Smarandache Curves and Applications According to Type-2 Bishop Frame in Euclidean 3-Space

2016·0.18 MB·

by  Suha Yilmaz

International J.Math. Combin. Vol.2(2016), 1-15
Smarandache Curves and Applications
According to Type-2 Bishop Frame in Euclidean 3-Space
Su¨ha Yılmaz
(DokuzEylu¨lUniversity,BucaEducationalFaculty,35150,Buca-Izmir,Turkey)
U¨mit Ziya Savcı
(CelalBayarUniversity,DepartmentofMathematicsEducation,45900,Manisa-Turkey)
E-mail: [email protected],[email protected]
Abstract: In this paper, we investigate Smarandache curves according to type-2 Bishop
frame in Euclidean 3- space and we give some differential geometric properties of Smaran-
dache curves. Also, some characterizations of Smarandache breadth curves in Euclidean 3-
space are presented. Besides, we illustrate examples of our results.
Key Words: Smarandache curves, Bishop frame, curvesof constant breadth.
AMS(2010): 53A05, 53B25, 53B30.
§1. Introduction
A regular curve in Euclidean 3-space, whose position vector is composed by Frenet frame
vectors on another regular curve, is called a Smarandache curve. M. Turgut and S. Yılmaz
have defined a special case of such curves and call it Smarandache TB curves in the space
2
E4 [10]. Moreover, special Smarandache curves have been investigated by some differential
1
geometric[6]. A.T.Alihas introducedsomespecialSmarandachecurvesinthe Euclideanspace
[2]. SpecialSmarandachecurvesaccordingtoSabbanframehavebeenstudiedby[5]. Besides,It
has been determined some specialSmarandache curves E3 by [12]. Curves of constant breadth
1
were introduced by L.Euler [3].
Weinvestigatepositionvectorofcurvesandsomecharacterizationscaseofconstantbreadth
according to type-2 Bishop frame in E3.
§2. Preliminaries
The Euclidean 3-space E3 proved with the standard flat metric given by
<,>=dx2+dx2+dx2
1 2 3
1ReceivedNovember26,2015,Accepted May6,2016.
2 Su¨haYılmazandU¨mitZiyaSavcı
where(x ,x ,x ) isrectangularcoordinatesystemofE3.Recallthat, the normofanarbitrary
1 2 3
vector a E3 given by a = √<a,a>. ϕ is called a unit speed curve if velocity vector υ of
∈ k k
ϕ satisfied υ =1
k k
The Bishopframe or paralleltransportframe is alternativeapproachto defining a moving
frame that is well defined even when the curve has vanishing second derivative. One can
express parallel transport of orthonormal frame along a curve simply by parallel transporting
each component of the frame [8]. The type-2 Bishop frame is expressed as
p
ξ 0 0 ε ξ
1 − 1 1
 ξp = 0 0 ε . ξ  (2.1)
2 − 2 2
 Bp   ε ε 0   B 
   1 2   
     
In order to investigate type-2 Bishop frame relation with Serret-Frenet frame, first we
p
B = τN =ε ξ +ε ξ (2.2)
1 1 2 2
−
Taking the norm of both sides, we have
dθ(s)
κ(s)= , τ(s)= ε2+ε2 (2.3)
ds 1 2
q
Moreover,we may express
ε (s)= τcosθ(s), ε (s)= τsinθ(s) (2.4)
1 2
− −
ε
2
By this way, we conclude θ(s) = Arctan . The frame ξ ,ξ ,B is properly oriented,
1 2
ε { }
1
s
and τ and θ(s)= κ(s)ds are polar coordinates for the curve α(s).
0
R
We write the tangent vector according to frame ξ ,ξ ,B as
1 2
{ }
T =sinθ(s)ξ cosθ(s)ξ
1 2
−
and differentiate with respect to s
Tp =κN = θp(s)(cosθ(s)ξ +sinθ(s)ξ )
1 2
(2.5)
p p
+sinθ(s)ξ cosθ(s)ξ
1− 2
p p
Substituting ξ = ε B and ξ = ε B in equation (2.5) we have
1 − 1 2 − 2
p
κN =θ(s)(cosθ(s)ξ +sinθ(s)ξ )
1 2
In the above equation let us take θp(s)=κ(s). So we immediately arrive at
N =cosθ(s)ξ +sinθ(s)ξ
1 2
SmarandacheCurvesandApplicationsAccordingtoType-2BishopFrameinEuclidean3-Space 3
Considering the obtained equations, the relation matrix between Serret-Frenet and the type-2
Bishop frame can be expressed
T sinθ(s) cosθ(s) 0 ξ
1
−
 N = cosθ(s) sinθ(s) 0 . ξ  (2.6)
2
 B   0 0 1   B 
     
     
§3. Smarandache Curves According to Type-2 Bishop Frame in E3
Letα=α(s)beaunitspeedregularcurveinE3 anddenoteby ξα,ξα,Bα themovingBishop
{ 1 2 }
frame along the curve α. The following Bishop formulae is given by
ξ·α = εαBα, ξ·α = εαBα, B·α =εαξα+εαξα
1 − 1 2 − 2 1 1 2 2
3.1 ξ ξ -Smarandache Curves
1 2
Definition 3.1 Letα=α(s) beaunitspeedregularcurvein E3 and ξα,ξα,Bα beitsmoving
{ 1 2 }
Bishop frame. ξ ξ -Smarandache curves can be defined by
1 2
1
β(s∗)= √2(ξ1α+ξ2α) (3.1)
Now, we can investigate Bishop invariants of ξ ξ -Smarandache curves according to α =
1 2
α(s). Differentiating (3.1.1) with respect to s, we get
dβ ds 1
β· = ∗ = − (εα+εα)Bα
ds · ds √2 1 2
∗
(3.2)
ds 1
T ∗ = − (εα+εα)Bα
β · ds √2 1 2
where
ds 1
∗ = (εα+εα) (3.3)
ds √2 1 2
The tangent vector of curve β can be written as follow;
T = Bα = (εαξα+εαξα) (3.4)
β − − 1 1 2 2
Differentiating (3.4) with respect to s, we obtain
dT ds
β ∗ =εαξα+εαξα (3.5)
ds · ds 1 1 2 2
∗
4 Su¨haYılmazandU¨mitZiyaSavcı
Substituting (3.3) in (3.5), we get
√2
Tp = (εαξα+εαξα)
β εα+εα 1 1 2 2
1 2
Then, the curvature and principal normal vector field of curve β are respectively,
√2
Tp =κ = (εα)2+(εα)2
β β εα+εα 1 2
(cid:13) (cid:13) 1 2q
(cid:13) (cid:13)
(cid:13) (cid:13) 1
N = (εαξα+εαξα)
β 1 1 2 2
(εα)2+(εα)2
1 2
q
On the other hand, we express
ξα ξα Bα
1 2
1
Bβ = det 0 0 1 .
(εα)2+(εα)2 −
1 2  εα εα 0 
q  1 2 
 
So, the binormal vector of curve β is
1
B = (εαξα εαξα)
β (εα)2+(εα)2 2 1 − 1 2
1 2
q
We differentiate (3.2) with respect to s in order to calculate the torsion of curve β
1
β·· = 1 [(εα)2+εαεα]ξα
√−2{ 1 1 2 1
+[εαεα+(εα)2]ξα+[ε·α+ε·α] Bα]
1 2 2 2 1 2 }
and similarly
1
·β··= − (δ ξα+δ ξα+δ Bα)
√2 1 1 2 2 3
where
δ = 3εαε·α+εαεα+2εαε·α-(εα)3-(εα)2εα
1 1 1 1 2 1 2 1 1 2
δ = 2ε·αεα+εαε·α+3εαε·α-εα(εα)2-(εα)3
2 1 2 1 2 2 2 1 2 2
δ = ε·α· +ε·α·
3 1 2
The torsion of curve β is
τ = εα1+εα2 [(εα+εα)(εαεα+(εα)2]δ [(εα+εα)((εα)2+εαεα)]δ
β 4√2[(εα)2+(εα)2]{ 1 2 1 2 2 1− 1 2 1 1 2 2}
1 2
3.2 ξ B-Smarandache Curves
1
Definition 3.2 Letα=α(s) beaunitspeedregularcurvein E3 and ξα,ξα,Bα beitsmoving
{ 1 2 }
SmarandacheCurvesandApplicationsAccordingtoType-2BishopFrameinEuclidean3-Space 5
Bishop frame. ξ B-Smarandache curves can be defined by
1
1
β(s )= (ξα+Bα) (3.6)
∗ √2 1
Now, we can investigate Bishop invariants of ξ B-Smarandache curves according to α =
1
α(s). Differentiating (3.6) with respect to s, we get
dβ ds 1
β· = ∗ = − (εαBα+εαξα+εαξα)
ds · ds √2 1 1 1 2 2
∗
(3.7)
ds 1
T ∗ = − ( εαBα+εαξα+εαξα)
β · ds √2 − 1 1 1 2 2
where
ds 2(εα)2+(εα)2
∗ = 1 2 (3.8)
ds s 2
The tangent vector of curve β can be written as follow;
1
T = (εαξα+εαξα εαBα) (3.9)
β 2(εα)2+(εα)2 1 1 2 2 − 1
1 2
q
Differentiating (3.9) with respect to s, we obtain
dT ds 1
β ∗= (µ ξα+µ ξα+µ Bα) (3.10)
ds ds 3 1 1 2 2 3
∗ 2(εα)2+(εα)2 2
1 2
h i
where
µ = εαεαε·α+ε·α(εα)2
1 1 2 2 1 2
µ = 2(εα)2εα-2εαε·αεα+2(εα)2εα-2(εα)3εα-εα(εα)3
2 2 2 1 1 2 1 2 1 2 1 2
µ = εαεαε·α 2(εα)4+(εα)2(εα)2 ε·α(εα)2
3 1 2 2 − 1 1 2 − 1 2
Substituting (3.8) in (3.10), we have
√2
Tp = (µ ξα+µ ξα+µ Bα)
β 2 1 1 2 2 3
2(εα)2+(εα)2
1 2
h i
Then, the first curvature and principal normal vector field of curve β are respectively
Tp =κ = √2 µ2+µ2+µ2
β β 2(εα)2+(εα)2 2 1 2 3
1 2
(cid:13)(cid:13) (cid:13)(cid:13) h i p
(cid:13) (cid:13) 1
N = (µ ξα+µ ξα+µ Bα)
β µ2+µ2+µ2 1 1 2 2 3
1 2 3
p
6 Su¨haYılmazandU¨mitZiyaSavcı
On the other hand, we get
1
B = [(µ εα+µ εα)ξα
β 2 1 3 2 1
µ2+µ2+µ2 2(εα)2+(εα)2
1 2 3 1 2
q
p
(µ ξα+µ ξα)ξα+(µ εα-µ εα)Bα]
− 1 1 3 1 2 2 1 1 2
We differentiate (3.7) with respect to s in order to calculate the torsion of curve β
β·· = 1 [-2(εα)2+ε·α]ξα
√−2{ 1 1 1
+[ εαεα+ε·α (εα)2]ξα ε·αBα
− 1 2 1 − 2 2 − 1 }
and similarly
1
·β··= − (Γ ξα+Γ ξα+Γ Bα)
√2 1 1 2 2 3
where
Γ = -6εαε·α+ε·α·+2(εα)3
1 1 1 1 1
Γ = -2ε·αεα-εαε·α+ε·α·-2εαεα+εα(εα)2-ε·αεα+(εα)3
2 1 2 1 2 2 2 2 1 2 2 2 2
Γ = -ε·α·
3 1
The torsion of curve β is
[2(εα)2+(εα)2]4
τ = - 1 2 [(-εαεα-ε·α+(εα)2)Γ
β 4√2(µ2+µ2+µ2){ 1 2 2 2 1
1 2 3
-2((εα)2 ε·α)Γ +(-εαεα-ε·α+(εα)2)Γ ]εα
1 − 1 2 1 2 2 2 3 1
-[(ε·α 2(εα)2)Γ +ε·αΓ ]εα
1 − 1 3 1 1 2}
3.3 ξ B-Smarandache Curves
2
Definition 3.3 Letα=α(s) beaunitspeedregularcurvein E3 and ξα,ξα,Bα beitsmoving
{ 1 2 }
Bishop frame. ξ B-Smarandache curves can be defined by
2
1
β(s )= (ξα+Bα) (3.11)
∗ √2 2
Now, we can investigate Bishop invariants of ξ B-Smarandache curves according to α =
2
α(s). Differentiating (3.11) with respect to s, we get
dβ ds
β· = ∗ =( εαBα+εαξα+εαξα)
ddss∗ · ds − 2 1 1 2 2 (3.12)
T ∗ =(εαξα+εαξα εαBα)
β · ds 1 1 2 2 − 2
SmarandacheCurvesandApplicationsAccordingtoType-2BishopFrameinEuclidean3-Space 7
where
ds (εα)2+2(εα)2
∗ = 1 2 (3.13)
ds s 2
The tangent vector of curve β can be written as follow;
εαξα+εαξα εαBα
T = 1 1 2 2 − 2 (3.14)
β
2(εα)2+(εα)2
1 2
q
Differentiating (3.14) with respect to s, we obtain
dT ds 1
β ∗= (η ξα+η ξα+η Bα) (3.15)
ds ds 3 1 1 2 2 3
∗ (εα)2+2(εα)2 2
1 2
h i
where
η = 2(ε·α(εα)2-εαεα)
1 1 2 1 2
η = (εα)2ε·α+(εα)2ε·α-εαε·αεα
2 2 2 1 1 1 1 2
η = (εα)2εα+ 2(εα)3-(εα)4-2(εα)4-3(εα)2(εα)2
3 1 2 2 1 1 1 2
Substituting (3.13) in (3.15), we have
√2
Tp = (η ξα+η ξα+η Bα)
β 2 1 1 2 2 3
2(εα)2+(εα)2
1 2
h i
Then, the first curvature and principal normal vector field of curve β are respectively
√2 η2+η2+η2
Tp =κ = 1 2 3
β β 2
(εαp)2+2(εα)2
(cid:13) (cid:13) 1 2
(cid:13) (cid:13)
(cid:13) (cid:13) h i
1
N = (η ξα+η ξα+η Bα)
β η2+η2+η2 1 1 2 2 3
1 2 3
p
On the other hand, we express
1
B = [(η εα+η εα)ξα
β 2 2 3 2 1
η2+η2+η2 (εα)2+2(εα)2
1 2 3 1 2
p q−(η1ξ2α+η3ξ1α)ξ2α+(η2εα1-η1εα2)Bα]
We differentiate (3.12) with respect to s in order to calculate the torsion of curve β
1
β·· = 1 [εαξα+ε·α (εα)2]ξα
√2{ 1 1 1 − 1 1
+[ε·α 2(εα)2]ξα ε·αBα
2 − 2 2 − 2 }
8 Su¨haYılmazandU¨mitZiyaSavcı
and similarly
1
·β··= (η ξα+η ξα+η Bα)
√2 1 1 2 2 3
where
η = -ε·αεα 5ε·αεα+ε·α·+(εα)2εα+(εα)3
1 1 2 − 1 1 1 1 2 1
η = -4ε·αεα+ε·α· +2εα
2 2 2 2 2
η = -ε·α·
3 2
The torsion of curve β is
[(εα)2+2(εα)2]4
τ = - 1 2 [ε·αη +(ε·α 2(εα)2)η ]εα
β 4√2(η2+η2+η2){ 2 2 2 − 2 3 1
1 2 3
+[2(εα)2η +(εαεα-ε·α+(εα)2)η
2 1 1 2 1 1 2
+(-εαεα+εα·)η ]εα
1 2 1 3 2}
3.4 ξ ξ B-Smarandache Curves
1 2
Definition 3.4 Letα=α(s) beaunitspeedregularcurvein E3 and ξα,ξα,Bα beitsmoving
{ 1 2 }
Bishop frame. ξαξ B-Smarandache curves can be defined by
1 2
1
β(s )= (ξα+ξα+Bα) (3.16)
∗ √3 1 2
Now, we can investigate Bishop invariants of ξαξ B-Smarandache curves according to
1 2
α=α(s). Differentiating (3.16) with respect to s, we get
dβ ds
β· = ∗ = 1 [(εα+εα)Bα εαξα εαξα]
ds · ds √3 1 2 − 1 1 − 2 2
∗
(3.17)
ds
T ∗ = 1 [(εα+εα)Bα εαξα εαξα)]
β · ds √3 1 2 − 1 1 − 2 2
where
ds 2[(εα)2+εαεα+(εα)2]
∗ = 1 1 2 2 (3.18)
ds s 3
The tangent vector of curve β can be written as follow;
εαξα+εαξα (εα+εα)Bα
T = 1 1 2 2 − 1 2 (3.19)
β
2[(εα)2+εαεα+(εα)2]
1 1 2 2
q
SmarandacheCurvesandApplicationsAccordingtoType-2BishopFrameinEuclidean3-Space 9
Differentiating (3.19) with respect to s, we get
dT ds (λ ξα+λ ξα+λ Bα)
β ∗= 1 1 2 2 3 (3.20)
ds ds 3
∗ 2√2 (εα)2+εαεα+(εα)2 2
1 1 2 2
h i
where
λ = [ε·α-2(εα)2-εαεα]u(s)-εα[2εαε·α+ε·αεα+εαεα+2ε·αεα]
1 1 1 1 2 1 1 1 1 2 1 2 2 2
λ = [ε·α-2(εα)2-εαεα]u(s)-εα[ε·α+εαε·α+2εαε·α]
2 2 2 1 2 2 1 1 2 2 2
λ = [-ε·α-ε·α]u(s)+εα[2εαε·α+3ε·αεα+εαε·α+2ε·αεα]
3 1 2 1 1 1 1 2 1 2 2 2
+ε·α[ε·α(εα)2+2(εα)2]
2 1 2 2
Substituting (3.18) in (3.20), we have
√3(λ ξα+λ ξα+λ Bα)
Tp = 1 1 2 2 3
β 2
4 (εα)2+εαεα+(εα)2
1 1 2 2
h i
Then, the first curvature and principal normal vector field of curve β are respectively
√3 λ2+λ2+λ2
Tp =κ = 1 2 3
β β 2
4 (εα)p2+εαεα+(εα)2
(cid:13) (cid:13) 1 1 2 2
(cid:13) (cid:13) (3.21)
(cid:13) (cid:13) h i
1
N = (λ ξα+λ ξα+λ Bα)
β λ2+λ2+λ2 1 1 2 2 3
1 2 3
p
On the other hand, we express
ξα ξα Bα
1 2
1
Bβ= det εα1 εα2 -(εα1+εα2) 
2[(εα)2+εαεα+(εα)2] λ2+λ2+λ2
q 1 1 2 2 · 1 2 3  λ1 λ2 λ3 
p  
So, the binormal vector field of curve β is
1
B = [(εα+εα)λ
β 2[(εα)2+εαεα+(εα)2] λ2+λ2+λ2{ 1 2 1
1 1 2 2 · 1 2 3
q
p
εαλ ]ξα+[ εαλ -(εα+εα)]ξα+[εαλ -εαλ ]Bα
− 2 3 1 − 1 3 1 2 2 1 2 2 1 }
We differentiate (3.20) with respect to s in order to calculate the torsion of curve β
β·· = - 1 [2(εα)2+εαξα-ε·α]ξα
√3{ 1 1 1 1 1
+[2(εα)2+εαεα-ε·α]ξα+[εα·+ε·α]Bα
2 1 2 2 2 1 2 }
10 Su¨haYılmazandU¨mitZiyaSavcı
and similarly
1
·β··=- (σ ξα+σ ξα+σ Bα)
√3 1 1 2 2 3
where
η = 4ε·αεα+3ε·αεα-ε·α·-2(εα)3-(εα)2εα
1 1 1 1 2 1 1 1 2
η = 5ε·αεα+ε·αεα+εαεα-ε·α·-2(εα)3-εα(εα)2
2 2 2 1 2 1 2 2 2 1 2
η = ε·α·+·ε·α
3 2 2
The torsion of curve β is
τ = -16[(εα1)2+εα1εα2+(εα2)2]2 [(2(εα)2+εαεα-ε·α)σ +(-ε·α-2(εα)2-εαεα)σ
β 9√3√λ2+λ2+λ2 { 2 1 2 2 1 2 1 1 2 2
1 2 3
+(2(εα)2+εαεα-ε·α)σ ]εα+[-ε·α-2ε·α+2(εα)2+εαεα)σ
2 1 2 2 3 1 1 2 2 1 2 1
+(-2(εα)2-εαεα+ε·α)σ +(2(εα)2+εαεα-ε·α)σ ]εα .
1 1 2 1 2 1 1 2 1 3 2}
§4. Smarandache Breadth Curves According to Type-2 Bishop Frame in E3
Aregularcurvewithmorethan2breadthsinEuclidean3-spaceiscalledSmarandachebreadth
curve.
Let α=α(s) be a Smarandache breadth curve. Moreover,let us suppose α=α(s) simple
closedspace-likecurveinthespaceE3. Thesecurveswillbedenotedby(C). Thenormalplane
at every point P on the curve meets the curve at a single point Q other than P.
We call the point Q the opposite point P. We consider a curve in the class Γ as in having
parallel tangents ξ and ξ opposite directions at opposite points α and α of the curves.
1 1∗ ∗
A simple closed curve having parallel tangents in opposite directions at opposite points
can be represented with respect to type-2 Bishop frame by the equation
α∗(s) = α(s)+λξ1+ϕξ2+ηB (4.1)
where λ(s),ϕ(s) and η(s) are arbitrary functions also α and α are opposite points.
∗
Differentiating both sides of (4.1) and considering type-2 Bishop equations, we have
dα ds dλ dϕ
∗ ∗
=ξ = ( +ηε +1)ξ +( +ηε )ξ
ds 1∗ ds ds 1 1 ds 2 2 (4.2)
dη
+( λε ϕε + )B
1 2
− − ds