 2 − 2 Definition 1 (see [18]). TN-Smarandache curves according to EG F the Frenet-Serret frame of the curve c = cðsÞ are given by Proposition 7 (see [19]). A ruled surface is developable if and ∗ 1 ! ! only if its Gaussian curvature vanishes. βðÞs ðÞs = pffiffiffi TðÞs + NðÞs : ð3Þ 2 Proposition 8 (see [19]). A regular surface is minimal if and Definition 2 (see [18]). TB-Smarandache curves according to only if its mean curvature vanishes. the Frenet-Serret frame of the curve c = cðsÞ are given by 3. Smarandache Ruled Surfaces according to 3 ∗ 1 ! ! γðÞs ðÞs = pffiffiffi TðÞs + BðÞs : ð4Þ Frenet-Serret Frame of a Regular Curve in E 2 In a first step, we start our section by giving the following new Definition 3 (see [18]). NB-Smarandache curves according to definitions of Smarandache ruled surfaces according to the 3 the Frenet-Serret frame of the curve c = cðsÞ are given by Frenet-Serret frame of a curve in E .

2 1 ! ! fi = ff δ ∗ = ffiffiffi + : 5 De nition 9. Let c cðsÞ be the C class di erentiable unit ðÞs ðÞs p NðÞs BðÞs ð Þ ! ! ! 2 speed curve whose Frenet-Serret apparatus is fTðsÞ, NðsÞ, B Abstract and Applied Analysis 3

ðsÞ, κðsÞ, τðsÞg in E3. The ruled surfaces generated by Smarandache ruled surface of Equation (10), at regular Smarandache curves according to Frenet-Serret of c = cðsÞ points, as follows: are as follows: 2κ2 + τ2 pffiffiffi 1 1 = − 2κτ + τ2 2, ð17Þ 1 ! ! ! E 2 v v ΨðÞs, v = pffiffiffi TðÞs + NðÞs + vBðÞs , ð10Þ 2 1 τ F = pffiffiffi , ð18Þ 2 1 ! ! ! 2 ΨðÞs, v = pffiffiffi TðÞs + BðÞs + vNðÞs , ð11Þ 2 1 G =1: ð19Þ 3 1 ! ! ! ΨðÞs, v = pffiffiffi NðÞs + BðÞs + vTðÞs : ð12Þ 2 Differentiating Equation (13) with respect to s and v, respectively, we get These ruled surfaces are called TN-Smarandache ruled sur- "#"# face, TB-Smarandache ruled surface, and NB-Smarandache κ′ κ −κ2 − τ2 + ′ 1Ψ = − ffiffiffi + κ ffiffiffi − τ ! + ffiffiffi k − τ′ ! ruled surface, according to the Frenet-Serret frame of the curve ss p p v T p v N = 2 2 2 c cðsÞ, respectively. "# τ′ κ ! + pffiffiffi + τ pffiffiffi − vτ B, Following of this section, we investigate theorems that 2 2 give necessary and sufficient conditions for Smarandache ruled surfaces (Equation (10)) to be developable and mini- ð20Þ mal. Also, we present an example with illustration for each Smarandache ruled surface. 1Ψ = −τ , 21 vs N ð Þ Theorem 10. The TN-Smarandache ruled surface of Equation 1Ψ =0: ð22Þ (10) is developable if and only if c = cðsÞ is a plane curve. vv

Theorem 11. The TN-Smarandache ruled surface of Equation Hence, from Equations (16) and (20), we get the (10) is minimal if and only if the curvature and the torsionp offfiffiffi components of the second fundamental form of the the curve c = cðsÞ satisfy the equation κðτ2 − κ′ − 2κ2Þ + v 2 TN-Smarandache ruled surface of Equation (10), at regu- ðκ′τ + 2κ2τÞ − v22κτ2 = 0. lar points, as follows: pffiffiffi pffiffiffi ÀÁÀÁ Proof. Differentiating the first line of Equation (10) with κ/ 2 − τ κ′ + κ2 / 2 − κτ + κκ2 + τ2 /2 1 v v respect to s and v, respectively, we get e = − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffi , κ2 − 2κτ + τ2 2 v v 1 κ ! κ ! τ ! ð23Þ Ψ = − pffiffiffi T + pffiffiffi − vτ N + pffiffiffi B, ð13Þ s 2 2 2 1 κτ f = − pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffi , ð24Þ 2 κ2 − 2κτ + τ2 2 1Ψ = !: ð14Þ v v v B 1 g =0: ð25Þ The crossproduct of these two vectors gives the normal vector on the TN-Smarandache ruled surface of Equation (10): From Equations (17) and (23), we get the Gaussian curvature and the mean curvature of the TN-Smarandache ruled κ κ surface of Equation (10), at regular points, as follows: 1Ψ × 1Ψ = ffiffiffi − τ ! + ffiffiffi !: 15 s v p v T p N ð Þ 2 2 2 32 κτ 1 = −4 5 , So under regularity condition, the unit normal takes K pffiffiffi pffiffiffi 2 κ2 − 2κτ + τ2 2 the form v v ffiffiffi 1 1 3 2 p 2 2 2 Ψ × Ψ 1 κ ! κ ! −κκ′ − 2κ + κτ + v 2 κ′τ +2κ τ − v 2κτ s v = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi − τ + ffiffiffi : 1 1 1 pffiffiffi p v T p N = , kkΨ × Ψ κ2 − 2κτ + τ2 2 2 2 H pffiffiffi 3/2 s v v v 4 κ2 − 2κτv + τ2v2 ð16Þ ð26Þ Making the norms for Equation (13), we get the components of the first fundamental form of the TN- which replies to both the above theorems. 4 Abstract and Applied Analysis

−5

−10 1.5 1 −0.4 −0.2 0.5 0 0.2 0.4 0 0.6 0.8 −0.5 1 1.2 −1 1.4

Figure 1: The TN-Smarandache ruled surface (Equation (27)).

Example 12. Let us consider the regular plane curve 1cðsÞ = which implies that the normal vector on the TB-Smaran- ð3s − s3,3s2,0Þ. The TN-Smarandache ruled surface accord- dache ruled surface of Equation (10) is ing to the Frenet-Serret frame of the curve 1cðsÞ is given by 2Ψ × 2Ψ = − τ! − κ: 30 0 1 s v v T v ð Þ 1 − s2 s B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − pffiffiffiffiffiffiffiffiffiffiffi C B 2 2 2 1+s2 C So under regularity condition, the unit normal is given by B ðÞ1 − s +4s C B C 1 B C ! ! 1Ψ , = ffiffiffi B 2 1 C 2Ψ × 2Ψ τ + κ ðÞs v p s s v = ∓ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT B : 31 2 B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − pffiffiffiffiffiffiffiffiffiffiffi C 2 2 ð Þ B 2 C Ψ × Ψ κ2 + τ2 B 1 − 2 2 +4 2 1+s C kks v ðÞ @ ðÞs s A 0 ð27Þ Making the norms for Equation (28), we get the compo- 0 1 fi 0 nents of the rst fundamental form of the TB-Smarandache B C ruled surface of Equation (10), at regular points: B C B 0 C B C κ − τ 2 ÀÁ + B C: 2 ðÞ2 2 2 32 vB 1 − +2 2 C E = + v κ − τ , ð Þ B rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis s C 2 @ ÂÃhiffiffiffiffiffiffiffiffiffiffiffi A 2 2 2 p 2 ðÞ1 − s +4s 1+s 2 κ − τ F = pffiffiffi , 33 2 ð Þ It is developable but not minimal at its regular points. 2 =1: 34 Figure 1 is the illustration of Equation (27) drawn for ðs, vÞ G ð Þ ∈ −2π,2π½ × −4, 4½. Differentiating Equation (28) with respect to s and v, Theorem 13. The TB-Smarandache ruled surface of Equation respectively, we get (10) is developable. 2 3 κκ− τ κ′ + τ′ ÀÁ Theorem 14. The TB-Smarandache ruled surface of Equation 2Ψ = − ðÞffiffiffi + κ′ ! + 4 ffiffiffi − κ2 − τ2 5! ss p v T p v N (10) is minimal if and only if the curve c = cðsÞ is a general helix. 2 2 ff τκðÞ− τ ! Proof. Di erentiating the second line of Equation (10) with + pffiffiffi + vτ′ B, respect to s and v, respectively, we get 2 ð35Þ κ − τ 2Ψ = − κ! + ffiffiffi ! + τ!, s v T p N v B ð28Þ 2 2Ψ = −κ! + τ!, ð36Þ vs T B

2Ψ = !, ð29Þ 2Ψ =0: 37 v N vv ð Þ Abstract and Applied Analysis 5

2.5

1.5

0.5

0 4 3 4 2 3 1 2 1 0 0 −1 −1 −2 −2 −3 −3 −4 −4

Figure 2: The TB-Smarandache ruled surface (Equation (42)).

Hence, from Equations (31) and (35), we get the Then, the TB-Smarandache ruled surface according to components of the second fundamental form of the TB- the Frenet-Serret frame of 2cðsÞ is Smarandache ruled surface of Equation (10) at regular points: 0 pffiffiffiffi pffiffiffiffi 1 0 pffiffiffiffi 1 cos 2 − sin 2 −cos 2 B s s C B s C ! B C B C 2 1 ffiffiffiffi ffiffiffiffi ffiffiffiffi τ Ψ s, v = B p p C + vB p C: κ2 ′ ðÞ2 B cos 2s + sin 2s C B −sin 2s C 38 @ A @ A 2 κ ð Þ e = ∓v pffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 1 1 κ2 + τ2 ð42Þ 2 =0, 39 f ð Þ It is developable and minimal at its regular points. Figure 2 is the illustration of Equation (42) drawn for ðs, vÞ 2 ∈ −2π,2π × −4, 4 g =0: ð40Þ ½ ½. Theorem 17. The NB-Smarandache ruled surface of Equation = From Equations (32) and (38), we get the Gaussian (10) is developable if and only if c cðsÞ is a plane curve. curvature and the mean curvature of the TB-Smarandache Theorem 18. ruled surface of Equation (10), at regular points, as The NB-Smarandache ruled surface of Equation follows: (10) is minimal if and only if the curvature and the torsion of the curve c = cðsÞ satisfy the equation 2 K =0, ! κ2τ − τ2 − κ2 pffiffiffi τ 2 2 ′ 2 2 ′ 2 2 2 τ pffiffiffi + v κ τ + 2κτ − κ τ − 2v κ τ = 0: κ ′ ð41Þ 2 κ 2 = ∓ , 43 H 3/2 ð Þ 2vðÞκ2 + τ2 Proof. Differentiating the third line of Equation (10) with which replies to both the above theorems. respect to s and v, respectively, we get Corollary 15. If cðsÞ is a general helix, the TB-Smarandache 3 κ ! τ ! τ ! Ψ = − pffiffiffi T + − pffiffiffi + vκ N + pffiffiffi B, ð44Þ ruled surface of Equation (10) is developable and minimal. s 2 2 2 2 =1/ Exampleffiffiffi 16.ffiffiffiffi Here,ffiffiffi we choseffiffiffiffi theffiffiffi regular curve cðsÞ ! p p p p p 3Ψ = : ð45Þ 2ðsin ð 2sÞ/ 2,− cos ð 2sÞ/ 2, sÞ as a general helix. v T 6 Abstract and Applied Analysis

0.25

0.2

0.15

0.1 −3 0.05 −2 25 20 −1 15 10 0 5 0 1 −5 −10 −15 2 −20 −25 3

Figure 3: The NB-Smarandache ruled surface (Equation (58)).

Realizing the crossproduct of those two vectors, we get 3 κ F = − pffiffiffi , 49 the normal vector on the NB-Smarandache ruled surface of 2 ð Þ Equation (10): 3 50 G =1: ð Þ 3 3 τ ! τ ! Ψ × Ψ = pffiffiffi N + pffiffiffi − vκ B: ð46Þ ff s v 2 2 Di erentiating Equation (44) with respect to s and v, respectively, we get "#"# So under regularity condition, the unit normal takes the ′ 2 2 ′ 3 κ − κτ 2 ! κ + τ + τ ! form Ψ = − pffiffiffi + vκ T + pffiffiffi + vκ′ N ss 2 2 3Ψ × 3Ψ 1 τ τ "# ð51Þ s v = rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi ! + ffiffiffi − κ ! : τ′ + τ2 3 3 p N p v B ! kkΨ × Ψ pffiffiffi 2 2 2 + pffiffiffi + vκτ B, s v τ2/2 + τ/ 2 − vκ 2

ð47Þ 3Ψ = −κ!, ð52Þ vs N Making the norms for Equation (44), we get the compo- 3Ψ =0: 53 nents of the first fundamental form of the NB-Smarandache vv ð Þ ruled surface of Equation (10), at regular points, as follows: Hence, from Equations (47) and (51), we get the κ2 + τ2 τ 2 components of the second fundamental form of the NB- 3 = + − ffiffiffi + κ , ð48Þ Smarandache ruled surface of Equation (10), at regular E 2 p v 2 points, as follows:

pffiffiffihipffiffiffi pffiffiffi hipffiffiffi τ/ 2 − κ2 + τ2 + τ′ / 2 + vκ′ + τ/ 2 − vκ τ′ + τ2 / 2 + vκτ 3 e = rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 54 pffiffiffi 2 ð Þ τ2/2 + τ/ 2 − vκ

3 κτ f = rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, pffiffiffi 2 ð55Þ τ2 + τ − 2vκ

3 g =0: ð56Þ Abstract and Applied Analysis 7

From Equations (48) and (54), we get the Gaussian cur- Data Availability vature and the mean curvature of the NB-Smarandache ruled surface of Equation (10), at regular points, as follows: No data were used to support the study.

2 32 pffiffiffi Conflicts of Interest 6 2κτ 7 3 = −4 5 , K pffiffiffi 2 The author declares that there are no conflicts of interest. τ2 + τ − 2vκ References hiÀÁffiffiffi hiffiffiffi 2 2 2 p ′ 2 2 ′ p 2 2 τ 2κ τ − 2τ − κ / 2 + v κ τ +2κτ − κ τ − 2v κ τ ff 3 =2 , [1] D. J. Struik, Lectures on Classical Di erential Geometry, Dover, H 3/2 pffiffiffi 2 AddisonWesley, 2nd edition, 1988. τ2 + τ − 2vκ [2] M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ, 1976. ð57Þ [3] S. Yilmaz and M. Turgut, “On the differential geometry of the curves in Minkowski spacetime,” International Journal of Con- which replies to both the above theorems. temporary Mathematical Sciences, vol. 27, no. 3, pp. 1343– 1349, 2008. Corollary 19. If the NB-Smarandache ruled surface of [4] S. Ouarab, A. O. Chahdi, and M. Izid, “Ruled surfaces with Equation (10) is developable, it is minimal. alternative moving frame in Euclidean 3-space,” International Journal of Mathematical Sciences and Engineering Applica- – Example 20. Let us consider the plane curve 3cðsÞ = ðs2, s3,0Þ. tions, vol. 12, no. 2, pp. 43 58, 2018. “ Thus, the NB-Smarandache ruled surface according to the [5] A. T. Ali, Position vectors of general helices in Euclidean 3- 3 space,” Bulletin of Mathematical Analysis and Applications, Frenet-Serret frame of cðsÞ is parametrized by vol. 3, no. 2, pp. 198–205, 2011. [6] A. T. Ali, “Position vectors of slant helice in Euclidean space 0 1 0 1 3 2 E ,” Journal of the Egyptian Mathematical Society, vol. 20, 2s 1 B C B C no. 1, pp. 1–6, 2012. B 6s C v B 2 C ΨðÞs, v = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B C + pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ 3s A: [7] S. Ouarab, A. O. Chahdi, and M. Izid, “Ruled surface generated 24+36ðÞs2 @ 6 2 A 4s2 +9s4 ” ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis by a curve lying on a regular surface and its characterizations, p 0 – 4s2 +9s4 Journal for Geometry and Graphics, vol. 24, no. 2, pp. 257 267, 2020. ð58Þ [8] G. Hu, H. Cao, J. Wu, and G. Wei, “Construction of developable surfaces using generalized C-Bézier bases with shape It is developable and minimal. Figure 3 is the illustration of parameters,” Computational and Applied Mathematics, Equation (58) which is drawn for ðs, vÞ ∈ −2π,2π½ × −4, 4½. vol. 39, no. 3, 2020. [9] T. G. Nelson, T. K. Zimmerman, S. P. Magleby, R. J. Lang, and 4. Conclusion L. L. Howell, “Developable mechanisms on developable surfaces,” Science Robotics, vol. 4, no. 27, article eaau5171, 2019. The main results of the present work assure the following: [10] P. Alegre, K. Arslan, A. Carriazo, C. E. Murathan, and G. Öztürk, “Some special types of developable ruled surface,” (i) The TN-Smarandache ruled surface according to the Hacettepe Journal of Mathematics and Statistics, vol. 39, Frenet-Serret frame is developable (resp., minimal) no. 3, pp. 319–325, 2010. if cðsÞ is a plane curve (resp., κ and τ, satisfy special [11] W. Y. Lam, “Minimal surfaces from infinitesimal deformations equation) of circle packings,” Advances in Mathematics, vol. 362, pp. 106–939, 2020. (ii) The NB-Smarandache ruled surface according to the [12] T. H. Colding and C. de Lellis, “The min-max construction of Frenet-Serret frame is developable (resp., minimal) minimal surfaces,” Surveys in Differential Geometry, vol. 8, if cðsÞ is a plane curve (resp., κ and τ, satisfy special no. 1, pp. 75–107, 2003. equation) [13] J. L. M. Barbosa and A. G. Colares, Minimal Surfaces in R3, Springer Verlag, Berlin heidelberg, 1986. (iii) The investigated theorems prove that developabil- [14] O. Bektas and S. Yuce, “Special Smarandache curves according ity and minimality conditions of the introduced to Darboux frame in Euclidean 3- space,” Romanian Journal of Smarandache ruled surfaces according to the Mathematics and Computer Science, vol. 3, no. 1, pp. 48–59, ff Frenet-Serret frame are related to di erential prop- 2013. erties of the reference curve. Therefore, we can easily [15] E. M. Solouma and W. M. Mahmoud, “On spacelike equiform- 2 constate that to construct a developable Smaran- Bishop Smarandache curves on S1,” Journal of the Egyptian dache ruled surface or a minimal Smarandache ruled Mathematical Society, vol. 27, no. 1, 2019. surface according to the Frenet-Serret frame, we just [16] M. Cetin, Y. Tuncer, and M. K. Karacan, “Smarandache curves need to make the right choice for the reference curve according to Bishop frame in Euclidean 3-space,” General c = cðsÞ Mathematics Notes, vol. 20, no. 2, pp. 50–66, 2014. 8 Abstract and Applied Analysis

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