Smarandache Curves of Curves lying on Lightlike Cone

2017·0.16 MB·

by  Tanju Kahraman

International J.Math. Combin. Vol.3(2017), 1-9
Smarandache Curves of Curves lying on Lightlike Cone in R3
1
Tanju Kahraman1 and Hasan Hu¨seyin U˘gurlu2
1. CelalBayarUniversity,DepartmentofMathematics,Manisa-Turkey.
2. GaziUniversity,GaziFacultyofEducation,MathematicsTeachingProgram,Ankara-Turkey.
E-mail: [email protected],[email protected]
Abstract: Inthispaper,weconsiderthenotionoftheSmarandachecurvesbyconsidering
the asymptotic orthonormal frames of curves lying fully on lightlike cone in Minkowski 3-
spaceR3. WegivetherelationshipsbetweenSmarandachecurvesandcurveslyingonlightlike
1
cone in R3.
1
Key Words: Minkowski 3-space, Smarandache curves,lightlike cone, curvatures.
AMS(2010): 53A35, 53B30, 53C50
§1. Introduction
In the study of the fundamental theory and the characterizations of space curves, the related
curvesforwhichthereexistcorrespondingrelationsbetweenthecurvesareveryinterestingand
an important problem. The most fascinating examples of such curves are associated curves
and special curves. Recently, a new special curve is called Smarandache curve is defined by
Turgut and Yılmaz in Minkowski space-time [9]. These curves are called Smarandache curves:
IfaregularcurveinEuclidean3-space,whosepositionvectoris composedbyFrenetvectorson
another regular curve, then the curve is called a Smarandache Curve. Then, Ali have studied
Smarandache curves in the Euclidean 3-space E3[1]. Kahraman and U˘gurlu have studied dual
2
Smarandachecurves of curves lying on unit dual sphereS˜ in dual space D3 [3] and they have
studied dual Smarandache curves of curves lying on unit dual hyperbolic sphere H˜2 in D3
0 1
[4]. Also, Kahraman,O¨nder and U˘gurlu have studied Blaschke approachto dual Smarandache
curves [2]
Inthispaper,weconsiderthenotionoftheSmarandachecurvesbymeansoftheasymptotic
orthonormal frames of curves lying fully on Lightlike cone in Minkowski 3-space R3. We show
1
the relationships between frames and curvatures of Smarandache curves and curves lying on
lightlike cone in R3.
1
§2. Preliminaries
The Minkowski 3-space R3 is the real vector space R3 provided with the standart flat metric
1
1ReceivedNovember25,2016,Accepted August2,2017.
2 TanjuKahramanandHasanHu¨seyinUg˘urlu
given by
, = dx2+dx2+dx2
h i − 1 2 3
where(x ,x ,x )isarectangularcoordinatesystemofIR3. Anarbitraryvector~v =(v ,v ,v )
1 2 3 1 1 2 3
in R3 can have one of three Lorentzian causal characters; it can be spacelike if ~v,~v > 0 or
1 h i
~v = 0, timelike if ~v,~v < 0 and null (lightlike) if ~v,~v = 0 and ~v = 0. Similarly, an arbitrary
h i h i 6
curve ~x=~x(s) can locally be spacelike, timelike or null (lightlike), if all of its velocity vectors
~x(s) are respectively spacelike, timelike or null (lightlike) [6, 7]. We say that a timelike vector
′
is future pointing or past pointing if the first compound of the vector is positive or negative,
respectively. Foranyvectors~a=(a ,a ,a )and~b=(b ,b ,b )inR3,inthemeaningofLorentz
1 2 3 1 2 3 1
vector product of~a and~b is defined by
e e e
1 2 3
− −
~a×~b=(cid:12)(cid:12) a1 a2 a3 (cid:12)(cid:12)=(a2b3−a3b2, a1b3−a3b1, a2b1− a1b2).
(cid:12) (cid:12)
(cid:12) b b b (cid:12)
(cid:12) 1 2 3 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Denoteby T~, N(cid:12)~, B~ themoving(cid:12)Frenetalongthecurvex(s)intheMinkowskispaceR3.
1
For an arbitrarynspacelikeocurve x(s) in the space R3, the following Frenet formulae are given
1
([8]),
T~ 0 κ 0 T~
′
 N~ = εκ 0 τ  N~ 
′ −
 B~   0 ετ 0  B~ 
 ′    
    
where T~,T~ =1, N~,N~ =ε= 1, B~,B~ = ε, T~,N~ = T~,B~ = N~,B~ =0andκ
± −
andτ aDrecurEvatureDandtoErsionofthespDacelikEecurvex(Ds)respEectiDvely[1E0]. HDere,εEdetermines
the kind of spacelike curve x(s). If ε = 1, then x(s) is a spacelike curve with spacelike first
principal normal N~ and timelike binormal B~. If ε = 1, then x(s) is a spacelike curve with
−
timelike principal normal N~ and spacelike binormal B~ [8]. Furthermore, for a timelike curve
x(s) in the space R3, the following Frenet formulae are given in as follows,
1
T~ 0 κ 0 T~
′
 N~ = κ 0 τ  N~ 
′
 B~   0 τ 0  B~ 
 ′   −  
    
where T~,T~ = 1, N~,N~ = B~,B~ = 1, T~,N~ = T~,B~ = N~,B~ = 0 and κ and τ
−
are curDvatureEand torsDion ofEthe tDimelikEe curveDx(s) rEespecDtivelyE[10].D E
Curves lying on lightlike cone are examined using moving asymptotic frame which is de-
noted by ~x, α~, ~y along the curve x(s) lying fully on lightlike cone in the Minkowski space
{ }
R3.
1
For an arbitrary curve x(s) lying on lightlike cone in R3, the following asymptotic frame
1
SmarandacheCurvesofCurveslyingonLightlikeConeinIR3 3
1
formulae are given by
~x 0 1 0 ~x
′
 α~ = κ 0 1  α~ 
′
−
 ~y   0 κ 0  ~y 
 ′   −  
    
where ~x,~x = ~y,~y = ~x,α~ = ~y,α~ =0, ~x,~y = α~,α~ =1 and κ is curvature function of
h i h i h i h i h i h i
curve α(s) [5].
§3. Smarandache Curves of Curves lying on Lightlike Cone in
Minkowski 3-Space R3
1
In this section, we firstdefine the four different type of the Smarandachecurvesofcurves lying
fullyonlightlikeconeinR3. Then,bytheaidofasymptoticframe,wegivethecharacterizations
1
between reference curve and its Smarandache curves.
3.1 Smarandache ~xα~-curves of curves lying on lightlike cone in R3
1
Definition 3.1 Let x = x(s) be a unit speed regular curve lying fully on lightlike cone and
~x,α~,~y be its moving asymptotic frame. The curve α defined by
1
{ }
α~ (s)=~x(s)+α~(s) (3.1)
1
is called the Smarandache ~xα~-curve of x and α fully lies on Lorentzian sphere S2.
1 1
Now,wecangivethe relationshipsbetweenxanditsSmarandache~xα~-curveα asfollows.
1
Theorem 3.1 Let x=x(s) be a unit speed regular curve lying on Lightlike cone in R3. Then
1
the relationships between the asymptotic frame of x and Frenet of its Smarandache ~xα~-curve
α are given by
1
κ 1 1
T~1  κ′+κ(√1κ−′2κ2κ+1) κ′+√21κ−24κκ2 2√κ1−−κ′2κ1  ~x
 N~1 = (1 −2κ)−2√A (1 2κ)−2√A (1 −2κ)2−√A  α~  (3.2)
 B~1   √−1−−21κ√A (1−−2κκ)3′/2√A (1κ−′−2+κκ)−3/22κ√2A  ~y 
 
where κ is curvature function of x(s) and A is
(2κ 1)(κ)2+(8κ 8κ2 2)κ 16κ4 24κ3+4κ2 2κ
′ ′
A= − − − − − − .
(1 2κ)4
−
Proof LettheFrenetofSmarandache~xα~-curvebe T~ ,N~ ,B~ . Sinceα~ (s)=~x(s)+α~(s)
1 1 1 1
n o
4 TanjuKahramanandHasanHu¨seyinUg˘urlu
and T~ =α~ / α~ , we have
1 ′1 k ′1k
κ 1 1
T~ = ~x(s)+ α~(s) ~y(s) (3.3)
1
√1 2κ √1 2κ − √1 2κ
− − −
where
ds 1 1
= and κ< .
ds1 √1 2κ 2
−
Since N~ =T~′ T~ , we get
1 1 1′
.(cid:13) (cid:13)
(cid:13) (cid:13)
κ (cid:13)κκ(cid:13) +κ 2κ2 κ +2κ 4κ2 2κ κ 1
N~ = ′− ′ − ~x(s)+ ′ − α~(s)+ − ′− ~y(s) (3.4)
1
(1 2κ)2√A (1 2κ)2√A (1 2κ)2√A
− − −
where
(2κ 1)(κ)2+(8κ 8κ2 2)κ 16κ4 24κ3+4κ2 2κ
′ ′
A= − − − − − − .
(1 2κ)4
−
Then from B~ =T~ N~ , we have
1 1 1
×
1 κ κ 2κ2+κ
B~ = − ~x(s)+ ′ α~(s)+ ′− ~y(s). (3.5)
1 √1−2κ√A (1 2κ)3/2√A (1 2κ)3/2√A
− −
From (3.3)-(3.5) we have (3.2).
2
Theorem 3.2 The curvature function κ of Smarandache ~xα~-curve α according to curvature
1 1
function of curve x is given by
√A
κ = . (3.6)
1 (1 2κ)2
−
Proof Since κ = T~ . Using the equation (3.3), we get the desired equality (3.6).
1 1′ 2
(cid:13) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
Corollary 3.1 If curve x is a line. Then Smarandache ~xα~-curve α is line.
1
Theorem3.3 Torsionτ ofSmarandache~xα~-curveα accordingtocurvaturefunctionofcurve
1 1
x is as follows
A A +2κA κκA κA 2κκA +4κ2A Aκ 2Aκ+2κκ A
′ ′ ′ ′ ′ ′ ′ ′ ′′
τ = − − − − − −
1 (1 2κ)3A2
−
κA Aκκ2+κκ A κ2A 6Aκκ 3Aκ′2
+ ′ − ′ ′ ′′ − ′ ′− ′− 1−2κ
(1 2κ)4A2
−
dB
1
Proof Sinceτ = ,N . Usingderivationoftheequation(3.5),weobtainthewanted
1 1
ds
(cid:28) 1 (cid:29)
equation.
2
SmarandacheCurvesofCurveslyingonLightlikeConeinIR3 5
1
Corollary 3.2 If Smarandache ~xα~-curve α is a plane curve. Then, we obtain
1
A A′+2κA′ κκ′A κ′A′ 2κκ′A′+4κ2A Aκ′ 2Aκ+2κκ′′A
− − − − − −
3Aκ2
(cid:0)+(1 2κ)+κ′A Aκκ′2+κ′κ′′A κ′2A′ 6Aκκ′ ′ =0. (cid:1)
− − − − − 1 2κ
−
3.2 Smarandache ~x~y-curves of curves lying on Lightlike cone in R3
1
In this section, we define the second type of Smarandache curves that is called Smarandache
~x~y-curve. Then, we give the relationships between the curve lying on lightlike cone and its
Smarandache ~x~y-curve.
Definition 3.2 Let x = x(s) be a unit speed regular curve lying fully on lightlike cone and
~x,α~,~y be its moving asymptotic frame. The curve α defined by
2
{ }
1
α~ (s)= (~x(s)+~y(s))
2
√2
is called the Smarandache ~x~y-curve of x and fully lies on Lorentzian sphere S2.
1
Now,we cangivethe relationshipsbetweenxandits Smarandache~x~y-curveα asfollows.
2
Theorem 3.4 Let x = x(s) be a unit speed regular curve lying on lightlike cone in R3. Then
1
the relationships between the asymptotic frame of x and Frenet of its Smarandache ~x~y-curve α
2
are given by
T~ 0 1 0 ~x
2
 N~2 = √κ2κ 0 √−12κ  α~ 
− −
 B~2   √12κ 0 √−κ2κ  ~y 
   − −  
where κ is curvature function of x(s).
Theorem 3.5 The curvature function κ of Smarandache ~x~y-curve α according to curvature
2 2
function of curve x is given
κ =√ 2κ.
2
−
Corollary 3.3 Curve x is a line if and only if Smarandache ~x~y-curve α is line.
2
Theorem 3.6 Torsion τ ofSmarandache~x~y-curveα according tocurvaturefunctionofcurve
2 2
x is as follows
2√2κ
′
τ = − .
2 4κ2 4κ3
−
Corollary 3.4 If Smarandache ~x~y-curve α is a plane curve. Then, curvature κ of curve x is
2
constant.
6 TanjuKahramanandHasanHu¨seyinUg˘urlu
3.3 Smarandache α~~y-curves of curves lying on lightlike cone in R3
1
In this section, we define the third type of Smarandache curves that is called Smarandache
α~~y-curve. Then, we give the relationships between the curve lying on lightlike cone and its
Smarandache α~~y-curve.
Definition 3.3 Let x = x(s) be a unit speed regular curve lying fully on lightlike cone and
~x,α~,~y be its moving asymptotic frame. The curve α defined by
3
{ }
α~ (s)=α~(s)+~y(s)
3
is called the Smarandache α~~y-curve of x and fully lies on Lorentzian sphere S2.
1
Now we can give the relationships between x and its Smarandacheα~~y-curve α as follows.
3
Theorem 3.7 Let x = x(s) be a unit speed regular curve lying on lightlike cone in R3. Then
1
the relationships between the asymptotic frame of x and Frenet of its Smarandache α~~y-curve α
3
are given by
T~3 √κ2κ 2κ √κ−2κ2κ √κ−212κ ~x
 N~3 = −κ4√+A−2κ3 −2κ2κ′+4κ√κA−′+2κ3−4κ2 κκ′−κ′√+A−κ3−2κ2  α~ 
 B~3   −3κ2κ′+√5Aκ√κ′−κ2κ−4+2κ4κ3−4κ2 √κA2√κ′κ−2κ−κ2′κ κ5−4κ4√+4Aκ√3+κ22κ−32κκ′−4κ2κ′  ~y 
where κ is curvature function of x(s) and A is
A=(4κ4 16κ3+16κ2)(κ′)2+κ′( 10κ5+38κ4 36κ3) 2κ7+12κ6 24κ5+16κ4.
− − − − −
Theorem 3.8 The curvature function κ of Smarandache α~~y-curve α according to curvature
3 3
function of curve x is given
(4κ4 16κ3+16κ2)(κ)2+κ( 10κ5+38κ4 36κ3) 2κ7+12κ6 24κ5+16κ4
κ = − ′ ′ − − − − .
3 (κ2 2κ)2
p −
Corollary 3.5 If curve x is a line. Then, Smarandache α~~y-curve α is line.
3
Theorem3.9 Torsionτ ofSmarandache α~~y-curveα according tocurvaturefunctionofcurve
3 3
x is as follows
τ = b1 κκ′−κ′+κ3−2κ2 + 2b2(κ−κ′) b3κ2
3 A(κ2 2κ) A − A
(cid:0) − (cid:1)
SmarandacheCurvesofCurveslyingonLightlikeConeinIR3 7
1
where b ,b and b are
1 2 3
b1 = 6κ(κ′)2+9κ2κ′+5(κ′)2+5κκ′′ 4κ3κ′ 8κκ′
− − −
A κκ κ
+(3κ2κ′ 5κκ′+κ4 4κ3+4κ2) ′ + ′− ′
− − 2A κ2 2κ
(cid:18) − (cid:19)
A κκ κ
b2 = (κ2−κ)κ′′+(2κ−1)(κ′)2−(κ2κ′−κκ′) 2A′ + κ2′−2κ′
(cid:18) − (cid:19)
b = (5κ2 16κ+12)κ2κ +(2κ 8)κ(κ)2+2κ3κ
3 ′ ′ ′′
− −
A κκ κ
(2κ3κ′ 4κ2κ′+κ5 4κ4+4κ3) ′ + ′− ′ .
− − − 2A κ2 2κ
(cid:18) − (cid:19)
Corollary 3.6 If Smarandache α~~y-curve α is a plane curve. Then, we obtain
3
b κκ κ +κ3 2κ2 +(κ2 2κ) 2b (κ κ) b κ2 =0.
1 ′ ′ 2 ′ 3
− − − − −
(cid:0) (cid:1) (cid:0) (cid:1)
3.4 Smarandache ~xα~~y-curves of curves lying on lightlike cone in R3
1
In this section, we define the fourth type of Smarandache curves that is called Smarandache
~xα~~y-curve. Then, we give the relationships between the curve lying on lightlike cone and its
Smarandache ~xα~~y-curve.
Definition 3.4 Let x = x(s) be a unit speed regular curve lying fully on Lightlike cone and
~x,α~,~y be its moving asymptotic frame. The curves α and α defined by
4 5
{ }
(i) α~ (s)= 1 (~x(s)+α~(s)+~y(s));
4 √3
(ii) α~ (s)= ~x(s)+α~(s)+~y(s)
5
−
2
are called the Smarandache ~xα~~y-curves of x and fully lies on Lorentzian sphere S and hyper-
1
bolic sphere H3.
0
Now, we can give the relationships between x and its Smarandache ~xα~~y-curve α on
4
Lorentzian sphere S2 as follows.
1
Theorem 3.10 Let x=x(s) be a unit speed regular curve lying on lightlike cone in R3. Then
1
the relationships between the asymptotic frame of x and Frenet of its Smarandache ~xα~~y-curve
α are given by
4
T~4 √κ2 κ4κ+1 √κ21−4κκ+1 √κ2−14κ+1 ~x
− − −
 N~4 = √2aa1c11+b21 √2a1bc11+b21 √2a1cc11+b21  α~ 
 B~4   √κ2−c41κ(+1−1√κ)+2ab11c1+b21 √κ2−4κc+1κ1+√a21a1c1+b21 √κ2−a14(κκ+−11√)−2ab11κc1+b21  ~y 
8 TanjuKahramanandHasanHu¨seyinUg˘urlu
where κ is curvature function of x(s) and a ,b ,c are
1 1 1
√3(κ 2κκ 5κ2+5κ3 κ4+κ)
′ ′
a = − − −
1 (κ2 4κ+1)2
−
√3(2κ+κ 8κ2+2κκ +2κ3)
′ ′
b = −
1 (κ2 4κ+1)2
−
√3( 2κ +κκ +5κ 5κ2+κ3 1)
′ ′
c = − − − .
1 (κ2 4κ+1)2
−
Theorem3.11 Thecurvaturefunctionκ ofSmarandache~xα~~y-curveα accordingtocurvature
4 4
function of curve x is given
κ = 2a c +b2.
4 1 1 1
q
Corollary 3.7 If curve x is a line. Then, Smarandache ~xα~~y-curve α is line.
4
Theorem 3.12 Torsion τ of Smarandache ~xα~~y-curve α according to curvature function of
4 4
curve x is as follows
a c +b b +a c
2 1 1 2 1 2
τ =
4
2a c +b2
1 1 1
where a ,b and c are p
2 2 2
a2 = √3(c′1−c′1κ−c1κ′+b′1+c1κ2+a1κ)−√3(c1−c1κ+b1)(cid:18)κκ2κ−′−4κ2+κ′1+ a′1c12a+1ac11c+′1b+21b1b′1(cid:19)
(κ2−4κ+1) 2a1c1+b21
b2 = √3(c′1κ+c1κ′+a′1+c1−c1κ+b1+a1κ−a1qκ2+b1κ2)−√3(c1κ+a1)(cid:18)κκ2κ−′−4κ2+κ′1 + a′1c12a+1ac11c+′1b+21b1b′1(cid:19)
(κ2−4κ+1) 2a1c1+b21
c2 = √3(a′1κ−a′1+a1κ′−b′1κ−b1κ′−c1κ−a1)−√3(aq1κ−a1−b1κ)(cid:18)κκ2κ−′−4κ2+κ′1+ a′1c12a+1ac11c+′1b+21b1b′1(cid:19).
(κ2−4κ+1) 2a1c1+b21
q
Corollary 3.8 If Smarandache ~xα~~y-curve α is a plane curve. Then, we obtain
4
a c +b b +a c =0.
2 1 1 2 1 2
Resultsofstatement(ii)canbe givenbyusingthe similarwaysusedforthe statement(i).
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