Special equiform Smarandache curves in Minkowski space-time

Journal of the Egyptian Mathematical Society 25 (2017) 319–325

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Special equiform Smarandache curves in Minkowski space-time E.M. Solouma a,b,∗ a

Al Imam Mohammad Ibn Saud Islamic University, College of Science, Department of Mathematics and Statistics, KSA, P. O. Box 90950, Riyadh 11623, Saudi Arabia b Department of Mathematics, Faculty of Science, Beni-Suef University, Egypt

a r t i c l e

i n f o

Article history: Received 10 March 2017 Revised 5 April 2017 Accepted 30 April 2017 Available online 23 May 2017

a b s t r a c t In this paper, we introduce special equiform Smarandache curves reference to the equiform Frenet frame of a curve ζ on a spacelike surface M in Minkowski 3-space E31 . Also, we study the equiform Frenet invariants of the spacial equiform Smarandache curves in E31 . Moreover, we give some properties to these curves when the curve ζ has constant curvature or it is a circular helix. Finally, we give an example to illustrate these curves.

MSC: 53B30 53C40 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 curve Equiform Frenet frame Minkowski space-time

1. Introduction

2. Preliminaries

A regular non-null curve in Minkowski space-time, whose position vector is composed by Frenet frame vectors on another regular curve, is called a Smarandache curve [1]. Recently special Smarandache curves have been studied by some authors [2–5].

In this work, we study special equiform Smarandache curves with reference to the equiform Frenet frame of a curve ζ on a spacelike surface M in Minkowski 3-space E31 . In Section 2, we clarify the basic conceptions of Minkowski 3-space E31 and give of equiform Fremet frame that will be used during this work.

Section 3 is delicate to the study of the special four equiform Smarandache curves, Tη, Tξ , ηξ and Tηξ -equiform Smarandache curves by being the connection with the first and second equiform curvature k1 (θ ), and k2 (θ ) of the equiform spacelike curve ζ in E31 .

Furthermore, we present some properties on the curves when the curve ζ has constant curvature or it is a circular helix. Finally, we give an example to clarify these curves. We hope these results will be helpful to mathematicians who are specialized on mathematical modeling.

The Minkowski 3-space E31 is the Euclidean 3-space E3 provided with the metric

G = −dz12 + dz22 + dz32 , where (z1 , z2 , z3 ) is a rectangular coordinate system of E31 . Any arbitrary vector ν ∈ E31 can have one of three Lorentzian clause depicts; it can be timelike if G (ν, ν ) < 0, spacelike if G (ν, ν ) > 0 or ν = 0, and lightlike if G (ν, ν ) = 0 and ν = 0. Similarly, any arbitrary curve ζ = ζ (s ) can be timelike, spacelike or lightlike if all of its velocity vectors ζ (s) are timelike, spacelike or lightlike, respectively.

Let ζ = ζ (s ) be a regular non-null curve parametrized by arclength in E31 and {t, n, b, κ , τ } be its Frenet invariants where {t, n, b}, κ and τ are the moving Frenet frame and the natural curvature functions respectively. If ζ is a spacelike curve with spacelike principal normal vector, then the Frenet formulas of the curve ζ can be given as [6–8]:

˙

t (s ) n˙ (s ) b˙ (s )

∗ Corresponding author at: Al Imam Mohammad Ibn Saud Islamic University, College of Science, Department of Mathematics and Statistics, KSA, P. O. Box 90950, Riyadh 11623, Saudi Arabia.

E-mail address: emadms74@gmail.com

=

0

−κ (s ) 0

κ (s ) 0

τ (s )

0 τ (s ) 0

t (s ) n (s ) , b( s )

(1)

d , G (t , t ) = G (n, n ) = −G (b, b) = 1, and G (t, n ) = ds G (t, b) = G (n, b) = 0. where

·=

http://dx.doi.org/10.1016/j.joems.2017.04.003 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/)

320

E.M. Solouma / Journal of the Egyptian Mathematical Society 25 (2017) 319â&#x20AC;&#x201C;325

DeďŹ nition 2.1. A surface M in the Minkowski 3-space E31 is said to be timelike, spacelike surface if, respectively the induced metric on the surface is a Lorentz metrica, positive deďŹ nite Riemannian metric. In other words, the normal vector on the timelike(spacelike) surface is a spacelike(timelike) vector [8].

Let Îś : I â&#x2020;&#x2019; E31 be a spacelike curve in Minkowski space E31 . We deďŹ ne the equiform parameter of Îś by θ = â&#x2C6;Ť Îş ds. Then, we have ds 1 Ď = , where Ď = is the radius of curvature of the curve Îś . Let dθ Îş F be a homothety with the center in the origin and the coeďŹ&#x192;cient Îź. If we put Μ¯ = F (Îś ), then it follows

sÂŻ = Îź s and Ď ÂŻ = Îź Ď , where sÂŻ is the arc-length parameter of Μ¯ and Ď ÂŻ the radius of curvature of this curve. Therefore, θ is an equiform invariant parameter of Îś [9]. From that point, we recall {T, Ρ, Ξ ,} be the moving equiform Frenet frame where T (θ ) = Ď t (s ), Ρ (θ ) = Ď n(s ) and Ξ (θ ) = Ď b(s ) are the equiform tangent vector, equiform principal normal vector and equiform binormal vector respectively. Additionally, the ďŹ rst and second equiform curvature of the curve dĎ Ď&#x201E; Îś = Îś (θ ) are deďŹ ned by k1 (θ ) = Ď Ë&#x2122; = and k2 (θ ) = . So, the ds Îş moving equiform Frenet frame of Îś = Îś (θ ) is given as [10]:

T (θ )

Ρ (θ ) = Ξ (θ )

k1 ( θ ) â&#x2C6;&#x2019;1 0

= d

1

k1 ( θ ) k2 ( θ )

0

k2 ( θ ) k1 ( θ )

T (θ )

Ρ (θ ) , Ξ (θ )

S12

â&#x2C6;&#x2C6; = {u

E31

:

â&#x2C6;&#x2019;u21

+

u22

+

u23

= 1.}

Let Îś = Îś (θ ) be a regular non-null curve parametrized by arclength in Minkowski 3-space E31 with its moving equiform Frenet frame {T, Ρ, Ξ ,}. Then TΡ, TΞ , ΡΞ and TΡΞ -equiform Smarandache curves of Îś are deďŹ ned, respectively as follows [11]:

= (θ â&#x2C6;&#x2014; ) = = (θ â&#x2C6;&#x2014; ) = = (θ â&#x2C6;&#x2014; ) = = (θ â&#x2C6;&#x2014; ) =

DeďŹ nition 3.2. Let Îś = Îś (θ ) be a regular equiform spacelike curve lying completely on a spacelike surface M in E31 with moving equiform Frenet frame {T, Ρ, Ξ }. Then TΡ-equiform Smarandache curves are deďŹ ned by

1 = (θ â&#x2C6;&#x2014; ) = â&#x2C6;&#x161; (T (θ ) + Ρ (θ )).

2

1 â&#x2C6;&#x161; (T (θ ) + Ρ (θ )), 2 1 â&#x2C6;&#x161; (T (θ ) + Ξ (θ )), 2 1 â&#x2C6;&#x161; (Ρ (θ ) + Ξ (θ )), 2 1 â&#x2C6;&#x161; (T (θ ) + Ρ (θ ) + Ξ (θ )).

3

3. Special equiform Smarandache curves in E13 In this section, we deďŹ ne the special equiform Smarandache curves reference to the equiform Frenet frame of a curve Îś in Minkowski 3-space E31 . Furthermore, we obtain the natural equiform curvature functions of the equiform Smarandache curves lying completely on pesdo-sphere S12 and give some properties on the curves when the curve Îś has constant curvature or it is a circular helix DeďŹ nition 3.1. A curve in Minkowski space-time, whose position vector is composed by Frenet frame vectors on another curve, is called a Smarandache curve.

As consequence with the above deďŹ nition, we introduce a special form of the equiform Smarandache curves in E31 in the following subsection

(3)

Theorem 3.1. Let Îś = Îś (s ) be a spacelike curve with spacelike principal normal vector in E31 . If Îś is a circular helix with Îş > 0, then TΡ-equiform Smarandache curve is also circular helix and its the natural curvature functions are satisďŹ ed the following equation,

â&#x2C6;&#x161;

Îş (θ ) = â&#x2C6;&#x2014;

2

:

â&#x2C6;&#x161; k2 = Âą 2,

Ď (k22 â&#x2C6;&#x2019; 2 )

â&#x2C6;&#x161; 2 2 2 k2 ( 2k2 + 1 ) â&#x2C6;&#x2019; ( k2 â&#x2C6;&#x2019; 1 ) k2 + k2 ( k2 + 2 ) â&#x2C6;&#x2014; Ď&#x201E; ( θ ) = 2 : Ď 2(k32 + 2 ) â&#x2C6;&#x161; 3 k2 = â&#x2C6;&#x2019; 2.

(4)

Proof. Let = (θ â&#x2C6;&#x2014; ) be a TΡ-equiform Smarandache curves reference to the equiform spacelike curve Îś = Îś (θ ). From Eq. (3) and using Eq. (2), we get

(2)

, G (T , T ) = G (Ρ, Ρ ) = â&#x2C6;&#x2019;G (Ξ , Ξ ) = Ď 2 , and dθ G ( T , Ρ ) = G ( T , Ξ ) = G ( Ρ , Ξ ) = 0.

The pseudo-Riemannian sphere with center at the origin and of radius r = 1 in the Minkowski 3-space E31 is a quadric deďŹ ned by where

3.1. TΡ-equiform Smarandache curves in E13

( θ â&#x2C6;&#x2014; )=

d dθ â&#x2C6;&#x2014; 1 = â&#x2C6;&#x161; ((k1 â&#x2C6;&#x2019; 1 )T (θ ) + (k1 + 1 )Ρ (θ ) + k2 Ξ (θ )), dθ â&#x2C6;&#x2014; dθ 2 (5)

hence

T (θ â&#x2C6;&#x2014; )=

1

((k1 â&#x2C6;&#x2019; 1 )T (θ )+(k1 +1 )Ρ (θ ) + k2 Ξ (θ )),

Ď 2k21 â&#x2C6;&#x2019; k22 â&#x2C6;&#x2019; 2 (6)

where

dθ â&#x2C6;&#x2014; = dθ

Ď 2k21 â&#x2C6;&#x2019; k22 â&#x2C6;&#x2019; 2 â&#x2C6;&#x161; 2

Now

dT = dθ â&#x2C6;&#x2014;

.

(7)

â&#x2C6;&#x161; 2

2 (Îť1 T (θ ) + Îť2 Ρ (θ ) + Îť3 Ξ (θ )), Ď 2 2k21 â&#x2C6;&#x2019; k22 â&#x2C6;&#x2019; 2

where

â&#x17D;§

2 2

2 â&#x17D;¨Îť1 = (k1 â&#x2C6;&#x2019; 1 )(2k1 k1 â&#x2C6;&#x2019; k2 k2 ) + (2k1 â&#x2C6;&#x2019; k2 â&#x2C6;&#x2019; 2 )(k1 + k1 â&#x2C6;&#x2019; 3k1 ),

2 2

Îť2 = (k1 + 1 )(2k1 k1 â&#x2C6;&#x2019; k2 k2 ) + (2k1 â&#x2C6;&#x2019; k2 â&#x2C6;&#x2019; 2 )(k1 + k21 + k22 + k1 â&#x2C6;&#x2019; 2 ), â&#x17D;Š Îť3 = k2 (2k1 k 1 â&#x2C6;&#x2019; k2 k 2 ) + (2k21 â&#x2C6;&#x2019; k22 â&#x2C6;&#x2019; 2 )(k 2 + 2k1 k2 ).

Then

2 Îť21 + Îť22 â&#x2C6;&#x2019; Îť23

dT

Îş (θ â&#x2C6;&#x2014; ) =

2 ,

dθ â&#x2C6;&#x2014; = 2 Ď 2k1 â&#x2C6;&#x2019; k22 â&#x2C6;&#x2019; 2 and

N (θ â&#x2C6;&#x2014; ) =

Ν1 T (θ ) + Ν2 Ρ (θ ) + Ν3 Ξ (θ ) .

Ď Îť21 + Îť22 â&#x2C6;&#x2019; Îť23

Also

B ( θ â&#x2C6;&#x2014; ) = where

1 {m1 T ( θ ) + m2 Ρ ( θ ) + m3 Ξ ( θ )}, p1

(8)

E.M. Solouma / Journal of the Egyptian Mathematical Society 25 (2017) 319–325

λ2 k2 − λ3 (k1 + 1 ), m2 = λ2 k2 − λ3 (k1 − 1 ), m3 = λ2 (k1 − 1 ) − λ1 (k1 + 1 )

Now

and p1 = ρ 2k21 − k22 − 2 λ21 + λ22 − λ23 .

Now, from Eq. (5)

where

m1 =

dT = dθ ∗

+ k22 + 2k1 − 1]η (θ ) + [k 2 + 2k1 k2 + k2 ]ξ (θ ) ,

Then

and thus

κ (θ ) = ∗

1 (θ ∗ ) = √

2

(β1 T (θ ) + β2 η (θ ) + β3 ξ (θ )),

⎧

2 ⎨β1 = k1 + 3k1 (k1 − 1 ) + k1 (k1 − 3 ) − k2 (k2 + 2 ), β2 = k 1 + k 1 + 3(k1 k 1 + k2 k 2 ) + 3k1 (k1 − 1 ) + k21 (3k1 + 1 ) − 1, ⎩ β3 = k 2 + k 2 + 3(k1 k 1 + k2 k 2 ) + 3k1 k2 (k1 + 1 ) − k2 .

Hence, we have

τ ( θ

√ 2 w1 + w2 + w3 )= 2 , ρ

21 + 22 − 23

N (θ ∗ ) = Also

B ( θ ∗ ) =

and

3 = k1 (2k1 + 1 ) − 2k 1 + k22 (k1 + 1 ) + 2.

Now, if κ and τ are non-zero constants, then the natural curvature functions κ , τ are also non-zero constants and satisfying Eq. (4) which means that the Tη-equiform Smarandache curve is circular heilx.

3.2. Tξ -equiform Smarandache curves in E13 Definition 3.3. Let ζ = ζ (θ ) be a regular equiform spacelike curve lying completely on a spacelike surface M in E31 with moving equiform Frenet frame {T, η, ξ }. Then Tξ -equiform Smarandache curves are defined by

1 = (θ ∗ ) = √ (T (θ ) + ξ (θ )).

2

(10)

Theorem 3.2. Let ζ = ζ (s ) be a spacelike curve with spacelike principal normal vector in E31 . If ζ is a circular helix with κ > 0, then Tξ -equiform Smarandache curve is contained in a palne and its curvature is satisfied the following equation,

:

k2 = −1. (11)

Proof. Let = (θ ∗ ) be a Tξ -equiform Smarandache curves of ζ = ζ (θ ). Then from Eq. (10), we have

1 (θ ∗ ) = √ (k1 T (θ ) + (k2 + 1 )η (θ ) + k1 ξ (θ )).

2 1

(k1 T (θ ) + (k2 + 1 )η (θ ) + k1 ξ (θ )),

(12) (13)

where

dθ ∗ = dθ

ρ ( k2 + 1 ) √ 2

.

+ [k 1 + k2 (2k2 + 1 )]ξ (θ ) ,

2 = k 1 k2 − k 2 (k1 − 1 ) − k1 k2 (k1 + 1 ),

ρ ( k2 + 1 )

1 [ ε 2 k 1 − ε 3 ( k 2 + 1 )] T ( θ ) + k 1 ( ε 1 − ε 3 ) η ( θ ) p2

1 = k 1 k2 − k 2 (k1 + 1 ) + k2 (k2 − k21 − k1 − 2 ),

T (θ ∗ ) =

ε1 T ( θ ) + ε2 η ( θ ) + ε3 ξ ( θ ) .

ρ ε12 + ε22 − ε32

w3 = (k 1 + k21 − 2k1 − 1 )[β2 k2 − β3 (k1 + 1 )],

ρ ( k2 + 1 )2

(15)

1 (θ ∗ ) = √ [k 1 + k21 − k2 + 1]T (θ ) + [k 2 + 2k1 (k2 + 2 )]η (θ ) 2

w2 = (k 2 + 2k1 k2 + k2 )[β1 (k1 + 1 ) − β2 (k1 − 1 )],

κ (θ ) =

,

where p2 = ρ (k2 + 1 ) ε12 + ε22 − ε32 .

Now, from Eq. (12) we have

w1 = (k 1 + k21 + k22 + 2k1 − 1 )[β3 (k1 − 1 ) − β1 k2 ],

√ 2 (1 − k22 )(k2 + 1 )2 − k22 (3k22 − 2 )

ρ ( k2 + 1 )3

+ [ ε 2 k 1 − ε 1 ( k 2 + 1 )] ξ ( θ ) ,

(9)

where

∗

√ 2 ε12 + ε22 − ε32

and

where

∗

√ 2 (ε T (θ ) + ε2 η (θ ) + ε3 ξ (θ )), ρ 2 ( k2 + 1 )3 1

⎧

⎨ε1 = (k2 + 1 )(k1 − k2 − 1 ) − k1 k2 , ε2 = k1 ( k2 + 1 )2 , ⎩ ε3 = (k2 + 1 )[k 1 + k2 (k2 + 1 ) + k22 − k21 ] − k1 k 2 .

1 (θ ∗ ) = √ [k 1 + k21 − 2k1 − 1]T (θ ) + [k 1 + k21 2

321

(14)

1 (θ ∗ ) = √ (δ1 T (θ ) + δ2 η (θ ) + δ3 ξ (θ )), 2 where

⎧ δ1 = k 1 − k 2 + 3k1 (k 1 − k2 ) + k1 (k21 − 1 ), ⎪ ⎪ ⎨δ = k + k + 3(k k + k k ) + k2 (2k + 3 ) 2 1 2 2 2 1 1 2 1 + k2 (2k22 + k2 − 1 ) + 1, ⎪ ⎪ ⎩ δ3 = k 1 + k 2 + k1 (k 1 + 3k2 ) + k2 (5k 2 + 4k1 k2 ).

Hence, we have ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪

2 ⎪ ⎪ (k1 + 2k2 + k2 ) δ1 (k2 + 1 ) − δ2 k1 + k1 (δ3 − δ1 ) ⎪ ⎪ √ ⎪ ⎨ ( k + 2k k + 2k ) + ( k + k2 − k + 1 ) δ k − δ ( k + 1 ) ⎪ ⎬ 2 1 2 1 2 2 1 3 2 2 1 1 ∗ τ ( θ ) = 2 ,

2 ρ ⎪ ⎪ k1 k 2 + (k2 + 1 )(2k21 − k 1 ) − k2 (2k22 + 3k2 + 1 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

2 ⎪ ⎪ ⎪ ⎪ + k1 [k21 − 2k2 (k2 + 1 ) + 1] ⎪ ⎪ ⎪ ⎪

2 ⎩ ⎭ 2 2

2 − k1 k2 + k2 − k1 − k1 ( k2 + 1 ) + k2 ( k2 + 1 ) − 1

(16) So, if κ and τ are non-zero constants, then κ is non-zero constant and satisfying Eq. (11), also τ = 0 which means that the Tξ -equiform Smarandache curve is contained in a plane.

3.3. ηξ -equiform Smarandache curves in E13 Definition 3.4. Let ζ = ζ (θ ) be a regular equiform spacelike curve lying completely on a spacelike surface M in E31 with moving equiform Frenet frame {T, η, ξ }. Then ηξ -equiform Smarandache curves are defined by

1 = (θ ∗ ) = √ 2

η (θ ) + ξ (θ ) .

(17)

Theorem 3.3. Let ζ = ζ (s ) be a spacelike curve with spacelike principal normal vector in E31 . If ζ is a circular helix with κ > 0, then

322

E.M. Solouma / Journal of the Egyptian Mathematical Society 25 (2017) 319â&#x20AC;&#x201C;325

ΡΞ -equiform Smarandache curve is also circular helix and its the natural curvature functions are satisďŹ ed the following equation,

â&#x2C6;&#x161; 2(k22 â&#x2C6;&#x2019; 1 )

Îş (θ â&#x2C6;&#x2014; ) =

â&#x2C6;&#x161; Ď 2 â&#x2C6;&#x2014; Ď&#x201E; ( θ ) = 2 : Ď k2

, (18)

k2 = 0.

3.4. TΡΞ -equiform Smarandache curves in E13 DeďŹ nition 3.5. Let Îś = Îś (θ ) be a regular equiform spacelike curve lying completely on a spacelike surface M in E31 with moving equiform Frenet frame {T, Ρ, Ξ }. Then TΡΞ -equiform Smarandache curves are deďŹ ned by

Proof. Let = (θ â&#x2C6;&#x2014; ) be a ΡΞ -equiform Smarandache curves of the curve Îś = Îś (θ ). From Eq. (17), we get

1 = (θ â&#x2C6;&#x2014; ) = â&#x2C6;&#x161; (T (θ ) + Ρ (θ ) + Ξ (θ )).

3

1 (θ â&#x2C6;&#x2014; ) = â&#x2C6;&#x161; (â&#x2C6;&#x2019;T (θ ) + (k1 + k2 )Ρ (θ ) + (k1 + k2 )Ξ (θ )), 2

Theorem 3.4. Let Îś = Îś (s ) be a spacelike curve with spacelike principal normal vector in E31 . If Îś is a circular helix with Îş > 0, then TΡΞ -equiform Smarandache curve is also circular helix and its the natural curvature functions are satisďŹ ed the following equation,

(19)

hence

1

T (θ â&#x2C6;&#x2014; ) =

Ď

(â&#x2C6;&#x2019;T (θ ) + (k1 + k2 )Ρ (θ ) + (k1 + k2 )Ξ (θ )),

(20)

â&#x2C6;&#x161; 3 2 ( 1 â&#x2C6;&#x2019; k2 ) Îş (θ ) = : |k1 | < 1, 2Ď â&#x2C6;&#x161; 2 k2 ( k2 + 1 ) 3 Ď&#x201E; ( θ â&#x2C6;&#x2014; ) = 2 : k2 = â&#x2C6;&#x2019;1, 3.

3Ď (k2 â&#x2C6;&#x2019; 3 )(k2 + 1 )2 â&#x2C6;&#x2014;

where

dθ â&#x2C6;&#x2014; Ď = â&#x2C6;&#x161; . dθ 2

Now

dT = dθ â&#x2C6;&#x2014;

â&#x2C6;&#x161;

Ď

(21)

(γ1 T (θ ) + γ2 Ρ (θ ) + γ3 Ξ (θ )), 2

1 (θ â&#x2C6;&#x2014; ) = â&#x2C6;&#x161; ((k1 â&#x2C6;&#x2019; 1 )T (θ ) + (k1 + k2 + 1 )Ρ (θ ) + (k1 + k2 )Ξ (θ )).

3

Îł1 = â&#x2C6;&#x2019;(k1 + k2 ), Îł2 = k 1 + k 2 + k2 (k1 + k2 ) â&#x2C6;&#x2019; 1, Îł3 = k 1 + k 2 + k2 (k1 + k2 ).

Then

Îş (θ â&#x2C6;&#x2014; ) = and

N (θ â&#x2C6;&#x2014; ) =

2 Îł12 + Îł22 â&#x2C6;&#x2019; Îł32 Ď

(26)

T (θ â&#x2C6;&#x2014; ) = ,

(22) where

γ1 T (θ ) + γ2 Ρ (θ ) + γ3 Ξ (θ ) .

Ď Îł12 + Îł22 â&#x2C6;&#x2019; Îł32

dθ â&#x2C6;&#x2014; = dθ

1

((k1 â&#x2C6;&#x2019; 1 )T (θ ) + (k1 + k2 + 1 )Ρ (θ )

Ď k21 + 2k2 + 2 + (k1 + k2 )Ξ (θ )), (27)

Ď k21 + 2k2 + 2 â&#x2C6;&#x161; 3

Now

Also

B ( θ ) =

1

{[(Îł2 â&#x2C6;&#x2019; Îł3 )(k1 + k2 )]T (θ )

Ď Îł +Îł â&#x2C6;&#x2019;Îł + [Îł3 + Îł1 (k1 + k2 )]Ρ (θ ) â&#x2C6;&#x2019; [Îł2 + Îł1 (k1 + k2 )]Ξ (θ )}.

2 1

2 2

2 3

From Eq. (19), we have

1 (θ â&#x2C6;&#x2014; ) = â&#x2C6;&#x161;

(25)

Proof. Let = (θ â&#x2C6;&#x2014; ) be a TΡΞ -equiform Smarandache curves of the curve Îś = Îś (θ ). Then from Eq. (24), we get

2

where

â&#x2C6;&#x2014;

(24)

{â&#x2C6;&#x2019;[2k1 + k2 ]T (θ ) + [k 1 + k 2 + (k1 + k2 )2 â&#x2C6;&#x2019; 1]Ρ (θ )

2 +[k 1 + k 2 + (k1 + k2 )2 ]Ξ (θ )},

and

1 (θ â&#x2C6;&#x2014; ) = â&#x2C6;&#x161; (Ď&#x2030;1 T (θ ) + Ď&#x2030;2 Ρ (θ ) + Ď&#x2030;3 Ξ (θ )), 2

dT = dθ â&#x2C6;&#x2014;

â&#x2C6;&#x161; 3

Ď 2 k21 + 2k2 + 2

.

(28)

2 (Ď&#x2021;1 T (θ ) + Ď&#x2021;2 Ρ (θ ) + Ď&#x2021;3 Ξ (θ )),

where

â&#x17D;§ Ď&#x2021;1 = (k1 â&#x2C6;&#x2019; 1 )(k1 k 1 + k 2 ) â&#x2C6;&#x2019; (k2 + 1 )(k21 + 2k2 + 2 ), â&#x17D;Ş â&#x17D;Ş â&#x17D;Ş 2

â&#x17D;Ş â&#x17D;¨Ď&#x2021;2 = (k1 + 2k2 + 2 )[k1 + k2 + k1 + k2 (k1 + k2 ) â&#x2C6;&#x2019; 1]

+ (k1 k1 + k2 )(k1 + k2 + 1 ), â&#x17D;Ş â&#x17D;Ş â&#x17D;Ş Ď&#x2021; = (k21 + 2k2 + 2 )[k 1 + k 2 + k2 (k1 + k2 + 1 )] â&#x17D;Ş â&#x17D;Š 3 + (k1 + k2 )(k1 k 1 + k 2 ).

Then

where

â&#x17D;§ â&#x17D;¨Ď&#x2030;1 = â&#x2C6;&#x2019; 3k 1 + 2k 2 + k1 (2k1 + k2 ) + (k1 + k2 )2 â&#x2C6;&#x2019; 1 , Ď&#x2030;2 = k 1 + k 2 â&#x2C6;&#x2019; 3k1 â&#x2C6;&#x2019; k2 + 3(k1 + k2 )(k 1 + k 2 ) + (k1 + k2 )3 , â&#x17D;Š Ď&#x2030;3 = k 1 + k 2 + 3(k1 + k2 )(k 1 + k 2 ) + (k1 + k2 )3 .

Hence, we have

â&#x17D;§ â&#x17D;Ť [k + k 2 + (k1 + k2 )2 â&#x2C6;&#x2019; 1][Ď&#x2030;3 + Ď&#x2030;1 (k1 + k2 )]â&#x17D;Ź â&#x2C6;&#x161; â&#x17D;¨ 1 2 â&#x2C6;&#x2019;[k 1 + k 2 + (k1 + k2 )2 ][Ď&#x2030;2 + Ď&#x2030;1 (k1 + k2 )] â&#x17D;Š â&#x17D;­ â&#x2C6;&#x2019;(Ď&#x2030;3 â&#x2C6;&#x2019; Ď&#x2030;2 )(k1 + k2 )(2k1 + k2 ) â&#x2C6;&#x2014;

Ď&#x201E; ( θ ) = . Ď 2 k21 â&#x2C6;&#x2019; k22 + 2(k 1 + k 2 ) (23) Then, if Îş and Ď&#x201E; are non-zero constants, then the natural curvature functions Îş , Ď&#x201E; are also non-zero constants and satisfying Eq. (18) which means that the ΡΞ -equiform Smarandache curve is circular heilx.

3 Ď&#x2021;12 + Ď&#x2021;22 â&#x2C6;&#x2019; Ď&#x2021;32 Îş (θ â&#x2C6;&#x2014; ) = 2 , Ď k21 + 2k2 + 2

(29)

and

N (θ â&#x2C6;&#x2014; ) =

Ď&#x2021;1 T (θ ) + Ď&#x2021;2 Ρ (θ ) + Ď&#x2021;3 Ξ (θ ) .

Ď Ď&#x2021;12 + Ď&#x2021;22 â&#x2C6;&#x2019; Ď&#x2021;32

Also

B ( θ â&#x2C6;&#x2014; ) =

1

Ď k21 + 2k2 + 2 Ď&#x2021;12 + Ď&#x2021;22 â&#x2C6;&#x2019; Ď&#x2021;32

Ă&#x2014; [â&#x2C6;&#x2019;Ď&#x2021;3 + (Ď&#x2021;2 â&#x2C6;&#x2019; Ď&#x2021;3 )(k1 + k2 )]T (θ ) + [Ď&#x2021;1 (k1 + k2 ) â&#x2C6;&#x2019; Ď&#x2021;1 (k1 â&#x2C6;&#x2019; 1 )]Ρ (θ ) + [Ď&#x2021;2 (k2 â&#x2C6;&#x2019; 1 )

â&#x2C6;&#x2019; Ď&#x2021;1 (k1 + k2 + 1 )]Ξ (θ ) ,

E.M. Solouma / Journal of the Egyptian Mathematical Society 25 (2017) 319â&#x20AC;&#x201C;325

323

Fig. 2. Equiform spacelike curve Μ = Μ (θ ) on S12 .

Fig. 1. Spacelike curve Îś = Îś (s ) on S12 .

From Eq. (26), we have

1 (θ â&#x2C6;&#x2014; ) = â&#x2C6;&#x161; {[k21 â&#x2C6;&#x2019; k1 â&#x2C6;&#x2019; k2 â&#x2C6;&#x2019; 1]T (θ ) 3 + [k 1 + k 2 + 2k1 + (k1 + k2 )2 â&#x2C6;&#x2019; 1]Ρ (θ ) + [k 1 + k 2 + k2 + (k1 + k2 )2 ]Ξ (θ )},

and thus

1 (θ â&#x2C6;&#x2014; ) = â&#x2C6;&#x161; (Ď&#x2020;1 T (θ ) + Ď&#x2020;2 Ρ (θ ) + Ď&#x2020;3 Ξ (θ )), 3 where

â&#x17D;§ Ď&#x2020;1 = 2k 1 (k1 + 1 ) â&#x2C6;&#x2019; 2k 2 + k1 (k21 â&#x2C6;&#x2019; k1 â&#x2C6;&#x2019; k2 â&#x2C6;&#x2019; 2 ) + 1, â&#x17D;Ş â&#x17D;Ş â&#x17D;Ş â&#x17D;¨Ď&#x2020;2 = k 1 + k 2 + 2k 1 + k2 (k2 â&#x2C6;&#x2019; 1 ) + 3(k1 + k2 )(k 1 + k 2 ) + k1 (2k21 + k1 â&#x2C6;&#x2019; 1 ) + (k1 + k2 )3 â&#x2C6;&#x2019; 1, â&#x17D;Ş

â&#x17D;Ş k 2 + k2 (3k1 â&#x2C6;&#x2019; 1 ) + 3(k1 + k2 )(k 1 + k 2 ) â&#x17D;Ş â&#x17D;ŠĎ&#x2020;3 = k1 + k2 + + + ( k1 + k2 )3 .

Hence, we have

Ď&#x201E; ( θ â&#x2C6;&#x2014; ) =

â&#x2C6;&#x161; 3 v1 + v2 + v3 , Ď 2 q2 + q22 â&#x2C6;&#x2019; q23

Fig. 3. The TΡ-equiform Smarandache curve (θ â&#x2C6;&#x2014; ) on S12 .

(30)

where

v 1 = ( â&#x2C6;&#x2019; k 1 â&#x2C6;&#x2019; k 2 â&#x2C6;&#x2019; 1 ) [ Ď&#x2020;2 ( k 1 + k 2 ) â&#x2C6;&#x2019; Ď&#x2020; 3 ( k 1 + k 2 â&#x2C6;&#x2019; 1 ) ] , v2 = [k 1 + k 2 + 2k1 + (k1 + k2 )2 â&#x2C6;&#x2019; 1][Ď&#x2020;3 (k1 â&#x2C6;&#x2019; 1 ) â&#x2C6;&#x2019; Ď&#x2020;1 (k1 + k2 )], v3 = [k 1 + k 2 + k2 + (k1 + k2 )2 ][Ď&#x2020;1 (k1 + k2 â&#x2C6;&#x2019; 1 ) â&#x2C6;&#x2019; Ď&#x2020;2 (k1 â&#x2C6;&#x2019; 1 )], q1 = (k1 + k2 )(2k1 â&#x2C6;&#x2019; k2 â&#x2C6;&#x2019; 1 ) â&#x2C6;&#x2019; [k 1 + k 2 + 2k1 + (k1 + k2 )2 ], q2 = â&#x2C6;&#x2019;(k1 + k2 )(k1 k2 + 1 ) â&#x2C6;&#x2019; (k1 â&#x2C6;&#x2019; 1 )(k 1 + k 2 + k2 ), q3 = (k1 â&#x2C6;&#x2019; 1 )[k 1 + k 2 + 2k1 + (k1 + k2 )2 ] + k1 (2k1 â&#x2C6;&#x2019; 3 ) + (k + 1 + k2 + 1 )[2(k2 + 1 ) â&#x2C6;&#x2019; k1 (k1 + 1 )] + 1. k21

Now, if Îş and Ď&#x201E; are non-zero constants, then the natural curvature functions Îş , Ď&#x201E; are also non-zero constants and satisfying Eq. (25) which means that the TΡ-equiform Smarandache curve is circular heilx.

4. Example

â&#x2C6;&#x161;

â&#x2C6;&#x161; â&#x2C6;&#x161; Let Îś (s ) = 3 s, s sin ( 3 ln s ), s cos ( 3 ln s ) be a unit speed spacelike curve parametrized by arc-length s with spacelike principal normal vector in E31 (see Fig. 1). Then it is easy to show that

â&#x17D;§ â&#x2C6;&#x161; â&#x2C6;&#x161; â&#x2C6;&#x161; â&#x2C6;&#x161; t (s ) = ( 3, sin ( 3 ln s ) + 3 cos ( 3 ln s ), â&#x17D;Ş â&#x17D;Ş â&#x17D;Ş â&#x2C6;&#x161; â&#x2C6;&#x161; â&#x2C6;&#x161; â&#x17D;Ş â&#x17D;Ş cos ( 3 ln s ) â&#x2C6;&#x2019; 3 sin ( 3 ln s ) ), â&#x17D;Ş â&#x17D;Ş â&#x2C6;&#x161; â&#x2C6;&#x161; â&#x2C6;&#x161; â&#x17D;Ş â&#x17D;Ş â&#x17D;Ş n(s ) = â&#x2C6;&#x161;12 (0, cos ( 3 ln s ) â&#x2C6;&#x2019; 3 sin ( 3 ln s ), â&#x17D;Ş â&#x17D;Ş â&#x17D;Ş â&#x17D;¨ â&#x2C6;&#x2019; sin (â&#x2C6;&#x161;3 ln s ) â&#x2C6;&#x2019; â&#x2C6;&#x161;3 cos (â&#x2C6;&#x161;3 ln s )), â&#x2C6;&#x161;

1 Ď = 2â&#x2C6;&#x161;s 3 , k1 = 2â&#x2C6;&#x161; , â&#x17D;ŞÎş = 2 s 3 , â&#x17D;Ş 3 â&#x17D;Ş â&#x2C6;&#x161; â&#x2C6;&#x161; â&#x17D;Ş â&#x17D;Şb(s ) = (2, 3 sin ( 3 ln s ) + 3 cos (â&#x2C6;&#x161;3 ln s ), â&#x17D;Ş 2 2 â&#x17D;Ş â&#x17D;Ş â&#x2C6;&#x161; â&#x17D;Ş â&#x17D;Ş 3 cos (â&#x2C6;&#x161;3 ln s ) â&#x2C6;&#x2019; 3 sin (â&#x2C6;&#x161;3 ln s )), â&#x17D;Ş 2 2 â&#x17D;Ş â&#x17D;Ş â&#x2C6;&#x161; â&#x17D;Š Ď&#x201E; = 3s , k2 = 23 .

324

E.M. Solouma / Journal of the Egyptian Mathematical Society 25 (2017) 319â&#x20AC;&#x201C;325

Fig. 4. The TΞ -equiform Smarandache curve (θ â&#x2C6;&#x2014; ) on S12 .

â&#x2C6;&#x161; Hence, the equiform parameter is θ = â&#x2C6;Ť Îş ds = 2 3 sâ&#x2C6;&#x161;+ c. Here we take c = 0, then we have s = eθ /2

â&#x2C6;&#x161; 3

and Ď =

equiform spacelike curve Îś is deďŹ ne as (see Fig. 2)

Μ (θ ) =

â&#x2C6;&#x161; â&#x2C6;&#x161; â&#x2C6;&#x161; 3 eθ /2 3 , eθ /2 3 sin

θ

, eθ /2

2

â&#x2C6;&#x161;

3

cos

2 3 eθ /â&#x2C6;&#x161; .

2 3

So the

θ

.

2

It easy to show that eθ /2 2

T (θ ) =

â&#x2C6;&#x161; 3

1 1, â&#x2C6;&#x161; sin 3

θ 2

+ cos

θ 1

, â&#x2C6;&#x161; cos 2 3

θ 2

â&#x2C6;&#x2019; sin

It is clear that T is an equiform spacelike vector. Also

Ρ (θ ) =

eθ /2 4

â&#x2C6;&#x161;

3

and

Ξ (θ ) =

eθ /2 4

â&#x2C6;&#x161; 3

1 0, â&#x2C6;&#x161; cos 3

4 â&#x2C6;&#x161; , sin 3

θ 2

θ 2

+

â&#x2C6;&#x2019; sin

â&#x2C6;&#x161;

θ â&#x2C6;&#x2019;1

, â&#x2C6;&#x161; sin 3

2

3 cos

θ 2

θ

θ

2

2

, cos

â&#x2C6;&#x2019;

â&#x2C6;&#x2019; cos

â&#x2C6;&#x161;

3 sin

θ

.

2

θ

,

2

θ

.

2

Then Ρ is an equiform spacelike vector and Ξ is an equiform timelike vector.

The TΡ-equiform Smarandache curve (θ â&#x2C6;&#x2014; ) of the curve Îś (θ ) is given by (see Fig. 3) â&#x2C6;&#x161;

â&#x2C6;&#x161; â&#x2C6;&#x161; â&#x2C6;&#x161; 6 eθ /2 3 θ (θ â&#x2C6;&#x2014; ) = 2 3, (2 3 + 1 ) cos 24 2

+ (2 â&#x2C6;&#x2019;

â&#x2C6;&#x161; 3 ) sin

θ 2

, (2 â&#x2C6;&#x2019;

â&#x2C6;&#x161;

3 ) cos

θ 2

â&#x2C6;&#x161; â&#x2C6;&#x2019; (2 3 + 1 ) sin

θ 2

Fig. 5. The ΡΞ -equiform Smarandache curve (θ â&#x2C6;&#x2014; ) on S12 .

The ΡΞ -equiform Smarandache curve (θ â&#x2C6;&#x2014; ) of the curve Îś (θ ) is given by (see Fig. 5)

.

The TΞ -equiform Smarandache curve (θ â&#x2C6;&#x2014; ) of the curve Îś (θ ) is given by (see Fig. 4) â&#x2C6;&#x161; (θ â&#x2C6;&#x2014; ) =

â&#x2C6;&#x161;

6 eθ /2 3 24

2 (2 +

â&#x2C6;&#x161; + (2 3 + 3 ) cos

â&#x2C6;&#x161;

3 ), ( 2 +

â&#x2C6;&#x161;

3 ) sin

2

θ

θ

2

2

â&#x2C6;&#x161; , (2 + 3 ) cos

(θ

θ â&#x2C6;&#x161; â&#x2C6;&#x2019; (2 3 + 3 ) sin

θ 2

.

â&#x2C6;&#x2014;

â&#x2C6;&#x161;

â&#x2C6;&#x161; 6 eθ /2 3 θ θ )= 1, cos , â&#x2C6;&#x2019; sin .

6 2 2

The TΡΞ -equiform Smarandache curve (θ â&#x2C6;&#x2014; ) of the curve Îś (θ ) is given by (see Fig. 6)

E.M. Solouma / Journal of the Egyptian Mathematical Society 25 (2017) 319â&#x20AC;&#x201C;325

325

Acknowledgment The author is grateful to the Editor in Chief and the referees for their efforts and constructive criticism which have improved the manuscript References

Fig. 6. The TΡΞ -equiform Smarandache curve (θ â&#x2C6;&#x2014; ) on S12 .

(θ ) = â&#x2C6;&#x2014;

â&#x2C6;&#x161;

9 eθ /2 18

â&#x2C6;&#x161; 3

2+

â&#x2C6;&#x161;

â&#x2C6;&#x161; 3, sin θ2 + (2 + 3 ) cos θ2 , cos θ2

â&#x2C6;&#x2019; (2 +

â&#x2C6;&#x161; 3 ) sin θ2 .

[1] C. Ashbacher, Smarandache geometries, Smarandache Notions J. 8 (1-3) (1997) 212â&#x20AC;&#x201C;215.

[2] Ă&#x2013;. Bektas¸ , S. YĂźce, Special Smarandache curves according to Darboux frame in euclidean 3- space, Romanian J. Math. Comput. Sci. 3 (2013) 48â&#x20AC;&#x201C;59.

[3] M. Ă&#x2021;etin, Y. Tunçer, M.K. Karacan, Smarandache curves according to bishop frame in euclidean 3- space, Gen. Math. Notes 20 (2) (2014) 50â&#x20AC;&#x201C;66.

[4] E.M. Solouma, Special Smarandache curves recording by curves on a spacelike surface in Minkowski space-time, PONTE J. 73 (2) (2017) 251â&#x20AC;&#x201C;263, doi:10.

21506/j.ponte.2017.2.20.

[5] K. Tas¸ kĂśprĂź, M. Tosun, Smarandache curves according to Sabban frame on, Boletim da Sociedade Paraneanse de Matematica 32 (1) (2014) 51â&#x20AC;&#x201C;59.

[6] M.P.D. Carmo, Differential geometry of curves and surfaces, Prentice Hall, Englewood Cliffs, NJ, 1976.

[7] R. LĂłpez, Differential geometry of curves and surfaces in Lorentzâ&#x20AC;&#x201C;Minkowski space, Int. Electron. J. Geometry 7 (7) (2014) 44â&#x20AC;&#x201C;107.

[8] B. Oâ&#x20AC;&#x2122;Neill, Semi-Riemannian geometry with applications to relativity, Academic press, New York, 1983.

[9] M.E. Aydin, Mahmut Ergut the equiform differential geometry of curves in 4-dimensional galilean space g4 , Stud. Univ. Babes-Bolyai Math. 58 (3) (2013) 399â&#x20AC;&#x201C;406.

[10] H.K. Elsayied, M. Elzawy, A. Elsharkawy, Equiform spacelike normal curves in Minkowski 3-space, in: Proceedings of the 5th International Conference of Mathematics & Information Sciences, Zewail City, Egypt, 2016, pp. 11â&#x20AC;&#x201C;13.

[11] M. Turgut, S. YÄąlmaz, Smarandache curves in Minkowski space-time, Int. J. Math. Comb. 3 (2008) 51â&#x20AC;&#x201C;55.