Smarandache Curves according to Curves on a Spacelike Surface in Minkowski 3-Space R 1 3

11
Research Article Smarandache Curves according to Curves on a Spacelike Surface in Minkowski 3-Space R 3 1 Ufuk Ozturk and Esra Betul Koc Ozturk Department of Mathematics, Faculty of Sciences, University of C ¸ ankırı Karatekin, 18100 C ¸ ankırı, Turkey Correspondence should be addressed to Ufuk Ozturk; [email protected] Received 4 August 2014; Accepted 23 September 2014; Published 16 October 2014 Academic Editor: Annalisa De Bonis Copyright © 2014 U. Ozturk and E. B. Koc Ozturk. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce Smarandache curves according to the Lorentzian Darboux frame of a curve on spacelike surface in Minkowski 3- space R 3 1 . Also, we obtain the Sabban frame and the geodesic curvature of the Smarandache curves and give some characterizations on the curves when the curve is an asymptotic curve or a principal curve. And we give an example to illustrate these curves. 1. Introduction In the theory of curves in the Euclidean and Minkowski spaces, one of the interesting problems is the characterization of a regular curve. In the solution of the problem, the curva- ture functions and of a regular curve have an effective role. It is known that the shape and size of a regular curve can be determined by using its curvatures and . Another approach to the solution of the problem is to consider the relationship between the corresponding Frenet vectors of two curves. For instance, Bertrand curves and Mannheim curves arise from this relationship. Another example is the Smarandache curves. ey are the objects of Smarandache geometry, that is, a geometry which has at least one Smarandachely denied axiom [1]. e axiom is said to be Smarandachely 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. By definition, if the position vector of a curve is composed by the Frenet frame’s vectors of another curve , then the curve is called a Smarandache curve [2]. Special Smarandache curves in the Euclidean and Minkowski spaces are studied by some authors [3–8]. For instance, the special Smarandache curves according to Darboux frame in E 3 are characterized in [9]. In this paper, we define Smarandache curves according to the Lorentzian Darboux frame of a curve on spacelike surface in Minkowski 3-space R 3 1 . Inspired by the previous papers we investigate the geodesic curvature and the Sabban frame’s vectors of Smarandache curves. In Section 2, we explain the basic concepts of Minkowski 3-space and give Lorentzian Darboux frame that will be used throughout the paper. Section 3 is devoted to the study of four Smarandache curves, T-Smarandache curve, T-Smarandache curve, - Smarandache curve, and T-Smarandache curve by consid- ering the relationship with invariants , (), and () of curve on spacelike surface in Minkowski 3-space R 3 1 . Also, we give some characterizations on the curves when the curve is an asymptotic curve or a principal curve. Finally, we illustrate these curves with an example. 2. Basic Concepts e Minkowski 3-space R 3 1 is the Euclidean 3-space R 3 provided with the standard flat metric given by ⟨⋅, ⋅⟩ = − 2 1 + 2 2 + 2 3 , (1) where ( 1 , 2 , 3 ) is a rectangular Cartesian coordinate sys- tem of R 3 1 . Since ⟨⋅, ⋅⟩ is an indefinite metric, recall that a nonzero vector x ∈ R 3 1 can have one of three Lorentzian causal characters; it can be spacelike if ⟨x, x⟩>0, timelike if ⟨x, x⟩<0, and null (lightlike) if ⟨x, x⟩=0. In particular, the norm (length) of a vector x ∈ R 3 1 is given by ‖x‖=√ |⟨x, x⟩| and two vectors x and y are said to be orthogonal, if ⟨x, y⟩=0. Hindawi Publishing Corporation Journal of Discrete Mathematics Volume 2014, Article ID 829581, 10 pages http://dx.doi.org/10.1155/2014/829581

Transcript of Smarandache Curves according to Curves on a Spacelike Surface in Minkowski 3-Space R 1 3

Page 1: Smarandache Curves according to Curves on a Spacelike Surface in Minkowski 3-Space R 1 3

Research ArticleSmarandache Curves according to Curves on a Spacelike Surfacein Minkowski 3-Space R3

1

Ufuk Ozturk and Esra Betul Koc Ozturk

Department of Mathematics Faculty of Sciences University of Cankırı Karatekin 18100 Cankırı Turkey

Correspondence should be addressed to Ufuk Ozturk ozturkufuk06gmailcom

Received 4 August 2014 Accepted 23 September 2014 Published 16 October 2014

Academic Editor Annalisa De Bonis

Copyright copy 2014 U Ozturk and E B Koc Ozturk This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We introduce Smarandache curves according to the Lorentzian Darboux frame of a curve on spacelike surface in Minkowski 3-spaceR3

1 Also we obtain the Sabban frame and the geodesic curvature of the Smarandache curves and give some characterizations

on the curves when the curve 120572 is an asymptotic curve or a principal curve And we give an example to illustrate these curves

1 Introduction

In the theory of curves in the Euclidean and Minkowskispaces one of the interesting problems is the characterizationof a regular curve In the solution of the problem the curva-ture functions 120581 and 120591 of a regular curve have an effective roleIt is known that the shape and size of a regular curve can bedetermined by using its curvatures 120581 and 120591 Another approachto the solution of the problem is to consider the relationshipbetween the corresponding Frenet vectors of two curvesFor instance Bertrand curves and Mannheim curves arisefrom this relationship Another example is the Smarandachecurves They are the objects of Smarandache geometry thatis a geometry which has at least one Smarandachely deniedaxiom [1] The axiom is said to be Smarandachely denied if itbehaves in at least two different ways within the same spaceSmarandache geometries are connected with the theory ofrelativity and the parallel universes

By definition if the position vector of a curve 120573 iscomposed by the Frenet framersquos vectors of another curve 120572then the curve 120573 is called a Smarandache curve [2] SpecialSmarandache curves in the Euclidean and Minkowski spacesare studied by some authors [3ndash8] For instance the specialSmarandache curves according to Darboux frame in E3 arecharacterized in [9]

In this paper we define Smarandache curves accordingto the Lorentzian Darboux frame of a curve on spacelikesurface in Minkowski 3-space R3

1 Inspired by the previous

papers we investigate the geodesic curvature and the Sabbanframersquos vectors of Smarandache curves In Section 2 weexplain the basic concepts of Minkowski 3-space and giveLorentzian Darboux frame that will be used throughout thepaper Section 3 is devoted to the study of four Smarandachecurves T120578-Smarandache curve T120585-Smarandache curve 120578120585-Smarandache curve and T120578120585-Smarandache curve by consid-ering the relationship with invariants 119896

119899 119896119892(119904) and 120591

119892(119904) of

curve on spacelike surface inMinkowski 3-spaceR31 Also we

give some characterizations on the curves when the curve 120572 isan asymptotic curve or a principal curve Finally we illustratethese curves with an example

2 Basic Concepts

The Minkowski 3-space R31is the Euclidean 3-space R3

provided with the standard flat metric given by

⟨sdot sdot⟩ = minus1198891199092

1+ 1198891199092

2+ 1198891199092

3 (1)

where (1199091 1199092 1199093) is a rectangular Cartesian coordinate sys-

tem of R31 Since ⟨sdot sdot⟩ is an indefinite metric recall that a

nonzero vector x isin R31can have one of three Lorentzian

causal characters it can be spacelike if ⟨x x⟩ gt 0 timelike if⟨x x⟩ lt 0 and null (lightlike) if ⟨x x⟩ = 0 In particular thenorm (length) of a vector x isin R3

1is given by x = radic|⟨x x⟩|

and two vectors x and y are said to be orthogonal if ⟨x y⟩ = 0

Hindawi Publishing CorporationJournal of Discrete MathematicsVolume 2014 Article ID 829581 10 pageshttpdxdoiorg1011552014829581

2 Journal of Discrete Mathematics

For any x = (1199091 1199092 1199093) and y = (119910

1 1199102 1199103) in the space R3

1

the pseudovector product of x and y is defined by

x times y = (minus11990921199103+ 11990931199102 11990931199101minus 11990911199103 11990911199102minus 11990921199101) (2)

Next recall that an arbitrary curve 120572 = 120572(119904) in E31

can locally be spacelike timelike or null (lightlike) if all ofits velocity vectors 120572

1015840(119904) are respectively spacelike timelike

or null (lightlike) for every 119904 isin 119868 [10] If 1205721015840(s) = 0

for every 119904 isin 119868 then 120572 is a regular curve in R31 A

spacelike (timelike) regular curve 120572 is parameterized bypseudoarclength parameter 119904 which is given by 120572 119868 sub R rarr

R31 then the tangent vector 120572

1015840(119904) along 120572 has unit length

that is ⟨1205721015840(119904) 1205721015840(119904)⟩ = 1 (⟨1205721015840(119904) 1205721015840(119904)⟩ = minus1) for all 119904 isin 119868

respectively

Remark 1 Let x = (1199091 1199092 1199093) y = (119910

1 1199102 1199103) and z =

(1199111 1199112 1199113) be vectors in R3

1 Then

(i) ⟨x times y z⟩ =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1199091

1199092

1199093

1199101

1199102

1199103

1199111

1199112

1199113

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(ii) x times (y times z) = minus ⟨x z⟩ y + ⟨x y⟩ z

(iii) ⟨x times y x times y⟩ = minus ⟨x x⟩ ⟨y y⟩ + ⟨x y⟩2

(3)

where times is the pseudovector product in the space R31

Lemma 2 In the Minkowski 3-spaceR31 the following proper-

ties are satisfied [10]

(i) two timelike vectors are never orthogonal(ii) two null vectors are orthogonal if and only if they are

linearly dependent(iii) timelike vector is never orthogonal to a null vector

Let 120601 119880 sub R2 rarr R31 120601(119880) = 119872 and 120574 119868 sub R rarr 119880

be a spacelike embedding and a regular curve respectivelyThen we have a curve 120572 on the surface 119872 which is definedby 120572(119904) = 120601(120574(119904)) and since 120601 is a spacelike embedding wehave a unit timelike normal vector field 120578 along the surface119872 which is defined by

120578 equiv120601119909times 120601119910

10038171003817100381710038171003817120601119909times 120601119910

10038171003817100381710038171003817

(4)

Since119872 is a spacelike surface we can choose a future directedunit timelike normal vector field 120578 along the surface 119872Hence we have a pseudoorthonormal frame T 120578 120585which iscalled the Lorentzian Darboux frame along the curve 120572where120585(119904) = T(119904)times120578(119904) is a unit spacelike vectorThe correspondingFrenet formulae of 120572 read

[

[

T10158401205781015840

1205851015840

]

]

= [

[

0 119896119899

119896119892

119896119899

0 120591119892

minus119896119892

120591119892

0

]

]

[

[

T120578

120585

]

]

(5)

where 119896119899(119904) = minus⟨T1015840(119904) 120578(119904)⟩ 119896

119892(119904) = ⟨T1015840(119904) 120585(119904)⟩ and

120591119892(119904) = minus⟨120585

1015840(119904) 120578(119904)⟩ are the asymptotic curvature the

geodesic curvature and the principal curvature of 120572 on thesurface 119872 in R3

1 respectively and 119904 is arclength parameter

of 120572 In particular the following relations hold

T times 120578 = 120585 120578 times 120585 = minusT 120585 times T = 120578 (6)

Both 119896119899and 119896

119892may be positive or negative Specifically 119896

119899

is positive if 120572 curves towards the normal vector 120578 and 119896119892is

positive if120572 curves towards the tangent normal vector 120585 Alsothe curve 120572 is characterized by 119896

119899 119896119892 and 120591

119892as follows

120572 is

an asymptotic curve iff 119896119899equiv 0

a geodesic curve iff 119896119892equiv 0

a principal curve iff 120591119892equiv 0

(7)

Since 120572 is a unit-speed curve is perpendicular to T but may have components in the normal and tangent normaldirections

= 119896119899120578 + 119896119892120585 (8)

These are related to the total curvature 120581 of 120572 by the formula

1205812=

2= 1198962

119892minus 1198962

119899 (9)

From (9) we can give the following proposition

Proposition 3 Let 119872 be a spacelike surface in R31 Let 120572 =

120572(119904) be regular unit speed curves lying fully with the LorentzianDarboux frame T 120578 120585 on the surface119872 inR3

1 There is not a

geodesic curve on 119872

The pseudosphere with center at the origin and of radius119903 = 1 in the Minkowski 3-space R3

1is a quadric defined by

1198782

1= isin R

3

1| minus1199092

1+ 1199092

2+ 1199092

3= 1 (10)

Let 120573 119868 sub R rarr 1198782

1be a curve lying fully in pseudosphere

1198782

1in R31 Then its position vector 120573 is a spacelike which

means that the tangent vector 119879120573

= 1205731015840 can be spacelike

timelike or null Depending on the causal character of 119879120573

we distinguish the following three cases [5]

Case 1 (119879120573is a unit spacelike vector) Thenwe have orthonor-

mal Sabban frame 120573(119904) 119879120573(119904) 120585120573(119904) along the curve120572 where

120585120573(119904) = minus120573(119904) times 119879

120573(119904) is the unit timelike vector The

corresponding Frenet formulae of 120573 according to the Sabbanframe read

[[

[

1205731015840

1198791015840

120573

1205851015840

120573

]]

]

=[[

[

0 1 0

minus1 0 minus119896119892 (119904)

0 minus119896119892(119904) 0

]]

]

[

[

120573

119879120573

120585120573

]

]

(11)

where 119896119892(119904) = det(120573(119904) 119879

120573(119904) 1198791015840

120573(119904)) is the geodesic curvature

of 120573 and 119904 is the arclength parameter of 120573 In particular thefollowing relations hold

120573 times 119879120573= minus120585120573 119879

120573times 120585120573= 120573 120585

120573times 120573 = 119879

120573 (12)

Journal of Discrete Mathematics 3

Case 2 (119879120573is a unit timelike vector) Hence we have ortho-

normal Sabban frame 120573(119904) 119879120573(119904) 120585120573(119904) along the curve 120573

where 120585120573(119904) = 120573(119904) times 119879

120573(119904) is the unit spacelike vector The

corresponding Frenet formulae of 120573 according to the Sabbanframe read

[[

[

1205731015840

1198791015840

120573

1205851015840

120573

]]

]

=[[

[

0 1 0

1 0 119896119892(119904)

0 119896119892(119904) 0

]]

]

[

[

120573

119879120573

120585120573

]

]

(13)

where 119896119892(119904) = det(120573(119904) 119879

120573(119904) 1198791015840

120573(119904)) is the geodesic curvature

of 120573 and 119904 is the arclength parameter of 120573 In particular thefollowing relations hold

120573 times 119879120573= 120585120573 119879

120573times 120585120573= 120573 120585

120573times 120573 = minus119879

120573 (14)

Case 3 (119879120573is a null vector) It is known that the only null

curves lying on pseudosphere 1198782

1are the null straight lines

which are the null geodesics

3 Smarandache Curves according toCurves on a SpacelikeSurface in Minkowski 3-Space R3

1

In the following section we define the Smarandache curvesaccording to the Lorentzian Darboux frame in Minkowski3-space Also we obtain the Sabban frame and the geodesiccurvature of the Smarandache curves lying on pseudosphere1198782

1and give some characterizations on the curves when the

curve 120572 is an asymptotic curve or a principal curve

Definition 4 Let 120572 = 120572(119904) be a spacelike curve lying fully onthe spacelike surface 119872 in R3

1with the moving Lorentzian

Darboux frame T 120578 120585 Then T120578-Smarandache curve of 120572is defined by

120573 (119904⋆(119904)) =

1

radic2

(119886T (119904) + 119887120578 (119904)) (15)

where 119886 119887 isin R0and 1198862minus 1198872= 2

Definition 5 Let 120572 = 120572(119904) be a spacelike curve lying fully onthe spacelike surface 119872 in R3

1with the moving Lorentzian

Darboux frame T 120578 120585Then T120585-Smarandache curve of 120572 isdefined by

120573 (119904⋆(119904)) =

1

radic2(119886T (s) + 119887120585 (119904)) (16)

where 119886 119887 isin R0and 1198862+ 1198872= 2

Definition 6 Let 120572 = 120572(119904) be a spacelike curve lying fully onthe spacelike surface 119872 in R3

1with the moving Lorentzian

Darboux frame T 120578 120585 Then 120578120585-Smarandache curve of 120572 isdefined by

120573 (119904⋆(119904)) =

1

radic2

(119886120578 (119904) + 119887120585 (119904)) (17)

where 119886 119887 isin R0and 1198872minus 1198862= 2

Definition 7 Let 120572 = 120572(119904) be a spacelike curve lying fully onthe spacelike surface 119872 in R3

1with the moving Lorentzian

Darboux frame T 120578 120585 Then T120578120585-Smarandache curve of 120572is defined by

120573 (119904⋆(119904)) =

1

radic3

(119886T (119904) + 119887120578 (119904) + 119888120585 (119904)) (18)

where 119886 119887 isin R0and 1198862minus 1198872+ 1198882= 3

Thus there are two following cases

Case 4 (120572 is an asymptotic curve) Then we have the follow-ing theorems

Theorem 8 Let 120572 = 120572(119904) be an asymptotic spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

(i) if 119886119896119892+119887120591119892

= 0 for all 119904 then the Sabban frame 120573T120573

120585120573 of the T120578 -Smarandache curve 120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[

[

119886

radic2

119887

radic2

0

0 0 120598

120598119887

radic2

120598119886

radic2

0

]]]]]

]

[

[

T120578

120585

]

]

(19)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892=

119886120591119892minus 119887119896119892

radic2

(20)

where 120598 = sign(119886119896119892+ 119887120591119892) for all 119904 and

119889119904lowast

119889119904= 120598

119886119896119892+ 119887120591119892

radic2

(21)

(ii) if 119886119896119892+119887120591119892= 0 for all 119904 then the Sabban frame 120573T

120573

120585120573 of the T120578-Smarandache curve 120573 is a null geodesic

Proof We assume that the curve 120572 is an asymptotic curveDifferentiating (15) with respect to 119904 and using (5) we obtain

1205731015840=

119889120573

119889119904

=1

radic2

(119886119896119892+ 119887120591119892) 120585

⟨1205731015840 1205731015840⟩ =

(119886119896119892+ 119887120591119892)2

2

(22)

Then there are two following cases

(i) If 119886119896119892+119887120591119892

= 0 for all 119904 since 1205731015840 = (119889120573119889119904lowast)(119889119904lowast119889119904)

then the tangent vectorT120573of the curve120573 is a spacelike

vector such that

T120573= 120598120585 (23)

4 Journal of Discrete Mathematics

where

119889119904lowast

119889119904= 120598

119886119896119892+ 119887120591119892

radic2

(24)

On the other hand from (15) and (23) it can be easilyseen that

120585120573= minus120573 times 119879

120573

= 120598119887

radic2

T + 120598119886

radic2

120578

(25)

is a unit timelike vectorConsequently the geodesic curvature 119896

119892of the curve

120573 = 120573(119904lowast) is given by

119896119892= det (120573 119879

120573T1015840120573)

=

119886120591119892minus 119887119896119892

radic2

(26)

From (15) (23) and (25) we obtain the Sabban frame120573T120573 120585120573 of 120573

(ii) If 119886119896119892+119887120591119892= 0 for all 119904 then 120573

1015840 is null So the tangentvector T

120573of the curve 120573 is a null vector It is known

that the only null curves lying on pseudosphere 11987821are

the null straight lines which are the null geodesics

In the theorems which follow in a similar way as inTheorem 8 we obtain the Sabban frame 120573T

120573 120585120573 and the

geodesic curvature 119896119892of a spacelike Smarandache curve We

omit the proofs of Theorems 9 10 and 11 since they areanalogous to the proof of Theorem 8

Theorem 9 Let 120572 = 120572(119904) be an asymptotic spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

(i) if 21198962119892minus (119887120591119892)2

= 0 for all 119904 then the Sabban frame 120573T120573 120585120573 of the T120585-Smarandache curve 120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[

[

119886

radic2

0119887

radic2minus119887119896119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

119887120591119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

119886119896119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

119887120591119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

minus2119896119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

119886119887120591119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

]]]]]]]]

]

times [

[

T120578

120585

]

]

(27)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= ((119886

211988731205913

119892minus 11988741205913

119892+ 41198871205911198921198962

119892) 1198961015840

119892

+ (11988741205912

119892119896119892minus 119886211988731205912

119892119896119892minus 4119887119896

3

119892) 1205911015840

119892

+ (21198861198872+ 2119886119887 minus 4119886) 120591

1198921198964

119892

minus (1198861198874+ 1198861198873) 1205913

1198921198962

119892+ 11988611988741205915

119892)

times ((41198962

119892minus 2(119887120591

119892)2

)

2

)

minus1

(28)

where 120598 = sign(21198962119892minus (119887120591119892)2) for all 119904 and

119889119904lowast

119889119904=

radic120598

21198962

119892minus (119887120591119892)2

2

(29)

(ii) if 119886119896119892+119887120591119892= 0 for all 119904 then the Sabban frame 120573T

120573

120585120573 of the T120578-Smarandache curve 120573 is a null geodesic

Theorem 10 Let 120572 = 120572(119904) be an asymptotic spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

(i) if (119887119896119892)2minus 21205912

119892= 0 for all 119904 then the Sabban frame 120573

T120573 120585120573 of the 120578120585-Smarandache curve 120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[[

[

0119886

radic2

119887

radic2minus119887119896119892

radic120598 ((119887119896119892)2

minus 21205912

119892)

119887120591119892

radic120598 ((119887119896119892)2

minus 21205912

119892)

119886120591119892

radic120598 ((119887119896119892)2

minus 21205912

119892)

minus2120591119892

radic2120598 ((119887119896119892)2

minus 21205912

119892)

1198872119896119892

radic2120598 ((119887119896119892)2

minus 21205912

119892)

minus119886119887119896119892

radic2120598 ((119887119896119892)2

minus 21205912

119892)

]]]]]]]]]

]

times [

[

T120578

120585

]

]

(30)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= ((4radic2119887120591

3

119892minus (radic2119886

21198873+ radic2119887

5) 1205911198921198962

119892) 1198961015840

119892

+ ((radic211988621198873+ radic2119887

5) 1198963

119892minus 4radic2119887120591

2

119892119896119892) 1205911015840

119892

minus radic211988611988741198965

119892+ 2radic2119886119887

41205912

1198921198963

119892

+ (4radic2119886 minus 4radic21198861198872) 1205914

119892119896119892)

times (2(11988721198962

119892minus 21205912

119892)2

)

minus1

(31)

Journal of Discrete Mathematics 5

where 120598 = sign((119887119896119892)2minus 21205912

119892) for all 119904 and

119889119904lowast

119889119904=

radic120598

(119887119896119892)2

minus 21205912

119892

2

(32)

(ii) if (119887119896119892)2minus 21205912

119892= 0 for all 119904 then the Sabban frame

120573T120573 120585120573 of the T120578-Smarandache curve 120573 is a null

geodesic

Theorem 11 Let 120572 = 120572(119904) be an asymptotic spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

(i) if 3(1198962119892minus1205912

119892)+(119887119896

119892+ 119886120591119892)2

= 0 for all 119904 then the Sabbanframe 120573T

120573 120585120573 of the T120578120585-Smarandache curve 120573 is

given by

120573 =119886T + 119887120578 + 119888120585

radic3

T120573=

minus119888119896119892T + 119888120591

119892120578 + (119886119896

119892+ 119887120591119892) 120585

radic120598 (3 (1198962119892minus 1205912119892) + (119887119896

119892+ 119886120591119892)2

)

120585120573= ((119887 (119886119896

119892+ 119887120591119892) minus 1198882120591119892)T

+ (1198882119896119892+ 119886 (119886119896

119892+ 119887120591119892)) 120578 minus (119886119888120591

119892+ 119887119888119896119892) 120585)

times (radic120598 (9 (1198962119892minus 1205912119892) + 3(119887119896

119892+ 119886120591119892)2

))

minus1

(33)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= ((119887 (119886119896

119892+ 119887120591119892) minus 1198882120591119892) 1198911

minus (1198882119896119892+ 119886 (119886119896

119892+ 119887120591119892)) 1198912

minus (119886119888120591119892+ 119887119888119896119892) 1198913)

times (radic3(3 (1198962

119892minus 1205912

119892) + (119887119896

119892+ 119886120591119892)2

)

2

)

minus1

(34)

where 120598 = sign(3(1198962119892minus 1205912

119892) + (119887119896

119892+ 119886120591119892)2) for all 119904 and

119889119904lowast

119889119904=

radic120598

(119887119896119892)2

minus 21205912

119892

2

1198911= ((119886

2119888 minus 3119888) 120591

2

119892+ 119886119887119888120591

119892119896119892) 1198961015840

119892

+ ((3119888 minus 1198862119888) 120591119892119896119892minus 119886119887119888119896

2

119892) 1205911015840

119892

+ (3119886 + 1198861198872) 1198964

119892+ (1198863+ 2119886119887

2minus 3119886) 120591

2

1198921198962

119892

+ (1198873+ 21198862119887 + 3119887) 120591

1198921198963

119892

+ (1198862119887 minus 3119887) 120591

3

119892119896119892

1198912= (119886119887119888120591

2

119892+ (1198872119888 + 3119888) 120591

119892119896119892) 1198961015840

119892

minus (119886119887119888120591119892119896119892+ (3119888 + 119887

2119888) 1198962

119892) 1205911015840

119892

+ (3119887 minus 1198862119887) 1205914

119892minus (1198873+ 21198862119887 + 3119887) 120591

2

1198921198962

119892

minus (1198861198872+ 3119886) 120591

1198921198963

119892

+ (3119886 minus 21198861198872minus 1198863) 1205913

119892119896119892

1198913= ((minus119886

3+ 3119886 + 119886119887

2) 1205912

119892+ (1198873+ 3119887 minus 119886

2119887) 120591119892119896119892) 1198961015840

119892

+ ((minus1198873minus 3119887 + 119886

2119887) 1198962

119892+ (1198863minus 1198861198872minus 3119886) 120591

119892119896119892) 1205911015840

119892

+ (3119888 + 1198872119888) 1198964

119892+ (3119888 minus 119886

2119888) 1205914

119892+ 2119886119887119888120591

1198921198963

119892

minus 21198861198871198881205913

119892119896119892+ (1198862119888 minus 6119888 minus 119887

2119888) 1205912

1198921198962

119892

(35)

(ii) if 3(1198962119892minus1205912

119892)+(119887119896

119892+ 119886120591119892)2= 0 for all 119904 then the Sabban

frame 120573T120573 120585120573 of the T120578-Smarandache curve 120573 is a

null geodesic

Case 5 (120572 is a principal curve) Then we have the followingtheorems

Theorem 12 Let 120572 = 120572(119904) be a principal spacelike curvelying fully on the spacelike surface 119872 in R3

1with the

moving Lorentzian Darboux frame T 120578 120585 Then the T120578-Smarandache curve 120573 is spacelike and the Sabban frame120573T120573 120585120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[[

[

119886

radic2

119887

radic2

0

119887119896119899

radic(119886119896119892)2

minus 21198962

119899

119886119896119899

radic(119886119896119892)2

minus 21198962

119899

119886119896119892

radic(119886119896119892)2

minus 21198962

119899

119886119887119896119892

radic2(119886119896119892)2

minus 41198962

119899

1198862119896119892

radic2(119886119896119892)2

minus 41198962

119899

minus2119896119899

radic2(119886119896119892)2

minus 41198962

119899

]]]]]]]]]

]

times [

[

T120578

120585

]

]

(36)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892=

1198861198871198961198921198921minus 11988621198961198921198922minus 21198961198991198923

radic2(11988621198962119892minus 21198962119899)2

(37)

6 Journal of Discrete Mathematics

where

119889119904lowast

119889119904=

radic(119886119896119892)2

minus 21198962

119899

2

1198921= 11988711988621198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198961015840

119899+ 11988631198964

119892

minus (1198863+ 2119886) 119896

2

1198921198962

119899+ 2119886119896

4

119899

1198922= minus11988631198962

1198921198961015840

119899+ 11988631198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198962

119899+ 2119887119896

4

119899

1198923= 2119886119896

2

1198991198961015840

119892minus 2119886119896

1198921198961198991198961015840

119899minus 11988711988621198963

119892119896119899+ 2119887119896

1198921198963

119899

(38)

Proof We assume that the curve 120572 is a principal curveDifferentiating (15) with respect to 119904 and using (5) we obtain

1205731015840=

119889120573

119889119904

=1

radic2

(119887119896119899T + 119886119896

119899120578 + 119886119896

119892120585)

⟨1205731015840 1205731015840⟩ =

(119886119896119892)2

minus 21198962

119899

2

(39)

where from (9) (119886119896119892)2minus 21198962

119899gt 0 for all 119904 Since 120573

1015840=

(119889120573119889119904lowast)(119889119904lowast119889119904) the tangent vector T

120573of the curve 120573 is a

spacelike vector such that

T120573=

1

radic(119886119896119892)2

minus 21198962119899

(119887119896119899T + 119886119896

119899120578 + 119886119896

119892120585)

(40)

where

119889119904lowast

119889119904=

radic(119886119896119892)2

minus 21198962

119899

2

(41)

On the other hand from (15) and (40) it can be easily seenthat

120585120573= minus120573 times 119879

120573

=1

radic2(119886119896119892)2

minus 41198962119899

(119886119887119896119892T + 1198862119896119892120578 minus 2119896

119899120585) (42)

is a unit timelike vectorConsequently the geodesic curvature 119896

119892of the curve 120573 =

120573(119904lowast) is given by

119896119892= det (120573 119879

120573T1015840120573)

=

1198861198871198961198921198921minus 11988621198961198921198922minus 21198961198991198923

radic2(11988621198962119892minus 21198962119899)2

(43)

where

1198921= 11988711988621198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198961015840

119899+ 11988631198964

119892

minus (1198863+ 2119886) 119896

2

1198921198962

119899+ 2119886119896

4

119899

1198922= minus11988631198962

1198921198961015840

119899+ 11988631198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198962

119899+ 2119887119896

4

119899

1198923= 2119886119896

2

1198991198961015840

119892minus 2119886119896

1198921198961198991198961015840

119899minus 11988711988621198963

119892119896119899+ 2119887119896

1198921198963

119899

(44)

From (15) (40) and (42) we obtain the Sabban frame120573T120573 120585120573 of 120573

In the theorems which follow in a similar way as inTheorem 12 we obtain the Sabban frame 120573T

120573 120585120573 and the

geodesic curvature 119896119892of a spacelike Smarandache curve

We omit the proofs of Theorems 13 and 15 since they areanalogous to the proof of Theorem 12

Theorem 13 Let 120572 = 120572(119904) be a principal spacelike curvelying fully on the spacelike surface 119872 in R3

1with the

moving Lorentzian Darboux frame T 120578 120585 Then the T120585-Smarandache curve 120573 is spacelike and the Sabban frame120573T120573 120585120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[[[

[

119886

radic2

0119887

radic2minus119887119896119892

radic21198962119892minus (119886119896119899)2

119886119896119899

radic21198962119892minus (119886119896119899)2

119886119896119892

radic21198962119892minus (119886119896119899)2

minus119886119887119896119899

radic41198962119892minus 2(119886119896

119899)2

2119896119892

radic41198962119892minus 2(119886119896

119899)2

minus1198862119896119899

radic41198962119892minus 2(119886119896

119899)2

]]]]]]]]]]

]

times [

[

T120578

120585

]

]

(45)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus

119886119887119896119899ℎ1+ 2119896119892ℎ2+ 1198862119896119899ℎ3

radic2(21198962119892minus (119886119896119899)2)2

(46)

where

119889119904lowast

119889119904=

radic21198962

119892minus (119886119896119899)2

2

ℎ1= minus119887119886

21198962

1198991198961015840

119892+ 11988711988621198961198921198961198991198961015840

119899+ 2119886119896

4

119892

minus 11988631198962

1198921198962

119899minus 2119886119896

2

1198921198962

119899+ 11988631198964

119899

ℎ2= 2119886119896

1198921198961198991198961015840

119892minus 2119886119896

2

1198921198961015840

119899minus 2119887119896

3

119892119896119899+ 11988711988621198961198921198963

119899

ℎ3= 11988631198962

1198991198961015840

119892minus 1198863119896119892119896n1198961015840

119899minus 2119887119896

4

119892+ 11988711988621198962

1198921198962

119899

(47)

Theorem 14 Let 120572 = 120572(119904) be a principal spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

Journal of Discrete Mathematics 7

(i) if 119886119896119899minus 119887119896g = 0 for all 119904 the 120578120585-Smarandache curve

120573 is spacelike and the Sabban frame 120573T120573 120585120573 is given

by

[

[

120573

T120573

120585120573

]

]

=

[[[[[

[

0119886

radic2

119887

radic2

120598 0 0

0 minus120598119887

radic2

120598119886

radic2

]]]]]

]

[

[

T120578

120585

]

]

(48)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus

119886119896119892+ 119887119896119899

radic2

(49)

where 120598 = sign(119886119896119899minus 119887119896119892) for all 119904 and

119889119904lowast

119889119904= 120598

119886119896119899minus 119887119896119892

radic2

(50)

(ii) if 119886119896119899minus 119887119896119892

= 0 for all 119904 then the Sabban frame120573T120573 120585120573 of the 120578120585-Smarandache curve 120573 is a null

geodesic

Proof We assume that the curve 120572 is a principal curveDifferentiating (17) with respect to 119904 and using (5) we obtain

1205731015840=

119889120573

119889119904lowast

119889119904lowast

119889119904

=1

radic2

(119886119896119899minus 119887119896119892)T

⟨1205731015840 1205731015840⟩ =

(119886119896119899minus 119887119896119892)2

2

(51)

Then there are two following cases

(i) If 119886119896119899minus119887119896119892

= 0 for all 119904 since 1205731015840 = (119889120573119889119904lowast)(119889119904lowast119889119904)

then we obtain that the unit tangent vector T120573of the

curve 120573 is a spacelike vector such that

T120573= 120598T (52)

where

119889119904lowast

119889119904= 120598

119886119896119899minus 119887119896119892

radic2

(53)

and 120598 = sign(119886119896119899minus 119887119896119892)

On the other hand from (17) and (52) it can be easilyseen that

120585120573= minus120573 times 119879

120573

= minus120598119887

radic2

120578 + 120598119886

radic2

120585

(54)

is a unit timelike vector

Consequently the geodesic curvature 119896119892of the curve

120573 = 120573(119904lowast) is given by

119896119892= det (120573 119879

120573T1015840120573)

= minus

119886119896119892+ 119887119896119899

radic2

(55)

From (17) (52) and (54) we obtain the Sabban frame120573T120573 120585120573 of 120573

(ii) If 119886119896119899minus119887119896119892= 0 for all 119904 then 120573

1015840 is null So the tangentvector T

120573of the curve 120573 is a null vector It is known

that the only null curves lying on pseudosphere 11987821are

the null straight lines which are the null geodesics

Theorem 15 Let 120572 = 120572(119904) be a principal spacelikecurve lying fully on the spacelike surface 119872 in R3

1with

the moving Lorentzian Darboux frame T 120578 120585 Then theT120578120585-Smarandache curve 120573 is spacelike and the Sabban frame120573T120573 120585120573 is given by

120573 =1

radic3

(119886T + 119887120578 + 119888120585)

T120573=

(119887119896119899minus 119888119896119892)T + 119886119896

119899120578 + 119886119896

119892120585

radic3 (1198962119899minus 1198962119892) + (119887119896

119892minus 119888119896119899)2

120585120573= ((119886119887119896

119892minus 119886119888119896119899)T + ((3 + 119887

2) 119896119892minus 119887119888119896119899) 120578

minus ((3 minus 1198882) 119896119899+ 119887119888119896119892) 120585)

times (radic9 (1198962119899minus 1198962119892) + 3(119887119896

119892minus 119888119896119899)2

)

minus1

(56)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus ((119886119887119896

119892minus 119886119888119896119899) 1198971minus ((3 + 119887

2) 119896119892minus 119887119888119896119899) 1198972

minus ((3 minus 1198882) 119896119899+ 119887119888119896119892) 1198973)

times (radic2(11988621198962

119892minus 21198962

119899)2

)

minus1

(57)

where

119889119904lowast

119889119904=

radic3 (1198962119899minus 1198962119892) + (119887119896

119892minus 119888119896119899)2

radic3

1198971= (3119886119887119896

2

119899minus 3119886119888119896

119892119896119899) 1198961015840

119892+ (3119886119888119896

2

119892minus 3119886119887119896

119892119896119899) 1198961015840

119899

+ (119887119896119899minus 119888119896119892) (119887119896119892minus 119888119896119899)3

+ 31198871198881198964

119892minus 3119887119888119896

4

119899

+ 3 (1198872+ 1198882) 1198961198921198963

119899minus 3 (119887

2+ 1198882) 1198963

119892119896119899

8 Journal of Discrete Mathematics

x

y

z

10

0

minus10

minus5

0

5minus10

0

10

Figure 1 The spacelike surface 120601(119904 119906)

1198972= ((3119887

2+ 31198882+ 9) 119896

2

119899minus 6119887119888119896

119892119896119899) 1198961015840

119892

minus ((31198872+ 31198882+ 9) 119896

119892119896119899minus 6119887119888119896

2

119892) 1198961015840

119899

minus (1198861198883+ 3119886119888) 119896

4

119899+ (3119886119887 + 3119886119887119888

2) 1198961198921198963

119899

+ (1198861198873minus 3119886119887) 119896

3

119892119896119899+ (3119886119888 minus 3119886119887

2119888) 1198962

1198921198962

119899

1198973= ((3119887

2+ 31198882minus 9) 119896

119892119896119899minus 6119887119888119896

2

119899) 1198961015840

119892

+ ((9 minus 31198872minus 31198882) 1198962

119892+ 6119887119888119896

119892119896119899) 1198961015840

119899

+ (1198861198873minus 3119886119887) 119896

4

119892minus (3119886119888 + 119886119888

3) 1198961198921198963

119899

+ (3119886119888 minus 31198861198872119888) 1198963

119892119896119899+ (3119886119887 + 3119886119887119888

2) 1198962

1198921198962

119899

(58)

Example 16 Let us define a spacelike ruled surface (seeFigure 1) in the Minkowski 3-space such as

120601 119880 sub R2997888rarr R

3

1

(119904 119906) 997888rarr 120601 (119904 119906) = 120572 (119904) + 119906e (119904)

120601 (119904 119906) = (minus119906 sinh 119904 119904 minus119906 cosh 119904)

(59)

where 119906 isin (minus1 1)

Then we get the Lorentzian Darboux frame T 120578 120585 alongthe curve 120572 as follows

T (119904) = (0 1 0)

120578 (119904) equiv1

radic1 minus 1199062(cosh 119904 minus119906 sinh 119904)

120585 (119904) =1

radic1 minus 1199062(minus sinh 119904 0 minus cosh 119904)

(60)

where 120585(119904) is spacelike vectors and 120578(119904) is a unit timelikevector

Moreover the geodesic curvature 119896119892(119904) the asymptotic

curvature 119896119899(119904) and the principal curvature 120591

119892(119904) of the curve

120572 have the form119896119892(119904) = ⟨T1015840 (119904) 120585 (119904)⟩ = 0

119896119899 (119904) = minus ⟨T1015840 (119904) 120578 (119904)⟩ = 0

120591119892 (119904) = minus ⟨120585

1015840(119904) 120578 (119904)⟩ = minus

1

1 minus 1199062

(61)

Taking 119886 = radic3 119887 = 1 and using (15) we obtain that theT120578-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 2(a))

120573 (119904⋆(119904)) = (

cosh 119904

radic1 minus 1199062radic3 minus

119906

radic1 minus 1199062sinh 119904

radic1 minus 1199062) (62)

Taking 119886 = 119887 = 1 and using (16) we obtain that theT120585-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 2(b))

120573 (119904⋆(119904)) = (minus

sinh 119904

radic1 minus 1199062 1 minus

cosh 119904

radic1 minus 1199062) (63)

Journal of Discrete Mathematics 9

x

y

z

minus20

minus10

0

10

20

minus20

minus10

0

10

20

minus20 minus10 0 10 20

20

0

(a)

x

y

z2010

0minus10

minus20

20

10

0

minus10

minus20

20

10

0

minus10

minus20

(b)

Figure 2 (a) The T120578-Smarandache curve 120573 on 1198782

1for 119906 = 1radic3 (b) The T120585-Smarandache curve 120573 on 119878

2

1for 119906 = 1radic3

x

y

z

20

20

10

0

0

minus10

minus20

minus20

20

0

minus20

(a)

x

y

z

20100minus10minus20

0

10

minus10

20

minus20

20

10

0

minus10

minus20

(b)

Figure 3 (a) The 120578120585-Smarandache curve 120573 on 1198782

1for 119906 = 1radic3 (b) The T120578120585-Smarandache curve 120573 on 119878

2

1for 119906 = 1radic3

Taking 119886 = radic3 119887 = 1 and using (17) we obtain that the120578120585-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 3(a))

120573 (119904⋆(119904)) = minus

1

radic1 minus 1199062

times (sinh 119904 minusradic3 cosh 119904radic3119906 cosh 119904 minusradic3 sinh 119904)

(64)

Taking 119886 = radic3 119887 = 1 and 119888 = 1 and using (18) we obtainthat the T120578120585-Smarandache curve 120573 of the curve 120572 is given by(see Figure 3(b))

120573 (119904⋆(119904)) =

1

radic1 minus 1199062

times (cosh 119904 minus sinh 119904

radic3 minus 31199062 minus 119906 sinh 119904 minus cosh 119904)

(65)

10 Journal of Discrete Mathematics

x

y

z

20

0

minus20

200

minus20

20

10

0

minus10

minus20

Figure 4 The Smarandache curves on 1198782

1for 119906 = 1radic3

Also the Smarandache curves on 1198782 of 120572 for 119906 = 1radic3 with

Figure 4 are shown

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C Ashbacher ldquoSmarandache geometriesrdquo SmarandacheNotions Journal vol 8 no 1ndash3 pp 212ndash215 1997

[2] M Turgut and S Yilmaz ldquoSmarandache curves in Minkowskispace-timerdquo International Journal of Mathematical Combina-torics vol 3 pp 51ndash55 2008

[3] A T Ali ldquoSpecial smarandache curves in the euclidean spacerdquoInternational Journal of Mathematical Combinatorics vol 2 pp30ndash36 2010

[4] E B Koc Ozturk U Ozturk K Ilarslan and E NesovicldquoOn pseudohyperbolical Smarandache curves in Minkowski 3-spacerdquo International Journal of Mathematics and MathematicalSciences vol 2013 Article ID 658670 7 pages 2013

[5] E B Koc Ozturk U Ozturk K Ilarslan and E Nesovic ldquoOnpseudospherical smarandache curves in Minkowski 3-spacerdquoJournal of Applied Mathematics vol 2014 Article ID 404521 14pages 2014

[6] K Taskopru and M Tosun ldquoSmarandache curves on 1198782rdquo

Boletim da Sociedade Paranaense de Matematica vol 32 no 1pp 51ndash59 2014

[7] T Korpinar and E Turhan ldquoA new approach on SmarandacheTN-curves in terms of spacelike biharmonic curves with atimelike binormal in the Lorentzian Heisenberg group 119867119890119894119904

3rdquoJournal of Vectorial Relativity vol 6 pp 8ndash15 2011

[8] T Korpinar and E Turhan ldquoCharacterization of SmarandacheM1M2-curves of spacelike biharmonicB-slant helices according

to Bishop frame in E(1 1)rdquo Advanced Modeling and Optimiza-tion vol 14 no 2 pp 327ndash333 2012

[9] O Bektas and S Yuce ldquoSpecial Smarandache Curves AccordingtoDarboux Frame in E3rdquoRomanian Journal ofMathematics andComputer Science vol 3 no 1 pp 48ndash59 2013

[10] B ONeill Semi-Riemannian Geometry with Applications toRelativity Academic Press New York NY USA 1983

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Smarandache Curves according to Curves on a Spacelike Surface in Minkowski 3-Space R 1 3

2 Journal of Discrete Mathematics

For any x = (1199091 1199092 1199093) and y = (119910

1 1199102 1199103) in the space R3

1

the pseudovector product of x and y is defined by

x times y = (minus11990921199103+ 11990931199102 11990931199101minus 11990911199103 11990911199102minus 11990921199101) (2)

Next recall that an arbitrary curve 120572 = 120572(119904) in E31

can locally be spacelike timelike or null (lightlike) if all ofits velocity vectors 120572

1015840(119904) are respectively spacelike timelike

or null (lightlike) for every 119904 isin 119868 [10] If 1205721015840(s) = 0

for every 119904 isin 119868 then 120572 is a regular curve in R31 A

spacelike (timelike) regular curve 120572 is parameterized bypseudoarclength parameter 119904 which is given by 120572 119868 sub R rarr

R31 then the tangent vector 120572

1015840(119904) along 120572 has unit length

that is ⟨1205721015840(119904) 1205721015840(119904)⟩ = 1 (⟨1205721015840(119904) 1205721015840(119904)⟩ = minus1) for all 119904 isin 119868

respectively

Remark 1 Let x = (1199091 1199092 1199093) y = (119910

1 1199102 1199103) and z =

(1199111 1199112 1199113) be vectors in R3

1 Then

(i) ⟨x times y z⟩ =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1199091

1199092

1199093

1199101

1199102

1199103

1199111

1199112

1199113

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(ii) x times (y times z) = minus ⟨x z⟩ y + ⟨x y⟩ z

(iii) ⟨x times y x times y⟩ = minus ⟨x x⟩ ⟨y y⟩ + ⟨x y⟩2

(3)

where times is the pseudovector product in the space R31

Lemma 2 In the Minkowski 3-spaceR31 the following proper-

ties are satisfied [10]

(i) two timelike vectors are never orthogonal(ii) two null vectors are orthogonal if and only if they are

linearly dependent(iii) timelike vector is never orthogonal to a null vector

Let 120601 119880 sub R2 rarr R31 120601(119880) = 119872 and 120574 119868 sub R rarr 119880

be a spacelike embedding and a regular curve respectivelyThen we have a curve 120572 on the surface 119872 which is definedby 120572(119904) = 120601(120574(119904)) and since 120601 is a spacelike embedding wehave a unit timelike normal vector field 120578 along the surface119872 which is defined by

120578 equiv120601119909times 120601119910

10038171003817100381710038171003817120601119909times 120601119910

10038171003817100381710038171003817

(4)

Since119872 is a spacelike surface we can choose a future directedunit timelike normal vector field 120578 along the surface 119872Hence we have a pseudoorthonormal frame T 120578 120585which iscalled the Lorentzian Darboux frame along the curve 120572where120585(119904) = T(119904)times120578(119904) is a unit spacelike vectorThe correspondingFrenet formulae of 120572 read

[

[

T10158401205781015840

1205851015840

]

]

= [

[

0 119896119899

119896119892

119896119899

0 120591119892

minus119896119892

120591119892

0

]

]

[

[

T120578

120585

]

]

(5)

where 119896119899(119904) = minus⟨T1015840(119904) 120578(119904)⟩ 119896

119892(119904) = ⟨T1015840(119904) 120585(119904)⟩ and

120591119892(119904) = minus⟨120585

1015840(119904) 120578(119904)⟩ are the asymptotic curvature the

geodesic curvature and the principal curvature of 120572 on thesurface 119872 in R3

1 respectively and 119904 is arclength parameter

of 120572 In particular the following relations hold

T times 120578 = 120585 120578 times 120585 = minusT 120585 times T = 120578 (6)

Both 119896119899and 119896

119892may be positive or negative Specifically 119896

119899

is positive if 120572 curves towards the normal vector 120578 and 119896119892is

positive if120572 curves towards the tangent normal vector 120585 Alsothe curve 120572 is characterized by 119896

119899 119896119892 and 120591

119892as follows

120572 is

an asymptotic curve iff 119896119899equiv 0

a geodesic curve iff 119896119892equiv 0

a principal curve iff 120591119892equiv 0

(7)

Since 120572 is a unit-speed curve is perpendicular to T but may have components in the normal and tangent normaldirections

= 119896119899120578 + 119896119892120585 (8)

These are related to the total curvature 120581 of 120572 by the formula

1205812=

2= 1198962

119892minus 1198962

119899 (9)

From (9) we can give the following proposition

Proposition 3 Let 119872 be a spacelike surface in R31 Let 120572 =

120572(119904) be regular unit speed curves lying fully with the LorentzianDarboux frame T 120578 120585 on the surface119872 inR3

1 There is not a

geodesic curve on 119872

The pseudosphere with center at the origin and of radius119903 = 1 in the Minkowski 3-space R3

1is a quadric defined by

1198782

1= isin R

3

1| minus1199092

1+ 1199092

2+ 1199092

3= 1 (10)

Let 120573 119868 sub R rarr 1198782

1be a curve lying fully in pseudosphere

1198782

1in R31 Then its position vector 120573 is a spacelike which

means that the tangent vector 119879120573

= 1205731015840 can be spacelike

timelike or null Depending on the causal character of 119879120573

we distinguish the following three cases [5]

Case 1 (119879120573is a unit spacelike vector) Thenwe have orthonor-

mal Sabban frame 120573(119904) 119879120573(119904) 120585120573(119904) along the curve120572 where

120585120573(119904) = minus120573(119904) times 119879

120573(119904) is the unit timelike vector The

corresponding Frenet formulae of 120573 according to the Sabbanframe read

[[

[

1205731015840

1198791015840

120573

1205851015840

120573

]]

]

=[[

[

0 1 0

minus1 0 minus119896119892 (119904)

0 minus119896119892(119904) 0

]]

]

[

[

120573

119879120573

120585120573

]

]

(11)

where 119896119892(119904) = det(120573(119904) 119879

120573(119904) 1198791015840

120573(119904)) is the geodesic curvature

of 120573 and 119904 is the arclength parameter of 120573 In particular thefollowing relations hold

120573 times 119879120573= minus120585120573 119879

120573times 120585120573= 120573 120585

120573times 120573 = 119879

120573 (12)

Journal of Discrete Mathematics 3

Case 2 (119879120573is a unit timelike vector) Hence we have ortho-

normal Sabban frame 120573(119904) 119879120573(119904) 120585120573(119904) along the curve 120573

where 120585120573(119904) = 120573(119904) times 119879

120573(119904) is the unit spacelike vector The

corresponding Frenet formulae of 120573 according to the Sabbanframe read

[[

[

1205731015840

1198791015840

120573

1205851015840

120573

]]

]

=[[

[

0 1 0

1 0 119896119892(119904)

0 119896119892(119904) 0

]]

]

[

[

120573

119879120573

120585120573

]

]

(13)

where 119896119892(119904) = det(120573(119904) 119879

120573(119904) 1198791015840

120573(119904)) is the geodesic curvature

of 120573 and 119904 is the arclength parameter of 120573 In particular thefollowing relations hold

120573 times 119879120573= 120585120573 119879

120573times 120585120573= 120573 120585

120573times 120573 = minus119879

120573 (14)

Case 3 (119879120573is a null vector) It is known that the only null

curves lying on pseudosphere 1198782

1are the null straight lines

which are the null geodesics

3 Smarandache Curves according toCurves on a SpacelikeSurface in Minkowski 3-Space R3

1

In the following section we define the Smarandache curvesaccording to the Lorentzian Darboux frame in Minkowski3-space Also we obtain the Sabban frame and the geodesiccurvature of the Smarandache curves lying on pseudosphere1198782

1and give some characterizations on the curves when the

curve 120572 is an asymptotic curve or a principal curve

Definition 4 Let 120572 = 120572(119904) be a spacelike curve lying fully onthe spacelike surface 119872 in R3

1with the moving Lorentzian

Darboux frame T 120578 120585 Then T120578-Smarandache curve of 120572is defined by

120573 (119904⋆(119904)) =

1

radic2

(119886T (119904) + 119887120578 (119904)) (15)

where 119886 119887 isin R0and 1198862minus 1198872= 2

Definition 5 Let 120572 = 120572(119904) be a spacelike curve lying fully onthe spacelike surface 119872 in R3

1with the moving Lorentzian

Darboux frame T 120578 120585Then T120585-Smarandache curve of 120572 isdefined by

120573 (119904⋆(119904)) =

1

radic2(119886T (s) + 119887120585 (119904)) (16)

where 119886 119887 isin R0and 1198862+ 1198872= 2

Definition 6 Let 120572 = 120572(119904) be a spacelike curve lying fully onthe spacelike surface 119872 in R3

1with the moving Lorentzian

Darboux frame T 120578 120585 Then 120578120585-Smarandache curve of 120572 isdefined by

120573 (119904⋆(119904)) =

1

radic2

(119886120578 (119904) + 119887120585 (119904)) (17)

where 119886 119887 isin R0and 1198872minus 1198862= 2

Definition 7 Let 120572 = 120572(119904) be a spacelike curve lying fully onthe spacelike surface 119872 in R3

1with the moving Lorentzian

Darboux frame T 120578 120585 Then T120578120585-Smarandache curve of 120572is defined by

120573 (119904⋆(119904)) =

1

radic3

(119886T (119904) + 119887120578 (119904) + 119888120585 (119904)) (18)

where 119886 119887 isin R0and 1198862minus 1198872+ 1198882= 3

Thus there are two following cases

Case 4 (120572 is an asymptotic curve) Then we have the follow-ing theorems

Theorem 8 Let 120572 = 120572(119904) be an asymptotic spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

(i) if 119886119896119892+119887120591119892

= 0 for all 119904 then the Sabban frame 120573T120573

120585120573 of the T120578 -Smarandache curve 120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[

[

119886

radic2

119887

radic2

0

0 0 120598

120598119887

radic2

120598119886

radic2

0

]]]]]

]

[

[

T120578

120585

]

]

(19)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892=

119886120591119892minus 119887119896119892

radic2

(20)

where 120598 = sign(119886119896119892+ 119887120591119892) for all 119904 and

119889119904lowast

119889119904= 120598

119886119896119892+ 119887120591119892

radic2

(21)

(ii) if 119886119896119892+119887120591119892= 0 for all 119904 then the Sabban frame 120573T

120573

120585120573 of the T120578-Smarandache curve 120573 is a null geodesic

Proof We assume that the curve 120572 is an asymptotic curveDifferentiating (15) with respect to 119904 and using (5) we obtain

1205731015840=

119889120573

119889119904

=1

radic2

(119886119896119892+ 119887120591119892) 120585

⟨1205731015840 1205731015840⟩ =

(119886119896119892+ 119887120591119892)2

2

(22)

Then there are two following cases

(i) If 119886119896119892+119887120591119892

= 0 for all 119904 since 1205731015840 = (119889120573119889119904lowast)(119889119904lowast119889119904)

then the tangent vectorT120573of the curve120573 is a spacelike

vector such that

T120573= 120598120585 (23)

4 Journal of Discrete Mathematics

where

119889119904lowast

119889119904= 120598

119886119896119892+ 119887120591119892

radic2

(24)

On the other hand from (15) and (23) it can be easilyseen that

120585120573= minus120573 times 119879

120573

= 120598119887

radic2

T + 120598119886

radic2

120578

(25)

is a unit timelike vectorConsequently the geodesic curvature 119896

119892of the curve

120573 = 120573(119904lowast) is given by

119896119892= det (120573 119879

120573T1015840120573)

=

119886120591119892minus 119887119896119892

radic2

(26)

From (15) (23) and (25) we obtain the Sabban frame120573T120573 120585120573 of 120573

(ii) If 119886119896119892+119887120591119892= 0 for all 119904 then 120573

1015840 is null So the tangentvector T

120573of the curve 120573 is a null vector It is known

that the only null curves lying on pseudosphere 11987821are

the null straight lines which are the null geodesics

In the theorems which follow in a similar way as inTheorem 8 we obtain the Sabban frame 120573T

120573 120585120573 and the

geodesic curvature 119896119892of a spacelike Smarandache curve We

omit the proofs of Theorems 9 10 and 11 since they areanalogous to the proof of Theorem 8

Theorem 9 Let 120572 = 120572(119904) be an asymptotic spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

(i) if 21198962119892minus (119887120591119892)2

= 0 for all 119904 then the Sabban frame 120573T120573 120585120573 of the T120585-Smarandache curve 120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[

[

119886

radic2

0119887

radic2minus119887119896119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

119887120591119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

119886119896119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

119887120591119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

minus2119896119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

119886119887120591119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

]]]]]]]]

]

times [

[

T120578

120585

]

]

(27)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= ((119886

211988731205913

119892minus 11988741205913

119892+ 41198871205911198921198962

119892) 1198961015840

119892

+ (11988741205912

119892119896119892minus 119886211988731205912

119892119896119892minus 4119887119896

3

119892) 1205911015840

119892

+ (21198861198872+ 2119886119887 minus 4119886) 120591

1198921198964

119892

minus (1198861198874+ 1198861198873) 1205913

1198921198962

119892+ 11988611988741205915

119892)

times ((41198962

119892minus 2(119887120591

119892)2

)

2

)

minus1

(28)

where 120598 = sign(21198962119892minus (119887120591119892)2) for all 119904 and

119889119904lowast

119889119904=

radic120598

21198962

119892minus (119887120591119892)2

2

(29)

(ii) if 119886119896119892+119887120591119892= 0 for all 119904 then the Sabban frame 120573T

120573

120585120573 of the T120578-Smarandache curve 120573 is a null geodesic

Theorem 10 Let 120572 = 120572(119904) be an asymptotic spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

(i) if (119887119896119892)2minus 21205912

119892= 0 for all 119904 then the Sabban frame 120573

T120573 120585120573 of the 120578120585-Smarandache curve 120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[[

[

0119886

radic2

119887

radic2minus119887119896119892

radic120598 ((119887119896119892)2

minus 21205912

119892)

119887120591119892

radic120598 ((119887119896119892)2

minus 21205912

119892)

119886120591119892

radic120598 ((119887119896119892)2

minus 21205912

119892)

minus2120591119892

radic2120598 ((119887119896119892)2

minus 21205912

119892)

1198872119896119892

radic2120598 ((119887119896119892)2

minus 21205912

119892)

minus119886119887119896119892

radic2120598 ((119887119896119892)2

minus 21205912

119892)

]]]]]]]]]

]

times [

[

T120578

120585

]

]

(30)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= ((4radic2119887120591

3

119892minus (radic2119886

21198873+ radic2119887

5) 1205911198921198962

119892) 1198961015840

119892

+ ((radic211988621198873+ radic2119887

5) 1198963

119892minus 4radic2119887120591

2

119892119896119892) 1205911015840

119892

minus radic211988611988741198965

119892+ 2radic2119886119887

41205912

1198921198963

119892

+ (4radic2119886 minus 4radic21198861198872) 1205914

119892119896119892)

times (2(11988721198962

119892minus 21205912

119892)2

)

minus1

(31)

Journal of Discrete Mathematics 5

where 120598 = sign((119887119896119892)2minus 21205912

119892) for all 119904 and

119889119904lowast

119889119904=

radic120598

(119887119896119892)2

minus 21205912

119892

2

(32)

(ii) if (119887119896119892)2minus 21205912

119892= 0 for all 119904 then the Sabban frame

120573T120573 120585120573 of the T120578-Smarandache curve 120573 is a null

geodesic

Theorem 11 Let 120572 = 120572(119904) be an asymptotic spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

(i) if 3(1198962119892minus1205912

119892)+(119887119896

119892+ 119886120591119892)2

= 0 for all 119904 then the Sabbanframe 120573T

120573 120585120573 of the T120578120585-Smarandache curve 120573 is

given by

120573 =119886T + 119887120578 + 119888120585

radic3

T120573=

minus119888119896119892T + 119888120591

119892120578 + (119886119896

119892+ 119887120591119892) 120585

radic120598 (3 (1198962119892minus 1205912119892) + (119887119896

119892+ 119886120591119892)2

)

120585120573= ((119887 (119886119896

119892+ 119887120591119892) minus 1198882120591119892)T

+ (1198882119896119892+ 119886 (119886119896

119892+ 119887120591119892)) 120578 minus (119886119888120591

119892+ 119887119888119896119892) 120585)

times (radic120598 (9 (1198962119892minus 1205912119892) + 3(119887119896

119892+ 119886120591119892)2

))

minus1

(33)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= ((119887 (119886119896

119892+ 119887120591119892) minus 1198882120591119892) 1198911

minus (1198882119896119892+ 119886 (119886119896

119892+ 119887120591119892)) 1198912

minus (119886119888120591119892+ 119887119888119896119892) 1198913)

times (radic3(3 (1198962

119892minus 1205912

119892) + (119887119896

119892+ 119886120591119892)2

)

2

)

minus1

(34)

where 120598 = sign(3(1198962119892minus 1205912

119892) + (119887119896

119892+ 119886120591119892)2) for all 119904 and

119889119904lowast

119889119904=

radic120598

(119887119896119892)2

minus 21205912

119892

2

1198911= ((119886

2119888 minus 3119888) 120591

2

119892+ 119886119887119888120591

119892119896119892) 1198961015840

119892

+ ((3119888 minus 1198862119888) 120591119892119896119892minus 119886119887119888119896

2

119892) 1205911015840

119892

+ (3119886 + 1198861198872) 1198964

119892+ (1198863+ 2119886119887

2minus 3119886) 120591

2

1198921198962

119892

+ (1198873+ 21198862119887 + 3119887) 120591

1198921198963

119892

+ (1198862119887 minus 3119887) 120591

3

119892119896119892

1198912= (119886119887119888120591

2

119892+ (1198872119888 + 3119888) 120591

119892119896119892) 1198961015840

119892

minus (119886119887119888120591119892119896119892+ (3119888 + 119887

2119888) 1198962

119892) 1205911015840

119892

+ (3119887 minus 1198862119887) 1205914

119892minus (1198873+ 21198862119887 + 3119887) 120591

2

1198921198962

119892

minus (1198861198872+ 3119886) 120591

1198921198963

119892

+ (3119886 minus 21198861198872minus 1198863) 1205913

119892119896119892

1198913= ((minus119886

3+ 3119886 + 119886119887

2) 1205912

119892+ (1198873+ 3119887 minus 119886

2119887) 120591119892119896119892) 1198961015840

119892

+ ((minus1198873minus 3119887 + 119886

2119887) 1198962

119892+ (1198863minus 1198861198872minus 3119886) 120591

119892119896119892) 1205911015840

119892

+ (3119888 + 1198872119888) 1198964

119892+ (3119888 minus 119886

2119888) 1205914

119892+ 2119886119887119888120591

1198921198963

119892

minus 21198861198871198881205913

119892119896119892+ (1198862119888 minus 6119888 minus 119887

2119888) 1205912

1198921198962

119892

(35)

(ii) if 3(1198962119892minus1205912

119892)+(119887119896

119892+ 119886120591119892)2= 0 for all 119904 then the Sabban

frame 120573T120573 120585120573 of the T120578-Smarandache curve 120573 is a

null geodesic

Case 5 (120572 is a principal curve) Then we have the followingtheorems

Theorem 12 Let 120572 = 120572(119904) be a principal spacelike curvelying fully on the spacelike surface 119872 in R3

1with the

moving Lorentzian Darboux frame T 120578 120585 Then the T120578-Smarandache curve 120573 is spacelike and the Sabban frame120573T120573 120585120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[[

[

119886

radic2

119887

radic2

0

119887119896119899

radic(119886119896119892)2

minus 21198962

119899

119886119896119899

radic(119886119896119892)2

minus 21198962

119899

119886119896119892

radic(119886119896119892)2

minus 21198962

119899

119886119887119896119892

radic2(119886119896119892)2

minus 41198962

119899

1198862119896119892

radic2(119886119896119892)2

minus 41198962

119899

minus2119896119899

radic2(119886119896119892)2

minus 41198962

119899

]]]]]]]]]

]

times [

[

T120578

120585

]

]

(36)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892=

1198861198871198961198921198921minus 11988621198961198921198922minus 21198961198991198923

radic2(11988621198962119892minus 21198962119899)2

(37)

6 Journal of Discrete Mathematics

where

119889119904lowast

119889119904=

radic(119886119896119892)2

minus 21198962

119899

2

1198921= 11988711988621198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198961015840

119899+ 11988631198964

119892

minus (1198863+ 2119886) 119896

2

1198921198962

119899+ 2119886119896

4

119899

1198922= minus11988631198962

1198921198961015840

119899+ 11988631198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198962

119899+ 2119887119896

4

119899

1198923= 2119886119896

2

1198991198961015840

119892minus 2119886119896

1198921198961198991198961015840

119899minus 11988711988621198963

119892119896119899+ 2119887119896

1198921198963

119899

(38)

Proof We assume that the curve 120572 is a principal curveDifferentiating (15) with respect to 119904 and using (5) we obtain

1205731015840=

119889120573

119889119904

=1

radic2

(119887119896119899T + 119886119896

119899120578 + 119886119896

119892120585)

⟨1205731015840 1205731015840⟩ =

(119886119896119892)2

minus 21198962

119899

2

(39)

where from (9) (119886119896119892)2minus 21198962

119899gt 0 for all 119904 Since 120573

1015840=

(119889120573119889119904lowast)(119889119904lowast119889119904) the tangent vector T

120573of the curve 120573 is a

spacelike vector such that

T120573=

1

radic(119886119896119892)2

minus 21198962119899

(119887119896119899T + 119886119896

119899120578 + 119886119896

119892120585)

(40)

where

119889119904lowast

119889119904=

radic(119886119896119892)2

minus 21198962

119899

2

(41)

On the other hand from (15) and (40) it can be easily seenthat

120585120573= minus120573 times 119879

120573

=1

radic2(119886119896119892)2

minus 41198962119899

(119886119887119896119892T + 1198862119896119892120578 minus 2119896

119899120585) (42)

is a unit timelike vectorConsequently the geodesic curvature 119896

119892of the curve 120573 =

120573(119904lowast) is given by

119896119892= det (120573 119879

120573T1015840120573)

=

1198861198871198961198921198921minus 11988621198961198921198922minus 21198961198991198923

radic2(11988621198962119892minus 21198962119899)2

(43)

where

1198921= 11988711988621198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198961015840

119899+ 11988631198964

119892

minus (1198863+ 2119886) 119896

2

1198921198962

119899+ 2119886119896

4

119899

1198922= minus11988631198962

1198921198961015840

119899+ 11988631198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198962

119899+ 2119887119896

4

119899

1198923= 2119886119896

2

1198991198961015840

119892minus 2119886119896

1198921198961198991198961015840

119899minus 11988711988621198963

119892119896119899+ 2119887119896

1198921198963

119899

(44)

From (15) (40) and (42) we obtain the Sabban frame120573T120573 120585120573 of 120573

In the theorems which follow in a similar way as inTheorem 12 we obtain the Sabban frame 120573T

120573 120585120573 and the

geodesic curvature 119896119892of a spacelike Smarandache curve

We omit the proofs of Theorems 13 and 15 since they areanalogous to the proof of Theorem 12

Theorem 13 Let 120572 = 120572(119904) be a principal spacelike curvelying fully on the spacelike surface 119872 in R3

1with the

moving Lorentzian Darboux frame T 120578 120585 Then the T120585-Smarandache curve 120573 is spacelike and the Sabban frame120573T120573 120585120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[[[

[

119886

radic2

0119887

radic2minus119887119896119892

radic21198962119892minus (119886119896119899)2

119886119896119899

radic21198962119892minus (119886119896119899)2

119886119896119892

radic21198962119892minus (119886119896119899)2

minus119886119887119896119899

radic41198962119892minus 2(119886119896

119899)2

2119896119892

radic41198962119892minus 2(119886119896

119899)2

minus1198862119896119899

radic41198962119892minus 2(119886119896

119899)2

]]]]]]]]]]

]

times [

[

T120578

120585

]

]

(45)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus

119886119887119896119899ℎ1+ 2119896119892ℎ2+ 1198862119896119899ℎ3

radic2(21198962119892minus (119886119896119899)2)2

(46)

where

119889119904lowast

119889119904=

radic21198962

119892minus (119886119896119899)2

2

ℎ1= minus119887119886

21198962

1198991198961015840

119892+ 11988711988621198961198921198961198991198961015840

119899+ 2119886119896

4

119892

minus 11988631198962

1198921198962

119899minus 2119886119896

2

1198921198962

119899+ 11988631198964

119899

ℎ2= 2119886119896

1198921198961198991198961015840

119892minus 2119886119896

2

1198921198961015840

119899minus 2119887119896

3

119892119896119899+ 11988711988621198961198921198963

119899

ℎ3= 11988631198962

1198991198961015840

119892minus 1198863119896119892119896n1198961015840

119899minus 2119887119896

4

119892+ 11988711988621198962

1198921198962

119899

(47)

Theorem 14 Let 120572 = 120572(119904) be a principal spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

Journal of Discrete Mathematics 7

(i) if 119886119896119899minus 119887119896g = 0 for all 119904 the 120578120585-Smarandache curve

120573 is spacelike and the Sabban frame 120573T120573 120585120573 is given

by

[

[

120573

T120573

120585120573

]

]

=

[[[[[

[

0119886

radic2

119887

radic2

120598 0 0

0 minus120598119887

radic2

120598119886

radic2

]]]]]

]

[

[

T120578

120585

]

]

(48)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus

119886119896119892+ 119887119896119899

radic2

(49)

where 120598 = sign(119886119896119899minus 119887119896119892) for all 119904 and

119889119904lowast

119889119904= 120598

119886119896119899minus 119887119896119892

radic2

(50)

(ii) if 119886119896119899minus 119887119896119892

= 0 for all 119904 then the Sabban frame120573T120573 120585120573 of the 120578120585-Smarandache curve 120573 is a null

geodesic

Proof We assume that the curve 120572 is a principal curveDifferentiating (17) with respect to 119904 and using (5) we obtain

1205731015840=

119889120573

119889119904lowast

119889119904lowast

119889119904

=1

radic2

(119886119896119899minus 119887119896119892)T

⟨1205731015840 1205731015840⟩ =

(119886119896119899minus 119887119896119892)2

2

(51)

Then there are two following cases

(i) If 119886119896119899minus119887119896119892

= 0 for all 119904 since 1205731015840 = (119889120573119889119904lowast)(119889119904lowast119889119904)

then we obtain that the unit tangent vector T120573of the

curve 120573 is a spacelike vector such that

T120573= 120598T (52)

where

119889119904lowast

119889119904= 120598

119886119896119899minus 119887119896119892

radic2

(53)

and 120598 = sign(119886119896119899minus 119887119896119892)

On the other hand from (17) and (52) it can be easilyseen that

120585120573= minus120573 times 119879

120573

= minus120598119887

radic2

120578 + 120598119886

radic2

120585

(54)

is a unit timelike vector

Consequently the geodesic curvature 119896119892of the curve

120573 = 120573(119904lowast) is given by

119896119892= det (120573 119879

120573T1015840120573)

= minus

119886119896119892+ 119887119896119899

radic2

(55)

From (17) (52) and (54) we obtain the Sabban frame120573T120573 120585120573 of 120573

(ii) If 119886119896119899minus119887119896119892= 0 for all 119904 then 120573

1015840 is null So the tangentvector T

120573of the curve 120573 is a null vector It is known

that the only null curves lying on pseudosphere 11987821are

the null straight lines which are the null geodesics

Theorem 15 Let 120572 = 120572(119904) be a principal spacelikecurve lying fully on the spacelike surface 119872 in R3

1with

the moving Lorentzian Darboux frame T 120578 120585 Then theT120578120585-Smarandache curve 120573 is spacelike and the Sabban frame120573T120573 120585120573 is given by

120573 =1

radic3

(119886T + 119887120578 + 119888120585)

T120573=

(119887119896119899minus 119888119896119892)T + 119886119896

119899120578 + 119886119896

119892120585

radic3 (1198962119899minus 1198962119892) + (119887119896

119892minus 119888119896119899)2

120585120573= ((119886119887119896

119892minus 119886119888119896119899)T + ((3 + 119887

2) 119896119892minus 119887119888119896119899) 120578

minus ((3 minus 1198882) 119896119899+ 119887119888119896119892) 120585)

times (radic9 (1198962119899minus 1198962119892) + 3(119887119896

119892minus 119888119896119899)2

)

minus1

(56)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus ((119886119887119896

119892minus 119886119888119896119899) 1198971minus ((3 + 119887

2) 119896119892minus 119887119888119896119899) 1198972

minus ((3 minus 1198882) 119896119899+ 119887119888119896119892) 1198973)

times (radic2(11988621198962

119892minus 21198962

119899)2

)

minus1

(57)

where

119889119904lowast

119889119904=

radic3 (1198962119899minus 1198962119892) + (119887119896

119892minus 119888119896119899)2

radic3

1198971= (3119886119887119896

2

119899minus 3119886119888119896

119892119896119899) 1198961015840

119892+ (3119886119888119896

2

119892minus 3119886119887119896

119892119896119899) 1198961015840

119899

+ (119887119896119899minus 119888119896119892) (119887119896119892minus 119888119896119899)3

+ 31198871198881198964

119892minus 3119887119888119896

4

119899

+ 3 (1198872+ 1198882) 1198961198921198963

119899minus 3 (119887

2+ 1198882) 1198963

119892119896119899

8 Journal of Discrete Mathematics

x

y

z

10

0

minus10

minus5

0

5minus10

0

10

Figure 1 The spacelike surface 120601(119904 119906)

1198972= ((3119887

2+ 31198882+ 9) 119896

2

119899minus 6119887119888119896

119892119896119899) 1198961015840

119892

minus ((31198872+ 31198882+ 9) 119896

119892119896119899minus 6119887119888119896

2

119892) 1198961015840

119899

minus (1198861198883+ 3119886119888) 119896

4

119899+ (3119886119887 + 3119886119887119888

2) 1198961198921198963

119899

+ (1198861198873minus 3119886119887) 119896

3

119892119896119899+ (3119886119888 minus 3119886119887

2119888) 1198962

1198921198962

119899

1198973= ((3119887

2+ 31198882minus 9) 119896

119892119896119899minus 6119887119888119896

2

119899) 1198961015840

119892

+ ((9 minus 31198872minus 31198882) 1198962

119892+ 6119887119888119896

119892119896119899) 1198961015840

119899

+ (1198861198873minus 3119886119887) 119896

4

119892minus (3119886119888 + 119886119888

3) 1198961198921198963

119899

+ (3119886119888 minus 31198861198872119888) 1198963

119892119896119899+ (3119886119887 + 3119886119887119888

2) 1198962

1198921198962

119899

(58)

Example 16 Let us define a spacelike ruled surface (seeFigure 1) in the Minkowski 3-space such as

120601 119880 sub R2997888rarr R

3

1

(119904 119906) 997888rarr 120601 (119904 119906) = 120572 (119904) + 119906e (119904)

120601 (119904 119906) = (minus119906 sinh 119904 119904 minus119906 cosh 119904)

(59)

where 119906 isin (minus1 1)

Then we get the Lorentzian Darboux frame T 120578 120585 alongthe curve 120572 as follows

T (119904) = (0 1 0)

120578 (119904) equiv1

radic1 minus 1199062(cosh 119904 minus119906 sinh 119904)

120585 (119904) =1

radic1 minus 1199062(minus sinh 119904 0 minus cosh 119904)

(60)

where 120585(119904) is spacelike vectors and 120578(119904) is a unit timelikevector

Moreover the geodesic curvature 119896119892(119904) the asymptotic

curvature 119896119899(119904) and the principal curvature 120591

119892(119904) of the curve

120572 have the form119896119892(119904) = ⟨T1015840 (119904) 120585 (119904)⟩ = 0

119896119899 (119904) = minus ⟨T1015840 (119904) 120578 (119904)⟩ = 0

120591119892 (119904) = minus ⟨120585

1015840(119904) 120578 (119904)⟩ = minus

1

1 minus 1199062

(61)

Taking 119886 = radic3 119887 = 1 and using (15) we obtain that theT120578-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 2(a))

120573 (119904⋆(119904)) = (

cosh 119904

radic1 minus 1199062radic3 minus

119906

radic1 minus 1199062sinh 119904

radic1 minus 1199062) (62)

Taking 119886 = 119887 = 1 and using (16) we obtain that theT120585-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 2(b))

120573 (119904⋆(119904)) = (minus

sinh 119904

radic1 minus 1199062 1 minus

cosh 119904

radic1 minus 1199062) (63)

Journal of Discrete Mathematics 9

x

y

z

minus20

minus10

0

10

20

minus20

minus10

0

10

20

minus20 minus10 0 10 20

20

0

(a)

x

y

z2010

0minus10

minus20

20

10

0

minus10

minus20

20

10

0

minus10

minus20

(b)

Figure 2 (a) The T120578-Smarandache curve 120573 on 1198782

1for 119906 = 1radic3 (b) The T120585-Smarandache curve 120573 on 119878

2

1for 119906 = 1radic3

x

y

z

20

20

10

0

0

minus10

minus20

minus20

20

0

minus20

(a)

x

y

z

20100minus10minus20

0

10

minus10

20

minus20

20

10

0

minus10

minus20

(b)

Figure 3 (a) The 120578120585-Smarandache curve 120573 on 1198782

1for 119906 = 1radic3 (b) The T120578120585-Smarandache curve 120573 on 119878

2

1for 119906 = 1radic3

Taking 119886 = radic3 119887 = 1 and using (17) we obtain that the120578120585-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 3(a))

120573 (119904⋆(119904)) = minus

1

radic1 minus 1199062

times (sinh 119904 minusradic3 cosh 119904radic3119906 cosh 119904 minusradic3 sinh 119904)

(64)

Taking 119886 = radic3 119887 = 1 and 119888 = 1 and using (18) we obtainthat the T120578120585-Smarandache curve 120573 of the curve 120572 is given by(see Figure 3(b))

120573 (119904⋆(119904)) =

1

radic1 minus 1199062

times (cosh 119904 minus sinh 119904

radic3 minus 31199062 minus 119906 sinh 119904 minus cosh 119904)

(65)

10 Journal of Discrete Mathematics

x

y

z

20

0

minus20

200

minus20

20

10

0

minus10

minus20

Figure 4 The Smarandache curves on 1198782

1for 119906 = 1radic3

Also the Smarandache curves on 1198782 of 120572 for 119906 = 1radic3 with

Figure 4 are shown

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C Ashbacher ldquoSmarandache geometriesrdquo SmarandacheNotions Journal vol 8 no 1ndash3 pp 212ndash215 1997

[2] M Turgut and S Yilmaz ldquoSmarandache curves in Minkowskispace-timerdquo International Journal of Mathematical Combina-torics vol 3 pp 51ndash55 2008

[3] A T Ali ldquoSpecial smarandache curves in the euclidean spacerdquoInternational Journal of Mathematical Combinatorics vol 2 pp30ndash36 2010

[4] E B Koc Ozturk U Ozturk K Ilarslan and E NesovicldquoOn pseudohyperbolical Smarandache curves in Minkowski 3-spacerdquo International Journal of Mathematics and MathematicalSciences vol 2013 Article ID 658670 7 pages 2013

[5] E B Koc Ozturk U Ozturk K Ilarslan and E Nesovic ldquoOnpseudospherical smarandache curves in Minkowski 3-spacerdquoJournal of Applied Mathematics vol 2014 Article ID 404521 14pages 2014

[6] K Taskopru and M Tosun ldquoSmarandache curves on 1198782rdquo

Boletim da Sociedade Paranaense de Matematica vol 32 no 1pp 51ndash59 2014

[7] T Korpinar and E Turhan ldquoA new approach on SmarandacheTN-curves in terms of spacelike biharmonic curves with atimelike binormal in the Lorentzian Heisenberg group 119867119890119894119904

3rdquoJournal of Vectorial Relativity vol 6 pp 8ndash15 2011

[8] T Korpinar and E Turhan ldquoCharacterization of SmarandacheM1M2-curves of spacelike biharmonicB-slant helices according

to Bishop frame in E(1 1)rdquo Advanced Modeling and Optimiza-tion vol 14 no 2 pp 327ndash333 2012

[9] O Bektas and S Yuce ldquoSpecial Smarandache Curves AccordingtoDarboux Frame in E3rdquoRomanian Journal ofMathematics andComputer Science vol 3 no 1 pp 48ndash59 2013

[10] B ONeill Semi-Riemannian Geometry with Applications toRelativity Academic Press New York NY USA 1983

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Page 3: Smarandache Curves according to Curves on a Spacelike Surface in Minkowski 3-Space R 1 3

Journal of Discrete Mathematics 3

Case 2 (119879120573is a unit timelike vector) Hence we have ortho-

normal Sabban frame 120573(119904) 119879120573(119904) 120585120573(119904) along the curve 120573

where 120585120573(119904) = 120573(119904) times 119879

120573(119904) is the unit spacelike vector The

corresponding Frenet formulae of 120573 according to the Sabbanframe read

[[

[

1205731015840

1198791015840

120573

1205851015840

120573

]]

]

=[[

[

0 1 0

1 0 119896119892(119904)

0 119896119892(119904) 0

]]

]

[

[

120573

119879120573

120585120573

]

]

(13)

where 119896119892(119904) = det(120573(119904) 119879

120573(119904) 1198791015840

120573(119904)) is the geodesic curvature

of 120573 and 119904 is the arclength parameter of 120573 In particular thefollowing relations hold

120573 times 119879120573= 120585120573 119879

120573times 120585120573= 120573 120585

120573times 120573 = minus119879

120573 (14)

Case 3 (119879120573is a null vector) It is known that the only null

curves lying on pseudosphere 1198782

1are the null straight lines

which are the null geodesics

3 Smarandache Curves according toCurves on a SpacelikeSurface in Minkowski 3-Space R3

1

In the following section we define the Smarandache curvesaccording to the Lorentzian Darboux frame in Minkowski3-space Also we obtain the Sabban frame and the geodesiccurvature of the Smarandache curves lying on pseudosphere1198782

1and give some characterizations on the curves when the

curve 120572 is an asymptotic curve or a principal curve

Definition 4 Let 120572 = 120572(119904) be a spacelike curve lying fully onthe spacelike surface 119872 in R3

1with the moving Lorentzian

Darboux frame T 120578 120585 Then T120578-Smarandache curve of 120572is defined by

120573 (119904⋆(119904)) =

1

radic2

(119886T (119904) + 119887120578 (119904)) (15)

where 119886 119887 isin R0and 1198862minus 1198872= 2

Definition 5 Let 120572 = 120572(119904) be a spacelike curve lying fully onthe spacelike surface 119872 in R3

1with the moving Lorentzian

Darboux frame T 120578 120585Then T120585-Smarandache curve of 120572 isdefined by

120573 (119904⋆(119904)) =

1

radic2(119886T (s) + 119887120585 (119904)) (16)

where 119886 119887 isin R0and 1198862+ 1198872= 2

Definition 6 Let 120572 = 120572(119904) be a spacelike curve lying fully onthe spacelike surface 119872 in R3

1with the moving Lorentzian

Darboux frame T 120578 120585 Then 120578120585-Smarandache curve of 120572 isdefined by

120573 (119904⋆(119904)) =

1

radic2

(119886120578 (119904) + 119887120585 (119904)) (17)

where 119886 119887 isin R0and 1198872minus 1198862= 2

Definition 7 Let 120572 = 120572(119904) be a spacelike curve lying fully onthe spacelike surface 119872 in R3

1with the moving Lorentzian

Darboux frame T 120578 120585 Then T120578120585-Smarandache curve of 120572is defined by

120573 (119904⋆(119904)) =

1

radic3

(119886T (119904) + 119887120578 (119904) + 119888120585 (119904)) (18)

where 119886 119887 isin R0and 1198862minus 1198872+ 1198882= 3

Thus there are two following cases

Case 4 (120572 is an asymptotic curve) Then we have the follow-ing theorems

Theorem 8 Let 120572 = 120572(119904) be an asymptotic spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

(i) if 119886119896119892+119887120591119892

= 0 for all 119904 then the Sabban frame 120573T120573

120585120573 of the T120578 -Smarandache curve 120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[

[

119886

radic2

119887

radic2

0

0 0 120598

120598119887

radic2

120598119886

radic2

0

]]]]]

]

[

[

T120578

120585

]

]

(19)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892=

119886120591119892minus 119887119896119892

radic2

(20)

where 120598 = sign(119886119896119892+ 119887120591119892) for all 119904 and

119889119904lowast

119889119904= 120598

119886119896119892+ 119887120591119892

radic2

(21)

(ii) if 119886119896119892+119887120591119892= 0 for all 119904 then the Sabban frame 120573T

120573

120585120573 of the T120578-Smarandache curve 120573 is a null geodesic

Proof We assume that the curve 120572 is an asymptotic curveDifferentiating (15) with respect to 119904 and using (5) we obtain

1205731015840=

119889120573

119889119904

=1

radic2

(119886119896119892+ 119887120591119892) 120585

⟨1205731015840 1205731015840⟩ =

(119886119896119892+ 119887120591119892)2

2

(22)

Then there are two following cases

(i) If 119886119896119892+119887120591119892

= 0 for all 119904 since 1205731015840 = (119889120573119889119904lowast)(119889119904lowast119889119904)

then the tangent vectorT120573of the curve120573 is a spacelike

vector such that

T120573= 120598120585 (23)

4 Journal of Discrete Mathematics

where

119889119904lowast

119889119904= 120598

119886119896119892+ 119887120591119892

radic2

(24)

On the other hand from (15) and (23) it can be easilyseen that

120585120573= minus120573 times 119879

120573

= 120598119887

radic2

T + 120598119886

radic2

120578

(25)

is a unit timelike vectorConsequently the geodesic curvature 119896

119892of the curve

120573 = 120573(119904lowast) is given by

119896119892= det (120573 119879

120573T1015840120573)

=

119886120591119892minus 119887119896119892

radic2

(26)

From (15) (23) and (25) we obtain the Sabban frame120573T120573 120585120573 of 120573

(ii) If 119886119896119892+119887120591119892= 0 for all 119904 then 120573

1015840 is null So the tangentvector T

120573of the curve 120573 is a null vector It is known

that the only null curves lying on pseudosphere 11987821are

the null straight lines which are the null geodesics

In the theorems which follow in a similar way as inTheorem 8 we obtain the Sabban frame 120573T

120573 120585120573 and the

geodesic curvature 119896119892of a spacelike Smarandache curve We

omit the proofs of Theorems 9 10 and 11 since they areanalogous to the proof of Theorem 8

Theorem 9 Let 120572 = 120572(119904) be an asymptotic spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

(i) if 21198962119892minus (119887120591119892)2

= 0 for all 119904 then the Sabban frame 120573T120573 120585120573 of the T120585-Smarandache curve 120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[

[

119886

radic2

0119887

radic2minus119887119896119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

119887120591119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

119886119896119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

119887120591119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

minus2119896119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

119886119887120591119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

]]]]]]]]

]

times [

[

T120578

120585

]

]

(27)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= ((119886

211988731205913

119892minus 11988741205913

119892+ 41198871205911198921198962

119892) 1198961015840

119892

+ (11988741205912

119892119896119892minus 119886211988731205912

119892119896119892minus 4119887119896

3

119892) 1205911015840

119892

+ (21198861198872+ 2119886119887 minus 4119886) 120591

1198921198964

119892

minus (1198861198874+ 1198861198873) 1205913

1198921198962

119892+ 11988611988741205915

119892)

times ((41198962

119892minus 2(119887120591

119892)2

)

2

)

minus1

(28)

where 120598 = sign(21198962119892minus (119887120591119892)2) for all 119904 and

119889119904lowast

119889119904=

radic120598

21198962

119892minus (119887120591119892)2

2

(29)

(ii) if 119886119896119892+119887120591119892= 0 for all 119904 then the Sabban frame 120573T

120573

120585120573 of the T120578-Smarandache curve 120573 is a null geodesic

Theorem 10 Let 120572 = 120572(119904) be an asymptotic spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

(i) if (119887119896119892)2minus 21205912

119892= 0 for all 119904 then the Sabban frame 120573

T120573 120585120573 of the 120578120585-Smarandache curve 120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[[

[

0119886

radic2

119887

radic2minus119887119896119892

radic120598 ((119887119896119892)2

minus 21205912

119892)

119887120591119892

radic120598 ((119887119896119892)2

minus 21205912

119892)

119886120591119892

radic120598 ((119887119896119892)2

minus 21205912

119892)

minus2120591119892

radic2120598 ((119887119896119892)2

minus 21205912

119892)

1198872119896119892

radic2120598 ((119887119896119892)2

minus 21205912

119892)

minus119886119887119896119892

radic2120598 ((119887119896119892)2

minus 21205912

119892)

]]]]]]]]]

]

times [

[

T120578

120585

]

]

(30)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= ((4radic2119887120591

3

119892minus (radic2119886

21198873+ radic2119887

5) 1205911198921198962

119892) 1198961015840

119892

+ ((radic211988621198873+ radic2119887

5) 1198963

119892minus 4radic2119887120591

2

119892119896119892) 1205911015840

119892

minus radic211988611988741198965

119892+ 2radic2119886119887

41205912

1198921198963

119892

+ (4radic2119886 minus 4radic21198861198872) 1205914

119892119896119892)

times (2(11988721198962

119892minus 21205912

119892)2

)

minus1

(31)

Journal of Discrete Mathematics 5

where 120598 = sign((119887119896119892)2minus 21205912

119892) for all 119904 and

119889119904lowast

119889119904=

radic120598

(119887119896119892)2

minus 21205912

119892

2

(32)

(ii) if (119887119896119892)2minus 21205912

119892= 0 for all 119904 then the Sabban frame

120573T120573 120585120573 of the T120578-Smarandache curve 120573 is a null

geodesic

Theorem 11 Let 120572 = 120572(119904) be an asymptotic spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

(i) if 3(1198962119892minus1205912

119892)+(119887119896

119892+ 119886120591119892)2

= 0 for all 119904 then the Sabbanframe 120573T

120573 120585120573 of the T120578120585-Smarandache curve 120573 is

given by

120573 =119886T + 119887120578 + 119888120585

radic3

T120573=

minus119888119896119892T + 119888120591

119892120578 + (119886119896

119892+ 119887120591119892) 120585

radic120598 (3 (1198962119892minus 1205912119892) + (119887119896

119892+ 119886120591119892)2

)

120585120573= ((119887 (119886119896

119892+ 119887120591119892) minus 1198882120591119892)T

+ (1198882119896119892+ 119886 (119886119896

119892+ 119887120591119892)) 120578 minus (119886119888120591

119892+ 119887119888119896119892) 120585)

times (radic120598 (9 (1198962119892minus 1205912119892) + 3(119887119896

119892+ 119886120591119892)2

))

minus1

(33)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= ((119887 (119886119896

119892+ 119887120591119892) minus 1198882120591119892) 1198911

minus (1198882119896119892+ 119886 (119886119896

119892+ 119887120591119892)) 1198912

minus (119886119888120591119892+ 119887119888119896119892) 1198913)

times (radic3(3 (1198962

119892minus 1205912

119892) + (119887119896

119892+ 119886120591119892)2

)

2

)

minus1

(34)

where 120598 = sign(3(1198962119892minus 1205912

119892) + (119887119896

119892+ 119886120591119892)2) for all 119904 and

119889119904lowast

119889119904=

radic120598

(119887119896119892)2

minus 21205912

119892

2

1198911= ((119886

2119888 minus 3119888) 120591

2

119892+ 119886119887119888120591

119892119896119892) 1198961015840

119892

+ ((3119888 minus 1198862119888) 120591119892119896119892minus 119886119887119888119896

2

119892) 1205911015840

119892

+ (3119886 + 1198861198872) 1198964

119892+ (1198863+ 2119886119887

2minus 3119886) 120591

2

1198921198962

119892

+ (1198873+ 21198862119887 + 3119887) 120591

1198921198963

119892

+ (1198862119887 minus 3119887) 120591

3

119892119896119892

1198912= (119886119887119888120591

2

119892+ (1198872119888 + 3119888) 120591

119892119896119892) 1198961015840

119892

minus (119886119887119888120591119892119896119892+ (3119888 + 119887

2119888) 1198962

119892) 1205911015840

119892

+ (3119887 minus 1198862119887) 1205914

119892minus (1198873+ 21198862119887 + 3119887) 120591

2

1198921198962

119892

minus (1198861198872+ 3119886) 120591

1198921198963

119892

+ (3119886 minus 21198861198872minus 1198863) 1205913

119892119896119892

1198913= ((minus119886

3+ 3119886 + 119886119887

2) 1205912

119892+ (1198873+ 3119887 minus 119886

2119887) 120591119892119896119892) 1198961015840

119892

+ ((minus1198873minus 3119887 + 119886

2119887) 1198962

119892+ (1198863minus 1198861198872minus 3119886) 120591

119892119896119892) 1205911015840

119892

+ (3119888 + 1198872119888) 1198964

119892+ (3119888 minus 119886

2119888) 1205914

119892+ 2119886119887119888120591

1198921198963

119892

minus 21198861198871198881205913

119892119896119892+ (1198862119888 minus 6119888 minus 119887

2119888) 1205912

1198921198962

119892

(35)

(ii) if 3(1198962119892minus1205912

119892)+(119887119896

119892+ 119886120591119892)2= 0 for all 119904 then the Sabban

frame 120573T120573 120585120573 of the T120578-Smarandache curve 120573 is a

null geodesic

Case 5 (120572 is a principal curve) Then we have the followingtheorems

Theorem 12 Let 120572 = 120572(119904) be a principal spacelike curvelying fully on the spacelike surface 119872 in R3

1with the

moving Lorentzian Darboux frame T 120578 120585 Then the T120578-Smarandache curve 120573 is spacelike and the Sabban frame120573T120573 120585120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[[

[

119886

radic2

119887

radic2

0

119887119896119899

radic(119886119896119892)2

minus 21198962

119899

119886119896119899

radic(119886119896119892)2

minus 21198962

119899

119886119896119892

radic(119886119896119892)2

minus 21198962

119899

119886119887119896119892

radic2(119886119896119892)2

minus 41198962

119899

1198862119896119892

radic2(119886119896119892)2

minus 41198962

119899

minus2119896119899

radic2(119886119896119892)2

minus 41198962

119899

]]]]]]]]]

]

times [

[

T120578

120585

]

]

(36)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892=

1198861198871198961198921198921minus 11988621198961198921198922minus 21198961198991198923

radic2(11988621198962119892minus 21198962119899)2

(37)

6 Journal of Discrete Mathematics

where

119889119904lowast

119889119904=

radic(119886119896119892)2

minus 21198962

119899

2

1198921= 11988711988621198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198961015840

119899+ 11988631198964

119892

minus (1198863+ 2119886) 119896

2

1198921198962

119899+ 2119886119896

4

119899

1198922= minus11988631198962

1198921198961015840

119899+ 11988631198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198962

119899+ 2119887119896

4

119899

1198923= 2119886119896

2

1198991198961015840

119892minus 2119886119896

1198921198961198991198961015840

119899minus 11988711988621198963

119892119896119899+ 2119887119896

1198921198963

119899

(38)

Proof We assume that the curve 120572 is a principal curveDifferentiating (15) with respect to 119904 and using (5) we obtain

1205731015840=

119889120573

119889119904

=1

radic2

(119887119896119899T + 119886119896

119899120578 + 119886119896

119892120585)

⟨1205731015840 1205731015840⟩ =

(119886119896119892)2

minus 21198962

119899

2

(39)

where from (9) (119886119896119892)2minus 21198962

119899gt 0 for all 119904 Since 120573

1015840=

(119889120573119889119904lowast)(119889119904lowast119889119904) the tangent vector T

120573of the curve 120573 is a

spacelike vector such that

T120573=

1

radic(119886119896119892)2

minus 21198962119899

(119887119896119899T + 119886119896

119899120578 + 119886119896

119892120585)

(40)

where

119889119904lowast

119889119904=

radic(119886119896119892)2

minus 21198962

119899

2

(41)

On the other hand from (15) and (40) it can be easily seenthat

120585120573= minus120573 times 119879

120573

=1

radic2(119886119896119892)2

minus 41198962119899

(119886119887119896119892T + 1198862119896119892120578 minus 2119896

119899120585) (42)

is a unit timelike vectorConsequently the geodesic curvature 119896

119892of the curve 120573 =

120573(119904lowast) is given by

119896119892= det (120573 119879

120573T1015840120573)

=

1198861198871198961198921198921minus 11988621198961198921198922minus 21198961198991198923

radic2(11988621198962119892minus 21198962119899)2

(43)

where

1198921= 11988711988621198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198961015840

119899+ 11988631198964

119892

minus (1198863+ 2119886) 119896

2

1198921198962

119899+ 2119886119896

4

119899

1198922= minus11988631198962

1198921198961015840

119899+ 11988631198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198962

119899+ 2119887119896

4

119899

1198923= 2119886119896

2

1198991198961015840

119892minus 2119886119896

1198921198961198991198961015840

119899minus 11988711988621198963

119892119896119899+ 2119887119896

1198921198963

119899

(44)

From (15) (40) and (42) we obtain the Sabban frame120573T120573 120585120573 of 120573

In the theorems which follow in a similar way as inTheorem 12 we obtain the Sabban frame 120573T

120573 120585120573 and the

geodesic curvature 119896119892of a spacelike Smarandache curve

We omit the proofs of Theorems 13 and 15 since they areanalogous to the proof of Theorem 12

Theorem 13 Let 120572 = 120572(119904) be a principal spacelike curvelying fully on the spacelike surface 119872 in R3

1with the

moving Lorentzian Darboux frame T 120578 120585 Then the T120585-Smarandache curve 120573 is spacelike and the Sabban frame120573T120573 120585120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[[[

[

119886

radic2

0119887

radic2minus119887119896119892

radic21198962119892minus (119886119896119899)2

119886119896119899

radic21198962119892minus (119886119896119899)2

119886119896119892

radic21198962119892minus (119886119896119899)2

minus119886119887119896119899

radic41198962119892minus 2(119886119896

119899)2

2119896119892

radic41198962119892minus 2(119886119896

119899)2

minus1198862119896119899

radic41198962119892minus 2(119886119896

119899)2

]]]]]]]]]]

]

times [

[

T120578

120585

]

]

(45)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus

119886119887119896119899ℎ1+ 2119896119892ℎ2+ 1198862119896119899ℎ3

radic2(21198962119892minus (119886119896119899)2)2

(46)

where

119889119904lowast

119889119904=

radic21198962

119892minus (119886119896119899)2

2

ℎ1= minus119887119886

21198962

1198991198961015840

119892+ 11988711988621198961198921198961198991198961015840

119899+ 2119886119896

4

119892

minus 11988631198962

1198921198962

119899minus 2119886119896

2

1198921198962

119899+ 11988631198964

119899

ℎ2= 2119886119896

1198921198961198991198961015840

119892minus 2119886119896

2

1198921198961015840

119899minus 2119887119896

3

119892119896119899+ 11988711988621198961198921198963

119899

ℎ3= 11988631198962

1198991198961015840

119892minus 1198863119896119892119896n1198961015840

119899minus 2119887119896

4

119892+ 11988711988621198962

1198921198962

119899

(47)

Theorem 14 Let 120572 = 120572(119904) be a principal spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

Journal of Discrete Mathematics 7

(i) if 119886119896119899minus 119887119896g = 0 for all 119904 the 120578120585-Smarandache curve

120573 is spacelike and the Sabban frame 120573T120573 120585120573 is given

by

[

[

120573

T120573

120585120573

]

]

=

[[[[[

[

0119886

radic2

119887

radic2

120598 0 0

0 minus120598119887

radic2

120598119886

radic2

]]]]]

]

[

[

T120578

120585

]

]

(48)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus

119886119896119892+ 119887119896119899

radic2

(49)

where 120598 = sign(119886119896119899minus 119887119896119892) for all 119904 and

119889119904lowast

119889119904= 120598

119886119896119899minus 119887119896119892

radic2

(50)

(ii) if 119886119896119899minus 119887119896119892

= 0 for all 119904 then the Sabban frame120573T120573 120585120573 of the 120578120585-Smarandache curve 120573 is a null

geodesic

Proof We assume that the curve 120572 is a principal curveDifferentiating (17) with respect to 119904 and using (5) we obtain

1205731015840=

119889120573

119889119904lowast

119889119904lowast

119889119904

=1

radic2

(119886119896119899minus 119887119896119892)T

⟨1205731015840 1205731015840⟩ =

(119886119896119899minus 119887119896119892)2

2

(51)

Then there are two following cases

(i) If 119886119896119899minus119887119896119892

= 0 for all 119904 since 1205731015840 = (119889120573119889119904lowast)(119889119904lowast119889119904)

then we obtain that the unit tangent vector T120573of the

curve 120573 is a spacelike vector such that

T120573= 120598T (52)

where

119889119904lowast

119889119904= 120598

119886119896119899minus 119887119896119892

radic2

(53)

and 120598 = sign(119886119896119899minus 119887119896119892)

On the other hand from (17) and (52) it can be easilyseen that

120585120573= minus120573 times 119879

120573

= minus120598119887

radic2

120578 + 120598119886

radic2

120585

(54)

is a unit timelike vector

Consequently the geodesic curvature 119896119892of the curve

120573 = 120573(119904lowast) is given by

119896119892= det (120573 119879

120573T1015840120573)

= minus

119886119896119892+ 119887119896119899

radic2

(55)

From (17) (52) and (54) we obtain the Sabban frame120573T120573 120585120573 of 120573

(ii) If 119886119896119899minus119887119896119892= 0 for all 119904 then 120573

1015840 is null So the tangentvector T

120573of the curve 120573 is a null vector It is known

that the only null curves lying on pseudosphere 11987821are

the null straight lines which are the null geodesics

Theorem 15 Let 120572 = 120572(119904) be a principal spacelikecurve lying fully on the spacelike surface 119872 in R3

1with

the moving Lorentzian Darboux frame T 120578 120585 Then theT120578120585-Smarandache curve 120573 is spacelike and the Sabban frame120573T120573 120585120573 is given by

120573 =1

radic3

(119886T + 119887120578 + 119888120585)

T120573=

(119887119896119899minus 119888119896119892)T + 119886119896

119899120578 + 119886119896

119892120585

radic3 (1198962119899minus 1198962119892) + (119887119896

119892minus 119888119896119899)2

120585120573= ((119886119887119896

119892minus 119886119888119896119899)T + ((3 + 119887

2) 119896119892minus 119887119888119896119899) 120578

minus ((3 minus 1198882) 119896119899+ 119887119888119896119892) 120585)

times (radic9 (1198962119899minus 1198962119892) + 3(119887119896

119892minus 119888119896119899)2

)

minus1

(56)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus ((119886119887119896

119892minus 119886119888119896119899) 1198971minus ((3 + 119887

2) 119896119892minus 119887119888119896119899) 1198972

minus ((3 minus 1198882) 119896119899+ 119887119888119896119892) 1198973)

times (radic2(11988621198962

119892minus 21198962

119899)2

)

minus1

(57)

where

119889119904lowast

119889119904=

radic3 (1198962119899minus 1198962119892) + (119887119896

119892minus 119888119896119899)2

radic3

1198971= (3119886119887119896

2

119899minus 3119886119888119896

119892119896119899) 1198961015840

119892+ (3119886119888119896

2

119892minus 3119886119887119896

119892119896119899) 1198961015840

119899

+ (119887119896119899minus 119888119896119892) (119887119896119892minus 119888119896119899)3

+ 31198871198881198964

119892minus 3119887119888119896

4

119899

+ 3 (1198872+ 1198882) 1198961198921198963

119899minus 3 (119887

2+ 1198882) 1198963

119892119896119899

8 Journal of Discrete Mathematics

x

y

z

10

0

minus10

minus5

0

5minus10

0

10

Figure 1 The spacelike surface 120601(119904 119906)

1198972= ((3119887

2+ 31198882+ 9) 119896

2

119899minus 6119887119888119896

119892119896119899) 1198961015840

119892

minus ((31198872+ 31198882+ 9) 119896

119892119896119899minus 6119887119888119896

2

119892) 1198961015840

119899

minus (1198861198883+ 3119886119888) 119896

4

119899+ (3119886119887 + 3119886119887119888

2) 1198961198921198963

119899

+ (1198861198873minus 3119886119887) 119896

3

119892119896119899+ (3119886119888 minus 3119886119887

2119888) 1198962

1198921198962

119899

1198973= ((3119887

2+ 31198882minus 9) 119896

119892119896119899minus 6119887119888119896

2

119899) 1198961015840

119892

+ ((9 minus 31198872minus 31198882) 1198962

119892+ 6119887119888119896

119892119896119899) 1198961015840

119899

+ (1198861198873minus 3119886119887) 119896

4

119892minus (3119886119888 + 119886119888

3) 1198961198921198963

119899

+ (3119886119888 minus 31198861198872119888) 1198963

119892119896119899+ (3119886119887 + 3119886119887119888

2) 1198962

1198921198962

119899

(58)

Example 16 Let us define a spacelike ruled surface (seeFigure 1) in the Minkowski 3-space such as

120601 119880 sub R2997888rarr R

3

1

(119904 119906) 997888rarr 120601 (119904 119906) = 120572 (119904) + 119906e (119904)

120601 (119904 119906) = (minus119906 sinh 119904 119904 minus119906 cosh 119904)

(59)

where 119906 isin (minus1 1)

Then we get the Lorentzian Darboux frame T 120578 120585 alongthe curve 120572 as follows

T (119904) = (0 1 0)

120578 (119904) equiv1

radic1 minus 1199062(cosh 119904 minus119906 sinh 119904)

120585 (119904) =1

radic1 minus 1199062(minus sinh 119904 0 minus cosh 119904)

(60)

where 120585(119904) is spacelike vectors and 120578(119904) is a unit timelikevector

Moreover the geodesic curvature 119896119892(119904) the asymptotic

curvature 119896119899(119904) and the principal curvature 120591

119892(119904) of the curve

120572 have the form119896119892(119904) = ⟨T1015840 (119904) 120585 (119904)⟩ = 0

119896119899 (119904) = minus ⟨T1015840 (119904) 120578 (119904)⟩ = 0

120591119892 (119904) = minus ⟨120585

1015840(119904) 120578 (119904)⟩ = minus

1

1 minus 1199062

(61)

Taking 119886 = radic3 119887 = 1 and using (15) we obtain that theT120578-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 2(a))

120573 (119904⋆(119904)) = (

cosh 119904

radic1 minus 1199062radic3 minus

119906

radic1 minus 1199062sinh 119904

radic1 minus 1199062) (62)

Taking 119886 = 119887 = 1 and using (16) we obtain that theT120585-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 2(b))

120573 (119904⋆(119904)) = (minus

sinh 119904

radic1 minus 1199062 1 minus

cosh 119904

radic1 minus 1199062) (63)

Journal of Discrete Mathematics 9

x

y

z

minus20

minus10

0

10

20

minus20

minus10

0

10

20

minus20 minus10 0 10 20

20

0

(a)

x

y

z2010

0minus10

minus20

20

10

0

minus10

minus20

20

10

0

minus10

minus20

(b)

Figure 2 (a) The T120578-Smarandache curve 120573 on 1198782

1for 119906 = 1radic3 (b) The T120585-Smarandache curve 120573 on 119878

2

1for 119906 = 1radic3

x

y

z

20

20

10

0

0

minus10

minus20

minus20

20

0

minus20

(a)

x

y

z

20100minus10minus20

0

10

minus10

20

minus20

20

10

0

minus10

minus20

(b)

Figure 3 (a) The 120578120585-Smarandache curve 120573 on 1198782

1for 119906 = 1radic3 (b) The T120578120585-Smarandache curve 120573 on 119878

2

1for 119906 = 1radic3

Taking 119886 = radic3 119887 = 1 and using (17) we obtain that the120578120585-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 3(a))

120573 (119904⋆(119904)) = minus

1

radic1 minus 1199062

times (sinh 119904 minusradic3 cosh 119904radic3119906 cosh 119904 minusradic3 sinh 119904)

(64)

Taking 119886 = radic3 119887 = 1 and 119888 = 1 and using (18) we obtainthat the T120578120585-Smarandache curve 120573 of the curve 120572 is given by(see Figure 3(b))

120573 (119904⋆(119904)) =

1

radic1 minus 1199062

times (cosh 119904 minus sinh 119904

radic3 minus 31199062 minus 119906 sinh 119904 minus cosh 119904)

(65)

10 Journal of Discrete Mathematics

x

y

z

20

0

minus20

200

minus20

20

10

0

minus10

minus20

Figure 4 The Smarandache curves on 1198782

1for 119906 = 1radic3

Also the Smarandache curves on 1198782 of 120572 for 119906 = 1radic3 with

Figure 4 are shown

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C Ashbacher ldquoSmarandache geometriesrdquo SmarandacheNotions Journal vol 8 no 1ndash3 pp 212ndash215 1997

[2] M Turgut and S Yilmaz ldquoSmarandache curves in Minkowskispace-timerdquo International Journal of Mathematical Combina-torics vol 3 pp 51ndash55 2008

[3] A T Ali ldquoSpecial smarandache curves in the euclidean spacerdquoInternational Journal of Mathematical Combinatorics vol 2 pp30ndash36 2010

[4] E B Koc Ozturk U Ozturk K Ilarslan and E NesovicldquoOn pseudohyperbolical Smarandache curves in Minkowski 3-spacerdquo International Journal of Mathematics and MathematicalSciences vol 2013 Article ID 658670 7 pages 2013

[5] E B Koc Ozturk U Ozturk K Ilarslan and E Nesovic ldquoOnpseudospherical smarandache curves in Minkowski 3-spacerdquoJournal of Applied Mathematics vol 2014 Article ID 404521 14pages 2014

[6] K Taskopru and M Tosun ldquoSmarandache curves on 1198782rdquo

Boletim da Sociedade Paranaense de Matematica vol 32 no 1pp 51ndash59 2014

[7] T Korpinar and E Turhan ldquoA new approach on SmarandacheTN-curves in terms of spacelike biharmonic curves with atimelike binormal in the Lorentzian Heisenberg group 119867119890119894119904

3rdquoJournal of Vectorial Relativity vol 6 pp 8ndash15 2011

[8] T Korpinar and E Turhan ldquoCharacterization of SmarandacheM1M2-curves of spacelike biharmonicB-slant helices according

to Bishop frame in E(1 1)rdquo Advanced Modeling and Optimiza-tion vol 14 no 2 pp 327ndash333 2012

[9] O Bektas and S Yuce ldquoSpecial Smarandache Curves AccordingtoDarboux Frame in E3rdquoRomanian Journal ofMathematics andComputer Science vol 3 no 1 pp 48ndash59 2013

[10] B ONeill Semi-Riemannian Geometry with Applications toRelativity Academic Press New York NY USA 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Smarandache Curves according to Curves on a Spacelike Surface in Minkowski 3-Space R 1 3

4 Journal of Discrete Mathematics

where

119889119904lowast

119889119904= 120598

119886119896119892+ 119887120591119892

radic2

(24)

On the other hand from (15) and (23) it can be easilyseen that

120585120573= minus120573 times 119879

120573

= 120598119887

radic2

T + 120598119886

radic2

120578

(25)

is a unit timelike vectorConsequently the geodesic curvature 119896

119892of the curve

120573 = 120573(119904lowast) is given by

119896119892= det (120573 119879

120573T1015840120573)

=

119886120591119892minus 119887119896119892

radic2

(26)

From (15) (23) and (25) we obtain the Sabban frame120573T120573 120585120573 of 120573

(ii) If 119886119896119892+119887120591119892= 0 for all 119904 then 120573

1015840 is null So the tangentvector T

120573of the curve 120573 is a null vector It is known

that the only null curves lying on pseudosphere 11987821are

the null straight lines which are the null geodesics

In the theorems which follow in a similar way as inTheorem 8 we obtain the Sabban frame 120573T

120573 120585120573 and the

geodesic curvature 119896119892of a spacelike Smarandache curve We

omit the proofs of Theorems 9 10 and 11 since they areanalogous to the proof of Theorem 8

Theorem 9 Let 120572 = 120572(119904) be an asymptotic spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

(i) if 21198962119892minus (119887120591119892)2

= 0 for all 119904 then the Sabban frame 120573T120573 120585120573 of the T120585-Smarandache curve 120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[

[

119886

radic2

0119887

radic2minus119887119896119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

119887120591119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

119886119896119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

119887120591119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

minus2119896119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

119886119887120591119892

radic120598 (21198962

119892minus (119887120591119892)

2

)

]]]]]]]]

]

times [

[

T120578

120585

]

]

(27)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= ((119886

211988731205913

119892minus 11988741205913

119892+ 41198871205911198921198962

119892) 1198961015840

119892

+ (11988741205912

119892119896119892minus 119886211988731205912

119892119896119892minus 4119887119896

3

119892) 1205911015840

119892

+ (21198861198872+ 2119886119887 minus 4119886) 120591

1198921198964

119892

minus (1198861198874+ 1198861198873) 1205913

1198921198962

119892+ 11988611988741205915

119892)

times ((41198962

119892minus 2(119887120591

119892)2

)

2

)

minus1

(28)

where 120598 = sign(21198962119892minus (119887120591119892)2) for all 119904 and

119889119904lowast

119889119904=

radic120598

21198962

119892minus (119887120591119892)2

2

(29)

(ii) if 119886119896119892+119887120591119892= 0 for all 119904 then the Sabban frame 120573T

120573

120585120573 of the T120578-Smarandache curve 120573 is a null geodesic

Theorem 10 Let 120572 = 120572(119904) be an asymptotic spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

(i) if (119887119896119892)2minus 21205912

119892= 0 for all 119904 then the Sabban frame 120573

T120573 120585120573 of the 120578120585-Smarandache curve 120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[[

[

0119886

radic2

119887

radic2minus119887119896119892

radic120598 ((119887119896119892)2

minus 21205912

119892)

119887120591119892

radic120598 ((119887119896119892)2

minus 21205912

119892)

119886120591119892

radic120598 ((119887119896119892)2

minus 21205912

119892)

minus2120591119892

radic2120598 ((119887119896119892)2

minus 21205912

119892)

1198872119896119892

radic2120598 ((119887119896119892)2

minus 21205912

119892)

minus119886119887119896119892

radic2120598 ((119887119896119892)2

minus 21205912

119892)

]]]]]]]]]

]

times [

[

T120578

120585

]

]

(30)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= ((4radic2119887120591

3

119892minus (radic2119886

21198873+ radic2119887

5) 1205911198921198962

119892) 1198961015840

119892

+ ((radic211988621198873+ radic2119887

5) 1198963

119892minus 4radic2119887120591

2

119892119896119892) 1205911015840

119892

minus radic211988611988741198965

119892+ 2radic2119886119887

41205912

1198921198963

119892

+ (4radic2119886 minus 4radic21198861198872) 1205914

119892119896119892)

times (2(11988721198962

119892minus 21205912

119892)2

)

minus1

(31)

Journal of Discrete Mathematics 5

where 120598 = sign((119887119896119892)2minus 21205912

119892) for all 119904 and

119889119904lowast

119889119904=

radic120598

(119887119896119892)2

minus 21205912

119892

2

(32)

(ii) if (119887119896119892)2minus 21205912

119892= 0 for all 119904 then the Sabban frame

120573T120573 120585120573 of the T120578-Smarandache curve 120573 is a null

geodesic

Theorem 11 Let 120572 = 120572(119904) be an asymptotic spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

(i) if 3(1198962119892minus1205912

119892)+(119887119896

119892+ 119886120591119892)2

= 0 for all 119904 then the Sabbanframe 120573T

120573 120585120573 of the T120578120585-Smarandache curve 120573 is

given by

120573 =119886T + 119887120578 + 119888120585

radic3

T120573=

minus119888119896119892T + 119888120591

119892120578 + (119886119896

119892+ 119887120591119892) 120585

radic120598 (3 (1198962119892minus 1205912119892) + (119887119896

119892+ 119886120591119892)2

)

120585120573= ((119887 (119886119896

119892+ 119887120591119892) minus 1198882120591119892)T

+ (1198882119896119892+ 119886 (119886119896

119892+ 119887120591119892)) 120578 minus (119886119888120591

119892+ 119887119888119896119892) 120585)

times (radic120598 (9 (1198962119892minus 1205912119892) + 3(119887119896

119892+ 119886120591119892)2

))

minus1

(33)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= ((119887 (119886119896

119892+ 119887120591119892) minus 1198882120591119892) 1198911

minus (1198882119896119892+ 119886 (119886119896

119892+ 119887120591119892)) 1198912

minus (119886119888120591119892+ 119887119888119896119892) 1198913)

times (radic3(3 (1198962

119892minus 1205912

119892) + (119887119896

119892+ 119886120591119892)2

)

2

)

minus1

(34)

where 120598 = sign(3(1198962119892minus 1205912

119892) + (119887119896

119892+ 119886120591119892)2) for all 119904 and

119889119904lowast

119889119904=

radic120598

(119887119896119892)2

minus 21205912

119892

2

1198911= ((119886

2119888 minus 3119888) 120591

2

119892+ 119886119887119888120591

119892119896119892) 1198961015840

119892

+ ((3119888 minus 1198862119888) 120591119892119896119892minus 119886119887119888119896

2

119892) 1205911015840

119892

+ (3119886 + 1198861198872) 1198964

119892+ (1198863+ 2119886119887

2minus 3119886) 120591

2

1198921198962

119892

+ (1198873+ 21198862119887 + 3119887) 120591

1198921198963

119892

+ (1198862119887 minus 3119887) 120591

3

119892119896119892

1198912= (119886119887119888120591

2

119892+ (1198872119888 + 3119888) 120591

119892119896119892) 1198961015840

119892

minus (119886119887119888120591119892119896119892+ (3119888 + 119887

2119888) 1198962

119892) 1205911015840

119892

+ (3119887 minus 1198862119887) 1205914

119892minus (1198873+ 21198862119887 + 3119887) 120591

2

1198921198962

119892

minus (1198861198872+ 3119886) 120591

1198921198963

119892

+ (3119886 minus 21198861198872minus 1198863) 1205913

119892119896119892

1198913= ((minus119886

3+ 3119886 + 119886119887

2) 1205912

119892+ (1198873+ 3119887 minus 119886

2119887) 120591119892119896119892) 1198961015840

119892

+ ((minus1198873minus 3119887 + 119886

2119887) 1198962

119892+ (1198863minus 1198861198872minus 3119886) 120591

119892119896119892) 1205911015840

119892

+ (3119888 + 1198872119888) 1198964

119892+ (3119888 minus 119886

2119888) 1205914

119892+ 2119886119887119888120591

1198921198963

119892

minus 21198861198871198881205913

119892119896119892+ (1198862119888 minus 6119888 minus 119887

2119888) 1205912

1198921198962

119892

(35)

(ii) if 3(1198962119892minus1205912

119892)+(119887119896

119892+ 119886120591119892)2= 0 for all 119904 then the Sabban

frame 120573T120573 120585120573 of the T120578-Smarandache curve 120573 is a

null geodesic

Case 5 (120572 is a principal curve) Then we have the followingtheorems

Theorem 12 Let 120572 = 120572(119904) be a principal spacelike curvelying fully on the spacelike surface 119872 in R3

1with the

moving Lorentzian Darboux frame T 120578 120585 Then the T120578-Smarandache curve 120573 is spacelike and the Sabban frame120573T120573 120585120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[[

[

119886

radic2

119887

radic2

0

119887119896119899

radic(119886119896119892)2

minus 21198962

119899

119886119896119899

radic(119886119896119892)2

minus 21198962

119899

119886119896119892

radic(119886119896119892)2

minus 21198962

119899

119886119887119896119892

radic2(119886119896119892)2

minus 41198962

119899

1198862119896119892

radic2(119886119896119892)2

minus 41198962

119899

minus2119896119899

radic2(119886119896119892)2

minus 41198962

119899

]]]]]]]]]

]

times [

[

T120578

120585

]

]

(36)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892=

1198861198871198961198921198921minus 11988621198961198921198922minus 21198961198991198923

radic2(11988621198962119892minus 21198962119899)2

(37)

6 Journal of Discrete Mathematics

where

119889119904lowast

119889119904=

radic(119886119896119892)2

minus 21198962

119899

2

1198921= 11988711988621198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198961015840

119899+ 11988631198964

119892

minus (1198863+ 2119886) 119896

2

1198921198962

119899+ 2119886119896

4

119899

1198922= minus11988631198962

1198921198961015840

119899+ 11988631198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198962

119899+ 2119887119896

4

119899

1198923= 2119886119896

2

1198991198961015840

119892minus 2119886119896

1198921198961198991198961015840

119899minus 11988711988621198963

119892119896119899+ 2119887119896

1198921198963

119899

(38)

Proof We assume that the curve 120572 is a principal curveDifferentiating (15) with respect to 119904 and using (5) we obtain

1205731015840=

119889120573

119889119904

=1

radic2

(119887119896119899T + 119886119896

119899120578 + 119886119896

119892120585)

⟨1205731015840 1205731015840⟩ =

(119886119896119892)2

minus 21198962

119899

2

(39)

where from (9) (119886119896119892)2minus 21198962

119899gt 0 for all 119904 Since 120573

1015840=

(119889120573119889119904lowast)(119889119904lowast119889119904) the tangent vector T

120573of the curve 120573 is a

spacelike vector such that

T120573=

1

radic(119886119896119892)2

minus 21198962119899

(119887119896119899T + 119886119896

119899120578 + 119886119896

119892120585)

(40)

where

119889119904lowast

119889119904=

radic(119886119896119892)2

minus 21198962

119899

2

(41)

On the other hand from (15) and (40) it can be easily seenthat

120585120573= minus120573 times 119879

120573

=1

radic2(119886119896119892)2

minus 41198962119899

(119886119887119896119892T + 1198862119896119892120578 minus 2119896

119899120585) (42)

is a unit timelike vectorConsequently the geodesic curvature 119896

119892of the curve 120573 =

120573(119904lowast) is given by

119896119892= det (120573 119879

120573T1015840120573)

=

1198861198871198961198921198921minus 11988621198961198921198922minus 21198961198991198923

radic2(11988621198962119892minus 21198962119899)2

(43)

where

1198921= 11988711988621198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198961015840

119899+ 11988631198964

119892

minus (1198863+ 2119886) 119896

2

1198921198962

119899+ 2119886119896

4

119899

1198922= minus11988631198962

1198921198961015840

119899+ 11988631198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198962

119899+ 2119887119896

4

119899

1198923= 2119886119896

2

1198991198961015840

119892minus 2119886119896

1198921198961198991198961015840

119899minus 11988711988621198963

119892119896119899+ 2119887119896

1198921198963

119899

(44)

From (15) (40) and (42) we obtain the Sabban frame120573T120573 120585120573 of 120573

In the theorems which follow in a similar way as inTheorem 12 we obtain the Sabban frame 120573T

120573 120585120573 and the

geodesic curvature 119896119892of a spacelike Smarandache curve

We omit the proofs of Theorems 13 and 15 since they areanalogous to the proof of Theorem 12

Theorem 13 Let 120572 = 120572(119904) be a principal spacelike curvelying fully on the spacelike surface 119872 in R3

1with the

moving Lorentzian Darboux frame T 120578 120585 Then the T120585-Smarandache curve 120573 is spacelike and the Sabban frame120573T120573 120585120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[[[

[

119886

radic2

0119887

radic2minus119887119896119892

radic21198962119892minus (119886119896119899)2

119886119896119899

radic21198962119892minus (119886119896119899)2

119886119896119892

radic21198962119892minus (119886119896119899)2

minus119886119887119896119899

radic41198962119892minus 2(119886119896

119899)2

2119896119892

radic41198962119892minus 2(119886119896

119899)2

minus1198862119896119899

radic41198962119892minus 2(119886119896

119899)2

]]]]]]]]]]

]

times [

[

T120578

120585

]

]

(45)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus

119886119887119896119899ℎ1+ 2119896119892ℎ2+ 1198862119896119899ℎ3

radic2(21198962119892minus (119886119896119899)2)2

(46)

where

119889119904lowast

119889119904=

radic21198962

119892minus (119886119896119899)2

2

ℎ1= minus119887119886

21198962

1198991198961015840

119892+ 11988711988621198961198921198961198991198961015840

119899+ 2119886119896

4

119892

minus 11988631198962

1198921198962

119899minus 2119886119896

2

1198921198962

119899+ 11988631198964

119899

ℎ2= 2119886119896

1198921198961198991198961015840

119892minus 2119886119896

2

1198921198961015840

119899minus 2119887119896

3

119892119896119899+ 11988711988621198961198921198963

119899

ℎ3= 11988631198962

1198991198961015840

119892minus 1198863119896119892119896n1198961015840

119899minus 2119887119896

4

119892+ 11988711988621198962

1198921198962

119899

(47)

Theorem 14 Let 120572 = 120572(119904) be a principal spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

Journal of Discrete Mathematics 7

(i) if 119886119896119899minus 119887119896g = 0 for all 119904 the 120578120585-Smarandache curve

120573 is spacelike and the Sabban frame 120573T120573 120585120573 is given

by

[

[

120573

T120573

120585120573

]

]

=

[[[[[

[

0119886

radic2

119887

radic2

120598 0 0

0 minus120598119887

radic2

120598119886

radic2

]]]]]

]

[

[

T120578

120585

]

]

(48)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus

119886119896119892+ 119887119896119899

radic2

(49)

where 120598 = sign(119886119896119899minus 119887119896119892) for all 119904 and

119889119904lowast

119889119904= 120598

119886119896119899minus 119887119896119892

radic2

(50)

(ii) if 119886119896119899minus 119887119896119892

= 0 for all 119904 then the Sabban frame120573T120573 120585120573 of the 120578120585-Smarandache curve 120573 is a null

geodesic

Proof We assume that the curve 120572 is a principal curveDifferentiating (17) with respect to 119904 and using (5) we obtain

1205731015840=

119889120573

119889119904lowast

119889119904lowast

119889119904

=1

radic2

(119886119896119899minus 119887119896119892)T

⟨1205731015840 1205731015840⟩ =

(119886119896119899minus 119887119896119892)2

2

(51)

Then there are two following cases

(i) If 119886119896119899minus119887119896119892

= 0 for all 119904 since 1205731015840 = (119889120573119889119904lowast)(119889119904lowast119889119904)

then we obtain that the unit tangent vector T120573of the

curve 120573 is a spacelike vector such that

T120573= 120598T (52)

where

119889119904lowast

119889119904= 120598

119886119896119899minus 119887119896119892

radic2

(53)

and 120598 = sign(119886119896119899minus 119887119896119892)

On the other hand from (17) and (52) it can be easilyseen that

120585120573= minus120573 times 119879

120573

= minus120598119887

radic2

120578 + 120598119886

radic2

120585

(54)

is a unit timelike vector

Consequently the geodesic curvature 119896119892of the curve

120573 = 120573(119904lowast) is given by

119896119892= det (120573 119879

120573T1015840120573)

= minus

119886119896119892+ 119887119896119899

radic2

(55)

From (17) (52) and (54) we obtain the Sabban frame120573T120573 120585120573 of 120573

(ii) If 119886119896119899minus119887119896119892= 0 for all 119904 then 120573

1015840 is null So the tangentvector T

120573of the curve 120573 is a null vector It is known

that the only null curves lying on pseudosphere 11987821are

the null straight lines which are the null geodesics

Theorem 15 Let 120572 = 120572(119904) be a principal spacelikecurve lying fully on the spacelike surface 119872 in R3

1with

the moving Lorentzian Darboux frame T 120578 120585 Then theT120578120585-Smarandache curve 120573 is spacelike and the Sabban frame120573T120573 120585120573 is given by

120573 =1

radic3

(119886T + 119887120578 + 119888120585)

T120573=

(119887119896119899minus 119888119896119892)T + 119886119896

119899120578 + 119886119896

119892120585

radic3 (1198962119899minus 1198962119892) + (119887119896

119892minus 119888119896119899)2

120585120573= ((119886119887119896

119892minus 119886119888119896119899)T + ((3 + 119887

2) 119896119892minus 119887119888119896119899) 120578

minus ((3 minus 1198882) 119896119899+ 119887119888119896119892) 120585)

times (radic9 (1198962119899minus 1198962119892) + 3(119887119896

119892minus 119888119896119899)2

)

minus1

(56)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus ((119886119887119896

119892minus 119886119888119896119899) 1198971minus ((3 + 119887

2) 119896119892minus 119887119888119896119899) 1198972

minus ((3 minus 1198882) 119896119899+ 119887119888119896119892) 1198973)

times (radic2(11988621198962

119892minus 21198962

119899)2

)

minus1

(57)

where

119889119904lowast

119889119904=

radic3 (1198962119899minus 1198962119892) + (119887119896

119892minus 119888119896119899)2

radic3

1198971= (3119886119887119896

2

119899minus 3119886119888119896

119892119896119899) 1198961015840

119892+ (3119886119888119896

2

119892minus 3119886119887119896

119892119896119899) 1198961015840

119899

+ (119887119896119899minus 119888119896119892) (119887119896119892minus 119888119896119899)3

+ 31198871198881198964

119892minus 3119887119888119896

4

119899

+ 3 (1198872+ 1198882) 1198961198921198963

119899minus 3 (119887

2+ 1198882) 1198963

119892119896119899

8 Journal of Discrete Mathematics

x

y

z

10

0

minus10

minus5

0

5minus10

0

10

Figure 1 The spacelike surface 120601(119904 119906)

1198972= ((3119887

2+ 31198882+ 9) 119896

2

119899minus 6119887119888119896

119892119896119899) 1198961015840

119892

minus ((31198872+ 31198882+ 9) 119896

119892119896119899minus 6119887119888119896

2

119892) 1198961015840

119899

minus (1198861198883+ 3119886119888) 119896

4

119899+ (3119886119887 + 3119886119887119888

2) 1198961198921198963

119899

+ (1198861198873minus 3119886119887) 119896

3

119892119896119899+ (3119886119888 minus 3119886119887

2119888) 1198962

1198921198962

119899

1198973= ((3119887

2+ 31198882minus 9) 119896

119892119896119899minus 6119887119888119896

2

119899) 1198961015840

119892

+ ((9 minus 31198872minus 31198882) 1198962

119892+ 6119887119888119896

119892119896119899) 1198961015840

119899

+ (1198861198873minus 3119886119887) 119896

4

119892minus (3119886119888 + 119886119888

3) 1198961198921198963

119899

+ (3119886119888 minus 31198861198872119888) 1198963

119892119896119899+ (3119886119887 + 3119886119887119888

2) 1198962

1198921198962

119899

(58)

Example 16 Let us define a spacelike ruled surface (seeFigure 1) in the Minkowski 3-space such as

120601 119880 sub R2997888rarr R

3

1

(119904 119906) 997888rarr 120601 (119904 119906) = 120572 (119904) + 119906e (119904)

120601 (119904 119906) = (minus119906 sinh 119904 119904 minus119906 cosh 119904)

(59)

where 119906 isin (minus1 1)

Then we get the Lorentzian Darboux frame T 120578 120585 alongthe curve 120572 as follows

T (119904) = (0 1 0)

120578 (119904) equiv1

radic1 minus 1199062(cosh 119904 minus119906 sinh 119904)

120585 (119904) =1

radic1 minus 1199062(minus sinh 119904 0 minus cosh 119904)

(60)

where 120585(119904) is spacelike vectors and 120578(119904) is a unit timelikevector

Moreover the geodesic curvature 119896119892(119904) the asymptotic

curvature 119896119899(119904) and the principal curvature 120591

119892(119904) of the curve

120572 have the form119896119892(119904) = ⟨T1015840 (119904) 120585 (119904)⟩ = 0

119896119899 (119904) = minus ⟨T1015840 (119904) 120578 (119904)⟩ = 0

120591119892 (119904) = minus ⟨120585

1015840(119904) 120578 (119904)⟩ = minus

1

1 minus 1199062

(61)

Taking 119886 = radic3 119887 = 1 and using (15) we obtain that theT120578-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 2(a))

120573 (119904⋆(119904)) = (

cosh 119904

radic1 minus 1199062radic3 minus

119906

radic1 minus 1199062sinh 119904

radic1 minus 1199062) (62)

Taking 119886 = 119887 = 1 and using (16) we obtain that theT120585-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 2(b))

120573 (119904⋆(119904)) = (minus

sinh 119904

radic1 minus 1199062 1 minus

cosh 119904

radic1 minus 1199062) (63)

Journal of Discrete Mathematics 9

x

y

z

minus20

minus10

0

10

20

minus20

minus10

0

10

20

minus20 minus10 0 10 20

20

0

(a)

x

y

z2010

0minus10

minus20

20

10

0

minus10

minus20

20

10

0

minus10

minus20

(b)

Figure 2 (a) The T120578-Smarandache curve 120573 on 1198782

1for 119906 = 1radic3 (b) The T120585-Smarandache curve 120573 on 119878

2

1for 119906 = 1radic3

x

y

z

20

20

10

0

0

minus10

minus20

minus20

20

0

minus20

(a)

x

y

z

20100minus10minus20

0

10

minus10

20

minus20

20

10

0

minus10

minus20

(b)

Figure 3 (a) The 120578120585-Smarandache curve 120573 on 1198782

1for 119906 = 1radic3 (b) The T120578120585-Smarandache curve 120573 on 119878

2

1for 119906 = 1radic3

Taking 119886 = radic3 119887 = 1 and using (17) we obtain that the120578120585-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 3(a))

120573 (119904⋆(119904)) = minus

1

radic1 minus 1199062

times (sinh 119904 minusradic3 cosh 119904radic3119906 cosh 119904 minusradic3 sinh 119904)

(64)

Taking 119886 = radic3 119887 = 1 and 119888 = 1 and using (18) we obtainthat the T120578120585-Smarandache curve 120573 of the curve 120572 is given by(see Figure 3(b))

120573 (119904⋆(119904)) =

1

radic1 minus 1199062

times (cosh 119904 minus sinh 119904

radic3 minus 31199062 minus 119906 sinh 119904 minus cosh 119904)

(65)

10 Journal of Discrete Mathematics

x

y

z

20

0

minus20

200

minus20

20

10

0

minus10

minus20

Figure 4 The Smarandache curves on 1198782

1for 119906 = 1radic3

Also the Smarandache curves on 1198782 of 120572 for 119906 = 1radic3 with

Figure 4 are shown

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C Ashbacher ldquoSmarandache geometriesrdquo SmarandacheNotions Journal vol 8 no 1ndash3 pp 212ndash215 1997

[2] M Turgut and S Yilmaz ldquoSmarandache curves in Minkowskispace-timerdquo International Journal of Mathematical Combina-torics vol 3 pp 51ndash55 2008

[3] A T Ali ldquoSpecial smarandache curves in the euclidean spacerdquoInternational Journal of Mathematical Combinatorics vol 2 pp30ndash36 2010

[4] E B Koc Ozturk U Ozturk K Ilarslan and E NesovicldquoOn pseudohyperbolical Smarandache curves in Minkowski 3-spacerdquo International Journal of Mathematics and MathematicalSciences vol 2013 Article ID 658670 7 pages 2013

[5] E B Koc Ozturk U Ozturk K Ilarslan and E Nesovic ldquoOnpseudospherical smarandache curves in Minkowski 3-spacerdquoJournal of Applied Mathematics vol 2014 Article ID 404521 14pages 2014

[6] K Taskopru and M Tosun ldquoSmarandache curves on 1198782rdquo

Boletim da Sociedade Paranaense de Matematica vol 32 no 1pp 51ndash59 2014

[7] T Korpinar and E Turhan ldquoA new approach on SmarandacheTN-curves in terms of spacelike biharmonic curves with atimelike binormal in the Lorentzian Heisenberg group 119867119890119894119904

3rdquoJournal of Vectorial Relativity vol 6 pp 8ndash15 2011

[8] T Korpinar and E Turhan ldquoCharacterization of SmarandacheM1M2-curves of spacelike biharmonicB-slant helices according

to Bishop frame in E(1 1)rdquo Advanced Modeling and Optimiza-tion vol 14 no 2 pp 327ndash333 2012

[9] O Bektas and S Yuce ldquoSpecial Smarandache Curves AccordingtoDarboux Frame in E3rdquoRomanian Journal ofMathematics andComputer Science vol 3 no 1 pp 48ndash59 2013

[10] B ONeill Semi-Riemannian Geometry with Applications toRelativity Academic Press New York NY USA 1983

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Smarandache Curves according to Curves on a Spacelike Surface in Minkowski 3-Space R 1 3

Journal of Discrete Mathematics 5

where 120598 = sign((119887119896119892)2minus 21205912

119892) for all 119904 and

119889119904lowast

119889119904=

radic120598

(119887119896119892)2

minus 21205912

119892

2

(32)

(ii) if (119887119896119892)2minus 21205912

119892= 0 for all 119904 then the Sabban frame

120573T120573 120585120573 of the T120578-Smarandache curve 120573 is a null

geodesic

Theorem 11 Let 120572 = 120572(119904) be an asymptotic spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

(i) if 3(1198962119892minus1205912

119892)+(119887119896

119892+ 119886120591119892)2

= 0 for all 119904 then the Sabbanframe 120573T

120573 120585120573 of the T120578120585-Smarandache curve 120573 is

given by

120573 =119886T + 119887120578 + 119888120585

radic3

T120573=

minus119888119896119892T + 119888120591

119892120578 + (119886119896

119892+ 119887120591119892) 120585

radic120598 (3 (1198962119892minus 1205912119892) + (119887119896

119892+ 119886120591119892)2

)

120585120573= ((119887 (119886119896

119892+ 119887120591119892) minus 1198882120591119892)T

+ (1198882119896119892+ 119886 (119886119896

119892+ 119887120591119892)) 120578 minus (119886119888120591

119892+ 119887119888119896119892) 120585)

times (radic120598 (9 (1198962119892minus 1205912119892) + 3(119887119896

119892+ 119886120591119892)2

))

minus1

(33)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= ((119887 (119886119896

119892+ 119887120591119892) minus 1198882120591119892) 1198911

minus (1198882119896119892+ 119886 (119886119896

119892+ 119887120591119892)) 1198912

minus (119886119888120591119892+ 119887119888119896119892) 1198913)

times (radic3(3 (1198962

119892minus 1205912

119892) + (119887119896

119892+ 119886120591119892)2

)

2

)

minus1

(34)

where 120598 = sign(3(1198962119892minus 1205912

119892) + (119887119896

119892+ 119886120591119892)2) for all 119904 and

119889119904lowast

119889119904=

radic120598

(119887119896119892)2

minus 21205912

119892

2

1198911= ((119886

2119888 minus 3119888) 120591

2

119892+ 119886119887119888120591

119892119896119892) 1198961015840

119892

+ ((3119888 minus 1198862119888) 120591119892119896119892minus 119886119887119888119896

2

119892) 1205911015840

119892

+ (3119886 + 1198861198872) 1198964

119892+ (1198863+ 2119886119887

2minus 3119886) 120591

2

1198921198962

119892

+ (1198873+ 21198862119887 + 3119887) 120591

1198921198963

119892

+ (1198862119887 minus 3119887) 120591

3

119892119896119892

1198912= (119886119887119888120591

2

119892+ (1198872119888 + 3119888) 120591

119892119896119892) 1198961015840

119892

minus (119886119887119888120591119892119896119892+ (3119888 + 119887

2119888) 1198962

119892) 1205911015840

119892

+ (3119887 minus 1198862119887) 1205914

119892minus (1198873+ 21198862119887 + 3119887) 120591

2

1198921198962

119892

minus (1198861198872+ 3119886) 120591

1198921198963

119892

+ (3119886 minus 21198861198872minus 1198863) 1205913

119892119896119892

1198913= ((minus119886

3+ 3119886 + 119886119887

2) 1205912

119892+ (1198873+ 3119887 minus 119886

2119887) 120591119892119896119892) 1198961015840

119892

+ ((minus1198873minus 3119887 + 119886

2119887) 1198962

119892+ (1198863minus 1198861198872minus 3119886) 120591

119892119896119892) 1205911015840

119892

+ (3119888 + 1198872119888) 1198964

119892+ (3119888 minus 119886

2119888) 1205914

119892+ 2119886119887119888120591

1198921198963

119892

minus 21198861198871198881205913

119892119896119892+ (1198862119888 minus 6119888 minus 119887

2119888) 1205912

1198921198962

119892

(35)

(ii) if 3(1198962119892minus1205912

119892)+(119887119896

119892+ 119886120591119892)2= 0 for all 119904 then the Sabban

frame 120573T120573 120585120573 of the T120578-Smarandache curve 120573 is a

null geodesic

Case 5 (120572 is a principal curve) Then we have the followingtheorems

Theorem 12 Let 120572 = 120572(119904) be a principal spacelike curvelying fully on the spacelike surface 119872 in R3

1with the

moving Lorentzian Darboux frame T 120578 120585 Then the T120578-Smarandache curve 120573 is spacelike and the Sabban frame120573T120573 120585120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[[

[

119886

radic2

119887

radic2

0

119887119896119899

radic(119886119896119892)2

minus 21198962

119899

119886119896119899

radic(119886119896119892)2

minus 21198962

119899

119886119896119892

radic(119886119896119892)2

minus 21198962

119899

119886119887119896119892

radic2(119886119896119892)2

minus 41198962

119899

1198862119896119892

radic2(119886119896119892)2

minus 41198962

119899

minus2119896119899

radic2(119886119896119892)2

minus 41198962

119899

]]]]]]]]]

]

times [

[

T120578

120585

]

]

(36)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892=

1198861198871198961198921198921minus 11988621198961198921198922minus 21198961198991198923

radic2(11988621198962119892minus 21198962119899)2

(37)

6 Journal of Discrete Mathematics

where

119889119904lowast

119889119904=

radic(119886119896119892)2

minus 21198962

119899

2

1198921= 11988711988621198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198961015840

119899+ 11988631198964

119892

minus (1198863+ 2119886) 119896

2

1198921198962

119899+ 2119886119896

4

119899

1198922= minus11988631198962

1198921198961015840

119899+ 11988631198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198962

119899+ 2119887119896

4

119899

1198923= 2119886119896

2

1198991198961015840

119892minus 2119886119896

1198921198961198991198961015840

119899minus 11988711988621198963

119892119896119899+ 2119887119896

1198921198963

119899

(38)

Proof We assume that the curve 120572 is a principal curveDifferentiating (15) with respect to 119904 and using (5) we obtain

1205731015840=

119889120573

119889119904

=1

radic2

(119887119896119899T + 119886119896

119899120578 + 119886119896

119892120585)

⟨1205731015840 1205731015840⟩ =

(119886119896119892)2

minus 21198962

119899

2

(39)

where from (9) (119886119896119892)2minus 21198962

119899gt 0 for all 119904 Since 120573

1015840=

(119889120573119889119904lowast)(119889119904lowast119889119904) the tangent vector T

120573of the curve 120573 is a

spacelike vector such that

T120573=

1

radic(119886119896119892)2

minus 21198962119899

(119887119896119899T + 119886119896

119899120578 + 119886119896

119892120585)

(40)

where

119889119904lowast

119889119904=

radic(119886119896119892)2

minus 21198962

119899

2

(41)

On the other hand from (15) and (40) it can be easily seenthat

120585120573= minus120573 times 119879

120573

=1

radic2(119886119896119892)2

minus 41198962119899

(119886119887119896119892T + 1198862119896119892120578 minus 2119896

119899120585) (42)

is a unit timelike vectorConsequently the geodesic curvature 119896

119892of the curve 120573 =

120573(119904lowast) is given by

119896119892= det (120573 119879

120573T1015840120573)

=

1198861198871198961198921198921minus 11988621198961198921198922minus 21198961198991198923

radic2(11988621198962119892minus 21198962119899)2

(43)

where

1198921= 11988711988621198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198961015840

119899+ 11988631198964

119892

minus (1198863+ 2119886) 119896

2

1198921198962

119899+ 2119886119896

4

119899

1198922= minus11988631198962

1198921198961015840

119899+ 11988631198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198962

119899+ 2119887119896

4

119899

1198923= 2119886119896

2

1198991198961015840

119892minus 2119886119896

1198921198961198991198961015840

119899minus 11988711988621198963

119892119896119899+ 2119887119896

1198921198963

119899

(44)

From (15) (40) and (42) we obtain the Sabban frame120573T120573 120585120573 of 120573

In the theorems which follow in a similar way as inTheorem 12 we obtain the Sabban frame 120573T

120573 120585120573 and the

geodesic curvature 119896119892of a spacelike Smarandache curve

We omit the proofs of Theorems 13 and 15 since they areanalogous to the proof of Theorem 12

Theorem 13 Let 120572 = 120572(119904) be a principal spacelike curvelying fully on the spacelike surface 119872 in R3

1with the

moving Lorentzian Darboux frame T 120578 120585 Then the T120585-Smarandache curve 120573 is spacelike and the Sabban frame120573T120573 120585120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[[[

[

119886

radic2

0119887

radic2minus119887119896119892

radic21198962119892minus (119886119896119899)2

119886119896119899

radic21198962119892minus (119886119896119899)2

119886119896119892

radic21198962119892minus (119886119896119899)2

minus119886119887119896119899

radic41198962119892minus 2(119886119896

119899)2

2119896119892

radic41198962119892minus 2(119886119896

119899)2

minus1198862119896119899

radic41198962119892minus 2(119886119896

119899)2

]]]]]]]]]]

]

times [

[

T120578

120585

]

]

(45)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus

119886119887119896119899ℎ1+ 2119896119892ℎ2+ 1198862119896119899ℎ3

radic2(21198962119892minus (119886119896119899)2)2

(46)

where

119889119904lowast

119889119904=

radic21198962

119892minus (119886119896119899)2

2

ℎ1= minus119887119886

21198962

1198991198961015840

119892+ 11988711988621198961198921198961198991198961015840

119899+ 2119886119896

4

119892

minus 11988631198962

1198921198962

119899minus 2119886119896

2

1198921198962

119899+ 11988631198964

119899

ℎ2= 2119886119896

1198921198961198991198961015840

119892minus 2119886119896

2

1198921198961015840

119899minus 2119887119896

3

119892119896119899+ 11988711988621198961198921198963

119899

ℎ3= 11988631198962

1198991198961015840

119892minus 1198863119896119892119896n1198961015840

119899minus 2119887119896

4

119892+ 11988711988621198962

1198921198962

119899

(47)

Theorem 14 Let 120572 = 120572(119904) be a principal spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

Journal of Discrete Mathematics 7

(i) if 119886119896119899minus 119887119896g = 0 for all 119904 the 120578120585-Smarandache curve

120573 is spacelike and the Sabban frame 120573T120573 120585120573 is given

by

[

[

120573

T120573

120585120573

]

]

=

[[[[[

[

0119886

radic2

119887

radic2

120598 0 0

0 minus120598119887

radic2

120598119886

radic2

]]]]]

]

[

[

T120578

120585

]

]

(48)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus

119886119896119892+ 119887119896119899

radic2

(49)

where 120598 = sign(119886119896119899minus 119887119896119892) for all 119904 and

119889119904lowast

119889119904= 120598

119886119896119899minus 119887119896119892

radic2

(50)

(ii) if 119886119896119899minus 119887119896119892

= 0 for all 119904 then the Sabban frame120573T120573 120585120573 of the 120578120585-Smarandache curve 120573 is a null

geodesic

Proof We assume that the curve 120572 is a principal curveDifferentiating (17) with respect to 119904 and using (5) we obtain

1205731015840=

119889120573

119889119904lowast

119889119904lowast

119889119904

=1

radic2

(119886119896119899minus 119887119896119892)T

⟨1205731015840 1205731015840⟩ =

(119886119896119899minus 119887119896119892)2

2

(51)

Then there are two following cases

(i) If 119886119896119899minus119887119896119892

= 0 for all 119904 since 1205731015840 = (119889120573119889119904lowast)(119889119904lowast119889119904)

then we obtain that the unit tangent vector T120573of the

curve 120573 is a spacelike vector such that

T120573= 120598T (52)

where

119889119904lowast

119889119904= 120598

119886119896119899minus 119887119896119892

radic2

(53)

and 120598 = sign(119886119896119899minus 119887119896119892)

On the other hand from (17) and (52) it can be easilyseen that

120585120573= minus120573 times 119879

120573

= minus120598119887

radic2

120578 + 120598119886

radic2

120585

(54)

is a unit timelike vector

Consequently the geodesic curvature 119896119892of the curve

120573 = 120573(119904lowast) is given by

119896119892= det (120573 119879

120573T1015840120573)

= minus

119886119896119892+ 119887119896119899

radic2

(55)

From (17) (52) and (54) we obtain the Sabban frame120573T120573 120585120573 of 120573

(ii) If 119886119896119899minus119887119896119892= 0 for all 119904 then 120573

1015840 is null So the tangentvector T

120573of the curve 120573 is a null vector It is known

that the only null curves lying on pseudosphere 11987821are

the null straight lines which are the null geodesics

Theorem 15 Let 120572 = 120572(119904) be a principal spacelikecurve lying fully on the spacelike surface 119872 in R3

1with

the moving Lorentzian Darboux frame T 120578 120585 Then theT120578120585-Smarandache curve 120573 is spacelike and the Sabban frame120573T120573 120585120573 is given by

120573 =1

radic3

(119886T + 119887120578 + 119888120585)

T120573=

(119887119896119899minus 119888119896119892)T + 119886119896

119899120578 + 119886119896

119892120585

radic3 (1198962119899minus 1198962119892) + (119887119896

119892minus 119888119896119899)2

120585120573= ((119886119887119896

119892minus 119886119888119896119899)T + ((3 + 119887

2) 119896119892minus 119887119888119896119899) 120578

minus ((3 minus 1198882) 119896119899+ 119887119888119896119892) 120585)

times (radic9 (1198962119899minus 1198962119892) + 3(119887119896

119892minus 119888119896119899)2

)

minus1

(56)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus ((119886119887119896

119892minus 119886119888119896119899) 1198971minus ((3 + 119887

2) 119896119892minus 119887119888119896119899) 1198972

minus ((3 minus 1198882) 119896119899+ 119887119888119896119892) 1198973)

times (radic2(11988621198962

119892minus 21198962

119899)2

)

minus1

(57)

where

119889119904lowast

119889119904=

radic3 (1198962119899minus 1198962119892) + (119887119896

119892minus 119888119896119899)2

radic3

1198971= (3119886119887119896

2

119899minus 3119886119888119896

119892119896119899) 1198961015840

119892+ (3119886119888119896

2

119892minus 3119886119887119896

119892119896119899) 1198961015840

119899

+ (119887119896119899minus 119888119896119892) (119887119896119892minus 119888119896119899)3

+ 31198871198881198964

119892minus 3119887119888119896

4

119899

+ 3 (1198872+ 1198882) 1198961198921198963

119899minus 3 (119887

2+ 1198882) 1198963

119892119896119899

8 Journal of Discrete Mathematics

x

y

z

10

0

minus10

minus5

0

5minus10

0

10

Figure 1 The spacelike surface 120601(119904 119906)

1198972= ((3119887

2+ 31198882+ 9) 119896

2

119899minus 6119887119888119896

119892119896119899) 1198961015840

119892

minus ((31198872+ 31198882+ 9) 119896

119892119896119899minus 6119887119888119896

2

119892) 1198961015840

119899

minus (1198861198883+ 3119886119888) 119896

4

119899+ (3119886119887 + 3119886119887119888

2) 1198961198921198963

119899

+ (1198861198873minus 3119886119887) 119896

3

119892119896119899+ (3119886119888 minus 3119886119887

2119888) 1198962

1198921198962

119899

1198973= ((3119887

2+ 31198882minus 9) 119896

119892119896119899minus 6119887119888119896

2

119899) 1198961015840

119892

+ ((9 minus 31198872minus 31198882) 1198962

119892+ 6119887119888119896

119892119896119899) 1198961015840

119899

+ (1198861198873minus 3119886119887) 119896

4

119892minus (3119886119888 + 119886119888

3) 1198961198921198963

119899

+ (3119886119888 minus 31198861198872119888) 1198963

119892119896119899+ (3119886119887 + 3119886119887119888

2) 1198962

1198921198962

119899

(58)

Example 16 Let us define a spacelike ruled surface (seeFigure 1) in the Minkowski 3-space such as

120601 119880 sub R2997888rarr R

3

1

(119904 119906) 997888rarr 120601 (119904 119906) = 120572 (119904) + 119906e (119904)

120601 (119904 119906) = (minus119906 sinh 119904 119904 minus119906 cosh 119904)

(59)

where 119906 isin (minus1 1)

Then we get the Lorentzian Darboux frame T 120578 120585 alongthe curve 120572 as follows

T (119904) = (0 1 0)

120578 (119904) equiv1

radic1 minus 1199062(cosh 119904 minus119906 sinh 119904)

120585 (119904) =1

radic1 minus 1199062(minus sinh 119904 0 minus cosh 119904)

(60)

where 120585(119904) is spacelike vectors and 120578(119904) is a unit timelikevector

Moreover the geodesic curvature 119896119892(119904) the asymptotic

curvature 119896119899(119904) and the principal curvature 120591

119892(119904) of the curve

120572 have the form119896119892(119904) = ⟨T1015840 (119904) 120585 (119904)⟩ = 0

119896119899 (119904) = minus ⟨T1015840 (119904) 120578 (119904)⟩ = 0

120591119892 (119904) = minus ⟨120585

1015840(119904) 120578 (119904)⟩ = minus

1

1 minus 1199062

(61)

Taking 119886 = radic3 119887 = 1 and using (15) we obtain that theT120578-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 2(a))

120573 (119904⋆(119904)) = (

cosh 119904

radic1 minus 1199062radic3 minus

119906

radic1 minus 1199062sinh 119904

radic1 minus 1199062) (62)

Taking 119886 = 119887 = 1 and using (16) we obtain that theT120585-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 2(b))

120573 (119904⋆(119904)) = (minus

sinh 119904

radic1 minus 1199062 1 minus

cosh 119904

radic1 minus 1199062) (63)

Journal of Discrete Mathematics 9

x

y

z

minus20

minus10

0

10

20

minus20

minus10

0

10

20

minus20 minus10 0 10 20

20

0

(a)

x

y

z2010

0minus10

minus20

20

10

0

minus10

minus20

20

10

0

minus10

minus20

(b)

Figure 2 (a) The T120578-Smarandache curve 120573 on 1198782

1for 119906 = 1radic3 (b) The T120585-Smarandache curve 120573 on 119878

2

1for 119906 = 1radic3

x

y

z

20

20

10

0

0

minus10

minus20

minus20

20

0

minus20

(a)

x

y

z

20100minus10minus20

0

10

minus10

20

minus20

20

10

0

minus10

minus20

(b)

Figure 3 (a) The 120578120585-Smarandache curve 120573 on 1198782

1for 119906 = 1radic3 (b) The T120578120585-Smarandache curve 120573 on 119878

2

1for 119906 = 1radic3

Taking 119886 = radic3 119887 = 1 and using (17) we obtain that the120578120585-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 3(a))

120573 (119904⋆(119904)) = minus

1

radic1 minus 1199062

times (sinh 119904 minusradic3 cosh 119904radic3119906 cosh 119904 minusradic3 sinh 119904)

(64)

Taking 119886 = radic3 119887 = 1 and 119888 = 1 and using (18) we obtainthat the T120578120585-Smarandache curve 120573 of the curve 120572 is given by(see Figure 3(b))

120573 (119904⋆(119904)) =

1

radic1 minus 1199062

times (cosh 119904 minus sinh 119904

radic3 minus 31199062 minus 119906 sinh 119904 minus cosh 119904)

(65)

10 Journal of Discrete Mathematics

x

y

z

20

0

minus20

200

minus20

20

10

0

minus10

minus20

Figure 4 The Smarandache curves on 1198782

1for 119906 = 1radic3

Also the Smarandache curves on 1198782 of 120572 for 119906 = 1radic3 with

Figure 4 are shown

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C Ashbacher ldquoSmarandache geometriesrdquo SmarandacheNotions Journal vol 8 no 1ndash3 pp 212ndash215 1997

[2] M Turgut and S Yilmaz ldquoSmarandache curves in Minkowskispace-timerdquo International Journal of Mathematical Combina-torics vol 3 pp 51ndash55 2008

[3] A T Ali ldquoSpecial smarandache curves in the euclidean spacerdquoInternational Journal of Mathematical Combinatorics vol 2 pp30ndash36 2010

[4] E B Koc Ozturk U Ozturk K Ilarslan and E NesovicldquoOn pseudohyperbolical Smarandache curves in Minkowski 3-spacerdquo International Journal of Mathematics and MathematicalSciences vol 2013 Article ID 658670 7 pages 2013

[5] E B Koc Ozturk U Ozturk K Ilarslan and E Nesovic ldquoOnpseudospherical smarandache curves in Minkowski 3-spacerdquoJournal of Applied Mathematics vol 2014 Article ID 404521 14pages 2014

[6] K Taskopru and M Tosun ldquoSmarandache curves on 1198782rdquo

Boletim da Sociedade Paranaense de Matematica vol 32 no 1pp 51ndash59 2014

[7] T Korpinar and E Turhan ldquoA new approach on SmarandacheTN-curves in terms of spacelike biharmonic curves with atimelike binormal in the Lorentzian Heisenberg group 119867119890119894119904

3rdquoJournal of Vectorial Relativity vol 6 pp 8ndash15 2011

[8] T Korpinar and E Turhan ldquoCharacterization of SmarandacheM1M2-curves of spacelike biharmonicB-slant helices according

to Bishop frame in E(1 1)rdquo Advanced Modeling and Optimiza-tion vol 14 no 2 pp 327ndash333 2012

[9] O Bektas and S Yuce ldquoSpecial Smarandache Curves AccordingtoDarboux Frame in E3rdquoRomanian Journal ofMathematics andComputer Science vol 3 no 1 pp 48ndash59 2013

[10] B ONeill Semi-Riemannian Geometry with Applications toRelativity Academic Press New York NY USA 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Smarandache Curves according to Curves on a Spacelike Surface in Minkowski 3-Space R 1 3

6 Journal of Discrete Mathematics

where

119889119904lowast

119889119904=

radic(119886119896119892)2

minus 21198962

119899

2

1198921= 11988711988621198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198961015840

119899+ 11988631198964

119892

minus (1198863+ 2119886) 119896

2

1198921198962

119899+ 2119886119896

4

119899

1198922= minus11988631198962

1198921198961015840

119899+ 11988631198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198962

119899+ 2119887119896

4

119899

1198923= 2119886119896

2

1198991198961015840

119892minus 2119886119896

1198921198961198991198961015840

119899minus 11988711988621198963

119892119896119899+ 2119887119896

1198921198963

119899

(38)

Proof We assume that the curve 120572 is a principal curveDifferentiating (15) with respect to 119904 and using (5) we obtain

1205731015840=

119889120573

119889119904

=1

radic2

(119887119896119899T + 119886119896

119899120578 + 119886119896

119892120585)

⟨1205731015840 1205731015840⟩ =

(119886119896119892)2

minus 21198962

119899

2

(39)

where from (9) (119886119896119892)2minus 21198962

119899gt 0 for all 119904 Since 120573

1015840=

(119889120573119889119904lowast)(119889119904lowast119889119904) the tangent vector T

120573of the curve 120573 is a

spacelike vector such that

T120573=

1

radic(119886119896119892)2

minus 21198962119899

(119887119896119899T + 119886119896

119899120578 + 119886119896

119892120585)

(40)

where

119889119904lowast

119889119904=

radic(119886119896119892)2

minus 21198962

119899

2

(41)

On the other hand from (15) and (40) it can be easily seenthat

120585120573= minus120573 times 119879

120573

=1

radic2(119886119896119892)2

minus 41198962119899

(119886119887119896119892T + 1198862119896119892120578 minus 2119896

119899120585) (42)

is a unit timelike vectorConsequently the geodesic curvature 119896

119892of the curve 120573 =

120573(119904lowast) is given by

119896119892= det (120573 119879

120573T1015840120573)

=

1198861198871198961198921198921minus 11988621198961198921198922minus 21198961198991198923

radic2(11988621198962119892minus 21198962119899)2

(43)

where

1198921= 11988711988621198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198961015840

119899+ 11988631198964

119892

minus (1198863+ 2119886) 119896

2

1198921198962

119899+ 2119886119896

4

119899

1198922= minus11988631198962

1198921198961015840

119899+ 11988631198961198921198961198991198961015840

119892minus 11988711988621198962

1198921198962

119899+ 2119887119896

4

119899

1198923= 2119886119896

2

1198991198961015840

119892minus 2119886119896

1198921198961198991198961015840

119899minus 11988711988621198963

119892119896119899+ 2119887119896

1198921198963

119899

(44)

From (15) (40) and (42) we obtain the Sabban frame120573T120573 120585120573 of 120573

In the theorems which follow in a similar way as inTheorem 12 we obtain the Sabban frame 120573T

120573 120585120573 and the

geodesic curvature 119896119892of a spacelike Smarandache curve

We omit the proofs of Theorems 13 and 15 since they areanalogous to the proof of Theorem 12

Theorem 13 Let 120572 = 120572(119904) be a principal spacelike curvelying fully on the spacelike surface 119872 in R3

1with the

moving Lorentzian Darboux frame T 120578 120585 Then the T120585-Smarandache curve 120573 is spacelike and the Sabban frame120573T120573 120585120573 is given by

[

[

120573

T120573

120585120573

]

]

=

[[[[[[[[[[

[

119886

radic2

0119887

radic2minus119887119896119892

radic21198962119892minus (119886119896119899)2

119886119896119899

radic21198962119892minus (119886119896119899)2

119886119896119892

radic21198962119892minus (119886119896119899)2

minus119886119887119896119899

radic41198962119892minus 2(119886119896

119899)2

2119896119892

radic41198962119892minus 2(119886119896

119899)2

minus1198862119896119899

radic41198962119892minus 2(119886119896

119899)2

]]]]]]]]]]

]

times [

[

T120578

120585

]

]

(45)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus

119886119887119896119899ℎ1+ 2119896119892ℎ2+ 1198862119896119899ℎ3

radic2(21198962119892minus (119886119896119899)2)2

(46)

where

119889119904lowast

119889119904=

radic21198962

119892minus (119886119896119899)2

2

ℎ1= minus119887119886

21198962

1198991198961015840

119892+ 11988711988621198961198921198961198991198961015840

119899+ 2119886119896

4

119892

minus 11988631198962

1198921198962

119899minus 2119886119896

2

1198921198962

119899+ 11988631198964

119899

ℎ2= 2119886119896

1198921198961198991198961015840

119892minus 2119886119896

2

1198921198961015840

119899minus 2119887119896

3

119892119896119899+ 11988711988621198961198921198963

119899

ℎ3= 11988631198962

1198991198961015840

119892minus 1198863119896119892119896n1198961015840

119899minus 2119887119896

4

119892+ 11988711988621198962

1198921198962

119899

(47)

Theorem 14 Let 120572 = 120572(119904) be a principal spacelike curvelying fully on the spacelike surface 119872 in R3

1with the moving

Lorentzian Darboux frame T 120578 120585 Then

Journal of Discrete Mathematics 7

(i) if 119886119896119899minus 119887119896g = 0 for all 119904 the 120578120585-Smarandache curve

120573 is spacelike and the Sabban frame 120573T120573 120585120573 is given

by

[

[

120573

T120573

120585120573

]

]

=

[[[[[

[

0119886

radic2

119887

radic2

120598 0 0

0 minus120598119887

radic2

120598119886

radic2

]]]]]

]

[

[

T120578

120585

]

]

(48)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus

119886119896119892+ 119887119896119899

radic2

(49)

where 120598 = sign(119886119896119899minus 119887119896119892) for all 119904 and

119889119904lowast

119889119904= 120598

119886119896119899minus 119887119896119892

radic2

(50)

(ii) if 119886119896119899minus 119887119896119892

= 0 for all 119904 then the Sabban frame120573T120573 120585120573 of the 120578120585-Smarandache curve 120573 is a null

geodesic

Proof We assume that the curve 120572 is a principal curveDifferentiating (17) with respect to 119904 and using (5) we obtain

1205731015840=

119889120573

119889119904lowast

119889119904lowast

119889119904

=1

radic2

(119886119896119899minus 119887119896119892)T

⟨1205731015840 1205731015840⟩ =

(119886119896119899minus 119887119896119892)2

2

(51)

Then there are two following cases

(i) If 119886119896119899minus119887119896119892

= 0 for all 119904 since 1205731015840 = (119889120573119889119904lowast)(119889119904lowast119889119904)

then we obtain that the unit tangent vector T120573of the

curve 120573 is a spacelike vector such that

T120573= 120598T (52)

where

119889119904lowast

119889119904= 120598

119886119896119899minus 119887119896119892

radic2

(53)

and 120598 = sign(119886119896119899minus 119887119896119892)

On the other hand from (17) and (52) it can be easilyseen that

120585120573= minus120573 times 119879

120573

= minus120598119887

radic2

120578 + 120598119886

radic2

120585

(54)

is a unit timelike vector

Consequently the geodesic curvature 119896119892of the curve

120573 = 120573(119904lowast) is given by

119896119892= det (120573 119879

120573T1015840120573)

= minus

119886119896119892+ 119887119896119899

radic2

(55)

From (17) (52) and (54) we obtain the Sabban frame120573T120573 120585120573 of 120573

(ii) If 119886119896119899minus119887119896119892= 0 for all 119904 then 120573

1015840 is null So the tangentvector T

120573of the curve 120573 is a null vector It is known

that the only null curves lying on pseudosphere 11987821are

the null straight lines which are the null geodesics

Theorem 15 Let 120572 = 120572(119904) be a principal spacelikecurve lying fully on the spacelike surface 119872 in R3

1with

the moving Lorentzian Darboux frame T 120578 120585 Then theT120578120585-Smarandache curve 120573 is spacelike and the Sabban frame120573T120573 120585120573 is given by

120573 =1

radic3

(119886T + 119887120578 + 119888120585)

T120573=

(119887119896119899minus 119888119896119892)T + 119886119896

119899120578 + 119886119896

119892120585

radic3 (1198962119899minus 1198962119892) + (119887119896

119892minus 119888119896119899)2

120585120573= ((119886119887119896

119892minus 119886119888119896119899)T + ((3 + 119887

2) 119896119892minus 119887119888119896119899) 120578

minus ((3 minus 1198882) 119896119899+ 119887119888119896119892) 120585)

times (radic9 (1198962119899minus 1198962119892) + 3(119887119896

119892minus 119888119896119899)2

)

minus1

(56)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus ((119886119887119896

119892minus 119886119888119896119899) 1198971minus ((3 + 119887

2) 119896119892minus 119887119888119896119899) 1198972

minus ((3 minus 1198882) 119896119899+ 119887119888119896119892) 1198973)

times (radic2(11988621198962

119892minus 21198962

119899)2

)

minus1

(57)

where

119889119904lowast

119889119904=

radic3 (1198962119899minus 1198962119892) + (119887119896

119892minus 119888119896119899)2

radic3

1198971= (3119886119887119896

2

119899minus 3119886119888119896

119892119896119899) 1198961015840

119892+ (3119886119888119896

2

119892minus 3119886119887119896

119892119896119899) 1198961015840

119899

+ (119887119896119899minus 119888119896119892) (119887119896119892minus 119888119896119899)3

+ 31198871198881198964

119892minus 3119887119888119896

4

119899

+ 3 (1198872+ 1198882) 1198961198921198963

119899minus 3 (119887

2+ 1198882) 1198963

119892119896119899

8 Journal of Discrete Mathematics

x

y

z

10

0

minus10

minus5

0

5minus10

0

10

Figure 1 The spacelike surface 120601(119904 119906)

1198972= ((3119887

2+ 31198882+ 9) 119896

2

119899minus 6119887119888119896

119892119896119899) 1198961015840

119892

minus ((31198872+ 31198882+ 9) 119896

119892119896119899minus 6119887119888119896

2

119892) 1198961015840

119899

minus (1198861198883+ 3119886119888) 119896

4

119899+ (3119886119887 + 3119886119887119888

2) 1198961198921198963

119899

+ (1198861198873minus 3119886119887) 119896

3

119892119896119899+ (3119886119888 minus 3119886119887

2119888) 1198962

1198921198962

119899

1198973= ((3119887

2+ 31198882minus 9) 119896

119892119896119899minus 6119887119888119896

2

119899) 1198961015840

119892

+ ((9 minus 31198872minus 31198882) 1198962

119892+ 6119887119888119896

119892119896119899) 1198961015840

119899

+ (1198861198873minus 3119886119887) 119896

4

119892minus (3119886119888 + 119886119888

3) 1198961198921198963

119899

+ (3119886119888 minus 31198861198872119888) 1198963

119892119896119899+ (3119886119887 + 3119886119887119888

2) 1198962

1198921198962

119899

(58)

Example 16 Let us define a spacelike ruled surface (seeFigure 1) in the Minkowski 3-space such as

120601 119880 sub R2997888rarr R

3

1

(119904 119906) 997888rarr 120601 (119904 119906) = 120572 (119904) + 119906e (119904)

120601 (119904 119906) = (minus119906 sinh 119904 119904 minus119906 cosh 119904)

(59)

where 119906 isin (minus1 1)

Then we get the Lorentzian Darboux frame T 120578 120585 alongthe curve 120572 as follows

T (119904) = (0 1 0)

120578 (119904) equiv1

radic1 minus 1199062(cosh 119904 minus119906 sinh 119904)

120585 (119904) =1

radic1 minus 1199062(minus sinh 119904 0 minus cosh 119904)

(60)

where 120585(119904) is spacelike vectors and 120578(119904) is a unit timelikevector

Moreover the geodesic curvature 119896119892(119904) the asymptotic

curvature 119896119899(119904) and the principal curvature 120591

119892(119904) of the curve

120572 have the form119896119892(119904) = ⟨T1015840 (119904) 120585 (119904)⟩ = 0

119896119899 (119904) = minus ⟨T1015840 (119904) 120578 (119904)⟩ = 0

120591119892 (119904) = minus ⟨120585

1015840(119904) 120578 (119904)⟩ = minus

1

1 minus 1199062

(61)

Taking 119886 = radic3 119887 = 1 and using (15) we obtain that theT120578-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 2(a))

120573 (119904⋆(119904)) = (

cosh 119904

radic1 minus 1199062radic3 minus

119906

radic1 minus 1199062sinh 119904

radic1 minus 1199062) (62)

Taking 119886 = 119887 = 1 and using (16) we obtain that theT120585-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 2(b))

120573 (119904⋆(119904)) = (minus

sinh 119904

radic1 minus 1199062 1 minus

cosh 119904

radic1 minus 1199062) (63)

Journal of Discrete Mathematics 9

x

y

z

minus20

minus10

0

10

20

minus20

minus10

0

10

20

minus20 minus10 0 10 20

20

0

(a)

x

y

z2010

0minus10

minus20

20

10

0

minus10

minus20

20

10

0

minus10

minus20

(b)

Figure 2 (a) The T120578-Smarandache curve 120573 on 1198782

1for 119906 = 1radic3 (b) The T120585-Smarandache curve 120573 on 119878

2

1for 119906 = 1radic3

x

y

z

20

20

10

0

0

minus10

minus20

minus20

20

0

minus20

(a)

x

y

z

20100minus10minus20

0

10

minus10

20

minus20

20

10

0

minus10

minus20

(b)

Figure 3 (a) The 120578120585-Smarandache curve 120573 on 1198782

1for 119906 = 1radic3 (b) The T120578120585-Smarandache curve 120573 on 119878

2

1for 119906 = 1radic3

Taking 119886 = radic3 119887 = 1 and using (17) we obtain that the120578120585-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 3(a))

120573 (119904⋆(119904)) = minus

1

radic1 minus 1199062

times (sinh 119904 minusradic3 cosh 119904radic3119906 cosh 119904 minusradic3 sinh 119904)

(64)

Taking 119886 = radic3 119887 = 1 and 119888 = 1 and using (18) we obtainthat the T120578120585-Smarandache curve 120573 of the curve 120572 is given by(see Figure 3(b))

120573 (119904⋆(119904)) =

1

radic1 minus 1199062

times (cosh 119904 minus sinh 119904

radic3 minus 31199062 minus 119906 sinh 119904 minus cosh 119904)

(65)

10 Journal of Discrete Mathematics

x

y

z

20

0

minus20

200

minus20

20

10

0

minus10

minus20

Figure 4 The Smarandache curves on 1198782

1for 119906 = 1radic3

Also the Smarandache curves on 1198782 of 120572 for 119906 = 1radic3 with

Figure 4 are shown

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C Ashbacher ldquoSmarandache geometriesrdquo SmarandacheNotions Journal vol 8 no 1ndash3 pp 212ndash215 1997

[2] M Turgut and S Yilmaz ldquoSmarandache curves in Minkowskispace-timerdquo International Journal of Mathematical Combina-torics vol 3 pp 51ndash55 2008

[3] A T Ali ldquoSpecial smarandache curves in the euclidean spacerdquoInternational Journal of Mathematical Combinatorics vol 2 pp30ndash36 2010

[4] E B Koc Ozturk U Ozturk K Ilarslan and E NesovicldquoOn pseudohyperbolical Smarandache curves in Minkowski 3-spacerdquo International Journal of Mathematics and MathematicalSciences vol 2013 Article ID 658670 7 pages 2013

[5] E B Koc Ozturk U Ozturk K Ilarslan and E Nesovic ldquoOnpseudospherical smarandache curves in Minkowski 3-spacerdquoJournal of Applied Mathematics vol 2014 Article ID 404521 14pages 2014

[6] K Taskopru and M Tosun ldquoSmarandache curves on 1198782rdquo

Boletim da Sociedade Paranaense de Matematica vol 32 no 1pp 51ndash59 2014

[7] T Korpinar and E Turhan ldquoA new approach on SmarandacheTN-curves in terms of spacelike biharmonic curves with atimelike binormal in the Lorentzian Heisenberg group 119867119890119894119904

3rdquoJournal of Vectorial Relativity vol 6 pp 8ndash15 2011

[8] T Korpinar and E Turhan ldquoCharacterization of SmarandacheM1M2-curves of spacelike biharmonicB-slant helices according

to Bishop frame in E(1 1)rdquo Advanced Modeling and Optimiza-tion vol 14 no 2 pp 327ndash333 2012

[9] O Bektas and S Yuce ldquoSpecial Smarandache Curves AccordingtoDarboux Frame in E3rdquoRomanian Journal ofMathematics andComputer Science vol 3 no 1 pp 48ndash59 2013

[10] B ONeill Semi-Riemannian Geometry with Applications toRelativity Academic Press New York NY USA 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Smarandache Curves according to Curves on a Spacelike Surface in Minkowski 3-Space R 1 3

Journal of Discrete Mathematics 7

(i) if 119886119896119899minus 119887119896g = 0 for all 119904 the 120578120585-Smarandache curve

120573 is spacelike and the Sabban frame 120573T120573 120585120573 is given

by

[

[

120573

T120573

120585120573

]

]

=

[[[[[

[

0119886

radic2

119887

radic2

120598 0 0

0 minus120598119887

radic2

120598119886

radic2

]]]]]

]

[

[

T120578

120585

]

]

(48)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus

119886119896119892+ 119887119896119899

radic2

(49)

where 120598 = sign(119886119896119899minus 119887119896119892) for all 119904 and

119889119904lowast

119889119904= 120598

119886119896119899minus 119887119896119892

radic2

(50)

(ii) if 119886119896119899minus 119887119896119892

= 0 for all 119904 then the Sabban frame120573T120573 120585120573 of the 120578120585-Smarandache curve 120573 is a null

geodesic

Proof We assume that the curve 120572 is a principal curveDifferentiating (17) with respect to 119904 and using (5) we obtain

1205731015840=

119889120573

119889119904lowast

119889119904lowast

119889119904

=1

radic2

(119886119896119899minus 119887119896119892)T

⟨1205731015840 1205731015840⟩ =

(119886119896119899minus 119887119896119892)2

2

(51)

Then there are two following cases

(i) If 119886119896119899minus119887119896119892

= 0 for all 119904 since 1205731015840 = (119889120573119889119904lowast)(119889119904lowast119889119904)

then we obtain that the unit tangent vector T120573of the

curve 120573 is a spacelike vector such that

T120573= 120598T (52)

where

119889119904lowast

119889119904= 120598

119886119896119899minus 119887119896119892

radic2

(53)

and 120598 = sign(119886119896119899minus 119887119896119892)

On the other hand from (17) and (52) it can be easilyseen that

120585120573= minus120573 times 119879

120573

= minus120598119887

radic2

120578 + 120598119886

radic2

120585

(54)

is a unit timelike vector

Consequently the geodesic curvature 119896119892of the curve

120573 = 120573(119904lowast) is given by

119896119892= det (120573 119879

120573T1015840120573)

= minus

119886119896119892+ 119887119896119899

radic2

(55)

From (17) (52) and (54) we obtain the Sabban frame120573T120573 120585120573 of 120573

(ii) If 119886119896119899minus119887119896119892= 0 for all 119904 then 120573

1015840 is null So the tangentvector T

120573of the curve 120573 is a null vector It is known

that the only null curves lying on pseudosphere 11987821are

the null straight lines which are the null geodesics

Theorem 15 Let 120572 = 120572(119904) be a principal spacelikecurve lying fully on the spacelike surface 119872 in R3

1with

the moving Lorentzian Darboux frame T 120578 120585 Then theT120578120585-Smarandache curve 120573 is spacelike and the Sabban frame120573T120573 120585120573 is given by

120573 =1

radic3

(119886T + 119887120578 + 119888120585)

T120573=

(119887119896119899minus 119888119896119892)T + 119886119896

119899120578 + 119886119896

119892120585

radic3 (1198962119899minus 1198962119892) + (119887119896

119892minus 119888119896119899)2

120585120573= ((119886119887119896

119892minus 119886119888119896119899)T + ((3 + 119887

2) 119896119892minus 119887119888119896119899) 120578

minus ((3 minus 1198882) 119896119899+ 119887119888119896119892) 120585)

times (radic9 (1198962119899minus 1198962119892) + 3(119887119896

119892minus 119888119896119899)2

)

minus1

(56)

and the geodesic curvature 119896119892of the curve 120573 reads

119896119892= minus ((119886119887119896

119892minus 119886119888119896119899) 1198971minus ((3 + 119887

2) 119896119892minus 119887119888119896119899) 1198972

minus ((3 minus 1198882) 119896119899+ 119887119888119896119892) 1198973)

times (radic2(11988621198962

119892minus 21198962

119899)2

)

minus1

(57)

where

119889119904lowast

119889119904=

radic3 (1198962119899minus 1198962119892) + (119887119896

119892minus 119888119896119899)2

radic3

1198971= (3119886119887119896

2

119899minus 3119886119888119896

119892119896119899) 1198961015840

119892+ (3119886119888119896

2

119892minus 3119886119887119896

119892119896119899) 1198961015840

119899

+ (119887119896119899minus 119888119896119892) (119887119896119892minus 119888119896119899)3

+ 31198871198881198964

119892minus 3119887119888119896

4

119899

+ 3 (1198872+ 1198882) 1198961198921198963

119899minus 3 (119887

2+ 1198882) 1198963

119892119896119899

8 Journal of Discrete Mathematics

x

y

z

10

0

minus10

minus5

0

5minus10

0

10

Figure 1 The spacelike surface 120601(119904 119906)

1198972= ((3119887

2+ 31198882+ 9) 119896

2

119899minus 6119887119888119896

119892119896119899) 1198961015840

119892

minus ((31198872+ 31198882+ 9) 119896

119892119896119899minus 6119887119888119896

2

119892) 1198961015840

119899

minus (1198861198883+ 3119886119888) 119896

4

119899+ (3119886119887 + 3119886119887119888

2) 1198961198921198963

119899

+ (1198861198873minus 3119886119887) 119896

3

119892119896119899+ (3119886119888 minus 3119886119887

2119888) 1198962

1198921198962

119899

1198973= ((3119887

2+ 31198882minus 9) 119896

119892119896119899minus 6119887119888119896

2

119899) 1198961015840

119892

+ ((9 minus 31198872minus 31198882) 1198962

119892+ 6119887119888119896

119892119896119899) 1198961015840

119899

+ (1198861198873minus 3119886119887) 119896

4

119892minus (3119886119888 + 119886119888

3) 1198961198921198963

119899

+ (3119886119888 minus 31198861198872119888) 1198963

119892119896119899+ (3119886119887 + 3119886119887119888

2) 1198962

1198921198962

119899

(58)

Example 16 Let us define a spacelike ruled surface (seeFigure 1) in the Minkowski 3-space such as

120601 119880 sub R2997888rarr R

3

1

(119904 119906) 997888rarr 120601 (119904 119906) = 120572 (119904) + 119906e (119904)

120601 (119904 119906) = (minus119906 sinh 119904 119904 minus119906 cosh 119904)

(59)

where 119906 isin (minus1 1)

Then we get the Lorentzian Darboux frame T 120578 120585 alongthe curve 120572 as follows

T (119904) = (0 1 0)

120578 (119904) equiv1

radic1 minus 1199062(cosh 119904 minus119906 sinh 119904)

120585 (119904) =1

radic1 minus 1199062(minus sinh 119904 0 minus cosh 119904)

(60)

where 120585(119904) is spacelike vectors and 120578(119904) is a unit timelikevector

Moreover the geodesic curvature 119896119892(119904) the asymptotic

curvature 119896119899(119904) and the principal curvature 120591

119892(119904) of the curve

120572 have the form119896119892(119904) = ⟨T1015840 (119904) 120585 (119904)⟩ = 0

119896119899 (119904) = minus ⟨T1015840 (119904) 120578 (119904)⟩ = 0

120591119892 (119904) = minus ⟨120585

1015840(119904) 120578 (119904)⟩ = minus

1

1 minus 1199062

(61)

Taking 119886 = radic3 119887 = 1 and using (15) we obtain that theT120578-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 2(a))

120573 (119904⋆(119904)) = (

cosh 119904

radic1 minus 1199062radic3 minus

119906

radic1 minus 1199062sinh 119904

radic1 minus 1199062) (62)

Taking 119886 = 119887 = 1 and using (16) we obtain that theT120585-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 2(b))

120573 (119904⋆(119904)) = (minus

sinh 119904

radic1 minus 1199062 1 minus

cosh 119904

radic1 minus 1199062) (63)

Journal of Discrete Mathematics 9

x

y

z

minus20

minus10

0

10

20

minus20

minus10

0

10

20

minus20 minus10 0 10 20

20

0

(a)

x

y

z2010

0minus10

minus20

20

10

0

minus10

minus20

20

10

0

minus10

minus20

(b)

Figure 2 (a) The T120578-Smarandache curve 120573 on 1198782

1for 119906 = 1radic3 (b) The T120585-Smarandache curve 120573 on 119878

2

1for 119906 = 1radic3

x

y

z

20

20

10

0

0

minus10

minus20

minus20

20

0

minus20

(a)

x

y

z

20100minus10minus20

0

10

minus10

20

minus20

20

10

0

minus10

minus20

(b)

Figure 3 (a) The 120578120585-Smarandache curve 120573 on 1198782

1for 119906 = 1radic3 (b) The T120578120585-Smarandache curve 120573 on 119878

2

1for 119906 = 1radic3

Taking 119886 = radic3 119887 = 1 and using (17) we obtain that the120578120585-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 3(a))

120573 (119904⋆(119904)) = minus

1

radic1 minus 1199062

times (sinh 119904 minusradic3 cosh 119904radic3119906 cosh 119904 minusradic3 sinh 119904)

(64)

Taking 119886 = radic3 119887 = 1 and 119888 = 1 and using (18) we obtainthat the T120578120585-Smarandache curve 120573 of the curve 120572 is given by(see Figure 3(b))

120573 (119904⋆(119904)) =

1

radic1 minus 1199062

times (cosh 119904 minus sinh 119904

radic3 minus 31199062 minus 119906 sinh 119904 minus cosh 119904)

(65)

10 Journal of Discrete Mathematics

x

y

z

20

0

minus20

200

minus20

20

10

0

minus10

minus20

Figure 4 The Smarandache curves on 1198782

1for 119906 = 1radic3

Also the Smarandache curves on 1198782 of 120572 for 119906 = 1radic3 with

Figure 4 are shown

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C Ashbacher ldquoSmarandache geometriesrdquo SmarandacheNotions Journal vol 8 no 1ndash3 pp 212ndash215 1997

[2] M Turgut and S Yilmaz ldquoSmarandache curves in Minkowskispace-timerdquo International Journal of Mathematical Combina-torics vol 3 pp 51ndash55 2008

[3] A T Ali ldquoSpecial smarandache curves in the euclidean spacerdquoInternational Journal of Mathematical Combinatorics vol 2 pp30ndash36 2010

[4] E B Koc Ozturk U Ozturk K Ilarslan and E NesovicldquoOn pseudohyperbolical Smarandache curves in Minkowski 3-spacerdquo International Journal of Mathematics and MathematicalSciences vol 2013 Article ID 658670 7 pages 2013

[5] E B Koc Ozturk U Ozturk K Ilarslan and E Nesovic ldquoOnpseudospherical smarandache curves in Minkowski 3-spacerdquoJournal of Applied Mathematics vol 2014 Article ID 404521 14pages 2014

[6] K Taskopru and M Tosun ldquoSmarandache curves on 1198782rdquo

Boletim da Sociedade Paranaense de Matematica vol 32 no 1pp 51ndash59 2014

[7] T Korpinar and E Turhan ldquoA new approach on SmarandacheTN-curves in terms of spacelike biharmonic curves with atimelike binormal in the Lorentzian Heisenberg group 119867119890119894119904

3rdquoJournal of Vectorial Relativity vol 6 pp 8ndash15 2011

[8] T Korpinar and E Turhan ldquoCharacterization of SmarandacheM1M2-curves of spacelike biharmonicB-slant helices according

to Bishop frame in E(1 1)rdquo Advanced Modeling and Optimiza-tion vol 14 no 2 pp 327ndash333 2012

[9] O Bektas and S Yuce ldquoSpecial Smarandache Curves AccordingtoDarboux Frame in E3rdquoRomanian Journal ofMathematics andComputer Science vol 3 no 1 pp 48ndash59 2013

[10] B ONeill Semi-Riemannian Geometry with Applications toRelativity Academic Press New York NY USA 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Smarandache Curves according to Curves on a Spacelike Surface in Minkowski 3-Space R 1 3

8 Journal of Discrete Mathematics

x

y

z

10

0

minus10

minus5

0

5minus10

0

10

Figure 1 The spacelike surface 120601(119904 119906)

1198972= ((3119887

2+ 31198882+ 9) 119896

2

119899minus 6119887119888119896

119892119896119899) 1198961015840

119892

minus ((31198872+ 31198882+ 9) 119896

119892119896119899minus 6119887119888119896

2

119892) 1198961015840

119899

minus (1198861198883+ 3119886119888) 119896

4

119899+ (3119886119887 + 3119886119887119888

2) 1198961198921198963

119899

+ (1198861198873minus 3119886119887) 119896

3

119892119896119899+ (3119886119888 minus 3119886119887

2119888) 1198962

1198921198962

119899

1198973= ((3119887

2+ 31198882minus 9) 119896

119892119896119899minus 6119887119888119896

2

119899) 1198961015840

119892

+ ((9 minus 31198872minus 31198882) 1198962

119892+ 6119887119888119896

119892119896119899) 1198961015840

119899

+ (1198861198873minus 3119886119887) 119896

4

119892minus (3119886119888 + 119886119888

3) 1198961198921198963

119899

+ (3119886119888 minus 31198861198872119888) 1198963

119892119896119899+ (3119886119887 + 3119886119887119888

2) 1198962

1198921198962

119899

(58)

Example 16 Let us define a spacelike ruled surface (seeFigure 1) in the Minkowski 3-space such as

120601 119880 sub R2997888rarr R

3

1

(119904 119906) 997888rarr 120601 (119904 119906) = 120572 (119904) + 119906e (119904)

120601 (119904 119906) = (minus119906 sinh 119904 119904 minus119906 cosh 119904)

(59)

where 119906 isin (minus1 1)

Then we get the Lorentzian Darboux frame T 120578 120585 alongthe curve 120572 as follows

T (119904) = (0 1 0)

120578 (119904) equiv1

radic1 minus 1199062(cosh 119904 minus119906 sinh 119904)

120585 (119904) =1

radic1 minus 1199062(minus sinh 119904 0 minus cosh 119904)

(60)

where 120585(119904) is spacelike vectors and 120578(119904) is a unit timelikevector

Moreover the geodesic curvature 119896119892(119904) the asymptotic

curvature 119896119899(119904) and the principal curvature 120591

119892(119904) of the curve

120572 have the form119896119892(119904) = ⟨T1015840 (119904) 120585 (119904)⟩ = 0

119896119899 (119904) = minus ⟨T1015840 (119904) 120578 (119904)⟩ = 0

120591119892 (119904) = minus ⟨120585

1015840(119904) 120578 (119904)⟩ = minus

1

1 minus 1199062

(61)

Taking 119886 = radic3 119887 = 1 and using (15) we obtain that theT120578-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 2(a))

120573 (119904⋆(119904)) = (

cosh 119904

radic1 minus 1199062radic3 minus

119906

radic1 minus 1199062sinh 119904

radic1 minus 1199062) (62)

Taking 119886 = 119887 = 1 and using (16) we obtain that theT120585-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 2(b))

120573 (119904⋆(119904)) = (minus

sinh 119904

radic1 minus 1199062 1 minus

cosh 119904

radic1 minus 1199062) (63)

Journal of Discrete Mathematics 9

x

y

z

minus20

minus10

0

10

20

minus20

minus10

0

10

20

minus20 minus10 0 10 20

20

0

(a)

x

y

z2010

0minus10

minus20

20

10

0

minus10

minus20

20

10

0

minus10

minus20

(b)

Figure 2 (a) The T120578-Smarandache curve 120573 on 1198782

1for 119906 = 1radic3 (b) The T120585-Smarandache curve 120573 on 119878

2

1for 119906 = 1radic3

x

y

z

20

20

10

0

0

minus10

minus20

minus20

20

0

minus20

(a)

x

y

z

20100minus10minus20

0

10

minus10

20

minus20

20

10

0

minus10

minus20

(b)

Figure 3 (a) The 120578120585-Smarandache curve 120573 on 1198782

1for 119906 = 1radic3 (b) The T120578120585-Smarandache curve 120573 on 119878

2

1for 119906 = 1radic3

Taking 119886 = radic3 119887 = 1 and using (17) we obtain that the120578120585-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 3(a))

120573 (119904⋆(119904)) = minus

1

radic1 minus 1199062

times (sinh 119904 minusradic3 cosh 119904radic3119906 cosh 119904 minusradic3 sinh 119904)

(64)

Taking 119886 = radic3 119887 = 1 and 119888 = 1 and using (18) we obtainthat the T120578120585-Smarandache curve 120573 of the curve 120572 is given by(see Figure 3(b))

120573 (119904⋆(119904)) =

1

radic1 minus 1199062

times (cosh 119904 minus sinh 119904

radic3 minus 31199062 minus 119906 sinh 119904 minus cosh 119904)

(65)

10 Journal of Discrete Mathematics

x

y

z

20

0

minus20

200

minus20

20

10

0

minus10

minus20

Figure 4 The Smarandache curves on 1198782

1for 119906 = 1radic3

Also the Smarandache curves on 1198782 of 120572 for 119906 = 1radic3 with

Figure 4 are shown

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C Ashbacher ldquoSmarandache geometriesrdquo SmarandacheNotions Journal vol 8 no 1ndash3 pp 212ndash215 1997

[2] M Turgut and S Yilmaz ldquoSmarandache curves in Minkowskispace-timerdquo International Journal of Mathematical Combina-torics vol 3 pp 51ndash55 2008

[3] A T Ali ldquoSpecial smarandache curves in the euclidean spacerdquoInternational Journal of Mathematical Combinatorics vol 2 pp30ndash36 2010

[4] E B Koc Ozturk U Ozturk K Ilarslan and E NesovicldquoOn pseudohyperbolical Smarandache curves in Minkowski 3-spacerdquo International Journal of Mathematics and MathematicalSciences vol 2013 Article ID 658670 7 pages 2013

[5] E B Koc Ozturk U Ozturk K Ilarslan and E Nesovic ldquoOnpseudospherical smarandache curves in Minkowski 3-spacerdquoJournal of Applied Mathematics vol 2014 Article ID 404521 14pages 2014

[6] K Taskopru and M Tosun ldquoSmarandache curves on 1198782rdquo

Boletim da Sociedade Paranaense de Matematica vol 32 no 1pp 51ndash59 2014

[7] T Korpinar and E Turhan ldquoA new approach on SmarandacheTN-curves in terms of spacelike biharmonic curves with atimelike binormal in the Lorentzian Heisenberg group 119867119890119894119904

3rdquoJournal of Vectorial Relativity vol 6 pp 8ndash15 2011

[8] T Korpinar and E Turhan ldquoCharacterization of SmarandacheM1M2-curves of spacelike biharmonicB-slant helices according

to Bishop frame in E(1 1)rdquo Advanced Modeling and Optimiza-tion vol 14 no 2 pp 327ndash333 2012

[9] O Bektas and S Yuce ldquoSpecial Smarandache Curves AccordingtoDarboux Frame in E3rdquoRomanian Journal ofMathematics andComputer Science vol 3 no 1 pp 48ndash59 2013

[10] B ONeill Semi-Riemannian Geometry with Applications toRelativity Academic Press New York NY USA 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Smarandache Curves according to Curves on a Spacelike Surface in Minkowski 3-Space R 1 3

Journal of Discrete Mathematics 9

x

y

z

minus20

minus10

0

10

20

minus20

minus10

0

10

20

minus20 minus10 0 10 20

20

0

(a)

x

y

z2010

0minus10

minus20

20

10

0

minus10

minus20

20

10

0

minus10

minus20

(b)

Figure 2 (a) The T120578-Smarandache curve 120573 on 1198782

1for 119906 = 1radic3 (b) The T120585-Smarandache curve 120573 on 119878

2

1for 119906 = 1radic3

x

y

z

20

20

10

0

0

minus10

minus20

minus20

20

0

minus20

(a)

x

y

z

20100minus10minus20

0

10

minus10

20

minus20

20

10

0

minus10

minus20

(b)

Figure 3 (a) The 120578120585-Smarandache curve 120573 on 1198782

1for 119906 = 1radic3 (b) The T120578120585-Smarandache curve 120573 on 119878

2

1for 119906 = 1radic3

Taking 119886 = radic3 119887 = 1 and using (17) we obtain that the120578120585-Smarandache curve 120573 of the curve 120572 is given by (seeFigure 3(a))

120573 (119904⋆(119904)) = minus

1

radic1 minus 1199062

times (sinh 119904 minusradic3 cosh 119904radic3119906 cosh 119904 minusradic3 sinh 119904)

(64)

Taking 119886 = radic3 119887 = 1 and 119888 = 1 and using (18) we obtainthat the T120578120585-Smarandache curve 120573 of the curve 120572 is given by(see Figure 3(b))

120573 (119904⋆(119904)) =

1

radic1 minus 1199062

times (cosh 119904 minus sinh 119904

radic3 minus 31199062 minus 119906 sinh 119904 minus cosh 119904)

(65)

10 Journal of Discrete Mathematics

x

y

z

20

0

minus20

200

minus20

20

10

0

minus10

minus20

Figure 4 The Smarandache curves on 1198782

1for 119906 = 1radic3

Also the Smarandache curves on 1198782 of 120572 for 119906 = 1radic3 with

Figure 4 are shown

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C Ashbacher ldquoSmarandache geometriesrdquo SmarandacheNotions Journal vol 8 no 1ndash3 pp 212ndash215 1997

[2] M Turgut and S Yilmaz ldquoSmarandache curves in Minkowskispace-timerdquo International Journal of Mathematical Combina-torics vol 3 pp 51ndash55 2008

[3] A T Ali ldquoSpecial smarandache curves in the euclidean spacerdquoInternational Journal of Mathematical Combinatorics vol 2 pp30ndash36 2010

[4] E B Koc Ozturk U Ozturk K Ilarslan and E NesovicldquoOn pseudohyperbolical Smarandache curves in Minkowski 3-spacerdquo International Journal of Mathematics and MathematicalSciences vol 2013 Article ID 658670 7 pages 2013

[5] E B Koc Ozturk U Ozturk K Ilarslan and E Nesovic ldquoOnpseudospherical smarandache curves in Minkowski 3-spacerdquoJournal of Applied Mathematics vol 2014 Article ID 404521 14pages 2014

[6] K Taskopru and M Tosun ldquoSmarandache curves on 1198782rdquo

Boletim da Sociedade Paranaense de Matematica vol 32 no 1pp 51ndash59 2014

[7] T Korpinar and E Turhan ldquoA new approach on SmarandacheTN-curves in terms of spacelike biharmonic curves with atimelike binormal in the Lorentzian Heisenberg group 119867119890119894119904

3rdquoJournal of Vectorial Relativity vol 6 pp 8ndash15 2011

[8] T Korpinar and E Turhan ldquoCharacterization of SmarandacheM1M2-curves of spacelike biharmonicB-slant helices according

to Bishop frame in E(1 1)rdquo Advanced Modeling and Optimiza-tion vol 14 no 2 pp 327ndash333 2012

[9] O Bektas and S Yuce ldquoSpecial Smarandache Curves AccordingtoDarboux Frame in E3rdquoRomanian Journal ofMathematics andComputer Science vol 3 no 1 pp 48ndash59 2013

[10] B ONeill Semi-Riemannian Geometry with Applications toRelativity Academic Press New York NY USA 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Smarandache Curves according to Curves on a Spacelike Surface in Minkowski 3-Space R 1 3

10 Journal of Discrete Mathematics

x

y

z

20

0

minus20

200

minus20

20

10

0

minus10

minus20

Figure 4 The Smarandache curves on 1198782

1for 119906 = 1radic3

Also the Smarandache curves on 1198782 of 120572 for 119906 = 1radic3 with

Figure 4 are shown

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C Ashbacher ldquoSmarandache geometriesrdquo SmarandacheNotions Journal vol 8 no 1ndash3 pp 212ndash215 1997

[2] M Turgut and S Yilmaz ldquoSmarandache curves in Minkowskispace-timerdquo International Journal of Mathematical Combina-torics vol 3 pp 51ndash55 2008

[3] A T Ali ldquoSpecial smarandache curves in the euclidean spacerdquoInternational Journal of Mathematical Combinatorics vol 2 pp30ndash36 2010

[4] E B Koc Ozturk U Ozturk K Ilarslan and E NesovicldquoOn pseudohyperbolical Smarandache curves in Minkowski 3-spacerdquo International Journal of Mathematics and MathematicalSciences vol 2013 Article ID 658670 7 pages 2013

[5] E B Koc Ozturk U Ozturk K Ilarslan and E Nesovic ldquoOnpseudospherical smarandache curves in Minkowski 3-spacerdquoJournal of Applied Mathematics vol 2014 Article ID 404521 14pages 2014

[6] K Taskopru and M Tosun ldquoSmarandache curves on 1198782rdquo

Boletim da Sociedade Paranaense de Matematica vol 32 no 1pp 51ndash59 2014

[7] T Korpinar and E Turhan ldquoA new approach on SmarandacheTN-curves in terms of spacelike biharmonic curves with atimelike binormal in the Lorentzian Heisenberg group 119867119890119894119904

3rdquoJournal of Vectorial Relativity vol 6 pp 8ndash15 2011

[8] T Korpinar and E Turhan ldquoCharacterization of SmarandacheM1M2-curves of spacelike biharmonicB-slant helices according

to Bishop frame in E(1 1)rdquo Advanced Modeling and Optimiza-tion vol 14 no 2 pp 327ndash333 2012

[9] O Bektas and S Yuce ldquoSpecial Smarandache Curves AccordingtoDarboux Frame in E3rdquoRomanian Journal ofMathematics andComputer Science vol 3 no 1 pp 48ndash59 2013

[10] B ONeill Semi-Riemannian Geometry with Applications toRelativity Academic Press New York NY USA 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Smarandache Curves according to Curves on a Spacelike Surface in Minkowski 3-Space R 1 3

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The Quantum Spheres and their Embedding into Quantum Minkowski Space-Time

The Quantum Spheres and their Embedding into Quantum Minkowski Space-Time

Florentin SMARANDACHE & Eugen EVU - viXra.org

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On the Exact Maximum Complexity of Minkowski Sums of Polytopes

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