Smarandache Curves in Terms of Sabban Frame of Spherical Indicatrix Curves

2015·0.29 MB·

by  Suleyman Senyurt

Gen. Math. Notes, Vol. 31, No. 2, December 2015, pp.1-15
ISSN 2219-7184; Copyright (cid:13)c ICSRS Publication, 2015
www.i-csrs.org
Available free online at http://www.geman.in
Smarandache Curves in Terms of Sabban Frame
of Spherical Indicatrix Curves
Su¨leyman S¸enyurt1 and Abdussamet C¸alı¸skan2
1,2Faculty of Arts and Sciences, Department of Mathematics
Ordu University, 52100, Ordu, Turkey
1E-mail: [email protected]
2E-mail: [email protected]
(Received: 4-11-14 / Accepted: 20-4-15)
Abstract
In this paper, we investigate special Smarandache curves in terms of Sab-
ban frame of spherical indicatrix curves and we give some characterization of
Smarandache curves. Besides, we illustrate examples of our results.
Keywords: Smarandache Curves, Sabban Frame, Geodesic Curvature,
Spherical Indicatrix Curves.
1 Introduction
A regular curve in Minkowski space-time, whose position vector is composed
by Frenet frame vectors on another regular curve, is called a Smarandache
curve [5]. Special Smarandache curves have been studied by some authors.
AhmadT.AlistudiedsomespecialSmarandachecurvesintheEuclideanspace.
¨
He studied Frenet-Serret invariants of a special case, [2]. Ozcan Bekta¸s and
Salim Yu¨ce studied some special smarandache curves according to Darboux
Frame in E3, [4]. Muhammed C¸etin, Yılmaz Tuncer and Kemal Karacan in-
vestigated special smarandache curves according to Bishop frame in Euclidean
3-Space and they gave some differential geometric properties of Smarandache
curves, [3]. MelihTurgutandSu¨haYılmazstudiedaspecialcaseofsuchcurves
and called it smarandache TB curves in the space E4, [5]. Nurten Bayrak,
2 1
¨
Ozcan Bekta¸s and Salim Yu¨ce studied some special smarandache curves in
E3, [6]. Kemal Tas.ko¨pru¨ , Murat Tosun studied special Smarandache curves
1
2 Su¨leyman S¸enyurt et al.
according to Sabban frame on S2, [7].
In this paper, we study special Smarandache curves such as TT ,
T
T (T∧T ), TT (T∧T ), NT , T (N∧T ), NT (N∧T ), BT , T (B∧T )
T T T T N N N N N B B B
and BT (B∧T ) created by Sabban frame, {T,T ,T ∧T }, {N,T ,N ∧T }
B B T T N N
and {B,T ,B ∧ T }, that belongs to spherical indicatrix of a α curve are
B B
defined. Besides we have found some results.
2 Problem Formulations
The Euclidean 3-space E3 be inner product given by
(cid:104),(cid:105) = x2 +x3 +x2
1 2 3
where (x ,x ,x ) ∈ E3. Let α : I → E3 be a unit speed curve denote by
1 2 3
{T,N,B} the moving Frenet frame . For an arbitrary curve α ∈ E3, with first
and second curvature, κ and τ respectively, the Frenet formulae is given by [1]

T(cid:48) = κN


N(cid:48) = −κT +τB (1)

B(cid:48) = −τN.
Accordingly, the spherical indicatrix curves of Frenet vectors are (T), (N) and
(B) respectively. These equations of curves are given by [10]

α (s) = T(s)
 T
α (s) = N(s) (2)
N

α (s) = B(s)
B
For any unit speed curve α : I → E3, the vector W is called Darboux vector
defined by
W = τ(s)T(s)+κ(s)B(s).
If we consider the normalization of the Darboux c = W we have
(cid:107)W(cid:107)
κ(s) τ(s)
cosϕ = ,sinϕ =
(cid:107)W(cid:107) (cid:107)W(cid:107)
and
c = sinϕT(s)+cosϕB(s)
where ∠(W,B) = ϕ.
Smarandache Curves in Terms of Sabban Frame... 3
Let γ : I → S2 be a unit speed spherical curve. We denote s as the arc-length
parameter of γ. Let us denote by

γ(s) = γ(s)


t(s) = γ(cid:48)(s) (3)

d(s) = γ(s)∧t(s).
We call t(s) a unit tangent vector of γ. {γ,t,d} frame is called the Sabban
frame of γ on S2 . Then we have the following spherical Frenet formulae of γ :

γ(cid:48) = t


t(cid:48) = −γ +κ d (4)
g

d(cid:48) = −κ t
g
where is called the geodesic curvature of κ on S2 and
g
κ = (cid:104)t(cid:48),d(cid:105) [8] (5)
g
3 Smarandache Curves in Terms of Sabban
Frame of Spherical Indicatrix Curves
In this section, we investigate Smarandache curves according to the Sabban
frame of Spherical Indicatrix Curves.
Let α (s) = T(s) be a unit speed regular spherical curves on S2. We denote
T
s as the arc-lenght parameter of tangents indicatrix (T)
T
α (s) = T(s) (6)
T
Differentiating (6), we have
dα ds
T T = T(cid:48)(s)
ds ds
T
and
ds
T
T = κN (7)
T
ds
From the equation (7)
T = N
T
and
T ∧T = B
T
4 Su¨leyman S¸enyurt et al.
From the equation (3)

T(s) = T(s)


T (s) = N(s)
T

T ∧T (s) = B(s)
T
is called the Sabban frame of tangents indicatrix (T). From the equation (5)
τ
κ = (cid:104)T(cid:48),T ∧T (cid:105) =⇒ κ =
g T T g κ
Then from the equation (4) we have the following spherical Frenet formulae of
(T):

T(cid:48) = T
 T
T(cid:48) = −T + τT ∧T (8)
T κ T

(T ∧T )(cid:48) = −τT
T κ T
Let α (s) = N(s) be a unit speed regular spherical curves on S2.We denote
N
s as the arc-lenght parameter of principal normals indicatrix (N)
N
α (s) = N(s) (9)
N
Differentiating (9), we have
T = −cosϕT +sinϕB
N
and
N ∧T = sinϕT +cosϕB.
N
From the equation (3)

N(s) = N(s)


T (s) = −cosϕT(s)+sinϕB(s)
N

N ∧T (s) = sinϕT(s)+cosϕB(s)
N
is called the Sabban frame of principal normals indicatrix (N). From the equa-
tion (5)
ϕ(cid:48)
κ =
g
(cid:107)W(cid:107)
Then from the equation (4) we have the following spherical Frenet formulae of
(N):

N(cid:48) = T
 N

T(cid:48) = −N + ϕ(cid:48) (N ∧T ) (10)
N (cid:107)W(cid:107) N
(N ∧T )(cid:48) = − ϕ(cid:48) T
N (cid:107)W(cid:107) N
Smarandache Curves in Terms of Sabban Frame... 5
Let α (s) = B(s) be a unit speed regular spherical curves on S2.We denote
B
s as the arc-lenght parameter of indicatrix (B)
B
α (s) = B(s) (11)
B
Differentiating (11), we have
T = −N
B
and
B ∧T = T
B
From the equation (3)

B(s) = B(s)


T (s) = −N(s)
B

(B ∧T )(s) = T(s)
B
is called the Sabban frame of binormals indicatrix (B). From the equation (5)
κ
κ =
g
τ
Then from the equation (4) we have the following spherical Frenet formulae of
(B):

B(cid:48) = T
 B
T (cid:48) = −B + κ(B ∧T ) (12)
B τ B

(B ∧T )(cid:48) = −κT
B τ B
i-) TT -Smarandache Curves
T
Let S2 be a unit sphere in E3 and suppose that the unit speed regular curve
α (s) = T(s) lying fully on S2. In this case, TT - Smarandache curve can be
T T
defined by
1
β (s∗) = √ (T +T ). (13)
1 T
2
Now we can compute Sabban invariants of TT - Smarandache curves. Differ-
T
entiating (13), we have
ds∗ 1 τ
T = √ (−T +N + B),
β1 ds 2 κ
where
(cid:114)
ds∗ 2+(τ)2
= κ . (14)
ds 2
Thus, the tangent vector of curve β is to be
1
1 τ
T = (−T +N + B). (15)
β1 (cid:112)2+(τ)2 κ
κ
6 Su¨leyman S¸enyurt et al.
Differentiating (15), we get
ds∗ 1
T(cid:48) = (λ T +λ N +λ B) (16)
β1 ds (cid:0)2+(τ)2(cid:1)32 1 2 3
κ
where
τ τ τ
(cid:0) (cid:1)(cid:48) (cid:0) (cid:1)2
λ = − −2
1
κ κ κ
τ τ τ τ
(cid:0) (cid:1)(cid:48) (cid:0) (cid:1)4 (cid:0) (cid:1)2
λ = − − −3 −2
2
κ κ κ κ
τ τ τ
(cid:0) (cid:1) (cid:0) (cid:1)3
λ = 2 + +2 .
3
κ κ κ
Substituting the equation (15) into equation (16), we reach
√
2
T(cid:48) = (λ T +λ N +λ B). (17)
β1 (2+(τ)2)2 1 2 3
κ
Considering the equations (13) and (15), it easily seen that
1 τ τ
(T ∧T ) = ( T − N +2B). (18)
T β1 (cid:112)4+2(τ)2 κ κ
κ
From the equation (17) and (18), the geodesic curvature of β (s∗) is
1
κ β1 = 1 (λ τ −λ τ +2λ ).
g (2+(τ)2)25 1κ 2κ 3
κ
ii-) T (T ∧T )-Smarandache Curves
T T
Similarly, T (T ∧T ) - Smarandache curve can be defined by
T T
1
β (s∗) = √ (T +T ∧T ). (19)
2 T T
2
In that case, the tangent vector of curve β is as follows
2
1 τ τ
T = (−T − N + B). (20)
β2 (cid:112)1+2(τ)2 κ κ
κ
Differentiating (20), it is obtained that
√
2
T(cid:48) = (λ T +λ N +λ B) (21)
β2 (1+2(τ)2)2 1 2 3
κ
where
Smarandache Curves in Terms of Sabban Frame... 7
τ τ τ τ
(cid:0) (cid:1)3 (cid:0) (cid:1)(cid:0) (cid:1)(cid:48)
λ = +2 +2
1
κ κ κ κ
τ τ τ
(cid:0) (cid:1)4 (cid:0) (cid:1)2
λ = −2 −3 − −1
2
κ κ κ
τ τ τ
(cid:0) (cid:1)(cid:48) (cid:0) (cid:1)4 (cid:0) (cid:1)2
λ = −2 − .
3
κ κ κ
Using the equations (19) and (20), we easily find
1 τ
(T ∧T ) = (2 T −N +B). (22)
T β2 (cid:112)2+4(τ)2 κ
κ
So, the geodesic curvature of β (s∗) is as follows
2
1 τ
κ β2 = (2λ −λ +λ ).
g (1+2(τ)2)52 1κ 2 3
κ
iii-) TT (T ∧T )-Smarandache Curves
T T
TT T ∧T - Smarandache curve can be defined by
T T
1
β (s∗) = √ (T +T +T ∧T ). (23)
3 T T
3
Differentiating (23), we have the tangent vector of curve β is
3
1 τ τ
T = (−T +(1− )N + B). (24)
β3 (cid:112)2(1− τ +(τ)2) κ κ
κ κ
Differentiating (24), it is obtained that
√
3
T(cid:48) = (λ T +λ N +λ B). (25)
β3 4(1− τ +(τ)2)2 1 2 3
κ κ
where
τ τ τ τ τ
(cid:0) (cid:1)(cid:48)(cid:0) (cid:1) (cid:0) (cid:1)3 (cid:0) (cid:1)2
λ = 2 −1 +2 −4 +4 −2
1
κ κ κ κ κ
τ τ τ τ τ τ
(cid:0) (cid:1)(cid:48)(cid:0) (cid:1) (cid:0) (cid:1)4 (cid:0) (cid:1)3 (cid:0) (cid:1)2 (cid:0) (cid:1)
λ = − +1 −2 +2 −4 +2 −2
2
κ κ κ κ κ κ
τ τ τ τ τ τ
(cid:0) (cid:1)(cid:48)(cid:0) (cid:1) (cid:0) (cid:1)4 (cid:0) (cid:1)3 (cid:0) (cid:1)2 (cid:0) (cid:1)
λ = 2− −2 +4 −4 +2 .
3
κ κ κ κ κ κ
Using the equations (23) and (24), we have
8 Su¨leyman S¸enyurt et al.
(2τ −1)T +(−1− τ)N +(2− τ)B
(T ∧T ) = κ √ κ κ · (26)
T β3 6(cid:112)1− τ +(τ)2
κ κ
So, the geodesic curvature of β (s∗) is
3
λ (2τ −1)+λ (−1− τ)+λ (2− τ)
κ β3 = 1 κ √ 2 κ 3 κ ·
g 4 2(1− τ +(τ)2)52
κ κ
iv-) NT -Smarandache Curves
N
NT - Smarandache curve can be defined by
N
1
ς (s∗) = √ (N +T ). (27)
1 N
2
Differentiating (27), we have the tangent vector of curve ς is
3
(−cosϕ+ ϕ(cid:48) sinϕ)T −N +(sinϕ+ ϕ(cid:48) cosϕ)B
(cid:107)W(cid:107) (cid:107)W(cid:107)
T = · (28)
ς1 (cid:113)2+( ϕ(cid:48) )2
(cid:107)W(cid:107)
Differentiating (28), we get
1
T(cid:48) = ((λ sinϕ−λ cosϕ)T +λ N +(λ sinϕ+λ cosϕ)B).
ς1 (2+( ϕ(cid:48) )2)2 3 2 1 2 3
(cid:107)W(cid:107)
(29)
where
ϕ(cid:48) ϕ(cid:48) ϕ(cid:48)
(cid:0) (cid:1)(cid:0) (cid:1)(cid:48) (cid:0) (cid:1)2
λ = − −2
1
(cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107)
ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48)
(cid:0) (cid:1)(cid:0) (cid:1)(cid:48) (cid:0) (cid:1)4 (cid:0) (cid:1)2
λ = − − −3 −2
2
(cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107)
ϕ(cid:48) ϕ(cid:48) ϕ(cid:48)
(cid:0) (cid:1)(cid:48) (cid:0) (cid:1)3 (cid:0) (cid:1)
λ = 2 + +2 .
3
(cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107)
Considering the equations (27) and (28), it easily seen that
(cid:0)2sinϕ+ ϕ(cid:48) cosϕ)T + ϕ(cid:48) N +(2cosϕ− ϕ(cid:48) sinϕ)B
(cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107)
(N ∧T ) = · (30)
N ς1 (cid:113)4+2( ϕ(cid:48) )2
(cid:107)W(cid:107)
The geodesic curvature of ς (s∗) is
1
Smarandache Curves in Terms of Sabban Frame... 9
(cid:0)λ ϕ(cid:48) −λ ϕ(cid:48) +2λ (cid:1)
κ ς1 = 1(cid:107)W(cid:107) 2(cid:107)W(cid:107) 3 ·
g 5
(2+( ϕ(cid:48) )2)2
(cid:107)W(cid:107)
v-) T (N ∧T )-Smarandache Curves
N N
T (N ∧T ) - Smarandache curve can be defined by
N N
1
ς (s∗) = √ (T +N ∧T ). (31)
2 N N
2
Differentiating (31), the tangent vector of curve ς is
2
( ϕ(cid:48) (sinϕ+cosϕ))T −N +( ϕ(cid:48) (cosϕ−sinϕ)B)
(cid:107)W(cid:107) (cid:107)W(cid:107)
T = · (32)
ς2 (cid:113)1+2( ϕ(cid:48) )2
(cid:107)W(cid:107)
Differentiating (32), it is obtained that
√
2
(cid:2) (cid:3)
T(cid:48) = (λ sinϕ−λ cosϕ)T +λ N +(λ sinϕ+λ cosϕ)B
ς2 (1+2( ϕ(cid:48) )2)2 3 2 1 2 3
(cid:107)W(cid:107)
(33)
where
ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48)
(cid:0) (cid:1) (cid:0) (cid:1)3 (cid:0) (cid:1)(cid:0) (cid:1)(cid:48)
λ = +2 +2
1
(cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107)
ϕ(cid:48) ϕ(cid:48) ϕ(cid:48)
(cid:0) (cid:1)4 (cid:0) (cid:1)4 (cid:0) (cid:1)
λ = −2 −3 − −1
2
(cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107)
ϕ(cid:48) ϕ(cid:48) ϕ(cid:48)
(cid:0) (cid:1)(cid:48) (cid:0) (cid:1)4 (cid:0) (cid:1)2
λ = −2 − .
3
(cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107)
Using the equations (31) and (32), we easily find
(sinϕ+cosϕ)T +2 ϕ(cid:48) N +(cosϕ−sinϕ)B
(cid:107)W(cid:107)
(N ∧T ) = · (34)
N ς2 (cid:113)2+4( ϕ(cid:48) )2
(cid:107)W(cid:107)
So, the geodesic curvature of ς (s∗) is as follows
2
(cid:0) ϕ(cid:48) λ −λ +λ (cid:1)
κ ς2 = (cid:107)W(cid:107) 1 2 3 ·
g 5
(1+2( ϕ(cid:48) )2)2
(cid:107)W(cid:107)
10 Su¨leyman S¸enyurt et al.
vi-) NT (N ∧T )-Smarandache Curves
N N
NT N ∧T - Smarandache curve can be defined by
N N
1
ς (s∗) = √ (N +T +N ∧T ). (35)
3 N N
3
Differentiating (35), the tangent vector of curve ς is
2
(cid:0)−cosϕ+ ϕ(cid:48) (cosϕ+sinϕ)(cid:1)T −N +(cid:0)sinϕ+ ϕ(cid:48) (cosϕ−sinϕ)(cid:1)B
(cid:107)W(cid:107) (cid:107)W(cid:107)
T = ·
ς3 (cid:114) (cid:16) (cid:17)
2 1− ϕ(cid:48) +(cid:0) ϕ(cid:48) (cid:1)2
(cid:107)W(cid:107) (cid:107)W(cid:107)
(36)
Differentiating (36), it is obtained that
√
3
(cid:2) (cid:3)
T(cid:48) = (−λ cosϕ+λ sinϕ)T +λ N +(λ cosϕ+λ sinϕ)B .
ς3 4(cid:16)1− ϕ(cid:48) +( ϕ(cid:48) )2(cid:17)2 2 3 1 3 2
(cid:107)W(cid:107) (cid:107)W(cid:107)
(37)
where
ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48)
(cid:0) (cid:1)(cid:48)(cid:0) (cid:1) (cid:0) (cid:1)3 (cid:0) (cid:1)2 (cid:0) (cid:1)
λ = 2 −1 +2 −4 +4 −2
1
(cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107)
ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48)
(cid:0) (cid:1)(cid:48)(cid:0) (cid:1) (cid:0) (cid:1)4 (cid:0) (cid:1)3 (cid:0) (cid:1)2 (cid:0) (cid:1)
λ = − +1 −2 +2 −4 +2 −2
2
(cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107)
ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48)
(cid:0) (cid:1)(cid:48)(cid:0) (cid:1) (cid:0) (cid:1)4 (cid:0) (cid:1)3 (cid:0) (cid:1)2 (cid:0) (cid:1)
λ = 2− −2 +4 −4 +2 .
3
(cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107)
Using the equations (35) and (36), we have
1 (cid:16)
(N ∧TN)ς3 = √6(cid:113)1− ϕ(cid:48) +( ϕ(cid:48) )2 (2sinϕ+cosϕ (38)
(cid:107)W(cid:107) (cid:107)W(cid:107)
ϕ(cid:48) ϕ(cid:48)
+ (cosϕ−sinϕ))T +(−1+2 )N
(cid:107)W(cid:107) (cid:107)W(cid:107)
ϕ(cid:48) (cid:17)
+(2cosϕ−sinϕ− (cosϕ−sinϕ))B
(cid:107)W(cid:107)
The geodesic curvature of ς (s∗) is
3