Gen. Math. Notes, Vol. 31, No. 2, December 2015, pp.1-15
c
ISSN 2219-7184; Copyright 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
Süleyman Şenyurt1 and Abdussamet Çalışkan2
1,2
Faculty of Arts and Sciences, Department of Mathematics
Ordu University, 52100, Ordu, Turkey
1
E-mail: senyurtsuleyman@hotmail.com
2
E-mail: abdussamet65@gmail.com
(Received: 4-11-14 / Accepted: 20-4-15)
Abstract
In this paper, we investigate special Smarandache curves in terms of Sabban 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.
Ahmad T.Ali studied some special Smarandache curves in the Euclidean space.
He studied Frenet-Serret invariants of a special case, [2]. Özcan Bektaş and
Salim Yüce studied some special smarandache curves according to Darboux
Frame in E 3 , [4]. Muhammed Çetin, Yılmaz Tuncer and Kemal Karacan investigated special smarandache curves according to Bishop frame in Euclidean
3-Space and they gave some differential geometric properties of Smarandache
curves, [3]. Melih Turgut and Süha Yılmaz studied a special case of such curves
and called it smarandache T B2 curves in the space E14 , [5]. Nurten Bayrak,
Özcan Bektaş and Salim Yüce studied some special smarandache curves in
E13 , [6]. Kemal Tas.köprü , Murat Tosun studied special Smarandache curves
2
Süleyman Şenyurt et al.
according to Sabban frame on S 2 , [7].
In this paper, we study special Smarandache curves such as T TT ,
TT (T ∧ TT ), T TT (T ∧ TT ), N TN , TN (N ∧ TN ), N TN (N ∧ TN ), BTB , TB (B ∧ TB )
and BTB (B ∧ TB ) created by Sabban frame, {T, TT , T ∧ TT }, {N, TN , N ∧ TN }
and {B, TB , B ∧ TB }, that belongs to spherical indicatrix of a α curve are
defined. Besides we have found some results.
2
Problem Formulations
The Euclidean 3-space E 3 be inner product given by
h, i = x21 + x32 + x23
where (x1 , x2 , x3 ) ∈ E 3 . Let α : I → E 3 be a unit speed curve denote by
{T, N, B} the moving Frenet frame . For an arbitrary curve α ∈ E 3 , with first
and second curvature, κ and τ respectively, the Frenet formulae is given by [1]

0

T = κN
N 0 = −κT + τ B

 0
B = −τ N.
(1)
Accordingly, the spherical indicatrix curves of Frenet vectors are (T ), (N ) and
(B) respectively. These equations of curves are given by [10]


αT (s) = T (s)
αN (s) = N (s)


αB (s) = B(s)
(2)
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 =
cos ϕ =
τ (s)
κ(s)
, sin ϕ =
kW k
kW k
and
c = sin ϕT (s) + cos ϕB(s)
where ∠(W, B) = ϕ.
W
kW k
we have
Smarandache Curves in Terms of Sabban Frame...
3
Let γ : I → S 2 be a unit speed spherical curve. We denote s as the arc-length
parameter of γ. Let us denote by


γ(s) = γ(s)
(3)
t(s) = γ 0 (s)


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 S 2 . Then we have the following spherical Frenet formulae of γ :

0

γ = t
(4)
t0 = −γ + κg d

 0
d = −κg t
where is called the geodesic curvature of κg on S 2 and
κg = ht0 , di [8]
3
(5)
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 αT (s) = T (s) be a unit speed regular spherical curves on S 2 . We denote
sT as the arc-lenght parameter of tangents indicatrix (T )
αT (s) = T (s)
(6)
Differentiating (6), we have
dαT dsT
= T 0 (s)
dsT ds
and
TT
dsT
= κN
ds
From the equation (7)
TT = N
and
T ∧ TT = B
(7)
4
Süleyman Şenyurt et al.
From the equation (3)


T (s) = T (s)
TT (s) = N (s)


T ∧ TT (s) = B(s)
is called the Sabban frame of tangents indicatrix (T). From the equation (5)
τ
κ
Then from the equation (4) we have the following spherical Frenet formulae of
(T ):

0

T = TT
(8)
TT0 = −T + κτ T ∧ TT


(T ∧ TT )0 = − κτ TT
κg = hTT0 , T ∧ TT i =⇒ κg =
Let αN (s) = N (s) be a unit speed regular spherical curves on S 2 .We denote
sN as the arc-lenght parameter of principal normals indicatrix (N )
αN (s) = N (s)
(9)
Differentiating (9), we have
TN = − cos ϕT + sin ϕB
and
N ∧ TN = sin ϕT + cos ϕB.
From the equation (3)


N (s) = N (s)
TN (s) = − cos ϕT (s) + sin ϕB(s)


N ∧ TN (s) = sin ϕT (s) + cos ϕB(s)
is called the Sabban frame of principal normals indicatrix (N). From the equation (5)
ϕ0
κg =
kW k
Then from the equation (4) we have the following spherical Frenet formulae of
(N ):

0

N = TN
ϕ0
TN0 = −N + kW
(N ∧ TN )
(10)
k


ϕ0
0
(N ∧ TN ) = − kW k TN
Smarandache Curves in Terms of Sabban Frame...
5
Let αB (s) = B(s) be a unit speed regular spherical curves on S 2 .We denote
sB as the arc-lenght parameter of indicatrix (B)
αB (s) = B(s)
(11)
Differentiating (11), we have
TB = −N
and
B ∧ TB = T
From the equation (3)


B(s) = B(s)
TB (s) = −N (s)


(B ∧ TB )(s) = T (s)
is called the Sabban frame of binormals indicatrix (B). From the equation (5)
κ
τ
Then from the equation (4) we have the following spherical Frenet formulae of
(B):

0

B = TB
(12)
TB 0 = −B + κτ (B ∧ TB )


(B ∧ TB )0 = − κτ TB
κg =
i-) T TT -Smarandache Curves
Let S 2 be a unit sphere in E 3 and suppose that the unit speed regular curve
αT (s) = T (s) lying fully on S 2 . In this case, T TT - Smarandache curve can be
defined by
1
β1 (s∗ ) = √ (T + TT ).
(13)
2
Now we can compute Sabban invariants of T TT - Smarandache curves. Differentiating (13), we have
ds∗
τ
1
= √ (−T + N + B),
ds
κ
2
where
r
2 + ( κτ )2
ds∗
=
.
ds
2
Thus, the tangent vector of curve β1 is to be
Tβ1
1
τ
Tβ1 = p
(−T + N + B).
τ 2
κ
2 + (κ)
(14)
(15)
6
Süleyman Şenyurt et al.
Differentiating (15), we get
Tβ0 1
ds∗
1
=
3 (λ1 T + λ2 N + λ3 B)
ds
2 + ( κτ )2 2
where
λ1 =
(16)
τ τ 0
τ 2
−
−2
κ κ
κ
λ2 = −
τ τ 0
τ 4
τ 2
−
−3
−2
κ κ
κ
κ
τ
τ 3
τ
+
+2 .
κ
κ
κ
Substituting the equation (15) into equation (16), we reach
√
2
Tβ0 1 =
(λ1 T + λ2 N + λ3 B).
(2 + ( κτ )2 )2
λ3 = 2
(17)
Considering the equations (13) and (15), it easily seen that
τ
τ
1
(T ∧ TT )β1 = p
( T − N + 2B).
τ 2 κ
κ
4 + 2( κ )
(18)
From the equation (17) and (18), the geodesic curvature of β1 (s∗ ) is
κg β1 =
1
5
τ 2 2
(2+( κ
) )
(λ1 κτ − λ2 κτ + 2λ3 ).
ii-) TT (T ∧ TT )-Smarandache Curves
Similarly, TT (T ∧ TT ) - Smarandache curve can be defined by
1
β2 (s∗ ) = √ (TT + T ∧ TT ).
2
(19)
In that case, the tangent vector of curve β2 is as follows
Tβ2 = p
1
τ
τ
(−T − N + B).
τ 2
κ
κ
1 + 2( κ )
Differentiating (20), it is obtained that
√
2
0
Tβ2 =
(λ1 T + λ2 N + λ3 B)
(1 + 2( κτ )2 )2
where
(20)
(21)
Smarandache Curves in Terms of Sabban Frame...
λ1 =
7
τ
τ 3
τ τ 0
+2
+2
κ
κ
κ κ
λ2 = −2
τ 2 τ
τ 4
−3
− −1
κ
κ
κ
τ 0
τ 4
τ 2
−2
−
.
κ
κ
κ
Using the equations (19) and (20), we easily find
λ3 =
1
τ
(T ∧ TT )β2 = p
(2 T − N + B).
τ 2
2 + 4( κ ) κ
(22)
So, the geodesic curvature of β2 (s∗ ) is as follows
κg β2 =
1
(1 + 2( κτ )2 )
5
2
(2λ1
τ
− λ2 + λ3 ).
κ
iii-) T TT (T ∧ TT )-Smarandache Curves
T TT T ∧ TT - Smarandache curve can be defined by
1
β3 (s∗ ) = √ (T + TT + T ∧ TT ).
3
(23)
Differentiating (23), we have the tangent vector of curve β3 is
Tβ3 = p
2(1 −
1
τ
κ
+
( κτ )2 )
τ
τ
(−T + (1 − )N + B).
κ
κ
Differentiating (24), it is obtained that
√
3
0
(λ1 T + λ2 N + λ3 B).
Tβ3 =
τ
4(1 − κ + ( κτ )2 )2
where
λ1 =
τ 0 τ
τ 3
τ 2
τ
2 −1 +2
−4
+4 −2
κ
κ
κ
κ
κ
λ2 = −
λ3 =
τ 0 τ
τ 4
τ 3
τ 2
τ
+1 −2
+2
−4
+2
−2
κ κ
κ
κ
κ
κ
τ 0
τ
τ 4
τ 3
τ 2
τ
2−
−2
+4
−4
+2
.
κ
κ
κ
κ
κ
κ
Using the equations (23) and (24), we have
(24)
(25)
8
Süleyman Şenyurt et al.
(T ∧ TT )β3
(2 κτ − 1)T + (−1 − κτ )N + (2 − κτ )B
√ p
=
·
6 1 − κτ + ( κτ )2
(26)
So, the geodesic curvature of β3 (s∗ ) is
κg β3 =
λ1 (2 κτ − 1) + λ2 (−1 − κτ ) + λ3 (2 − κτ )
·
√
5
4 2(1 − κτ + ( κτ )2 ) 2
iv-) N TN -Smarandache Curves
N TN - Smarandache curve can be defined by
1
ς1 (s∗ ) = √ (N + TN ).
2
(27)
Differentiating (27), we have the tangent vector of curve ς3 is
(− cos ϕ +
Tς1 =
ϕ0
kW k
sin ϕ)T − N + (sin ϕ +
q
ϕ0 2
2 + ( kW
)
k
ϕ0
kW k
cos ϕ)B
·
(28)
Differentiating (28), we get
Tς01 =
1
0
ϕ
(2 + ( kW
)2 )
k
2 ((λ3
sin ϕ − λ2 cos ϕ)T + λ1 N + (λ2 sin ϕ + λ3 cos ϕ)B).
(29)
where
λ1 =
ϕ0 ϕ0 0
ϕ0 2
−
−2
kW k kW k
kW k
λ2 = −
ϕ0 ϕ0 0
ϕ0 4
ϕ0 2
−
−3
−2
kW k kW k
kW k
kW k
λ3 = 2
ϕ0 0
ϕ0 3
ϕ0 +
+2
.
kW k
kW k
kW k
Considering the equations (27) and (28), it easily seen that
(N ∧ TN )ς1 =
2 sin ϕ +
ϕ0
kW k
0
ϕ
cos ϕ)T + kW
N + (2 cos ϕ −
k
q
ϕ0 2
4 + 2( kW
)
k
The geodesic curvature of ς1 (s∗ ) is
ϕ0
kW k
sin ϕ)B
· (30)
9
Smarandache Curves in Terms of Sabban Frame...
0
ς1
κg =
0
ϕ
ϕ
λ1 kW
− λ2 kW
+ 2λ3
k
k
5
0
·
ϕ
)2 ) 2
(2 + ( kW
k
v-) TN (N ∧ TN )-Smarandache Curves
TN (N ∧ TN ) - Smarandache curve can be defined by
1
ς2 (s∗ ) = √ (TN + N ∧ TN ).
2
(31)
Differentiating (31), the tangent vector of curve ς2 is
0
0
ϕ
ϕ
( kW
(sin ϕ + cos ϕ))T − N + ( kW
(cos ϕ − sin ϕ)B)
k
k
q
Tς2 =
·
ϕ0 2
1 + 2( kW
)
k
(32)
Differentiating (32), it is obtained that
√
Tς02 =
2
(λ
sin
ϕ
−
λ
cos
ϕ)T
+
λ
N
+
(λ
sin
ϕ
+
λ
cos
ϕ)B
3
2
1
2
3
ϕ0 2 2
))
(1 + 2( kW
k
(33)
where
λ1 =
ϕ0 ϕ0 0
ϕ0 ϕ0 3
+2
+2
kW k
kW k
kW k kW k
λ2 = −2
λ3 =
ϕ0 4
ϕ0 4
ϕ0 −3
−
−1
kW k
kW k
kW k
ϕ0 0
ϕ0 4
ϕ0 2
−2
−
.
kW k
kW k
kW k
Using the equations (31) and (32), we easily find
0
ϕ
(sin ϕ + cos ϕ)T + 2 kW
N + (cos ϕ − sin ϕ)B
k
q
(N ∧ TN )ς2 =
·
ϕ0 2
2 + 4( kW
)
k
So, the geodesic curvature of ς2 (s∗ ) is as follows
κg ς2 =
ϕ0
λ
kW k 1
− λ2 + λ3
0
5
ϕ
(1 + 2( kW
)2 ) 2
k
·
(34)
10
Süleyman Şenyurt et al.
vi-) N TN (N ∧ TN )-Smarandache Curves
N TN N ∧ TN - Smarandache curve can be defined by
1
ς3 (s∗ ) = √ (N + TN + N ∧ TN ).
3
(35)
Differentiating (35), the tangent vector of curve ς2 is
− cos ϕ +
Tς3 =
ϕ0
(cos ϕ
kW k
+ sin ϕ) T − N + sin ϕ +
r ϕ0
ϕ0 2
2 1 − kW
+
k
kW k
ϕ0
(cos ϕ
kW k
− sin ϕ) B
·
(36)
Differentiating (36), it is obtained that
Tς03
= 4 1−
√
3
ϕ0
kW k
0
ϕ
+ ( kW
)2
k
2 (−λ2 cos ϕ + λ3 sin ϕ)T + λ1 N + (λ3 cos ϕ + λ2 sin ϕ)B .
(37)
where
λ1 =
ϕ0
ϕ0 3
ϕ0 2
ϕ0 ϕ0 0
2
−1 +2
−4
+4
−2
kW k
kW k
kW k
kW k
kW k
λ2 = −
λ3 =
ϕ0 4
ϕ0 0 ϕ0
ϕ0 3
ϕ0 2
ϕ0 +1 −2
+2
−4
+2
−2
kW k kW k
kW k
kW k
kW k
kW k
ϕ0 0
ϕ0 ϕ0 4
ϕ0 3
ϕ0 2
ϕ0 2−
−2
+4
−4
+2
.
kW k
kW k
kW k
kW k
kW k
kW k
Using the equations (35) and (36), we have
(N ∧ TN )ς3 = √ q
6 1−
1
ϕ0
kW k
+
ϕ0 2
( kW
)
k
(2 sin ϕ + cos ϕ
ϕ0
ϕ0
(cos ϕ − sin ϕ))T + (−1 + 2
)N
kW k
kW k
ϕ0
+(2 cos ϕ − sin ϕ −
(cos ϕ − sin ϕ))B
kW k
+
The geodesic curvature of ς3 (s∗ ) is
(38)
Smarandache Curves in Terms of Sabban Frame...
κςg3 =
1
√
4 2(1 −
5 [λ1 2
0
ϕ0
kW k
0
ϕ
+ ( kW
)2 ) 2
k
ϕ +λ3 2 −
].
kW k
11
ϕ0
ϕ0
− 1 + λ2 (−1 −
)
kW k
kW k
vii-) BTB -Smarandache Curves
BTB - Smarandache curve can be defined by
1
η1 (s∗ ) = √ (B + TB ).
2
(39)
Differentiating (39), the tangent vector of curve η1 is to be
1
Tη1 = q
2+
κ
T −N −B .
τ
(40)
(λ3 T − λ2 N + λ1 B).
(41)
κ 2
τ
Differentiating (40), we get
Tη01
√
2
=
2+
2
κ 2
τ
where
λ1 =
κ 0 κ κ 2
−
−2
τ τ
τ
λ2 = −2 − 3
κ 2
κ 4
κ κ 0
−
−
τ
τ
τ τ
κ 0
κ 3
κ
+
+2
.
τ
τ
τ
Considering the equations (39) and (40), it easily seen that
λ3 = 2
1
(B ∧ TB )η1 = q
4+2
κ
κ 2T + N + B .
τ
τ
κ 2
τ
∗
So, the geodesic curvature of η1 (s ) is
κg η1 =
1
κ
κ
λ
1 − λ2 + 2λ3 .
5
2 2 τ
τ
2 + κτ
(42)
12
Süleyman Şenyurt et al.
viii-) TB (B ∧ TB )-Smarandache Curves
TB (B ∧ TB ) - Smarandache curve can be defined by
1
η2 (s∗ ) = √ (TB + B ∧ TB ).
2
Differentiating (43), the tangent vector of curve η2 is as follows
Tη2 = p
κ
1
κ
T + N −B .
κ 2 τ
τ
1 + 2( τ )
Differentiating (44), it is obtained that
√
2
0
(λ3 T − λ2 N + λ1 B)
Tη2 =
(1 + 2( κτ )2 )2
where
λ1 =
(43)
(44)
(45)
κ
κ κ 0
κ 3
+2
+2
τ
τ
τ τ
λ2 = −2
κ 4
κ 2
κ
−3
−
−1
τ
τ
τ
κ 4
κ 2
κ 0
−2
−
.
τ
τ
τ
Using the equations (43) and (44), we easily find
λ3 =
κ 1
T +N +2 B .
(B ∧ TB )η2 = p
κ 2
τ
2 + 4( τ )
(46)
So, the geodesic curvature of η2 (s∗ ) is as follows
κg η2 =
1
(1 +
5
2( κτ )2 ) 2
κ
(2 λ1 − λ2 + λ3 ).
τ
ix-) BTB (B ∧ TB )-Smarandache Curves
BTB B ∧ TB - Smarandache curve can be defined by
1
η3 (s∗ ) = √ (B + TB + B ∧ TB ).
3
Differentiating (47), the tangent vector of curve η3 is
Tη3
= r 2 1−
1
κ
τ
κ
+
κ
T + (−1 + )N − B
τ
τ
κ 2
τ
(47)
(48)
Smarandache Curves in Terms of Sabban Frame...
Differentiating (48), it is obtained that
√
3
Tη03 = 2 (λ3 T − λ2 N + λ1 B).
4 1 − κτ + ( κτ )2
13
(49)
where
λ1 =
κ 0 κ
κ 3
κ 2
κ
2 −1 +2
−4
+4 −2
τ
τ
τ
τ
τ
λ2 = −
λ3 =
κ 0 κ
κ 4
κ 3
κ 2
κ
+1 −2
+2
−4
+2
−2
τ τ
τ
τ
τ
τ
κ 0
κ
κ 4
κ 3
κ 2
κ
2−
−2
+4
−4
+2
.
τ
τ
τ
τ
τ
τ
Using the equations (47) and (48), we have
(B ∧ TB )η3 =
(2 − κτ )T + (1 + κτ )N + (−1 + 2 κτ )B
√ p
·
6 1 − κτ + ( κτ )2
(50)
The geodesic curvature of η3 (s∗ ) is
κηg3
λ1 (2 κτ − 1) + λ2 (−1 − κτ ) + λ3 (2 − κτ )
·
=
√
5
4 2(1 − κτ + ( κτ )2 ) 2
Example
Let us consider the unit speed spherical curve:
α(s) = {
9
1
9
1
6
sin 16s −
sin 36s, −
cos 16s +
cos 36s,
sin 10s}.
208
117
208
117
65
In terms of definitions, we obtain Spherical indicatrix curves (T), (N), (B),
(see Figure 1) and Smarandache curves according to Sabban frame on S 2 ,
T TT , TT (T ∧ TT ), T TT (T ∧ TT ), N TN , TN (N ∧ TN ), N TN (N ∧ TN ), BTB ,
TB (B ∧ TB ), BTB (B ∧ TB ), (see Figure 2, 3, 4).
14
Figure 1: (T )
Figure 2: T TT
Süleyman Şenyurt et al.
(N )
(B)
TT (T ∧ TT )
T TT (T ∧ TT )
Figure 3: N TN
TN (N ∧ TN )
N TN (N ∧ TN )
Figure 4: BTB
TB (B ∧ TB )
BTB (B ∧ TB )
Smarandache Curves in Terms of Sabban Frame...
15
References
[1] M.P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice
Hall, Englewood Cliffs, (1976).
[2] A.T. Ali, Special Smarandache curves in the Euclidean space, International Journal of Mathematical Combinatorics, 2(2010), 30-36.
[3] M. Çetin, Y. Tunçer and M.K. Karacn, Smarandache curves according to
Bishop frame in Euclidean 3- space, Gen. Math. Notes, 20(2014), 50-66.
[4] Ö. Bektaş and S. Yüce, Special Smarandache curves according to Darboux frame in Euclidean 3- space, Romanian Journal of Mathematics and
Computer Science, 3(2013), 48-59.
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