Some Special Curves Belonging to Mannheim Curves Pair

2017·0.39 MB·

by  Süleyman Şenyurt

SOME SPECIAL CURVES BELONGING TO MANNHEIM CURVES PAIR
Süleyman Şenyurt*1, Yasin Altun2, Ceyda Cevahir3and Hüseyin Kocayiğit4
*1,2,3 Ordu University, Faculty of Art and Science, Ordu, Turkey
4Manisa Celal Bayar University, Faculty of Education, Manisa,Turkey
* Correspondig Author: E-mail: [email protected]

In this paper, we investigate special Smarandache curves with regard to
Sabban frame for Mannheim partner curve spherical indicatrix. We created
Sabban frame belonging to this curves. It was explained Smarandache curves
position vector is consisted by Sabban vectors belonging to this curves. Then,
we calculated geodesic curvatures of this Smarandache curves. Found results
were expressed depending on the Mannheim curve.

Key Words: Mannheim curve pair, Smarandache curve, Sabban frame,
Geodesic curvature.
1. Introduction
In differential geometry, special curves have an important role. One of these curves Mannheim
curves. Mannheim curve was firstly defined by A. Mannheim in 1878. Any curve can be a Mannheim

curve if and only if = ,  is a nonzere constant, where curvature of curve is  and curvature
2 2
of torsion is . After a time, Mannheim curve was redefined by Liu and Wang. According to this new
definition, when first curves principal normal vector and second curves binormal vector are linearly
dependent, first curve was named as Mannheim curve, and second curve was named as Mannheim
partner curve [8]. We can found many studies in literature related to Mannheim curves [4, 9]. A regular
curve in Minkowski space-time, whose position vector is composed by Frenet frame vectors regular
curve is called a Smarandache curve [15]. Special Smarandache curves have been studied by some
authors [1, 2, 5, 10, 11]. K. Taşköprü, M. Tosun studied special Smarandache curves according to
Sabban frame on S2 [14]. S. Şenyurt and A. Çalışkan investigated special Smarandache curves in terms
of Sabban frame of spherical indicatrix curves and they gave some characterization of Smarandache
curves [3]. We investigated special Smarandache curves belonging to Sabban frame drawn on the
surface of the sphere by Darboux vector of involute and Bertrand partner curves [12, 13]. Let
:I E3 be a unit speed curve, we defined the quantities of the Frenet frame and Frenet formulae,
respectively [7],
(s)
T(s)=(s),N(s)= ,B(s)=T(s)N(s), (1.1)
(s)
T(s)=(s)N(s),N(s)=(s)T(s)(s)B(s),B(s)=(s)N(s).
(1.2)
Let , be Mannheim pair curve and Frenet apparatus {T(s),N(s),B(s),(s),(s)} and
{T(s),N(s),B(s),(s),(s)} are respectively. The relation between the Frenet apparatus are as
follows, [9],
 
T =cosT sinB, N =sinT cosB,B = N,  = ,  =  (1.3)
W 

1
Let :I S2 be a unit speed spherical curve. We denote s as the arc-length parameter of . Let us
denote by
(s)=(s), t(s)=(s), d(s)=(s)t(s)
(1.4)
{(s),t(s),d(s)} frame is called the Sabban frame of  on S2. Then we have the following
spherical Frenet formulae of 
(s)=t(s), t(s)=(s) (s)d(s), d(s)= (s)t(s)
g g (1.5)
where  is called the geodesic curvature of the curve  on S2 which is
g
 (s)=t(s),d(s). (1.6)
g
2. Main Results
In this section, we investigated special Smarandache curves such as created by Sabban frame,
{T,T ,T T }, {N,T ,N T } and {B,T ,BT }. We will find some results. These
     
T T N N B B
results will be expressed depending on the Mannheim curve. Let’s find results on this Smarandache
curves.  (s )=T(s),  (s )= N(s) and  (s )= B(s) be a regular spherical
  
 T  N  B
T N B
curves on S2The Sabban frames of spherical indicatrix belonging to Mannheim partner curve are as
follows:
T =T,T = N,TT = B, (2.1)
 
T T
N = N, T =cosTsinB, NT =sinTcosB, (2.2)
 
N N
B = B, T =N, BT =T. (2.3)
 
B B
From the equation (1.5), the spherical Frenet formulae of (T), (N) and (B) are, respectively,
 
T'=T , T '=T TT , (TT )= T , (2.4)
T T  T T  T
' '
N'=T ,T '=N NT ,(NT )=  T , (2.5)
N N W N N W N
 
B'=T ,T '=B BT ,(BT )= T . (2.6)
B B  B B  B
Using the equation (1.6) the geodesic curvatures of (T), (N) and (B) are,
   '  
T N B
 = , = and = . (2.7)
g  g W g 
Definition 2.1. Let (T) curve be of , T and T be unit vector of (T). In this case -
 1
T
Smarandache curve can be defined by
1
(s)= (TT ). (2.8)
1 
2 T
Theorem 2.1. The 1 geodesic curvature belonging to -Smarandache curve of the Mannheim
g 1
1 W W
curve is 1 = (    2), where
g W 5  1  2 3
(2( )2)2

2
W W W W W W W
 =2( )2 ( )( ),  =23( )2 ( )4 ( )( )
1 2
      

W W W
 =2( )( )3( ).
3
  
1
Proof:. (s )= (TT ) and from the equation (2.1), we can write
1  
T 2 T
1
(s)= (TN). (2.9)
1
2
Differentiating (2.9), T (s) is

1
1 
T (s)= (T N B). (2.10)
1  
2( )2

Considering the equations (2.9) and (2.10), with ease seen that
1  
(T )(s)= ( T N2B). (2.11)
1 1   
42( )2

Differentiating (2.10) equation, T ' vector is

1
2
T '(s)= (TNB), (2.12)
1  1 2 3
(2( )2)2

where coefficients are
   
=2( )2 ( )( ),

1   

    
 =23( )2 ( )4 ( )( ), (2.13)
2    

   
 =2( )( )3( ).
 3   
From the equation (1.6), (2.11) and (2.12), 1 geodesic curvature of (s) is
g 1
1  

1 = (   2). (2.14)
g  5  1  2 3
(2( )2)2

Substituting the equations (1.3) into equation (2.9), (2.10), (2.11) and (2.12), Sabban apparatus of the
-Smarandache curve for Mannheim curve are
1
1
(s)= ((sincos)T (cossin)B),
1
2
(sincos) W (cossin)
T (s)= T  N  B,

1 W 2 22 W 2 22 W 2 22

W (cossin) 2 W (cossin)
(T )(s)= T  N  B,
1 
1 2 W 2 42 2 W 2 42 2 W 2 42
3
()4 2(cos sin) ()4 2 ()4 2( cossin)
T '(s)= 1 2 T  3 N 2 1 B
1 (W 222)2 (W 222)2 (W 222)2

and 1 geodesic curvature is
g
1 W W
1 = (    2).
g W 5  1  2 3
(2( )2)2


Definition 2.2. Let (T) curve be of , T and TT be unit vector of (T). In this case  -
  2
T T
Smarandache curve can be defined by
1
(s)= (T TT ). (2.15)
2  
2 T T
Theorem 2.2. The 2 geodesic curvature belonging to  -Smarandache curve of the Mannheim
g 2
1 W
curve is 2 = (2    ), where
g W 5  1 2 3
(12( )2)2

W W W W W W W
 =( )2( )32( )( ),  =13( )2 2( )4 ( )
1 2
      

W W W
 =( )2 2( )4 ( ).
3
  

Proof: From the equations (1.3), (2.1) and (2.15), we can write
1
(s)= (sinT N cosB).
2
2
Herein, if Sabban apparatus calculate

cos W sin W sin W cos
T (s)= T  N B,

2 2 W 22 2 W 22 2 W 22
()4 2( sin cos) ()4 2 ()4 2( cos sin)
T '(s)= 2 1 T  3 N  2 1 B,
2 2(W 2 2)2 (2 W 2 2)2 (2 W 2 2)2
2 W cossin  2 W sincos
( T )(s)= T  N  B,
2 
2 4 W 2 22 4 W 2 22 4 W 2 22

 2 geodesic curvature is
g
1 W
2 = (2    ).
g W 5  1 2 3
(12( )2)2

Definition 2.3. Let (T) curve be of , T, T and TT be unit vector of (T). In this case
 
T T
-Smarandache curve can be defined by
3
1
(s)= (TT TT ). (2.16)
3  
3 T T
4
Theorem 2.3 The 3 geodesic curvature belonging to -Smarandache curve of the Mannheim
g 3
1   
curve is 3 = ((2 1) (1 ) (2 )), where
g   5 W 1 W 2 W 3
4 2(1 ( )2)2
W W
 W W W W W
 =24( )4( )2 2( )3( )(2 1)
1     

 W W W W W W
 =22( )4( )2 ( )32( )4 ( )(1 )
2      

 W W W W W W
 =2( )4( )2 4( )32( )4 ( )(2 ).
3      

Proof: From the equations (1.3),(2.1) and (2.16), we can write
1
(s)= ((sincos)T N (cossin)B).
3
3
Then if Sabban apparatus calculate
cos(W )sin W sin(W )cos
T (s)= T  N B,

3 2(W 2 W 2) 2(W 2 W 2) 2(W 2 W 2)
()4 3( sincos) ()4 3
T '(s)= 2 1 T  3 N
3 4(W 2  W 2)2 4(W 2  W 2)2
()4 3( cossin)
 2 1 B,
4(W 2  W 2)2

(2 W )cos(W )sin 2 W
( T )(s)= T  N
3 
3 6 W 2 6 W 62 6 W 2 6 W 62
(2 W )sin(W )cos
 B,
6 W 2 6 W 62

 3 geodesic curvature is
g
1 W W W
3 = ((2 1) (1 ) (2 )).
g W W 5  1  2  3
4 2(1 ( )2)2
 
Definition 2.4. Let (N) curve be of , N and T be unit vector of (N). In this case  -
 4
N
Smarandache curve can be defined by
1
(s)= (NT ). (2.17)
4 
4 N
Theorem 2.4. The 4 geodesic curvature belonging to  -Smarandache curve of the Mannheim
g 4
1 W 

curve is 4 = (  2), where =( ) csc and
g (22)52 1 2 3 ()2  W 2 
 =22 ,  =232 4 ,  =23.
1 2 3

5
Proof: From the equations (1.3), (2.2) and (2.17), we can say

2  W 2 sincos W sin 2  W 2 cos
(s)= T  N  B.
4
22 2 W 2 22 2 W 2 22 2 W 2
If Sabban apparatus calculate
(W )cos W 2 2 sin  W
T (s)= T  N

4 W 2 2 22 W 2 2 22
(W )sin W 2 2 cos
 B,
W 2 2 22
( W ) 2cos 2 W 2 22sin ( W ) 2
T '(s)= 3 2 1 T  3 2 N

4 (22)2 W 2 2 (22)2 W 2 2
( W ) 2sin 2 W 2 22 cos
 2 3 1 B,
(22)2 W 2 2

(2 W )cos W 2 2 sin 2W
( T )(s)= T  N
4 
4 422 W 2 2 422 W 2 2
(2 W )sin W 2 2 cos
 B,
422 W 2 2

 4 geodesic curvature is
g
1

 4 = (  2).
g 5 1 2 3
(22)2
Definition 2.5. Let (N) curve be of , T and N T be unit vector of (N). In this case 
  5
N N
-Smarandache curve can be defined by
1
(s)= (T NT ). (2.18)
5  
2 N N
Theorem 2.5. The 5 geodesic curvature belonging to -Smarandache curve of the Mannheim
g 5
1
curve is 5 = (2   ), where
g 5 1 2 3
(22)2
 =232,  =132 24 ,  =2 24 
1 2 3

Proof: From the equations (1.3),(2.2) and (2.18), we can write

(W )cos  W ( W )sin
(s)= T  N  B.
5
22 2 W 2 22 2 W 2 22 2 W 2

6
Here, if Sabban apparatus calculate
(W )cos W 2 2 sin ( W )
T (s)= T  N

5 122 W 2 2 122 W 2 2
(W )sin W 2 2 cos
 B,
122 W 2 2
( W ) 2cos 2 W 2 22 sin ( W ) 2
3 2 1 3 2
T '(s)= T  N

5 (122)2 2  W 2 (122)2 2  W 2
( W ) 2sin 2 W 2 22 cos
2 3 1
 B,
(122)2 2  W 2
(W )cos2 W 2 2 sin  W
( T )(s)= T  N
5 
5 242 W 2 2 242 W 2 2
2 W 2 2 cos(W )sin
 B,
242 W 2 2

 5 geodesic curvature is
g
1

5 = (2   ).
g 5 1 2 3
(22)2

Definition 2.6 Let (N) curve be of , N, T and N T be unit vector of (N). In this
 
N N
case -Smarandache curve can be defined by
6
1
(s)= (NT NT ). (2.19)
6  
3 N N
Theorem 2.6 The 6 geodesic curvature belonging to -Smarandache curve of the Mannheim
g 6
(21) (1) (2)
curve is 6 = 1 2 3 ,where
g 5
4 2(12)2
 =2442 232(21),  =2242 2324 (1),
1 2

 =242 4324 (2).
3

Proof: From the equations (1.3), (2.2) and (2.19), we can write
(W )cos 2  W 2 sin  W
(s)= T  N
6
32 3 W 2 32 3 W 2
( W )sin 2  W 2 cos
 B.
32 3 W 2
Herein, if Sabban apparatus calculate
7
(W (1))cos W 2 2 sin (1) W
T (s)= T  N

6 2(12) W 2 2 2(12) W 2 2
((1)W )sin W 2 2 cos
 B,
2(12) W 2 2
( W ) 3cos 3 W 232 sin ( W ) 3
T '(s)= 3 2 1 T  3 2 N
6 4(12)2 2 W 2 4(12)2 2 W 2
( W ) 3sin 3 W 2 32 cos
 2 3 1 B,
4(12)2 2  W 2
((2) W (1))cos(21) W 2 2 sin
( T )(s)= T
6 
6 6662 W 2 2
(2)(1) W
 N
6662 W 2 2
(21) W 2 2 cos((2) W (1))sin
 B,
6662 W 2 2

 6 geodesic curvature is
g
(21) (1) (2)
6 = 1 2 3.
g 5
4 2(12)2

Definition 2.7. Let (B) curve be of , T and B T be unit vector of (B). In this case -
  7
B B
Smarandache curve can be defined by
1
(s)= (T TT ). (2.20)
7  
2 B B
Theorem 2.7. The 7 geodesic curvature belonging to -Smarandache curve of the Mannheim
g 7
1 
curve is 7 = (2    ), where
g  5 W 1 2 3
(12( )2)2
W
      
 =( )( )32( )( ),  =13( )22( )4( )
1 W W W W 2 W W W

  
 =( )22( )4( ).
3 W W W

Proof: From the equations (1.3), (2.3) and (2.20), we can write
1
(s)= (sinT N cosB).
7
2
If Sabban apparatus calculate
8
cos W sin W sin W cos
T (s)= T  N  B,

7 2 W 2 2 2 W 2 2 2 W 2 2
4 4 4
W 2( cos sin) W 2 W 2( cossin)
3 2 1 2 3
T '(s)= T  N  B,
7 (2 W 2 2)2 (2 W 2 2)2 (2 W 2 2)2
2 W cossin  cos2 W sin
( T )(s)= T  N  B,
7 
7 4 W 2 22 4 W 2 22 4 W 2 22

 7 geodesic curvature is
g
1  
7 = (    2).
g  5 W 1 W 2 3
(2( )2)2
W
Definition 2.8. Let (B) curve be of , B, T and B T be unit vector of (B). In this case
 
B B
-Smarandache curve can be defined by
8
1
(s)= (BT BT ). (2.21)
8  
3 B B
Theorem 2.8. The 8 geodesic curvature belonging to -Smarandache curve of the Mannheim
g 8
1   
curve is 8 = ((2 1) (1 ) (2 ) ), where
g   5 W 1 W 2 W 3
4 2(1 ( )2)2
W W
      
 =24 4( )( )22( )3( )(2 1)
1 W W W W W W

      
 =22 4( )2( )32( )4( )(1 )
2 W W W W W W

      
 =2 4( )24( )32( )4( )(2 ).
 3 W W W W W W
Proof: From the equations (1.3), (2.3) and (2.21), we can write
1
(s)= ((cossin)T N (cossin)B).
8
3
Here if Sabban apparatus calculate
cos( W )sin W sin( W )cos
T (s)= T  N  B,

8 2(W 2  W 2) 2(W 2  W 2) 2(W 2  W 2)
4 4 4
W 3( cos sin) W 3 W 3( cos sin)
T '(s)= 3 2 T  1 N 2 3 B,
8 4(W 2 W 2)2 4(W 2 W 2)2 4(W 2 W 2)2

(2 W )cos(W )sin 2 W
( T )(s)= T  N
8 
8 6 W 2 6 W 62 6 W 2 6 W 62
(W )cos(2 W )sin
 B,
6 W 2 6 W 62

8 geodesic curvature is
g
9
1   
8 = ((2 1) (1 ) (2 ) ).
g   5 W 1 W 2 W 3
4 2(1 ( )2)2
W W
3. Conclusion
In this study, we studied Mannheim Partner Curves and Smarandache curves, which are
well known in Differential Geometry. The Sabban frames of the differential curves drawn by
the Frenet vectors of the Mannheim Partner Curve on the unit sphere surface were calculated.
Then Smarandache curves were defined with the help of these frames. Finally, geodesic
curvatures of these curves were calculated according to Sabban frames.
References
[1] A. T. Ali, Special Smarandache curves in the Euclidian Space, International Journal of
Mathematical Combinatorics 2 (2010), pp.30-36
[2] Ö. Bektaş, S. Yüce, Special Smarandache curves according to Darboux Frame in Euclidean 3-
Space, Romanian Journal of Mathematics and Computer Sciencel, 3 (2013), 1, pp. 48-59
[3] A. Çalışkan S. Şenyurt, Smarandache curves in terms of Sabban frame of spherical indicatrix
curves, Gen. Math. Notes 31 (2015), 2, pp.1-15
[4] A. Çalışkan , S.Şenyurt, NC-Smarandache curves of Mannheim curve couple according to
Frenet Frame, International Journal of Mathematical Combinatorics 1(2015), pp. 1-13.
[5] M. Çetin, Y. Tuncer, M. K. Karacan, Smarandache curves according to Bishop frame in
Euclidean 3-Space, Gen. Math. Notes, 20 (2014), pp. 50-66
[6] W. Fenchel, On the Differential Geometry of closed space curves, Bulletin of the American
Mathematical Society, 57 (1951), pp.44-54
[7] H. H. Hacısalihoğlu, Differential Geometry, Academic Press Inc. Ankara, 1994.
[8] H. Liu, F. Wang, Mannheim partner curves in 3-space, Journal of Geometry, 88 (2008) ,(1-2),
pp. 120-126
[9] K. Orbay, E. Kasap, On Mannheim partner curves in , International Journal of Physical
Sciences, 4 (2009), 5, pp. 261-264
[10] M. Elzawy, Smarandache curves in Euclidian 4-space E4, Journal of the Eqyptian
Mathematical Society (in press), DOI: http://dx.doi.org/10.1016/j.joems
[11] S. Senyurt, S. Sivas, An application of Smarandache curve, University of Ordu Journal of
Science and Technology, 3 (2013), 1, pp.46-60
[12] S. Şenyurt, Y. Altun, C. Cevahir, Smarandache Curves According to Sabban Frame of Fixed
Pole Curve Belonging to The Bertrand Curves Pair, AIP Conf. Proc. 1726 (2016)
doi:10.1063/1.4945871.
[13] S. Şenyurt , Y. Altun , C. Cevahir, On The Darboux Vector Belonging to Involute Curve A
Different View, Mathematical Sciences and Applications E-Notes, 4 (2016), 2, pp. 121-130.
[14] K. Taşköprü, M. Tosun, Smarandache Curves on S2, Boletim da Sociedade Paranaense de
Matematica 3 Srie., 32 (2014), 1, pp. 51-59
[15] M. Turgut, S. Ylmaz, Smarandache curves in Minkowski space-time, International Journal of
Mathematical Combinatorics, 3 (2008) pp.51-55
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