Computation of Smarandache curves according to Darboux frame in Minkowski 3-space

Journal of the Egyptian Mathematical Society 25 (2017) 382–390

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Journal of the Egyptian Mathematical Society journal homepage: www.elsevier.com/locate/joems

Original Article

Computation of Smarandache curves according to Darboux frame in Minkowski 3-space H.S. Abdel-Aziz a, M. Khalifa Saad a,b,∗ a b

Math. Dept., Faculty of Science, Sohag University, 82524 Sohag, Egypt Math. Dept., Faculty of Science, Islamic University in Madinah, Saudi Arabia

a r t i c l e

i n f o

Article history: Received 6 April 2017 Revised 9 May 2017 Accepted 28 May 2017 Available online 7 June 2017

a b s t r a c t In this paper, we study Smarandache curves according to Darboux frame in the three-dimensional Minkowski space E13 . Using the usual transformation between Frenet and Darboux frames, we investigate some special Smarandache curves for a given timelike curve lying fully on a timelike surface. Finally, we defray a computational example to conﬁrm our main results.

MSC: 53A04 53A35 53C50

Keywords: Smarandache curves Timelike curves Timelike surfaces Darboux frame Minkowski 3-space

1. Introduction In the study of the fundamental theory and the characterizations of space curves, the corresponding relations between the curves are very interesting and important problem. Among all space curves, Smarandache curves have special emplacement regarding their properties, because of this they deserve especial attention in Euclidean geometry as well as in other geometries. It is known that Smarandache geometry is a geometry which has at least one Smarandache denied axiom [1]. An axiom is said to be Smarandache denied, if it behaves in at least two different ways within the same space. Smarandache geometries are connected with the theory of relativity and the parallel universes. Smarandache curves are the objects of Smarandache geometry. By deﬁnition, if the position vector of a curve δ is composed by Frenet frame’s vectors of another curve β , then the curve δ is called a Smarandache curve [2]. In three-dimensional curve theory, each unit speed curve with at least four continuous derivatives, one can associate three mutually orthogonal unit vector ﬁelds T, N and B

∗

Corresponding author. E-mail addresses: habdelaziz2005@yahoo.com (H.S. Abdel-Aziz), mohamed_khalifa77@science.sohag.edu.eg (M. Khalifa Saad).

(Frenet frame vectors) are respectively, the tangent, the principal normal and the binormal vector fields. In the light of the existing studies in the field of geometry, many interesting results on Smarandache curves have been obtained by many mathematicians [3–9]. Turgut and Yilmaz have introduced a particular circumstance of such curves, they entitled it Smarandache TB2 curves in the space E41 [2]. They studied special Smarandache curves which are defined by the tangent and second binormal vector fields. In [4], the author has illustrated certain special Smarandache curves in the Euclidean space. Special Smarandache curves in such a manner that Smarandache curves TN1 , TN2 , N1 N2 and TN1 N2 with respect to Bishop frame in Euclidean 3-space have been seeked for by Çetin et al.[6]. Furthermore, they worked differential geometric properties of these special curves and checked out first and second curvatures of these curves. Also, they get the centers of the curvature spheres and osculating spheres of Smarandache curves. Recently, H.S. Abdel-Aziz and M. Khalifa Saad have studied special Smarandache curves of an arbitrary curve such as TN, TB and TNB with respect to Frenet frame in the three-dimensional Galilean and pseudo-Galilean spaces [3,7]. The main goal of this article is to introduce and describe some special Smarandache curves in E13 for a given timelike surface and a timelike curve lying fully on it with reference to its Darboux

http://dx.doi.org/10.1016/j.joems.2017.05.004 1110-256X/© 2017 Egyptian Mathematical Society. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382–390

frame. We looking forward to that our results will be helpful to researchers who are specialized on mathematical modeling and other applications. 2. Basic concepts Let us ﬁrst recall the basic notions from Lorentz geometry [10]. Let R3 = {(x1 , x2 , x3 ) | x1 , x2 , x3 ∈ R }be a 3-dimensional vector space, and let x = (x1 , x2 , x3 ) and y = (y1 , y2 , y3 ) be two vectors in R3 . The Lorentz scalar product of x and y is deﬁned by

x , y L = x1 y1 + x2 y2 − x3 y3 . E13 = (R3 , x, y L ) is called 3-dimensional Lorentzian space, Minkowski 3-space or 3-dimensional Semi-Euclidean space. The arbitrary vector x in E13 can have one of three Lorentzian causal characters; it can be spacelike, timelike and lightlike (null) if x, x L > 0 or x = 0, x, x L < 0 and x, x L = 0 and x = 0, respectively. Similarly, a curve r, locally parameterized by r = r (s ) : I ⊂ R −→ E13 where s is pseudo arclength parameter, is called a spacelike curve if r (s), r (s) L > 0, timelike if r (s), r (s) L < 0 and lightlike if r (s ), r (s ) L = 0 and r (s) = 0 for all s ∈ I. The two vectors x = (x1 , x2 , x3 ), y = (y1 , y2 , y3 ) ∈ E13 are orthogonal if and only if x , y L = 0. Also, for any x, y ∈ E13 , Lorentzian vector product of x and y is deﬁned by

e1 x ×L y = x 1 y1

e2 x2 y2

−e3 x3 . y3

Deﬁnition 1. A surface M in the Minkowski 3-space E13 is said to be spacelike, timelike surface if, respectively the induced metric on the surface is a positive deﬁnite Riemannian metric, Lorentz metric. In other words, the normal vector on the spacelike (timelike) surface is a timelike (spacelike) vector [10].

Consider a timelike curve r = r (s ) in E13 , parameterized by its arc length s and lying fully on an oriented timelike surface M. Let T, N, B be the tangent, principal normal and binormal vector ﬁelds along r(s). Then, Frenet frame is given by

=

0

κ

0

0 −τ

κ

0

τ

0

T N , B

(1)

where a prime denotes differentiation with respect to s. For this frame the following are satisfying

T, T = −1, B, B = N, N = 1, T, N = T, B = N, B = 0 . The coeﬃcients κ and τ are the curve’s curvature and torsion. Let {T, P, U} be the Darboux frame of r(s) with T as the tangent vector of r and U is the unit normal to the surface M and P = T × U, then the usual transformation between Frenet and Darboux frames takes the form [10,11]:

T P U

=

1 0 0

0 cosh θ sinh θ

T P U

κg

0

κg κN

=

κN −τg

0

τg

0

T P , U

0 sinh θ cosh θ

T N , B

(3)

where κ N , κ g and τ g are the normal curvature, geodesic curvature and geodesic torsion, respectively. In the differential geometry of surfaces, for a curve r = r (s ) lying on a surface M the following are well-known [11] (i) r(s) is a geodesic curve if and only if κg = 0. (ii) r(s) is an asymptotic line if and only if κN = 0. (iii) r(s) is a principal line if and only if τg = 0. Definition 2. A regular curve in Minkowski 3-space, whose position vector is composed by Frenet frame vectors on another regular curve, is called a Smarandache curve [2]. In the following, we investigate Smarandache curves TP, TU, PU and TPU and study some of their properties which represent the main results. 3.1. TP−Smarandache curves From the Definition 2, the TP− Smarandache curve of r can be defined by

(4)

2

Differentiating Eq. (4) with respect to s, we obtain

α =

dα ds¯ 1 = √ (κg T+κg P + (κN − τg )U ), ds¯ ds 2

with the parameterization

ds¯ 1 = √ (κN − τg ), ds 2

(5)

and then

T¯ α =

1

(κN − τg )

(κg T+κg P + (κN −τ g )U ).

(6)

Differentiating Eq. (6) with respect to s and using Eq. (5), we get

3. Smarandache curves of a timelike curve on a timelike surface

T N B

1

The norm of a vector x ∈ E13 is given by x L = | x, x L |. We denote by {T, N, B} the moving Frenet frame along the curve r(s) in the Minkowski space E13 , where the vectors T, N, B are called the vectors of the tangent, principal normal and the binormal of r, respectively. Consider now the following deﬁnition that we are needed throughout this study.

where θ is the angle between the vectors P and N. Therefore, the Darboux frame of r(s) is given as follows

α (s¯) = √ (T + P ).

383

(2)

√ dT¯ α − 2 = (ω1 T + ω2 P + ω3 U ), ds¯ (κN − τg )3

where

ω1 = τg κg − κN κg + (κN − τg ) κg + κg 2 + κN 2 − κN τg , ω2 = τg κg − κN κg + (κN − τg ) κg + κg 2 + (κN − τg )τg , ω3 = κg (κN − τg )2 . The curvature and torsion of α are given as follows

dT¯ α 1 2 2 2 = κ¯ α = ds¯ (κ − τg )3 2 −ω1 + ω2 − ω3 , N and

¯α = N

ω1 T + ω2 P + ω3 U . −ω12 + ω22 − ω32

On the other hand, we express

¯ = B¯ α = − T¯ × N

−1

(κN − τg ) −ω12 + ω22 − ω32

(7)

384

H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382â€“390

T Îşg Ď‰1

P

Îşg Ď‰2

âˆ’U (ÎşN âˆ’Ď„ g ) .

Ď‰3

3. If Îą is a principal line, the following hold

Îşg âˆ’ 2ÎşN 2Îşg + 3Îşg 2 + ÎşN 2 ÎşÂŻ Îą = , ÎşN 3 2 Îşg 2 ÎşN 3 ÎşN Îş g + Îş N 23Îşg + 52 ÎşN4 âˆ’ÎşN ÎşN Îşg 5Îşg + 2 Îşg + ÎşN + Îşg + Îşg ÎşN Ď„ÂŻÎą = . âˆš 2 2 4ÎşN Îşg âˆ’ 2ÎşN 2Îşg + 3Îşg 2 + ÎşN 2

So, the binormal vector of Îą is

BÂŻ Îą =

âˆ’1

(Ď‰ÂŻ 1 T + Ď‰ÂŻ 2 P + Ď‰ÂŻ 3 U ), (ÎşN âˆ’ Ď„g ) âˆ’Ď‰12 + Ď‰22 âˆ’ Ď‰32

where

Ď‰ÂŻ 1 = (Ď„g âˆ’ ÎşN )Ď‰2 + Îşg Ď‰3 , Ď‰ÂŻ 2 = (ÎşN âˆ’ Ď„g )Ď‰1 âˆ’ Îşg Ď‰3 , Ď‰ÂŻ 3 = Îşg Ď‰1 âˆ’ Îşg Ď‰2 . We consider the derivatives Îą ,

1

Îą = âˆš

4ÎşN

3.2. TUâˆ’Smarandache curves

Îą with respect to s as follows

(Îşg âˆ’ Ď„g ÎşN + ÎşN2 + Îşg2 )T + (Îşg âˆ’ Ď„g2 + Ď„g ÎşN + Îşg2 )P +(âˆ’Ď„g + ÎşN âˆ’ Îşg Ď„g + Îşg ÎşN )U ,

Let r = r (s ) be a timelike curve lying on an oriented timelike surface M in Minkowski 3-space E13 . Using DeďŹ nition 2, the TUâˆ’ Smarandache curve of r is given by

1

Î˛ (sÂŻ) = âˆš (T + U ),

2

(9)

2

it leads to

1

(ÎşN T+(Îşg + Ď„g )P+ÎşN U ).

Îą = âˆš (Îť1 T + Îť2 P + Îť3 U ),

TÂŻ Î˛ =

where

By taking the derivative of Eq. (10) with respect to s, we have

2

Îť1 = âˆ’2Ď„g ÎşN + ÎşN (3ÎşN âˆ’ Ď„g ) + Îşg 3Îşg + Îşg 2 + ÎşN 2 âˆ’ Ď„g 2 + Îşg , Îť2 = Ď„g (ÎşN âˆ’ 3Ď„g ) + 2ÎşN Ď„g + Îşg 3Îşg + Îşg 2 + ÎşN 2 âˆ’ Ď„g 2 + Îşg ,

(8)

âˆ’3Ď„ g 2 Îşg âˆ’ 3ÎşN 2 Îşg âˆ’ ÎşN (ÎşN âˆ’ Ď„ g ) 2 Ă— 3Îşg + 5Îşg + (ÎşN âˆ’ Ď„2g )Ď„g 2 , +Ď„g 6ÎşN Îşg + (ÎşN âˆ’ Ď„g ) 3Îşg + 5Îşg + ÎşN âˆ’ ÎşN Ď„g +(ÎşN âˆ’ Ď„g )

5Îşg Îşg (ÎşN âˆ’ Ď„g ) + 2Îşg 3 (ÎşN âˆ’ Ď„g ) + (ÎşN âˆ’ Ď„ g )Îşg . +Îşg 2(ÎşN âˆ’ Ď„g )2 (ÎşN + Ď„g ) âˆ’ Ď„g + ÎşN

Thus we can state the following Corollary. Corollary 1. Let Îą (sÂŻ) be a timelike curve lies on a timelike surface M in Minkowski 3-space E13 , then 1. If Îą is a geodesic curve, then the equations

Ď„g 2 âˆ’ ÎşN 2 , (ÎşN âˆ’ Ď„g )2 (ÎşN âˆ’ Ď„g )5 Ď„g ÎşN âˆ’ ÎşN Ď„g Ď„ÂŻÎą = âˆ’ , âˆš 4 2 Ď„g 2 âˆ’ ÎşN 2 ÎşÂŻ Îą =

2

are hold. 2. If Îą is an asymptotic line, the following hold

Îşg âˆ’ Ď„g 2Îşg + 3Îşg 2 âˆ’ Ď„g 2 ÎşÂŻ Îą = , Ď„g 3 âˆ’3Ď„g 2 Îşg âˆ’ Ď„g 3Îşg + 5Îşg 2 Ď„g 4 Ď„g +Ď„g Ď„g Îşg 5Îşg + 2Îşg 2 âˆ’ 2Ď„g 2 + Îşg + Îşg Ď„g Ď„ÂŻÎą = . âˆš 4 2 2Ď„g Îşg âˆ’ Ď„g 2Îşg + 3Îşg 2 âˆ’ Ď„g 2 2 2Ď„g

3 =

where

Ď‡2 =

âˆš dTÂŻ Î˛ 2 ( 1 T + 2 P + 3 U ), = 2 dsÂŻ âˆ’2ÎşN + (Îşg + Ď„g )2 2

âŽ› 4 âŽž Îşg + 3Îşg 3 Ď„g + ÎşN (Îşg + Ď„g )2 âˆ’ Îşg 2 ÎşN 2 âˆ’ 3Ď„g 2 âŽ , 1 = âŽ? +ÎşN âˆ’2ÎşN 3 âˆ’ (Îşg + Ď„g )Ď„g + ÎşN Ď„g 2 +Îşg âˆ’(Îşg + Ď„g )ÎşN + Ď„g 3 2 = ÎşN âˆ’2(Îşg + Ď„g )ÎşN + (Îşg + Ď„g ) 2ÎşN âˆ’ 2ÎşN 2 + (Îşg + Ď„g )2 ,

In the light of the above, the torsion of Îą is given by

Ď‡1 =

(10)

where

Îť3 = âˆ’Ď„g Îşg + ÎşN Îşg + (ÎşN âˆ’ Ď„g ) 2Îşg + Îşg 2 + ÎşN 2 âˆ’ Ď„g 2 âˆ’Ď„g + ÎşN .

(ÎşN âˆ’ Ď„g ) Ď‡1 Ď„ÂŻÎą = âˆ’ âˆš , 2 2 ÎşÂŻ Îą2 Ď‡2

âˆ’2ÎşN 2 + (Îşg + Ď„g )2

âˆ’Îşg ÎşN (Îşg + Ď„g ) âˆ’ Ď„g ÎşN (Îşg + Ď„g ) + ÎşN (Îşg + Ď„ g )2 . 2 âˆ’ âˆ’ÎşN + Ď„g (Îşg + Ď„g ) âˆ’2ÎşN 2 + (Îşg + Ď„g )2

Therefore, from aforementioned equations, the curvature functions

ÎşÂŻ Î˛ , Ď„ÂŻÎ˛ are expressed as follows

dTÂŻ Î˛ 1 = ÎşÂŻ Î˛ = 2 âˆ’ 12 + 22 âˆ’ 32 , 2 2 2 dsÂŻ âˆ’2ÎşN + (Îşg + Ď„g ) T + 2 P + 3 U . 1 âˆ’ 12 + 22 âˆ’ 32

ÂŻÎ˛ = N

Besides, the binormal vector of Î˛ is

BÂŻ Î˛ =

âˆ’1 âˆ’2ÎşN + (Îşg + Ď„g )2 2

âˆ’ 12 + 22 âˆ’ 32

( ÂŻ1 T + ÂŻ2 P + ÂŻ3 U ),

where

ÂŻ1 = ÎşN 3 + (Îşg + Ď„g ) 1 , ÂŻ2 = ÎşN 1 âˆ’ ÎşN 3 , ÂŻ3 = (Îşg + Ď„g ) 1 âˆ’ ÎşN 2 . After differentiate Î˛ with respect to s, we get

(ÎşN + Ď„g Îşg + ÎşN2 + Îşg2 )T + (Îşg + Ď„g + Ď„g ÎşN + ÎşN Îşg )P , +(ÎşN âˆ’ Îşg Ď„g âˆ’ Ď„g2 + ÎşN2 )U 2

1 Î˛ = âˆš

similarly,

1

Î˛ = âˆš (Îź1 T + Îź2 P + Îź3 U ), 2

where

Îź1 = 2Ď„g Îşg + Îşg (3Îşg + Ď„g ) + ÎşN 3ÎşN + Îşg 2 + ÎşN 2 âˆ’ Ď„g 2 + ÎşN ,

H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382â€“390

Îź2 = (Îşg + Ď„g )ÎşN + (Îşg + Ď„g ) 2ÎşN + Îşg 2 + ÎşN 2 âˆ’ Ď„g 2 + Îşg + Ď„g , Îź3 = âˆ’2Îşg Ď„g âˆ’ Ď„g (Îşg + 3Ď„g ) + ÎşN 3ÎşN + Îşg 2 + ÎşN 2 âˆ’ Ď„g 2 + ÎşN .

385

where

âŽž 3 Îşg 2 Îşg ÎşN 2 âŽ 1 = âŽ?âˆ’ÎşN 4ÎşN Îşg + 7Îş N Îşg 3 + 2 Îşg 3 Îşg 2 + ÎşN 2 + 2ÎşN + Îşg 2 + 2ÎşN 2 Îş . g âˆ’Îşg Îşg 2 âˆ’ 2ÎşN 2 ÎşN + Îşg ÎşN 3Îşg 2 + 4ÎşN 2 + ÎşN 7Îşg 2 ÎşN + 2ÎşN âŽ›

Following that, the torsion of Î˛ is obtained as

1

Ď„ÂŻÎ˛ = âˆš

2 2ÎşÂŻ Î˛2

(Îž1 + Îž2 âˆ’ Îž3 ),

where

Îž1 =

(11)

Assume that Îł = Îł (sÂŻ) is a timelike curve lying fully on M in E13 . Let {T, P, U} be Darboux frame of Îł . Then by DeďŹ nition 2, the PUâˆ’ Smarandache curve of Îł is identiďŹ ed by

âˆ’ÎşN 2ÎşN + Îşg 2 + 2ÎşN 2 âˆ’ Ď„g 2 (Îşg + Ď„g )ÎşN + (Îşg + Ď„g ) 2ÎşN + Îşg 2 + ÎşN 2 âˆ’ Ď„g 2 +Îşg + Ď„g ,

1

Îł (sÂŻ) = âˆš (P + U ).

(Îşg + Ď„g )ÎşN + (Îşg + Ď„g ) ÎşN + 2ÎşN 2 âˆ’ Ď„g (Îşg + Ď„ g ) Îž2 = 2Ď„g Îşg + Îşg (3Îşg + Ď„g ) + ÎşN 3ÎşN + Îşg 2 + ÎşN 2 âˆ’ Ď„g 2 + ÎşN , âˆ’(Îşg + Ď„g )ÎşN + (Îşg + Ď„g ) ÎşN + Îşg (Îşg + Ď„g ) Îž3 = âˆ’2Îşg Ď„g âˆ’ Ď„g (Îşg + 3Ď„g ) + ÎşN 3ÎşN + Îşg 2 + ÎşN 2 âˆ’ Ď„g 2 + ÎşN . Thus we can give the following Corollary. Corollary 2. Let Î˛ (sÂŻ) be a timelike curve lies on M in Minkowski 3-space E13 , then 1. If Î˛ is a geodesic curve, the following are satisďŹ ed

2Ď‡ÂŻ 1 , Ď„g 2 âˆ’ 2ÎşN 2 3

4Îş âˆ’ 2Ď„g 2 ÎşN 2 âˆ’ 2ÎşN 2 Ď„g 2 âˆ’ 8 ÎşN 4 Ď„g 2 âˆ’ 2Ď„g ÎşN Ď„g 3 Ď‡ÂŻ 1 = N , +5Îş 2 Ď„ 4 âˆ’ Ď„ 6 + 2Îş Ď„ 2Ď„ Îş + Ď„ 3 6

Ď‡ÂŻ 2

Ď„ÂŻÎ˛ = âˆš

4 2 Ď‡ÂŻ 3

where

âŽ›

g

and

((Îşg + ÎşN )T+Ď„g Pâˆ’Ď„g U ) . TÂŻ Îł = âˆ’(Îşg + ÎşN )2

(12)

Differentiating Eq. (12) with respect to s, we have

âˆš dTÂŻ Îł 2 = (Îś1 T + Îś2 P + Îś3 U ), dsÂŻ (Îşg + ÎşN )3

where

Îś1 = Ď„g ÎşN2 âˆ’ Îşg2 ,

Îś3 = âˆ’(Îşg + ÎşN )Ď„g + (Îşg + ÎşN ) Ď„g âˆ’ ÎşN (Îşg + ÎşN ) + Ď„g 2 .

where

N

2

Îś2 = (Îşg + ÎşN )Ď„g âˆ’ (Îşg + ÎşN ) Ď„g + Îşg (Îşg + ÎşN ) âˆ’ Ď„g 2 ,

ÎşÂŻ Î˛ =

3.3. PUâˆ’Smarandache curves

g

N g

The curvature of Îł is determined by

dTÂŻ Îł 1 2 2 2 = ÎşÂŻ Îł = dsÂŻ (Îşg + ÎşN )3 2 âˆ’Îś1 + Îś2 âˆ’ Îś3 .

g

g N

,

ÎşN Ď„ g âˆ’ 2ÎşN âˆ’ 2ÎşN 2 + Ď„ 2Îş + Îş 2 âˆ’ Ď„ 2 + Ď„ 2

âŽž

âŽœ âŽ&#x; Ď„g Îş N g g N âŽœ âŽ&#x; N g Îş + Ď„ Îş + 2Îş 2 âˆ’ Ď„ 2 âŽœ âŽ&#x; + Ď„ g g N N g N âŽœ âŽ&#x;, Ď‡ÂŻ 2 = âŽœ + Îş 2 âˆ’ Ď„ 2 + Îş âŽ&#x; Îş 3 Îş g N N N N âŽœ âŽ&#x; âŽ?âˆ’ ÎşN Ď„g âˆ’ Ď„g ÎşN âˆ’3Ď„g Ď„g + ÎşN 3ÎşN + ÎşN 2 âˆ’ Ď„g 2 âŽ +ÎşN âŽž âŽ› 6 4ÎşN âˆ’ 2Ď„g 2 ÎşN 2 âˆ’ 2ÎşN 2 Ď„g 2 4 2 3 âˆ’8ÎşN Ď„g âˆ’ 2Ď„g ÎşN Ď„g âŽ&#x; 1 âŽœ Ď‡ÂŻ 3 = 2 . âŽ? +5ÎşN 2 Ď„g 4 âˆ’ Ď„g 6 âŽ Ď„g âˆ’ 2ÎşN 2 3 +2ÎşN Ď„g 2Ď„g ÎşN + Ď„g 3 2. If Î˛ is an asymptotic line, then

âˆ’2 Îşg 6 + 4Îşg 5 Ď„g + 7Îşg 4 Ď„g 2 + 8Îşg 3 Ď„g 3 + 7Îşg 2 Ď„g 4 + 4Îşg Ď„g 5 + Ď„g 6 ÎşÂŻ Î˛ = , (Îşg + Ď„g )6 (Îşg + Ď„g )9 Ď„g Îşg âˆ’ Îşg Ď„g . Ď„ÂŻÎ˛ = âˆš âˆ’4 2 Îşg 6 + 4Îşg 5 Ď„g + 7Îşg 4 Ď„g 2 + 8Îşg 3 Ď„g 3 + 7Îşg 2 Ď„g 4 + 4Îşg Ď„g 5 + Ď„g 6 3. If Î˛ is a principal line, the following are clariďŹ ed

2 4ÎşN 6 âˆ’ 2ÎşN 2 Îşg 2 âˆ’ Îşg 6 + 2Îşg Îşg 3 ÎşN âˆ’ 2 Îşg 2 ÎşN 2 +Îşg 4 ÎşN 2 âˆ’ 2ÎşN Îşg Îşg 3 âˆ’ 2Îşg ÎşN ÎşÂŻ Î˛ = , Îşg 2 âˆ’ 2ÎşN 2 3 2 Îşg âˆ’ 2ÎşN 2 3 1 , Ď„ÂŻÎ˛ = âˆš 4ÎşN 6 âˆ’ 2ÎşN 2 Îşg 2 âˆ’ Îşg 6 + 2Îşg Îşg 3 ÎşN âˆ’ 2 Îşg 2 ÎşN 2 4 2 +Îşg 4 ÎşN 2 âˆ’ 2ÎşN Îşg Îşg 3 âˆ’ 2Îşg ÎşN

Further, we deďŹ ne the principal normal and the binormal vectors as follows

ÂŻÎł = N

Îś T + Îś2 P + Îś3 U , 1 âˆ’Îś12 + Îś22 âˆ’ Îś32

BÂŻ Îł =

âˆ’1 âˆ’(Îşg + ÎşN )2

âˆ’Îś12 + Îś22 âˆ’ Îś32

ÎśÂŻ1 T + ÎśÂŻ2 P + ÎśÂŻ3 U ,

where

ÎśÂŻ1 = Ď„g Îś3 + Ď„g Îś2 , ÎśÂŻ2 = âˆ’Ď„g Îś1 âˆ’ (Îşg + Ď„ g )Îś3 , ÎśÂŻ3 = Ď„g Îś1 âˆ’ (Îşg + Ď„ g )Îś2 . If we differentiate Îł to get Îł , Îł , then we obtain the torsion of Îł as follows

1

Îł = âˆš

(Îşg + ÎşN + Ď„g Îşg âˆ’ Ď„g ÎşN )T + (Ď„g + Îşg2 + ÎşN Îşg âˆ’ Ď„g 2 )P +(âˆ’Ď„g + ÎşN Îşg + ÎşN2 âˆ’ Ď„g 2 )U , 2

1

Îł = âˆš (Î˝1 T + Î˝2 P + Î˝3 U ), 2

where

Î˝1 = 2Ď„g (Îşg âˆ’ ÎşN ) + (Îşg âˆ’ ÎşN )Ď„g + (Îşg + ÎşN ) Îşg 2 + ÎşN 2 âˆ’ Ď„g 2 +Îşg + ÎşN ,

Î˝2 = 2ÎşN Îşg + Îşg (3Îşg + ÎşN ) + Ď„g âˆ’3Ď„g + Îşg 2 + ÎşN 2 âˆ’ Ď„g 2 + Ď„g ,

386

H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382â€“390

3.4. TPUâˆ’Smarandache curves

Î˝3 = 2Îşg ÎşN + ÎşN (Îşg + 3ÎşN ) âˆ’ 3Ď„g + Îşg 2 + ÎşN 2 Ď„g + Ď„g 3 âˆ’ Ď„g , 1 Ď„ÂŻÎł = âˆš 2 ( 1 + 2 + 3 ), 2 2ÎşÂŻ Îł where

âŽ›

1

(13)

âŽž

1

Î´ (sÂŻ) = âˆš (T + P + U ).

âˆ’ âˆ’Ď„g (Îşg + ÎşN ) + ÎşN (Îşg + ÎşN )2 âŽœ âŽ&#x; âˆ’(Îşg + ÎşN )Ď„g âˆ’ 2Îşg Ď„g 2 âŽœ âŽ&#x;, = âŽ? âŽ 2Îş N Îşg +2Îşg (3Îş2g + ÎşN2) +Ď„g âˆ’3Ď„g + Îşg + ÎşN âˆ’ Ď„g + Ď„g

âŽ›

3

This implies to

( (Îşg + ÎşN )T+(Îşg + Ď„g )P+(ÎşN âˆ’ Ď„g )U ) . TÂŻ Î´ = âˆ’2(Îşg + ÎşN )(ÎşN âˆ’ Ď„g )

âŽž

Ď„g (Îş g + ÎşN ) âˆ’ (Îşg + ÎşN )Ď„g 2 2 âŽœ âŽ&#x; âŽ&#x;, +Îşg ( Îşg + ÎşN ) âˆ’ 2Ď„g 2 = âŽœ âŽ? âŽ âˆ’2Îşg ÎşN âˆ’ ÎşN (Îşg + 3ÎşN ) +Ď„g 3Ď„g + Îşg 2 + ÎşN 2 âˆ’ Ď„g 2 + Ď„g âŽž âŽ› 2Ď„g + Îşg 2 âˆ’ ÎşN 2 Ď„g âˆ’ âŽ 3 = âŽ? 2Ď„g (Îşg âˆ’ ÎşN ) + (Îşg âˆ’ ÎşN )Ď„g . +(Îşg + ÎşN ) Îşg 2 + ÎşN 2 âˆ’ Ď„g 2 + Îşg + ÎşN âˆ’

âˆš dTÂŻ Î´ 3 = (Îž1 T + Îž2 P + Îž3 U ), dsÂŻ 4(Îşg + ÎşN )2 (ÎşN âˆ’ Ď„g )2

where

Corollary 3. Let Îł (sÂŻ) be a timelike curve lies on M in Minkowski 3space E13 , then 1. If Îł is a geodesic curve, the curvature and torsion of Îł are, respectively

ÎşÂŻ Îł =

2ÎşN 3 2Ď„g âˆ’ ÎşN 2 âˆ’ 4ÎşN ÎşN 2 Ď„g âˆ’ 2ÎşN 4Ď„g âˆ’ ÎşN 2

ÎşN

âŽ›

Ď„g 2 + 8ÎşN Ď„g 3

5

âŽž 2 g N2 Ď„g âˆ’ 3ÎşN3 ÎşN Ď„g âŽœ N g ÎşN Ď„g âˆ’ 2Ď„g + 2 Ď„g âŽ&#x; 1 âŽœ +Ď„ Îş 3Îş 2 + 2Ď„ 2 âŽ&#x; Ď„ÂŻÎł = âˆš g N âŽœ âŽ&#x;, g N 2 2 4 âŽ? +Ď„ âˆ’Îş 3 + 2Îş Ď„ 2 âˆ’ 2Îş âŽ g N N g N +ÎşN 2 âˆ’ÎşN Ď„g + Ď„g ÎşN

4 =

2ÎşN

3

âˆ’2Ď„ 2 Îş

2Ď„g âˆ’ ÎşN

2

ÎşÂŻ Îł =

5

2 2

6

Ď„g 2 + 8Îşg Ď„g 3

Îşg 5

âŽ›

BÂŻ Î´ =

Îž T + Îž2 P + Îž3 U . 1 âˆ’Îž12 + Îž22 âˆ’ Îž32

ÎžÂŻ1 T + ÎžÂŻ2 P + ÎžÂŻ3 U , 2 2 2 âˆ’2(Îşg + ÎşN )(ÎşN âˆ’ Ď„g ) âˆ’Îž1 + Îž2 âˆ’ Îž3 âˆ’1

where

.

3Îşg 2 Îşg Ď„g 10 Ď„g 2 + 7Ď„g Îş g 2 + 2Îşg 4 âˆ’2 Ď„g + 3Îşg 2 Ď„g 2 + 4Ď„g 4

âŽœ +Ď„ âˆ’Îş âˆ’ 2Ď„g + Îşg 2 Îşg g g âŽœ 5 = âŽœ âŽ? âˆ’Îşg Îşg 2 âˆ’ 4Ď„g 2 Ď„g + Îşg Ď„g 3Îşg 2 âˆ’ 2Ď„g 2 2 +Ď„g 7Îşg Ď„g âˆ’ 2Ď„g 3 + 2Ď„g 2Îşg 3 2Ď„g + Îşg 2 âˆ’ 4Îşg Îşg 2 Ď„g âˆ’ 2Îşg 4Ď„g + 3Îşg 2 Ď„g 2 + 8Îşg Ď„g 3 6 = Îşg 5 3. If Îł is a principal line, the following hold

Îşg 2 âˆ’ ÎşN 2 , (Îşg + ÎşN )2 (Îşg + ÎşN )3 ÎşN Îşg âˆ’ Îşg ÎşN

Ď„ÂŻÎł = . âˆš 2 ( Îşg 2 âˆ’ Îş N 2 ) 2 2 ÎşÂŻ Îł =

ÂŻÎ´ = N

,

where

2

(Îşg +ÎşN )2

(15)

and

Also, the binormal vector of Î´ is given by

1

âŽž Ď„g (Îşg + ÎşN )(Îşg + 2ÎşN âˆ’ Ď„g ) âŽœ âˆ’ÎşN (Îşg + 2 ÎşN âˆ’ Ď„g )(Îşg + Ď„g ) + (ÎşN âˆ’ Ď„g ) âŽ&#x; Îž2 = âˆ’(Îşg + ÎşN )âŽ? âŽ , Îşg (Îşg + 2ÎşN âˆ’ Ď„g ) +2(Îşg + ÎşN )(Îşg + ÎşN âˆ’ Ď„g )(Îşg + Ď„g ) )

dTÂŻ Î´ 1 = ÎşÂŻ Î´ = 3 âˆ’Îž12 + Îž22 âˆ’ Îž32 , dsÂŻ 4(Îşg + ÎşN )2 (ÎşN âˆ’ Ď„g )2

âˆ’ 4ÎşN ÎşN 2 Ď„g âˆ’ 2ÎşN 4Ď„g âˆ’ ÎşN 2 Ď„g 2 + 8ÎşN Ď„g 3 . ÎşN 5

2Îşg 3 2Ď„g + Îşg 2 âˆ’ 4Îşg Îşg 2 Ď„g âˆ’ 2Îşg 4Ď„g + 3Îşg 2

Ď„ÂŻÎł = âˆš

âŽ›

Then, the curvature and principal normal vector of Î´ are, respectively

2. If Îł is an asymptotic line, we get

Îž1 = (Îşg + ÎşN ) âˆ’Ď„g (Îşg + ÎşN ) + ÎşN (Îşg + Ď„g ) âˆ’(ÎşN âˆ’ Ď„g ) Îşg + 2 Îşg 2 + ÎşN 2 + (Îşg âˆ’ ÎşN )Ď„g ,

Îž3 = (ÎşN âˆ’ Ď„g ) Ď„g (Îşg + ÎşN ) âˆ’ ÎşN (Îşg + Ď„g ) âˆ’(ÎşN âˆ’ Ď„g ) âˆ’Îşg + 2(Îşg + ÎşN )(Îşg + ÎşN + Ď„g ) .

,

+Îş Ď„

where

(14)

Differentiate Eq. (14) with respect to s and use Eq. (3), we obtain

From the above calculations, we can introduce the following result:

Let r = r (s ) be a timelike curve lying on a timelike surface M in Minkowski 3-space E13 and {T, P, U} be the Darboux frame of r(s). According to the deďŹ nition of Smarandache curve, the TPUâˆ’ Smarandache curve of r is expressed as

âŽž âŽ&#x; âŽ&#x; âŽ&#x;, âŽ

ÎžÂŻ1 = (Îşg + Ď„g )Îž3 âˆ’ (ÎşN âˆ’ Ď„g )Îž2 , ÎžÂŻ2 = (ÎşN âˆ’ Ď„g )Îž1 âˆ’ (Îşg + ÎşN )Îž3 , ÎžÂŻ3 = (Îşg + Ď„g )Îž1 âˆ’ (Îşg + ÎşN )Îž2 . The derivatives Î´ , Î´ of Î´ are

(Îşg + ÎşN + ÎşN2 + Îşg2 + Ď„g Îşg âˆ’ Ď„g ÎşN )T 2 2 +(Îşg + Ď„g + Îşg + ÎşN Îşg + ÎşN Ď„g âˆ’ Ď„g )P , 3 +(âˆ’Ď„ + Îş + Îş Îş + Ď„ Îş + Îş 2 âˆ’ Ď„ 2 )U g g g N g g N N

1 Î´ = âˆš

1

Î´ = âˆš (Îˇ1 T + Îˇ2 P + Îˇ3 U ), 3

where

Îˇ1 = 2Ď„g (Îşg âˆ’ ÎşN ) + 3ÎşN ÎşN âˆ’ ÎşN Ď„g + Îşg (3Îşg + Ď„g ) +(Îşg + ÎşN ) Îşg 2 + ÎşN 2 âˆ’ Ď„g 2 + Îşg + ÎşN , Îˇ2 = Îşg (3Îşg + ÎşN ) + Ď„g (ÎşN âˆ’ 3Ď„g )

H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382â€“390

387

Fig. 1. The timelike curve r(u) on the timelike surface M.

Fig. 2. TP and TUâˆ’ Smarandache curves.

+(Îşg + Ď„g ) 2ÎşN + Îşg 2 + ÎşN 2 âˆ’ Ď„g 2 + Îşg + Ď„g ,

âŽ› âŽœ âŽœ 10 = âŽœ âŽ?

Îˇ3 = ÎşN (Îşg + 3ÎşN ) âˆ’ Ď„g (Îşg + 3Ď„g ) +(ÎşN âˆ’ Ď„g ) 2Îşg + Îşg 2 + ÎşN 2 âˆ’ Ď„g 2 âˆ’ Ď„g + ÎşN . In the light of these derivatives, the curveâ€™s torsion of Î´ can be computed as follows

Ď„ÂŻÎ´ = âˆš 3

7 + 8 , ÎşÎ´ ((ÎşN âˆ’ Ď„g )( 9 + 2 10 ))

(16)

3ÂŻ2

where

âŽ›

2 âˆ’ÎşN (3Îşg + 4ÎşN âˆ’ Ď„g )(Îşg + Ď„g )

âŽž

âˆ’Ď„g 2 3Îşg 2 âˆ’ 4Îşg ÎşN + ÎşN 2 + 2(5Îşg + ÎşN )Ď„g âŽœ âŽ&#x; âŽ§ âŽŤâŽ&#x; âŽœ Îşg

(5Îşg âˆ’ 2ÎşN + 3Ď„g ) âŽ&#x;, 7 = âŽœ âŽ¨ âŽŹ âŽœ âŽ&#x; 2 Îşg ( Îşg âˆ’ 2 ÎşN ) âŽ?âˆ’2ÎşN (ÎşN âˆ’ Ď„g ) +(Îşg + Ď„g ) âŽ +(3Îşg + ÎşN )Ď„g âˆ’ Ď„g 2 âŽŠ âŽ +Îşg + Ď„g âŽ›

âŽž 2 2 ÎşN 3Îşg + 2ÎşN + 6Îşg Ď„g + Ď„g âŽœ âŽ&#x; Îşg (5Îşg + Ď„g ) 8 = 2Ď„g âŽ? âŽ , +(Îşg + ÎşN )(2Îşg âˆ’ Ď„g )(Îşg âˆ’ ÎşN + Ď„g ) +(ÎşN âˆ’ Ď„g ) + Îşg + ÎşN

2

Îşg (2ÎşN âˆ’ Ď„g ) ' 9 = , +2Îşg âˆ’(ÎşN âˆ’ Ď„g ) âˆ’2Îşg + Îşg (4ÎşN âˆ’ Ď„g ) âˆ’ Ď„g + ÎşN +Ď„g (ÎşN + Ď„g ) &

+Îşg

2Îşg 4 (ÎşN âˆ’ Ď„g ) + 2Îşg 3 (ÎşN âˆ’ Ď„g )Ď„g +2Îşg 2 (ÎşN âˆ’ Ď„g )2 (ÎşN + Ď„g ) âˆ’ Ď„g + ÎşN +ÎşN âˆ’ÎşN Îşg + Ď„g + Ď„g Îşg + ÎşN 2ÎşN 3 Ď„g âˆ’ 2ÎşN 2 Ď„g 2 + ÎşN âˆ’2Ď„g 3 âˆ’ Ď„g + ÎşN 3 +2Ď„g Ď„g âˆ’ Ď„g + ÎşN

âŽž âŽ&#x; âŽ&#x; . âŽ&#x; âŽ

Corollary 4. Let Î´ (sÂŻ) be a timelike curve lies on M in Minkowski 3space E13 , then 1. If Î´ is a geodesic curve, the curvature and torsion can be expressed as follows

( ÎşÂŻ Î´ =

8

3 1 , ÎşN 3 (ÎşN âˆ’ Ď„g )3

Ď„ÂŻÎ´ = âˆš 2 2 , 3 3 ÎşÂŻ Î´ where

1 =

Ď„ g 2 ÎşN 2 âˆ’ 2Ď„g ÎşN 2 ÎşN âˆ’ 2(ÎşN âˆ’ Ď„g ) Ď„g + ÎşN 2 Ď„g 2 âˆ’ 4ÎşN (ÎşN âˆ’ Ď„g )2 Ď„g 2 âˆ’4ÎşN 2 (ÎşN âˆ’ Ď„g )3 (ÎşN + Ď„g )

,

388

H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382â€“390

âŽž ÎşN 2 Ď„g (âˆ’4ÎşN + Ď„ g ) âˆ’ Ď„g 2 ÎşN (ÎşN + 2Ď„ g ) âŽœ âŽ&#x; âˆ’2ÎşN (ÎşN âˆ’ Ď„g ) (ÎşN âˆ’ Ď„g )Ď„g 2 + Ď„g âŽ&#x;. 2 = âŽœ âŽ? âŽ +22 ÎşN (Îş2N âˆ’ Ď„g ) âˆ’ÎşN Ď„g + Ď„g ÎşN + 2Ď„g ÎşN 2ÎşN + Ď„g + (ÎşN âˆ’ Ď„g ) ÎşN (ÎşN âˆ’ Ď„g )Ď„g + ÎşN âŽ›

2. If Î´ is an asymptotic line, we have

( (

1 2

ÎşÂŻ Î´ =

3 2

3 , Îşg 3 Ď„g 3

2

Ď„ g 2 Îşg 2 âˆ’ 2Ď„g Îşg Ď„g Îşg + 2Îşg 2 âˆ’ 2 Ď„g +Ď„g Îşg 2 + 4Îşg 2 (Îşg + Ď„g )2 + 4Îşg Îşg 2 âˆ’ Ď„g 2 âˆ’Ď„g 2 Îşg (3Îşg + 10Ď„g ) âˆ’ 2Ď„g Ď„g (Îşg (2Îşg âˆ’ Ď„g )(Îşg + Ď„g ) +Îşg (5Îşg + Ď„g ) + Îşg +Ď„g Îşg + 2Îşg (Îşg + Ď„g ) Ď„g Îşg + 2Îşg 2 âˆ’ 2Ď„g 2 + 2Ď„g

4 = âŽ?

U=

2

âŽ›

,

3 8

ÎşÂŻ Î´ =

cosh(3u ) cosh(u ) + (1 + v ) cosh(2u ) âˆ’ âˆš âˆš 2 2

( B1, B2, B3 ) , 1

where

âˆš B1 = âˆ’ 2(1 + v ) cosh(u ) + 2 sinh(u )2 , âˆš B2 = 1 + 2v + cosh(2u ) âˆ’ 2 sinh(u ), âˆš B3 = âˆ’1 âˆ’ 2v sinh(u ) + sinh(2u ).

âŽž âŽ .

3. If Î´ is a principal line, we obtain

( (

v+

and

3 =

A2 =

âˆš âˆ’ 2v sinh(u ) + sinh(2u ), sinh(3u ) sinh(u ) A3 = 1 + 2v + cosh(2u ) + âˆš + (1 + v ) sinh(2u ) âˆ’ , âˆš 2 2

Ď„ÂŻÎ´ = âˆš 4 2 , 3 3ÎşÂŻ Î´ where

âˆš âˆš A1 = 2 2(1 + v ) cosh(u ) + sinh(u ) 2 âˆ’ 2 cosh(u ) + 2v sinh(u ) ,

According to Eq. (3), the geodesic curvature Îş g , the normal curvature Îş N and the geodesic torsion Ď„ g of the curve r(u) are

5 , ÎşN 3 (Îşg + ÎşN )3

1

Îşg = T , P =

âˆ’ÎşN 2 Îşg (3Îşg + 4ÎşN ) 1 Ď„ÂŻÎ´ = âˆš +2ÎşN ÎşN âˆ’5Îşg Îşg âˆ’ 2Îşg 3 + 2 Îşg + 2Îşg 2 ÎşN âˆ’ Îşg , 3 3 + Îş 6

2

âˆ’2 sinh(u ) 2v +

âˆš 2 sinh(u )

âˆš + cosh(u ) 2 + 4v âˆ’ 2 2v sinh(u ) ,

N

where

âŽž âŽ› 2 2 ÎşN Îşg + ÎşN 2 Îşg + 2Îşg 2 2 + 4Îşg Îşg ÎşN âŽ , 5 = âŽ? âˆ’8Îşg 2 ÎşN 2 âˆ’ 8Îşg ÎşN 3 âˆ’ 4ÎşN 4 âˆ’2ÎşN Îşg ÎşN Îşg + 2Îşg (Îşg + ÎşN ) âŽ›

âŽž ÎşN Îşg 2 + 4Îşg Îşg ( Îşg âˆ’ 2ÎşN ) 6 = âŽ? +4Îş g 2 Îşg 2 + ÎşN 2 âˆ’ 2Îş N Îşg âŽ . +2 Îşg + Îşg (2Îşg + ÎşN ) ÎşN 4. Computational example

ÎşN = U , T

âŽœ âŽœ âŽœ âŽœ âŽœ âŽœ 1 âŽœ = 3/2 âŽœ 2 ( 2 ) âŽœ âŽœ âŽœ âŽœ âŽœ âŽ?

In this example, we construct some special Smarandache curves (TP, TU, PU and TPU) of a timelike curve which lies on a timelike surface. Moreover, using Mathematica program, we compute their differential geometric properties (Figs. 1â€“3). Suppose we are given a timelike ruled surface represented as

M (u, v ) = r (u ) + vQ (u ), where the timelike base curve is given by

âˆš

r (u ) = u, and

Q (u ) =

2 cosh u,

âˆš

âˆš 2 sinh u ,

2 cosh u, 1,

âˆš 2 sinh u ,

is the ruling vector of M. The triples of Darboux frame can be computed as follows

âˆš

T = 1, P=

2 sinh u,

( A1, A2, A3 ) , 1

âˆš 2 cosh u ,

âŽ›

âˆš 4 2 âˆ’4 + v âˆ’2 + 3v + 6v2 âˆ’4 âˆš (10 + 3v(7 + 4v )) cosh(u ) +3 2v(7 + 4v(5 + 2v )) cosh(2u ) âˆ’2(16 âˆš + v(43 + 24v )) cosh(3u ) +4 2(4 + 3v ) cosh(4u ) âˆ’2(4 + 3v ) cosh(5u ) âˆš âˆ’ 2v cosh(6u ) âˆ’ 4(5 +âˆš3v(2 + v(5 + 4v ))) sinh(u ) + 2(17 +12v(5 + 4v )) sinh(2u ) âˆ’ 2 9 + 22v + 6v2 sinh(3u ) âˆš +2 2(8 + 11v ) sinh âˆš(4u ) + 2(âˆ’3 + 2v ) sinh(5u ) + 2 sinh(6u )

âŽž âˆš 1 âˆ’ 2v âˆ’ 2 âˆ’4 + v + 4v2 âŽœ cosh(u ) + 2(3 + 5v ) cosh(2u ) âŽ&#x; âŽœ âˆš âŽ&#x; 1 âŽœ + 2(âˆ’2 + v ) cosh(3u ) + cosh(4u ) âŽ&#x; âˆš = P , U = âŽœ âŽ&#x;. 2 âŽœ âˆ’ 2(7 + 4v(2 + v )) sinh(u ) âŽ&#x; âŽ? +2(2 + v(3 + 2v )) sinh(2u ) âŽ âˆš âˆ’ 2(3 + 4v ) sinh(3u ) âˆ’ v sinh(4u ) âŽ›

Ď„g

where

âŽ› âŽœ 2 = âŽ?

âˆš 5 + 12v(1 + v ) + 2 2âˆšcosh(u ) +4(1 + 3vâˆš ) cosh(2u ) âˆ’ 2 2 cosh(3u ) + cosh(4u ) âˆ’ 2 2(âˆš 1 + 6v ) sinh(u ) + 4 sinh(2u ) âˆ’2 2 sinh(3u )

âŽž âŽ&#x; âŽ

If we choose u = 0 and v = 0, the curvatures (Îş g , Îş N and Ď„ g ) are

(

2 2 where Îşg = , ÎşN = âˆ’2 , 5 5 âˆš âˆš 1 + 2v + cosh(2u ) âˆ’ 2 sinh(u ) 2 + 2(1 + v ) cosh(u ) âˆ’ 2 sinh2 (u ) 2 1 = , 2 âˆš TPâˆ’ Smarandache curve âˆ’ 1 + 2v sinh(u ) âˆ’ sinh(2u )

Ď„g =

âˆš 1 8+2 2 . 10

âŽž âŽ&#x; âŽ&#x; âŽ&#x; âŽ&#x; âŽ&#x; âŽ&#x; âŽ&#x; âŽ&#x;, âŽ&#x; âŽ&#x; âŽ&#x; âŽ&#x; âŽ&#x; âŽ

H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382–390

389

Fig. 3. PU and TPU− Smarandache curves.

The TP−Smarandache curve can be computed as

PU− Smarandache curve The PU−Smarandache curve can be computed as

γPU = (γ1 , γ2 , γ3 ), √ √ 2(1 + v ) cosh(u ) + sinh(u ) 2 − 2 cosh(u ) + 2(1 + v ) sinh(u ) γ1 = , √ 2 1 √ √ √ 2 + 6v + 2 cosh(u ) + 2(2 + v ) cosh(2u ) − 2 cosh(3u ) − 2 2(1 + v ) sinh(u ) + 2 sinh(2u ) γ2 = , √ 2 2 1 √ √ 4v + 2 cosh(2u ) + 2(1 − 2v ) sinh(u ) + 2(2 + v ) sinh(2u ) − 2 sinh(3u ) γ3 = . √ 2 2 1 αTP = (α1 , α2 , α3 ),

In this case, we get (u = 0 and v = 0)

where

α1 α2

√ √ 2 2(1 + v ) cosh(u ) + sinh(u ) 2 − 2 cosh(u ) + 2v sinh(u ) 1 = √ 1+ , 1 2 √ √ (u ) + (1 + v ) cosh (2u ) − cosh √(3u ) − v + cosh 2v sinh(u ) + sinh(2u ) √ 1 2 2 = 2 sinh(u ) + √ , 1 2

α3

1 + 2v + cosh(2u ) + 1 = √ cosh(u ) + 2

sinh √ (u ) 2

+ (1 + v ) sinh(2u ) −

1

Therefore, we get (u = 0 and v = 0)

κ¯ α = 0.793, τ¯α = 0.455. TU− Smarandache curve For this curve, we get

βTU = (β1 , β2 , β3 ),

sinh √(3u ) 2

β1 β2 β3

after some calculations, we obtain (u = 0 and v = 0)

κ¯ β = 2.965, τ¯β = 0.199.

.

κ¯ γ = 2.204,

τ¯γ = 0.113.

TPU− Smarandache curve The TPU−Smarandache curve is given by

(δ , δ , δ ) δTPU = 1√ 2 3 , 2 3 1

where

√ 2 1 − 2(1 + v ) cosh(u ) + 2 sinh (u ) = √ 1+ , 1 2 √ 1 √ 1 + 2v + cosh(2u ) − 2 sinh(u ) = √ 2 sinh(u ) + , 1 2 √ 1 √ −1 − 2v sinh(u ) + sinh(2u ) = √ 2 cosh(u ) + , 1 2

δ1 =

2

√

⎛ √ δ2 = ⎝ δ3 =

√ 2(1 + v ) cosh(u ) + 2 sinh(u ) + 2 sinh(u )2 + 2v sinh(u )2 +2 1 − 2 sinh(2u )

√ 2 cosh (u ) + 2(2 + v ) cosh(2u ) − 2 cosh (3u ) 1 + 3v + √ sinh(2u ) + sinh(u ) +2 − 2(1 + v ) + 2

⎞ ⎠,

√ 4v + 2 cosh(2u ) √ + 2(1 − 2v ) sinh(u ) + 2(2 + v ) sinh(2u ) − 2 sinh(3u ) + 2 2 cosh(u )

it follows that (u = 0 and v = 0)

κ¯ δ = 1.319, τ¯δ = 0.549.

,

390

H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382–390

5. Conclusion In the present paper, we have studied special curves called Smarandache curves according to Darboux frame in the threedimensional Minkowski space E13 . These curves are composed using Frenet frame vectors of another curve. Moreover, some results for the meaning curves are obtained. Finally, for conﬁrming our results, a computational example is given and plotted. Acknowledgment The authors are very grateful to referees for the useful suggestions and remarks for the revised version. References [1] C. Ashbacher, Smarandache geometries, Smarandache Notions J. 8 (1–3) (1997) 212–215. [2] M. Turgut, S. Yilmaz, Smarandache curves in Minkowski space-time, Int. J. Math.Comb. 3 (2008) 51–55.

[3] H.S. Abdel-Aziz, M.K. Saad, Smarandache curves of some special curves in the Galilean 3-space, Honam Math. J. 37 (2) (2015) 253–264. [4] A.T. Ali, Special Smarandache curves in the Euclidean space, Int. J. Math.Comb. 2 (2010) 30–36. [5] O. Bektas, S. Yuce, Smarandache curves according to Darboux frame in Euclidean space, Rom. J. Math. Comput.Sci. 3 (1) (2013) 48–59. [6] M. Çetin, Y. Tunçer, M.K. Karacan, Smarandache curves according to Bishop frame in Euclidean space, Gen. Math. Notes 20 (2) (2014) 50–66. [7] M. Khalifa Saad, Spacelike and timelike admissible Smarandache curves in pseudo-Galilean space, J. Egypt. Math. Soc. 24 (2016) 416–423. [8] M. Çetin, H. Kocayig˘ it, On the quaternionic Smarandache curves in Euclidean 3- space, Int. J. Contemp. Math. Sci. 8 (3) (2013) 139–150. [9] K.T. prü, M. Tosun, Smarandache curves according to Sabban frame on S2 , Boletim da Sociedade Paraneanse de Matematica 32 (1) (2014) 51–59. [10] B. O’Neil, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, 1983. [11] M.P.D. Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ, 1976.

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