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

© 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/)

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&#x20AC;&#x201C;390

T

Îşg Ď&#x2030;1

P

Îşg Ď&#x2030;2

â&#x2C6;&#x2019;U

(ÎşN â&#x2C6;&#x2019;Ď&#x201E; g ) .

Ď&#x2030;3

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

Îşg â&#x2C6;&#x2019; 2ÎşN 2Îşg + 3Îşg 2 + ÎşN 2 κ¯ Îą = , ÎşN 3

2

Îşg 2 ÎşN 3 ÎşN Îş g + Îş N 23Îşg + 52

ÎşN4 â&#x2C6;&#x2019;ÎşN ÎşN Îşg 5Îşg + 2 Îşg + ÎşN + Îşg + Îşg ÎşN

Ď&#x201E;ÂŻÎą = . â&#x2C6;&#x161; 2 2 4ÎşN Îşg â&#x2C6;&#x2019; 2ÎşN 2Îşg + 3Îşg 2 + ÎşN 2

So, the binormal vector of Îą is

BÂŻ Îą =

â&#x2C6;&#x2019;1

(Ď&#x2030;ÂŻ 1 T + Ď&#x2030;ÂŻ 2 P + Ď&#x2030;ÂŻ 3 U ), (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) â&#x2C6;&#x2019;Ď&#x2030;12 + Ď&#x2030;22 â&#x2C6;&#x2019; Ď&#x2030;32

where

Ď&#x2030;ÂŻ 1 = (Ď&#x201E;g â&#x2C6;&#x2019; ÎşN )Ď&#x2030;2 + Îşg Ď&#x2030;3 , Ď&#x2030;ÂŻ 2 = (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )Ď&#x2030;1 â&#x2C6;&#x2019; Îşg Ď&#x2030;3 , Ď&#x2030;ÂŻ 3 = Îşg Ď&#x2030;1 â&#x2C6;&#x2019; Îşg Ď&#x2030;2 .

We consider the derivatives Îą ,

1

Îą = â&#x2C6;&#x161;

4ÎşN

3.2. TUâ&#x2C6;&#x2019;Smarandache curves

Îą with respect to s as follows

(Îşg â&#x2C6;&#x2019; Ď&#x201E;g ÎşN + ÎşN2 + Îşg2 )T + (Îşg â&#x2C6;&#x2019; Ď&#x201E;g2 + Ď&#x201E;g ÎşN + Îşg2 )P

+(â&#x2C6;&#x2019;Ď&#x201E;g + ÎşN â&#x2C6;&#x2019; Îşg Ď&#x201E;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 TUâ&#x2C6;&#x2019; Smarandache curve of r is given by

1

β (sÂŻ) = â&#x2C6;&#x161; (T + U ),

2

(9)

2

it leads to

1

1

(ÎşN T+(Îşg + Ď&#x201E;g )P+ÎşN U ).

Îą = â&#x2C6;&#x161; (Îť1 T + Îť2 P + Îť3 U ),

T¯ β =

where

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

2

Îť1 = â&#x2C6;&#x2019;2Ď&#x201E;g ÎşN + ÎşN (3ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) + Îşg 3Îşg + Îşg 2 + ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 + Îşg ,

Îť2 = Ď&#x201E;g (ÎşN â&#x2C6;&#x2019; 3Ď&#x201E;g ) + 2ÎşN Ď&#x201E;g + Îşg 3Îşg + Îşg 2 + ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 + Îşg ,

(8)

â&#x2C6;&#x2019;3Ď&#x201E; g 2 Îşg â&#x2C6;&#x2019; 3ÎşN 2 Îşg â&#x2C6;&#x2019; ÎşN (ÎşN â&#x2C6;&#x2019; Ď&#x201E; g )

2 Ă&#x2014; 3Îşg + 5Îşg + (ÎşN â&#x2C6;&#x2019; Ď&#x201E;2g )Ď&#x201E;g 2 ,

+Ď&#x201E;g 6ÎşN Îşg + (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) 3Îşg + 5Îşg + ÎşN â&#x2C6;&#x2019; ÎşN Ď&#x201E;g +(ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )

5Îşg Îşg (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) + 2Îşg 3 (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) + (ÎşN â&#x2C6;&#x2019; Ď&#x201E; g )Îşg . +Îşg 2(ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )2 (ÎşN + Ď&#x201E;g ) â&#x2C6;&#x2019; Ď&#x201E;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&#x201E;g 2 â&#x2C6;&#x2019; ÎşN 2 , (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )2

(ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )5 Ď&#x201E;g ÎşN â&#x2C6;&#x2019; ÎşN Ď&#x201E;g

Ď&#x201E;ÂŻÎą = â&#x2C6;&#x2019; , â&#x2C6;&#x161; 4 2 Ď&#x201E;g 2 â&#x2C6;&#x2019; ÎşN 2 κ¯ Îą =

2

are hold.

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

Îşg â&#x2C6;&#x2019; Ď&#x201E;g 2Îşg + 3Îşg 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 κ¯ Îą = , Ď&#x201E;g 3

â&#x2C6;&#x2019;3Ď&#x201E;g 2 Îşg â&#x2C6;&#x2019; Ď&#x201E;g 3Îşg + 5Îşg 2 Ď&#x201E;g 4

Ď&#x201E;g +Ď&#x201E;g Ď&#x201E;g Îşg 5Îşg + 2Îşg 2 â&#x2C6;&#x2019; 2Ď&#x201E;g 2 + Îşg + Îşg Ď&#x201E;g

Ď&#x201E;ÂŻÎą = . â&#x2C6;&#x161; 4 2 2Ď&#x201E;g Îşg â&#x2C6;&#x2019; Ď&#x201E;g 2Îşg + 3Îşg 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 2 2Ď&#x201E;g

3 =

where

Ď&#x2021;2 =

â&#x2C6;&#x161; dTÂŻ β 2 ( 1 T + 2 P + 3 U ), = 2 dsÂŻ â&#x2C6;&#x2019;2ÎşN + (Îşg + Ď&#x201E;g )2 2

â&#x17D;&#x203A; 4

â&#x17D;&#x17E; Îşg + 3Îşg 3 Ď&#x201E;g + ÎşN (Îşg + Ď&#x201E;g )2 â&#x2C6;&#x2019; Îşg 2 ÎşN 2 â&#x2C6;&#x2019; 3Ď&#x201E;g 2 â&#x17D; , 1 = â&#x17D;? +ÎşN â&#x2C6;&#x2019;2ÎşN 3 â&#x2C6;&#x2019; (Îşg + Ď&#x201E;g )Ď&#x201E;g + ÎşN Ď&#x201E;g 2

+Îşg â&#x2C6;&#x2019;(Îşg + Ď&#x201E;g )ÎşN + Ď&#x201E;g 3

2 = ÎşN â&#x2C6;&#x2019;2(Îşg + Ď&#x201E;g )ÎşN + (Îşg + Ď&#x201E;g ) 2ÎşN â&#x2C6;&#x2019; 2ÎşN 2 + (Îşg + Ď&#x201E;g )2 ,

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

Ď&#x2021;1 =

(10)

where

Îť3 = â&#x2C6;&#x2019;Ď&#x201E;g Îşg + ÎşN Îşg + (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) 2Îşg + Îşg 2 + ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 â&#x2C6;&#x2019;Ď&#x201E;g + ÎşN .

(ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) Ď&#x2021;1 Ď&#x201E;ÂŻÎą = â&#x2C6;&#x2019; â&#x2C6;&#x161; , 2 2 κ¯ Îą2 Ď&#x2021;2

â&#x2C6;&#x2019;2ÎşN 2 + (Îşg + Ď&#x201E;g )2

â&#x2C6;&#x2019;Îşg ÎşN (Îşg + Ď&#x201E;g ) â&#x2C6;&#x2019; Ď&#x201E;g ÎşN (Îşg + Ď&#x201E;g ) + ÎşN (Îşg + Ď&#x201E; g )2 .

2 â&#x2C6;&#x2019; â&#x2C6;&#x2019;ÎşN + Ď&#x201E;g (Îşg + Ď&#x201E;g ) â&#x2C6;&#x2019;2ÎşN 2 + (Îşg + Ď&#x201E;g )2

Therefore, from aforementioned equations, the curvature functions

κ¯ β , Ď&#x201E;¯β are expressed as follows

dT¯ β

1

=

κ¯ β = 2 â&#x2C6;&#x2019; 12 + 22 â&#x2C6;&#x2019; 32 , 2 2 2 dsÂŻ â&#x2C6;&#x2019;2ÎşN + (Îşg + Ď&#x201E;g ) T + 2 P + 3 U .

1 â&#x2C6;&#x2019; 12 + 22 â&#x2C6;&#x2019; 32

¯β = N

Besides, the binormal vector of β is

B¯ β =

â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;2ÎşN + (Îşg + Ď&#x201E;g )2 2

â&#x2C6;&#x2019; 12 + 22 â&#x2C6;&#x2019; 32

( ÂŻ1 T + ÂŻ2 P + ÂŻ3 U ),

where

ÂŻ1 = ÎşN 3 + (Îşg + Ď&#x201E;g ) 1 , ÂŻ2 = ÎşN 1 â&#x2C6;&#x2019; ÎşN 3 , ÂŻ3 = (Îşg + Ď&#x201E;g ) 1 â&#x2C6;&#x2019; ÎşN 2 .

After differentiate β with respect to s, we get

(ÎşN + Ď&#x201E;g Îşg + ÎşN2 + Îşg2 )T + (Îşg + Ď&#x201E;g + Ď&#x201E;g ÎşN + ÎşN Îşg )P , +(ÎşN â&#x2C6;&#x2019; Îşg Ď&#x201E;g â&#x2C6;&#x2019; Ď&#x201E;g2 + ÎşN2 )U 2

1 β = â&#x2C6;&#x161;

similarly,

1

β = â&#x2C6;&#x161; (Îź1 T + Îź2 P + Îź3 U ), 2

where

Îź1 = 2Ď&#x201E;g Îşg + Îşg (3Îşg + Ď&#x201E;g ) + ÎşN 3ÎşN + Îşg 2 + ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 + ÎşN ,

H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382â&#x20AC;&#x201C;390

Îź2 = (Îşg + Ď&#x201E;g )ÎşN + (Îşg + Ď&#x201E;g ) 2ÎşN + Îşg 2 + ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 + Îşg + Ď&#x201E;g ,

Îź3 = â&#x2C6;&#x2019;2Îşg Ď&#x201E;g â&#x2C6;&#x2019; Ď&#x201E;g (Îşg + 3Ď&#x201E;g ) + ÎşN 3ÎşN + Îşg 2 + ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 + ÎşN .

385

where

â&#x17D;&#x17E; 3 Îşg 2 Îşg ÎşN 2

â&#x17D; 1 = â&#x17D;?â&#x2C6;&#x2019;ÎşN 4ÎşN Îşg + 7Îş N Îşg 3 + 2 Îşg 3 Îşg 2 + ÎşN 2 + 2ÎşN + Îşg 2 + 2ÎşN 2 Îş . g â&#x2C6;&#x2019;Îşg Îşg 2 â&#x2C6;&#x2019; 2ÎşN 2 ÎşN + Îşg ÎşN 3Îşg 2 + 4ÎşN 2 + ÎşN 7Îşg 2 ÎşN + 2ÎşN â&#x17D;&#x203A;

Following that, the torsion of β is obtained as

1

Ď&#x201E;¯β = â&#x2C6;&#x161;

2 2κ¯ β2

(Ξ1 + Ξ2 â&#x2C6;&#x2019; Ξ3 ),

where

Ξ1 =

(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 PUâ&#x2C6;&#x2019; Smarandache curve of Îł is identiďŹ ed by

â&#x2C6;&#x2019;ÎşN 2ÎşN + Îşg 2 + 2ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g 2

(Îşg + Ď&#x201E;g )ÎşN + (Îşg + Ď&#x201E;g ) 2ÎşN + Îşg 2 + ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 +Îşg + Ď&#x201E;g ,

1

Îł (sÂŻ) = â&#x2C6;&#x161; (P + U ).

(Îşg + Ď&#x201E;g )ÎşN + (Îşg + Ď&#x201E;g ) ÎşN + 2ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g (Îşg + Ď&#x201E; g )

Ξ2 = 2Ď&#x201E;g Îşg + Îşg (3Îşg + Ď&#x201E;g ) + ÎşN 3ÎşN + Îşg 2 + ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 + ÎşN ,

â&#x2C6;&#x2019;(Îşg + Ď&#x201E;g )ÎşN + (Îşg + Ď&#x201E;g ) ÎşN + Îşg (Îşg + Ď&#x201E;g )

Ξ3 = â&#x2C6;&#x2019;2Îşg Ď&#x201E;g â&#x2C6;&#x2019; Ď&#x201E;g (Îşg + 3Ď&#x201E;g ) + ÎşN 3ÎşN + Îşg 2 + ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 + ÎşN .

Thus we can give the following Corollary.

Corollary 2. Let β (sÂŻ) be a timelike curve lies on M in Minkowski 3-space E13 , then 1. If β is a geodesic curve, the following are satisďŹ ed

2Ď&#x2021;ÂŻ 1 , Ď&#x201E;g 2 â&#x2C6;&#x2019; 2ÎşN 2 3

4Îş â&#x2C6;&#x2019; 2Ď&#x201E;g 2 ÎşN 2 â&#x2C6;&#x2019; 2ÎşN 2 Ď&#x201E;g 2 â&#x2C6;&#x2019; 8 ÎşN 4 Ď&#x201E;g 2 â&#x2C6;&#x2019; 2Ď&#x201E;g ÎşN Ď&#x201E;g 3 Ď&#x2021;ÂŻ 1 = N , +5Îş 2 Ď&#x201E; 4 â&#x2C6;&#x2019; Ď&#x201E; 6 + 2Îş Ď&#x201E; 2Ď&#x201E; Îş + Ď&#x201E; 3 6

Ď&#x2021;ÂŻ 2

Ď&#x201E;¯β = â&#x2C6;&#x161;

4 2 Ď&#x2021;ÂŻ 3

where

â&#x17D;&#x203A;

g

and

((Îşg + ÎşN )T+Ď&#x201E;g Pâ&#x2C6;&#x2019;Ď&#x201E;g U ) .

TÂŻ Îł =

â&#x2C6;&#x2019;(Îşg + ÎşN )2

(12)

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

â&#x2C6;&#x161; dTÂŻ Îł 2 = (Îś1 T + Îś2 P + Îś3 U ), dsÂŻ (Îşg + ÎşN )3

where

Îś1 = Ď&#x201E;g ÎşN2 â&#x2C6;&#x2019; Îşg2 ,

Îś3 = â&#x2C6;&#x2019;(Îşg + ÎşN )Ď&#x201E;g + (Îşg + ÎşN ) Ď&#x201E;g â&#x2C6;&#x2019; ÎşN (Îşg + ÎşN ) + Ď&#x201E;g 2 .

where

N

2

Îś2 = (Îşg + ÎşN )Ď&#x201E;g â&#x2C6;&#x2019; (Îşg + ÎşN ) Ď&#x201E;g + Îşg (Îşg + ÎşN ) â&#x2C6;&#x2019; Ď&#x201E;g 2 ,

κ¯ β =

3.3. PUâ&#x2C6;&#x2019;Smarandache curves

g

N g

The curvature of Îł is determined by

dTÂŻ Îł

1 2 2 2 = κ¯ Îł = dsÂŻ (Îşg + ÎşN )3 2 â&#x2C6;&#x2019;Îś1 + Îś2 â&#x2C6;&#x2019; Îś3 .

g

g N

,

ÎşN Ď&#x201E; g â&#x2C6;&#x2019; 2ÎşN â&#x2C6;&#x2019; 2ÎşN 2

+ Ď&#x201E; 2Îş + Îş 2 â&#x2C6;&#x2019; Ď&#x201E; 2 + Ď&#x201E; 2

â&#x17D;&#x17E;

â&#x17D;&#x153; â&#x17D;&#x; Ď&#x201E;g Îş N g g N â&#x17D;&#x153; â&#x17D;&#x; N g Îş + Ď&#x201E; Îş + 2Îş 2 â&#x2C6;&#x2019; Ď&#x201E; 2 â&#x17D;&#x153; â&#x17D;&#x; + Ď&#x201E; g g N N g N â&#x17D;&#x153; â&#x17D;&#x;,

Ď&#x2021;ÂŻ 2 = â&#x17D;&#x153; + Îş 2 â&#x2C6;&#x2019; Ď&#x201E; 2 + Îş â&#x17D;&#x; Îş 3 Îş g N N N N â&#x17D;&#x153; â&#x17D;&#x;

â&#x17D;?â&#x2C6;&#x2019; ÎşN Ď&#x201E;g â&#x2C6;&#x2019; Ď&#x201E;g ÎşN â&#x2C6;&#x2019;3Ď&#x201E;g Ď&#x201E;g + ÎşN 3ÎşN + ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 â&#x17D;

+ÎşN â&#x17D;&#x17E; â&#x17D;&#x203A; 6 4ÎşN â&#x2C6;&#x2019; 2Ď&#x201E;g 2 ÎşN 2 â&#x2C6;&#x2019; 2ÎşN 2 Ď&#x201E;g 2 4 2

3 â&#x2C6;&#x2019;8ÎşN Ď&#x201E;g â&#x2C6;&#x2019; 2Ď&#x201E;g ÎşN Ď&#x201E;g â&#x17D;&#x; 1 â&#x17D;&#x153; Ď&#x2021;ÂŻ 3 = 2 . â&#x17D;? +5ÎşN 2 Ď&#x201E;g 4 â&#x2C6;&#x2019; Ď&#x201E;g 6 â&#x17D; Ď&#x201E;g â&#x2C6;&#x2019; 2ÎşN 2 3 +2ÎşN Ď&#x201E;g 2Ď&#x201E;g ÎşN + Ď&#x201E;g 3 2. If β is an asymptotic line, then

â&#x2C6;&#x2019;2 Îşg 6 + 4Îşg 5 Ď&#x201E;g + 7Îşg 4 Ď&#x201E;g 2 + 8Îşg 3 Ď&#x201E;g 3 + 7Îşg 2 Ď&#x201E;g 4 + 4Îşg Ď&#x201E;g 5 + Ď&#x201E;g 6 κ¯ β = , (Îşg + Ď&#x201E;g )6

(Îşg + Ď&#x201E;g )9 Ď&#x201E;g Îşg â&#x2C6;&#x2019; Îşg Ď&#x201E;g . Ď&#x201E;¯β = â&#x2C6;&#x161; â&#x2C6;&#x2019;4 2 Îşg 6 + 4Îşg 5 Ď&#x201E;g + 7Îşg 4 Ď&#x201E;g 2 + 8Îşg 3 Ď&#x201E;g 3 + 7Îşg 2 Ď&#x201E;g 4 + 4Îşg Ď&#x201E;g 5 + Ď&#x201E;g 6 3. If β is a principal line, the following are clariďŹ ed

2 4ÎşN 6 â&#x2C6;&#x2019; 2ÎşN 2 Îşg 2 â&#x2C6;&#x2019; Îşg 6 + 2Îşg Îşg 3 ÎşN â&#x2C6;&#x2019; 2 Îşg 2 ÎşN 2

+Îşg 4 ÎşN 2 â&#x2C6;&#x2019; 2ÎşN Îşg Îşg 3 â&#x2C6;&#x2019; 2Îşg ÎşN

κ¯ β = , Îşg 2 â&#x2C6;&#x2019; 2ÎşN 2 3 2

Îşg â&#x2C6;&#x2019; 2ÎşN 2 3 1 ,

Ď&#x201E;¯β = â&#x2C6;&#x161; 4ÎşN 6 â&#x2C6;&#x2019; 2ÎşN 2 Îşg 2 â&#x2C6;&#x2019; Îşg 6 + 2Îşg Îşg 3 ÎşN â&#x2C6;&#x2019; 2 Îşg 2 ÎşN 2 4 2 +Îşg 4 ÎşN 2 â&#x2C6;&#x2019; 2ÎşN Îşg Îşg 3 â&#x2C6;&#x2019; 2Îşg ÎşN

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

ÂŻÎł = N

Îś T + Îś2 P + Îś3 U , 1 â&#x2C6;&#x2019;Îś12 + Îś22 â&#x2C6;&#x2019; Îś32

BÂŻ Îł =

â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;(Îşg + ÎşN )2

â&#x2C6;&#x2019;Îś12 + Îś22 â&#x2C6;&#x2019; Îś32

Μ¯1 T + Μ¯2 P + Μ¯3 U ,

where

Μ¯1 = Ď&#x201E;g Îś3 + Ď&#x201E;g Îś2 , Μ¯2 = â&#x2C6;&#x2019;Ď&#x201E;g Îś1 â&#x2C6;&#x2019; (Îşg + Ď&#x201E; g )Îś3 , Μ¯3 = Ď&#x201E;g Îś1 â&#x2C6;&#x2019; (Îşg + Ď&#x201E; g )Îś2 .

If we differentiate Îł to get Îł , Îł , then we obtain the torsion of Îł as follows

1

Îł = â&#x2C6;&#x161;

(Îşg + ÎşN + Ď&#x201E;g Îşg â&#x2C6;&#x2019; Ď&#x201E;g ÎşN )T + (Ď&#x201E;g + Îşg2 + ÎşN Îşg â&#x2C6;&#x2019; Ď&#x201E;g 2 )P

+(â&#x2C6;&#x2019;Ď&#x201E;g + ÎşN Îşg + ÎşN2 â&#x2C6;&#x2019; Ď&#x201E;g 2 )U , 2

1

Îł = â&#x2C6;&#x161; (ν1 T + ν2 P + ν3 U ), 2

where

ν1 = 2Ď&#x201E;g (Îşg â&#x2C6;&#x2019; ÎşN ) + (Îşg â&#x2C6;&#x2019; ÎşN )Ď&#x201E;g + (Îşg + ÎşN ) Îşg 2 + ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 +Îşg + ÎşN ,

ν2 = 2ÎşN Îşg + Îşg (3Îşg + ÎşN ) + Ď&#x201E;g â&#x2C6;&#x2019;3Ď&#x201E;g + Îşg 2 + ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 + Ď&#x201E;g ,

386

H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382â&#x20AC;&#x201C;390

3.4. TPUâ&#x2C6;&#x2019;Smarandache curves

ν3 = 2Îşg ÎşN + ÎşN (Îşg + 3ÎşN ) â&#x2C6;&#x2019; 3Ď&#x201E;g + Îşg 2 + ÎşN 2 Ď&#x201E;g + Ď&#x201E;g 3 â&#x2C6;&#x2019; Ď&#x201E;g , 1 Ď&#x201E;ÂŻÎł = â&#x2C6;&#x161; 2 ( 1 + 2 + 3 ), 2 2κ¯ Îł where

â&#x17D;&#x203A;

1

(13)

â&#x17D;&#x17E;

1

δ (sÂŻ) = â&#x2C6;&#x161; (T + P + U ).

â&#x2C6;&#x2019; â&#x2C6;&#x2019;Ď&#x201E;g (Îşg + ÎşN ) + ÎşN (Îşg + ÎşN )2

â&#x17D;&#x153; â&#x17D;&#x; â&#x2C6;&#x2019;(Îşg + ÎşN )Ď&#x201E;g â&#x2C6;&#x2019; 2Îşg Ď&#x201E;g 2 â&#x17D;&#x153; â&#x17D;&#x;,

=

â&#x17D;? â&#x17D; 2Îş N Îşg +2Îşg (3Îş2g + ÎşN2)

+Ď&#x201E;g â&#x2C6;&#x2019;3Ď&#x201E;g + Îşg + ÎşN â&#x2C6;&#x2019; Ď&#x201E;g + Ď&#x201E;g

â&#x17D;&#x203A;

3

This implies to

( (Îşg + ÎşN )T+(Îşg + Ď&#x201E;g )P+(ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )U ) .

T¯ δ =

â&#x2C6;&#x2019;2(Îşg + ÎşN )(ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )

â&#x17D;&#x17E;

Ď&#x201E;g (Îş g + ÎşN ) â&#x2C6;&#x2019; (Îşg + ÎşN )Ď&#x201E;g 2 2 â&#x17D;&#x153; â&#x17D;&#x; â&#x17D;&#x;, +Îşg ( Îşg + ÎşN ) â&#x2C6;&#x2019; 2Ď&#x201E;g 2 = â&#x17D;&#x153; â&#x17D;? â&#x17D; â&#x2C6;&#x2019;2Îşg ÎşN â&#x2C6;&#x2019; ÎşN (Îşg + 3ÎşN )

+Ď&#x201E;g 3Ď&#x201E;g + Îşg 2 + ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 + Ď&#x201E;g â&#x17D;&#x17E; â&#x17D;&#x203A;

2Ď&#x201E;g + Îşg 2 â&#x2C6;&#x2019; ÎşN 2 Ď&#x201E;g â&#x2C6;&#x2019; â&#x17D; 3 = â&#x17D;?

2Ď&#x201E;g (Îşg â&#x2C6;&#x2019; ÎşN ) + (Îşg â&#x2C6;&#x2019; ÎşN )Ď&#x201E;g

. +(Îşg + ÎşN ) Îşg 2 + ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 + Îşg + ÎşN â&#x2C6;&#x2019;

â&#x2C6;&#x161; dTÂŻ δ 3 = (Ξ1 T + Ξ2 P + Ξ3 U ), dsÂŻ 4(Îşg + ÎşN )2 (ÎşN â&#x2C6;&#x2019; Ď&#x201E;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

κ¯ γ =

2ÎşN 3 2Ď&#x201E;g â&#x2C6;&#x2019; ÎşN 2 â&#x2C6;&#x2019; 4ÎşN ÎşN 2 Ď&#x201E;g â&#x2C6;&#x2019; 2ÎşN 4Ď&#x201E;g â&#x2C6;&#x2019; ÎşN 2

ÎşN

â&#x17D;&#x203A;

Ď&#x201E;g 2 + 8ÎşN Ď&#x201E;g 3

5

â&#x17D;&#x17E; 2 g N2 Ď&#x201E;g â&#x2C6;&#x2019; 3ÎşN3 ÎşN Ď&#x201E;g â&#x17D;&#x153; N g ÎşN Ď&#x201E;g â&#x2C6;&#x2019; 2Ď&#x201E;g + 2 Ď&#x201E;g â&#x17D;&#x; 1 â&#x17D;&#x153; +Ď&#x201E; Îş 3Îş 2 + 2Ď&#x201E; 2 â&#x17D;&#x; Ď&#x201E;ÂŻÎł = â&#x2C6;&#x161; g N â&#x17D;&#x153; â&#x17D;&#x;, g N 2 2 4 â&#x17D;? +Ď&#x201E; â&#x2C6;&#x2019;Îş 3 + 2Îş Ď&#x201E; 2 â&#x2C6;&#x2019; 2Îş â&#x17D; g N N g

N +ÎşN 2 â&#x2C6;&#x2019;ÎşN Ď&#x201E;g + Ď&#x201E;g ÎşN

4 =

2ÎşN

3

â&#x2C6;&#x2019;2Ď&#x201E; 2 Îş

2Ď&#x201E;g â&#x2C6;&#x2019; ÎşN

2

κ¯ γ =

5

2 2

6

Ď&#x201E;g 2 + 8Îşg Ď&#x201E;g 3

Îşg 5

â&#x17D;&#x203A;

B¯ δ =

Ξ T + Ξ2 P + Ξ3 U .

1 â&#x2C6;&#x2019;Ξ12 + Ξ22 â&#x2C6;&#x2019; Ξ32

Ξ¯1 T + Ξ¯2 P + Ξ¯3 U ,

2 2 2 â&#x2C6;&#x2019;2(Îşg + ÎşN )(ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) â&#x2C6;&#x2019;Ξ1 + Ξ2 â&#x2C6;&#x2019; Ξ3 â&#x2C6;&#x2019;1

where

.

3Îşg 2 Îşg Ď&#x201E;g 10 Ď&#x201E;g 2 + 7Ď&#x201E;g Îş g 2 + 2Îşg 4 â&#x2C6;&#x2019;2 Ď&#x201E;g + 3Îşg 2 Ď&#x201E;g 2 + 4Ď&#x201E;g 4

â&#x17D;&#x153; +Ď&#x201E; â&#x2C6;&#x2019;Îş â&#x2C6;&#x2019; 2Ď&#x201E;g + Îşg 2 Îşg g g â&#x17D;&#x153; 5 = â&#x17D;&#x153;

â&#x17D;? â&#x2C6;&#x2019;Îşg Îşg 2 â&#x2C6;&#x2019; 4Ď&#x201E;g 2 Ď&#x201E;g + Îşg Ď&#x201E;g 3Îşg 2 â&#x2C6;&#x2019; 2Ď&#x201E;g 2 2

+Ď&#x201E;g 7Îşg Ď&#x201E;g â&#x2C6;&#x2019; 2Ď&#x201E;g 3 + 2Ď&#x201E;g

2Îşg 3 2Ď&#x201E;g + Îşg 2 â&#x2C6;&#x2019; 4Îşg Îşg 2 Ď&#x201E;g â&#x2C6;&#x2019; 2Îşg 4Ď&#x201E;g + 3Îşg 2 Ď&#x201E;g 2 + 8Îşg Ď&#x201E;g 3 6 = Îşg 5 3. If Îł is a principal line, the following hold

Îşg 2 â&#x2C6;&#x2019; ÎşN 2 , (Îşg + ÎşN )2

(Îşg + ÎşN )3 ÎşN Îşg â&#x2C6;&#x2019; Îşg ÎşN

Ď&#x201E;ÂŻÎł = . â&#x2C6;&#x161; 2 ( Îşg 2 â&#x2C6;&#x2019; Îş N 2 ) 2 2 κ¯ Îł =

¯δ = N

,

where

2

(Îşg +ÎşN )2

(15)

and

Also, the binormal vector of δ is given by

1

â&#x17D;&#x17E; Ď&#x201E;g (Îşg + ÎşN )(Îşg + 2ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) â&#x17D;&#x153; â&#x2C6;&#x2019;ÎşN (Îşg + 2 ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )(Îşg + Ď&#x201E;g ) + (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) â&#x17D;&#x; Ξ2 = â&#x2C6;&#x2019;(Îşg + ÎşN )â&#x17D;? â&#x17D; , Îşg (Îşg + 2ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) +2(Îşg + ÎşN )(Îşg + ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )(Îşg + Ď&#x201E;g ) )

dT¯ δ

1

= κ¯ δ = 3 â&#x2C6;&#x2019;Ξ12 + Ξ22 â&#x2C6;&#x2019; Ξ32 , dsÂŻ 4(Îşg + ÎşN )2 (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )2

â&#x2C6;&#x2019; 4ÎşN ÎşN 2 Ď&#x201E;g â&#x2C6;&#x2019; 2ÎşN 4Ď&#x201E;g â&#x2C6;&#x2019; ÎşN 2 Ď&#x201E;g 2 + 8ÎşN Ď&#x201E;g 3 . ÎşN 5

2Îşg 3 2Ď&#x201E;g + Îşg 2 â&#x2C6;&#x2019; 4Îşg Îşg 2 Ď&#x201E;g â&#x2C6;&#x2019; 2Îşg 4Ď&#x201E;g + 3Îşg 2

Ď&#x201E;ÂŻÎł = â&#x2C6;&#x161;

â&#x17D;&#x203A;

Then, the curvature and principal normal vector of δ are, respectively

2. If Îł is an asymptotic line, we get

Ξ1 = (Îşg + ÎşN ) â&#x2C6;&#x2019;Ď&#x201E;g (Îşg + ÎşN ) + ÎşN (Îşg + Ď&#x201E;g )

â&#x2C6;&#x2019;(ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) Îşg + 2 Îşg 2 + ÎşN 2 + (Îşg â&#x2C6;&#x2019; ÎşN )Ď&#x201E;g ,

Ξ3 = (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) Ď&#x201E;g (Îşg + ÎşN ) â&#x2C6;&#x2019; ÎşN (Îşg + Ď&#x201E;g )

â&#x2C6;&#x2019;(ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) â&#x2C6;&#x2019;Îşg + 2(Îşg + ÎşN )(Îşg + ÎşN + Ď&#x201E;g ) .

,

+Îş Ď&#x201E;

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 TPUâ&#x2C6;&#x2019; Smarandache curve of r is expressed as

â&#x17D;&#x17E; â&#x17D;&#x; â&#x17D;&#x; â&#x17D;&#x;, â&#x17D;

Ξ¯1 = (Îşg + Ď&#x201E;g )Ξ3 â&#x2C6;&#x2019; (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )Ξ2 , Ξ¯2 = (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )Ξ1 â&#x2C6;&#x2019; (Îşg + ÎşN )Ξ3 , Ξ¯3 = (Îşg + Ď&#x201E;g )Ξ1 â&#x2C6;&#x2019; (Îşg + ÎşN )Ξ2 .

The derivatives δ , δ of δ are

(Îşg + ÎşN + ÎşN2 + Îşg2 + Ď&#x201E;g Îşg â&#x2C6;&#x2019; Ď&#x201E;g ÎşN )T

2 2 +(Îşg + Ď&#x201E;g + Îşg + ÎşN Îşg + ÎşN Ď&#x201E;g â&#x2C6;&#x2019; Ď&#x201E;g )P , 3 +(â&#x2C6;&#x2019;Ď&#x201E; + Îş + Îş Îş + Ď&#x201E; Îş + Îş 2 â&#x2C6;&#x2019; Ď&#x201E; 2 )U g g g N g g N N

1 δ = â&#x2C6;&#x161;

1

δ = â&#x2C6;&#x161; (Ρ1 T + Ρ2 P + Ρ3 U ), 3

where

Ρ1 = 2Ď&#x201E;g (Îşg â&#x2C6;&#x2019; ÎşN ) + 3ÎşN ÎşN â&#x2C6;&#x2019; ÎşN Ď&#x201E;g + Îşg (3Îşg + Ď&#x201E;g )

+(Îşg + ÎşN ) Îşg 2 + ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 + Îşg + ÎşN , Ρ2 = Îşg (3Îşg + ÎşN ) + Ď&#x201E;g (ÎşN â&#x2C6;&#x2019; 3Ď&#x201E;g )

H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382â&#x20AC;&#x201C;390

387

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

Fig. 2. TP and TUâ&#x2C6;&#x2019; Smarandache curves.

+(Îşg + Ď&#x201E;g ) 2ÎşN + Îşg 2 + ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 + Îşg + Ď&#x201E;g ,

â&#x17D;&#x203A; â&#x17D;&#x153; â&#x17D;&#x153; 10 = â&#x17D;&#x153; â&#x17D;?

Ρ3 = ÎşN (Îşg + 3ÎşN ) â&#x2C6;&#x2019; Ď&#x201E;g (Îşg + 3Ď&#x201E;g )

+(ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) 2Îşg + Îşg 2 + ÎşN 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 â&#x2C6;&#x2019; Ď&#x201E;g + ÎşN .

In the light of these derivatives, the curveâ&#x20AC;&#x2122;s torsion of δ can be computed as follows

Ď&#x201E;¯δ = â&#x2C6;&#x161; 3

7 + 8 , κδ ((ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )( 9 + 2 10 ))

(16)

3ÂŻ2

where

â&#x17D;&#x203A;

2 â&#x2C6;&#x2019;ÎşN (3Îşg + 4ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )(Îşg + Ď&#x201E;g )

â&#x17D;&#x17E;

â&#x2C6;&#x2019;Ď&#x201E;g 2 3Îşg 2 â&#x2C6;&#x2019; 4Îşg ÎşN + ÎşN 2 + 2(5Îşg + ÎşN )Ď&#x201E;g â&#x17D;&#x153; â&#x17D;&#x; â&#x17D;§ â&#x17D;Ťâ&#x17D;&#x; â&#x17D;&#x153;

Îşg

(5Îşg â&#x2C6;&#x2019; 2ÎşN + 3Ď&#x201E;g ) â&#x17D;&#x;, 7 = â&#x17D;&#x153; â&#x17D;¨ â&#x17D;Ź

â&#x17D;&#x153; â&#x17D;&#x; 2 Îşg ( Îşg â&#x2C6;&#x2019; 2 ÎşN ) â&#x17D;?â&#x2C6;&#x2019;2ÎşN (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) +(Îşg + Ď&#x201E;g ) â&#x17D; +(3Îşg + ÎşN )Ď&#x201E;g â&#x2C6;&#x2019; Ď&#x201E;g 2 â&#x17D;Š â&#x17D;­

+Îşg + Ď&#x201E;g â&#x17D;&#x203A;

â&#x17D;&#x17E; 2 2 ÎşN 3Îşg + 2ÎşN + 6Îşg Ď&#x201E;g + Ď&#x201E;g

â&#x17D;&#x153; â&#x17D;&#x; Îşg (5Îşg + Ď&#x201E;g ) 8 = 2Ď&#x201E;g â&#x17D;? â&#x17D; , +(Îşg + ÎşN )(2Îşg â&#x2C6;&#x2019; Ď&#x201E;g )(Îşg â&#x2C6;&#x2019; ÎşN + Ď&#x201E;g ) +(ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )

+ Îşg + ÎşN

2

2

Îşg (2ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )

' 9 = , +2Îşg â&#x2C6;&#x2019;(ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) â&#x2C6;&#x2019;2Îşg + Îşg (4ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) â&#x2C6;&#x2019; Ď&#x201E;g + ÎşN +Ď&#x201E;g (ÎşN + Ď&#x201E;g ) &

+Îşg

2Îşg 4 (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) + 2Îşg 3 (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )Ď&#x201E;g +2Îşg 2 (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )2 (ÎşN + Ď&#x201E;g ) â&#x2C6;&#x2019; Ď&#x201E;g + ÎşN

+ÎşN â&#x2C6;&#x2019;ÎşN Îşg + Ď&#x201E;g + Ď&#x201E;g Îşg + ÎşN

2ÎşN 3 Ď&#x201E;g â&#x2C6;&#x2019; 2ÎşN 2 Ď&#x201E;g 2 + ÎşN â&#x2C6;&#x2019;2Ď&#x201E;g 3 â&#x2C6;&#x2019; Ď&#x201E;g + ÎşN 3

+2Ď&#x201E;g Ď&#x201E;g â&#x2C6;&#x2019; Ď&#x201E;g + ÎşN

â&#x17D;&#x17E; â&#x17D;&#x; â&#x17D;&#x; . â&#x17D;&#x; â&#x17D;

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

( κ¯ δ =

8

3 1 , ÎşN 3 (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )3

Ď&#x201E;¯δ = â&#x2C6;&#x161; 2 2 , 3 3 κ¯ δ where

1 =

Ď&#x201E; g 2 ÎşN 2 â&#x2C6;&#x2019; 2Ď&#x201E;g ÎşN

2 ÎşN â&#x2C6;&#x2019; 2(ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) Ď&#x201E;g + ÎşN 2 Ď&#x201E;g 2 â&#x2C6;&#x2019; 4ÎşN (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )2 Ď&#x201E;g 2 â&#x2C6;&#x2019;4ÎşN 2 (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )3 (ÎşN + Ď&#x201E;g )

,

388

H.S. Abdel-Aziz, M. Khalifa Saad / Journal of the Egyptian Mathematical Society 25 (2017) 382â&#x20AC;&#x201C;390

â&#x17D;&#x17E; ÎşN 2 Ď&#x201E;g (â&#x2C6;&#x2019;4ÎşN + Ď&#x201E; g ) â&#x2C6;&#x2019; Ď&#x201E;g 2 ÎşN (ÎşN + 2Ď&#x201E; g ) â&#x17D;&#x153; â&#x17D;&#x; â&#x2C6;&#x2019;2ÎşN (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )Ď&#x201E;g 2 + Ď&#x201E;g

â&#x17D;&#x;.

2 = â&#x17D;&#x153;

â&#x17D;? â&#x17D; +22 ÎşN (Îş2N â&#x2C6;&#x2019; Ď&#x201E;g ) â&#x2C6;&#x2019;ÎşN Ď&#x201E;g + Ď&#x201E;g ÎşN + 2Ď&#x201E;g ÎşN 2ÎşN + Ď&#x201E;g + (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g ) ÎşN (ÎşN â&#x2C6;&#x2019; Ď&#x201E;g )Ď&#x201E;g + ÎşN â&#x17D;&#x203A;

2. If δ is an asymptotic line, we have

( (

1 2

κ¯ δ =

3 2

3 , Îşg 3 Ď&#x201E;g 3

2

Ď&#x201E; g 2 Îşg 2 â&#x2C6;&#x2019; 2Ď&#x201E;g Îşg Ď&#x201E;g Îşg + 2Îşg 2 â&#x2C6;&#x2019; 2 Ď&#x201E;g

+Ď&#x201E;g Îşg 2 + 4Îşg 2 (Îşg + Ď&#x201E;g )2 + 4Îşg Îşg 2 â&#x2C6;&#x2019; Ď&#x201E;g 2 â&#x2C6;&#x2019;Ď&#x201E;g 2 Îşg (3Îşg + 10Ď&#x201E;g ) â&#x2C6;&#x2019; 2Ď&#x201E;g Ď&#x201E;g (Îşg (2Îşg â&#x2C6;&#x2019; Ď&#x201E;g )(Îşg + Ď&#x201E;g ) +Îşg (5Îşg + Ď&#x201E;g ) + Îşg

+Ď&#x201E;g Îşg + 2Îşg (Îşg + Ď&#x201E;g ) Ď&#x201E;g Îşg + 2Îşg 2 â&#x2C6;&#x2019; 2Ď&#x201E;g 2 + 2Ď&#x201E;g

4 = â&#x17D;?

U=

2

â&#x17D;&#x203A;

,

3 8

κ¯ δ =

cosh(3u ) cosh(u ) + (1 + v ) cosh(2u ) â&#x2C6;&#x2019; â&#x2C6;&#x161; â&#x2C6;&#x161; 2 2

( B1, B2, B3 ) , 1

where

â&#x2C6;&#x161; B1 = â&#x2C6;&#x2019; 2(1 + v ) cosh(u ) + 2 sinh(u )2 , â&#x2C6;&#x161; B2 = 1 + 2v + cosh(2u ) â&#x2C6;&#x2019; 2 sinh(u ), â&#x2C6;&#x161; B3 = â&#x2C6;&#x2019;1 â&#x2C6;&#x2019; 2v sinh(u ) + sinh(2u ).

â&#x17D;&#x17E; â&#x17D; .

3. If δ is a principal line, we obtain

( (

v+

and

3 =

A2 =

â&#x2C6;&#x161; â&#x2C6;&#x2019; 2v sinh(u ) + sinh(2u ), sinh(3u ) sinh(u ) A3 = 1 + 2v + cosh(2u ) + â&#x2C6;&#x161; + (1 + v ) sinh(2u ) â&#x2C6;&#x2019; , â&#x2C6;&#x161; 2 2

Ď&#x201E;¯δ = â&#x2C6;&#x161; 4 2 , 3 3κ¯ δ where

â&#x2C6;&#x161; â&#x2C6;&#x161;

A1 = 2 2(1 + v ) cosh(u ) + sinh(u ) 2 â&#x2C6;&#x2019; 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&#x201E; g of the curve r(u) are

5 , ÎşN 3 (Îşg + ÎşN )3

1

Îşg = T , P =

â&#x2C6;&#x2019;ÎşN 2 Îşg (3Îşg + 4ÎşN )

1 Ď&#x201E;¯δ = â&#x2C6;&#x161; +2ÎşN ÎşN â&#x2C6;&#x2019;5Îşg Îşg â&#x2C6;&#x2019; 2Îşg 3 + 2 Îşg + 2Îşg 2 ÎşN â&#x2C6;&#x2019; Îşg , 3 3 + Îş 6

2

â&#x2C6;&#x2019;2 sinh(u ) 2v +

â&#x2C6;&#x161;

2 sinh(u )

â&#x2C6;&#x161;

+ cosh(u ) 2 + 4v â&#x2C6;&#x2019; 2 2v sinh(u ) ,

N

where

â&#x17D;&#x17E; â&#x17D;&#x203A; 2 2

ÎşN Îşg + ÎşN 2 Îşg + 2Îşg 2 2 + 4Îşg Îşg ÎşN â&#x17D; , 5 = â&#x17D;? â&#x2C6;&#x2019;8Îşg 2 ÎşN 2 â&#x2C6;&#x2019; 8Îşg ÎşN 3 â&#x2C6;&#x2019; 4ÎşN 4

â&#x2C6;&#x2019;2ÎşN Îşg ÎşN Îşg + 2Îşg (Îşg + ÎşN ) â&#x17D;&#x203A;

â&#x17D;&#x17E; ÎşN Îşg 2 + 4Îşg Îşg ( Îşg â&#x2C6;&#x2019; 2ÎşN ) 6 = â&#x17D;? +4Îş g 2 Îşg 2 + ÎşN 2 â&#x2C6;&#x2019; 2Îş N Îşg â&#x17D; . +2 Îşg + Îşg (2Îşg + ÎşN ) ÎşN 4. Computational example

ÎşN = U , T

â&#x17D;&#x153; â&#x17D;&#x153; â&#x17D;&#x153; â&#x17D;&#x153; â&#x17D;&#x153; â&#x17D;&#x153; 1 â&#x17D;&#x153; = 3/2 â&#x17D;&#x153; 2 ( 2 ) â&#x17D;&#x153; â&#x17D;&#x153; â&#x17D;&#x153; â&#x17D;&#x153; â&#x17D;&#x153; â&#x17D;?

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&#x20AC;&#x201C;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

â&#x2C6;&#x161;

r (u ) = u, and

Q (u ) =

2 cosh u,

â&#x2C6;&#x161;

â&#x2C6;&#x161;

2 sinh u ,

2 cosh u, 1,

â&#x2C6;&#x161;

2 sinh u ,

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

â&#x2C6;&#x161;

T = 1, P=

2 sinh u,

( A1, A2, A3 ) , 1

â&#x2C6;&#x161;

2 cosh u ,

â&#x17D;&#x203A;

â&#x2C6;&#x161; 4 2 â&#x2C6;&#x2019;4 + v â&#x2C6;&#x2019;2 + 3v + 6v2 â&#x2C6;&#x2019;4 â&#x2C6;&#x161; (10 + 3v(7 + 4v )) cosh(u ) +3 2v(7 + 4v(5 + 2v )) cosh(2u ) â&#x2C6;&#x2019;2(16 â&#x2C6;&#x161; + v(43 + 24v )) cosh(3u ) +4 2(4 + 3v ) cosh(4u ) â&#x2C6;&#x2019;2(4 + 3v ) cosh(5u ) â&#x2C6;&#x161; â&#x2C6;&#x2019; 2v cosh(6u ) â&#x2C6;&#x2019; 4(5 +â&#x2C6;&#x161;3v(2 + v(5 + 4v ))) sinh(u ) + 2(17

+12v(5 + 4v )) sinh(2u ) â&#x2C6;&#x2019; 2 9 + 22v + 6v2 sinh(3u ) â&#x2C6;&#x161; +2 2(8 + 11v ) sinh â&#x2C6;&#x161;(4u ) + 2(â&#x2C6;&#x2019;3 + 2v ) sinh(5u ) + 2 sinh(6u )

â&#x17D;&#x17E; â&#x2C6;&#x161; 1 â&#x2C6;&#x2019; 2v â&#x2C6;&#x2019; 2 â&#x2C6;&#x2019;4 + v + 4v2 â&#x17D;&#x153; cosh(u ) + 2(3 + 5v ) cosh(2u ) â&#x17D;&#x; â&#x17D;&#x153; â&#x2C6;&#x161; â&#x17D;&#x; 1 â&#x17D;&#x153; + 2(â&#x2C6;&#x2019;2 + v ) cosh(3u ) + cosh(4u ) â&#x17D;&#x;

â&#x2C6;&#x161; = P , U = â&#x17D;&#x153; â&#x17D;&#x;.

2 â&#x17D;&#x153; â&#x2C6;&#x2019; 2(7 + 4v(2 + v )) sinh(u ) â&#x17D;&#x; â&#x17D;? +2(2 + v(3 + 2v )) sinh(2u ) â&#x17D; â&#x2C6;&#x161; â&#x2C6;&#x2019; 2(3 + 4v ) sinh(3u ) â&#x2C6;&#x2019; v sinh(4u ) â&#x17D;&#x203A;

Ď&#x201E;g

where

â&#x17D;&#x203A; â&#x17D;&#x153; 2 = â&#x17D;?

â&#x2C6;&#x161; 5 + 12v(1 + v ) + 2 2â&#x2C6;&#x161;cosh(u ) +4(1 + 3vâ&#x2C6;&#x161; ) cosh(2u ) â&#x2C6;&#x2019; 2 2 cosh(3u ) + cosh(4u ) â&#x2C6;&#x2019; 2 2(â&#x2C6;&#x161; 1 + 6v ) sinh(u ) + 4 sinh(2u ) â&#x2C6;&#x2019;2 2 sinh(3u )

â&#x17D;&#x17E; â&#x17D;&#x; â&#x17D;

If we choose u = 0 and v = 0, the curvatures (Îş g , Îş N and Ď&#x201E; g ) are

(

(

2 2 where Îşg = , ÎşN = â&#x2C6;&#x2019;2 ,

5 5

â&#x2C6;&#x161; â&#x2C6;&#x161; 1 + 2v + cosh(2u ) â&#x2C6;&#x2019; 2 sinh(u ) 2 + 2(1 + v ) cosh(u ) â&#x2C6;&#x2019; 2 sinh2 (u ) 2 1 = ,

2 â&#x2C6;&#x161; TPâ&#x2C6;&#x2019; Smarandache curve â&#x2C6;&#x2019; 1 + 2v sinh(u ) â&#x2C6;&#x2019; sinh(2u )

Ď&#x201E;g =

â&#x2C6;&#x161; 1 8+2 2 .

10

â&#x17D;&#x17E; â&#x17D;&#x; â&#x17D;&#x; â&#x17D;&#x; â&#x17D;&#x; â&#x17D;&#x; â&#x17D;&#x; â&#x17D;&#x; â&#x17D;&#x;, â&#x17D;&#x; â&#x17D;&#x; â&#x17D;&#x; â&#x17D;&#x; â&#x17D;&#x; â&#x17D;

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.

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