On The Sabban Frame Belonging to Involute-Evolute Curves

2017·0.4 MB·

by  Süleyman Şenyurt

ON THE SABBAN FRAME BELONGING TO INVOLUTE-EVOLUTE CURVES

Süleyman ŞENYURT, Yasin ALTUN, Ceyda CEVAHİR and Hüseyin KOCAYİĞİT
*1,2,3 Ordu University, Faculty of Art and Science, Ordu, Turkey
4Manisa Celal Bayar University, Faculty of Education, Manisa,Turkey
* Corresponding Author; E-mail: [email protected]

In this article, we investigate special Smarandache curves with regard to
Sabban frame of involute curve. We created Sabban frame belonging to
spherical indicatrix of involute curve. It was explained Smarandache curves
position vector is consisted by Sabban vectors belonging to spherical
indicatrix. Then, we calculated geodesic curvatures of this Smarandache
curves. The results found for each curve was given depend on evolute curve.
The example related to the subject were given and their figures were drawn
with Mapple program.

Key Words: Geodesic curvature, Involute-evolute curves, Sabban frame,
Smarandache curves

1. Introduction and Preliminaries
The involute of the curve is decent known by the mathematicians especially the differential geometry
scientists. There are many essential consequences and properties of curves. Involute curves examined
by some authors[3, 6]. Frenet vectors of a curve were taken as the position vector and the regular curve
drawn by the new vector is identified. This curve were called the Smarandache curve [10]. Special
Smarandache curves examined by certain writers [1, 2, 5, 7, 8, 9]. K. Taşköprü, M. Tosun studious
particular Smarandache curves belonging to Sabban frame on S2 [11]. Şenyurt and Çalışkan
investigated particular Smarandache curves belonging to Sabban frame of spherical indicatrix curves
and they gave some characterization of Smarandache curves [4]. Let :I E3 be a unit speed curve
and the quantities {T,N,B,,} are collectively Frenet-Serret apparatus of this curve. The Frenet
formulae are also well known as, respectively [6]
T(s)=(s)N(s), N(s)=(s)T(s)(s)B(s), B(s)=(s)N(s). (1.1)
Let  :I E3 be the C2-class differentiable curve and {T (s),N (s),B (s), (s),(s)} is
1 1 1 1 1 1
Frenet- serret apparatus of the  involute curve, then [3]
1
T = N, N =cosT sinB, B =sinT cosB. (1.2)
1 1 1
where (W,B)=. For the curvatures and the torsions we have
W ()
 = ,  = . (1.3)
1 |cs| 1 |cs|W 2
 W  2  W 2
sin = , cos = ,'=( ) (1.4)
1 2  W 2 1 2  W 2 1 2  W 2 W
Let :I S2 be a unit speed spherical curve. We symbolize s as the arc-length parameter of .
Let’s give
(s)=(s),t(s)=(s),d(s)=(s)t(s) (1.5)
{(s),t(s),d(s)} frame is denominated the Sabban frame of  on S2. The spherical Frenet formulae
of  is as follows
(s)=t(s), t(s)=(s)(s)d(s), d(s)=(s)t(s) (1.6)
g g
where  is denominated the geodesic curvature of the curve  on S2 which is
g
 (s)=t(s),d(s) [11]. (1.7)
g
2. On The Sabban Frame Belonging To İnvolute-Evolute Curves
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
1 1 1 1 1 1
1 1 1 1 1 1
results will be expressed depending on the evolute curve. Let’s find results on this Smarandache curves.
Let  (s)=T (s),  (s)= N (s) and  (s)= B (s) be a regular spherical curves on S2. The
T 1 N 1 B 1
1 1 1
Sabban frames of spherical indicatrix belonging to involute curve are as follows:
T =T , T = N , T T = B , (2.1)
T T
1 1 1 1 1
1 1
N = N , T =cosT sinB , N T =sinT cosB, (2.2)
N N 1
1 1 1 1 1 1 1 1 1 1
1 1
B = B , T =N , B T =T . (2.3)
B B
1 1 1 1 1
1 1
From the equation (1.6), the spherical Frenet formulae of (T ), (N ) and (B ) are as follows,
1 1 1
respectively
 
T '=T , T '= T  1 T T , (T T )=  1 T , (2.4)
1 T T 1  1 T 1 T  T
1 1 1 1 1
1 1
' '
N '=T , T '=N  1 N T , (N T )= 1 T , (2.5)
N N N N N
1 1 1 1 W 1 1 1 1 W 1
1 1
 
B '=T , T '=B  1 B T , (B T )= 1 T . (2.6)
1 B B 1  1 B 1 B  B
1 1 1 1 1
1 1
Using the equation (1.7), the geodesic curvatures of (T ), (N ) and (B ) are
1 1 1
T  N ' B 
1 = 1 ,  1 = 1 , 1 = 1 . (2.7)
g  g W g 
1 1 1
Definition 2.1. Let (T ) spherical curve be of  , T and T be Sabban vectors of (T ). In the fact
T
1 1 1 1
1
-Smarandache curve can be identified by
1
1  
(s)= T T . (2.8)
1 2 1 T 
1
Theorem 2.1. The geodesic curvature according to -Smarandache curve of the involute curve is
1
4
1 = 1  2, (2.9)
g  5 1 1 1 2 1 3
22 2 2
1 1
where coefficients are
2  2 4 
             
=2 1   1   1 ,  =23 1   1   1   1 ,
1       2        
              (2.10)
1 1 1 1 1 1 1
3 
     
 =2 1  1   1 
3      
     
1 1 1
1  
Proof: (s)= T T  or from the equation (2.1), we can write
1 2 1 T 
1
1  
(s)= T N . (2.11)
1
2 1 1
If taken derivative of the equation (2.11), T vector is

1
1   (2.12)
T (s)= T N B .
1 2 2 1 1 1 1 1 1
1 1
Considering the equations (2.11) and (2.12), we have
1  
(T )(s)= T N 2B . (2.13)
1 
1 42 22 1 1 1 1 1 1
1 1
Using the equation (2.12), T ' vector is

1
24
 
T '(s)= 1 T N B . (2.14)
1 2222 1 1 2 1 3 1
1 1

From the equation (2.13) and (2.14), 1 geodesic curvature of (s) is
g 1
4
1 = 1  2.
g  5 1 1 1 2 1 3
22 2 2
1 1
Corollary 2.1. The geodesic curvature belonging to -Smarandache curve of the evolute curve is
1
6
W 2
1 = (   ), (2.15)
g 5 1 2 W 3
(2 W 2 2)2
where coefficients are i
      
 =2( )2 ( )( ),  =23( )2 ( )4 ( )( ),
1 2
W W W W W W W
(2.16)
  
 =2( )( )3( )
3
W W W
Proof: From the equations (1.2) and (2.11), we calculate
1
(s)= (cosT N sinB). (2.17)
1
2
If taken derivative of the this expression, T vector is,

1
sin W cos W cos W sin
T = T  N B. (2.18)

1 2 W 2 2 2 W 2 2 2 W 2 2
If made cross product from the equations (2.17) and (2.18), we have
2 W sincos  2 W cossin
T = T  N B. (2.19)
1 
1 4 W 2 22 4 W 2 22 4 W 2 22
From the equation (2.18), T ' vector is

1
4 4 4
W (sin cos) 2 W  2 W ( cos sin) 2
T '= 3 2 T 1 N 3 2 B. (2.20)
1 (2 W 22)2 (2 W 22)2 (2 W 22)2

If made inner product from the equations (2.19) and (2.20), 1 geodesic curvature is found like
g
equations (2.15).
The proofs of the subsequent theorems and corollaries belonging to  , ,  ,  ,  ,  , 
2 3     
1 2 3 1 2
and  - Smarandache curves will be similar to the theorem 2.1 and corollaries 2.1.

3
Definition 2.2. Let (T ) spherical curve be of  , T and T T be Sabban vectors of (T ). In
T T
1 1 1 1
1 1
the fact  -Smarandache curve can be identified by
2
1
(s)= (T T T ). (2.21)
2 T T
2 1
1 1
Theorem 2.2. Let  be involute of . The geodesic curvature and belonging to 1
(s)= (N B ),
1 2 2 1 1
 -Smarandache curve of involute curve is,
2
4
2 = 1 (2   ) (2.22)
g 5 1 1 1 2 3
(2 22)2
1 1
where coefficients are
         
 = 1 2( 1)32( 1)( 1),  =13( 1)22( 1)4( 1),  =( 1)22( 1)4( 1). (2.23)
1     2    3   
1 1 1 1 1 1 1 1 1 1
Corollary 2.2. Let  be involute of . The geodesic curvature belonging to
1
1
 = ((sincos)T (sincos)B)  -Smarandache curve of evolute curve is,
2 2
2
4
W
2 = (2  W   ) (2.24)
g 5 1 2 3
(W 2 2()2)2
where coefficients are
      
 =( )( )32( )( ), =13( )2 2( )4 ( ),
1 2
W W W W W W W
(2.25)
  
 =( )2 2( )4 ( )
3
W W W
Definition 2.3. Let (T ) spherical curve be of  , T , T and T T be Sabban vectors of (T ).
T T
1 1 1 1 1
1 1
In the fact -Smarandache curve can be identified by
3
1
(s)= (T T T T ) (2.26)
3 T T
3 1 1
1 1
1
Theorem 2.3. Let  be involute of . The geodesic curvature belonging to (s)= (T N B )
1 3 3 1 1 1
-Smarandache curve of involute curve is,
3
4
3 = 1 ((2 )( ) (2 )) (2.27)
g 5 1 1 1 1 1 2 1 1 3
4 2(2  2)2
1 1 1 1
where coefficients are
    
=24( 1 )( 1 )2 2( 1 )3( 1 )(2 1 1),
1     
1 1 1 1 1
     
 =22( 1 )4( 1 )2 2( 1 )32( 1 )4 ( 1 )(1 1),
2      
1 1 1 1 1 1
     
 =2( 1 )4( 1 )2 4( 1 )32( 1 )4 ( 1 )(2 1 ) (2.28)
3      
1 1 1 1 1 1
Corollary 2.3. Let  be involute of . The geodesic curvature belonging to
1
1
(s)= ((sincos)T N (sincos)B), -Smarandache curve of evolute curve is,
3 3
3
4
W
3 = ((2 W ) (W ) (2 W )) (2.29)
g 5 1 2 3
4 2(W 2  W 2)2
where coefficients are
     
 =24( )4( )( )2 2( )3( )(2 )
1 W W W W W W 1
     
 =22( )4( )2 ( )32( )4 ( )(1 )
2 W W W W W W
     
 =2( )4( )2 4( )32( )4 ( )(2 ) (2.30)
3 W W W W W W 1
Definition 2.4. Let (N ) spherical curve be of  , N , T be Sabban vectors of (N ). In the fact
N
1 1 1 1
1
 -Smarandache curve can be identified by

1
1
 (s)= (N T ) (2.31)
 N
2 1
1 1
Theorem 2.4. Let  be involute of . The geodesic curvature belonging to
1
1
 (s)= (cosT N sinB )  -Smarandache curve of involute curve is,
 1 1 
2 1 1 1
1 1
 W 4  
 1 = 1  2 W  (2.32)
g 5 1 1 1 2 1 3
(2 W 2 ()2)2
1 1
where coefficients are
      
 =2( 1 )2 ( 1 )( 1 ),  =23( 1 )2 ( 1 )4 ( 1 )( 1 ),
1 W W W 2 W W W W
1 1 1 1 1 1 1 (2.33)
  
 =2( 1 )( 1 )3( 1 )
3 W W W
1 1 1
Corollary 2.4. Let  be involute of . The geodesic curvature Sabban apparatus belonging to
1
sin 2  W 2 cos W cos 2  W 2 sin
 (s)= T  N  B
1 22 2 W 2 22 2 W 2 22 2 W 2
 -Smarandache curve of evolute curve is,

1
 1
 1 = (  2) (2.34)
g 5 1 2 3
(22)2
' 
where in the event of = 1 =( )cos(cs) coefficients are
W 2  W 2
1
 =22 ,  =232 4 ,  =23. (2.35)
1 2 3
Definition 2.5. Let (N ) spherical curve be of  , T and N T be Sabban vectors of (N ).
N N
1 1 1 1
1 1
In the fact  -Smarandache curve can be identified by

2
1
 (s)= (T N T ) (2.36)
 N N
2 1
2 1 1
Theorem 2.5. Let  be involute of . The geodesic curvature belonging to
1
1
 (s)= ((sin cos)T (sin cos)B ),  -Smarandache curve of involute curve is ,
 
2 2 1 1 1 1 1 1 2
4
 W

 2 = 1 (2' W   W ) (2.37)
g 2 5 1 1 1 2 1 3
(W 2()2)2
1 1
where coefficients are
     
=24( 1 )( 1 )22( 1 )3( 1)(2 1 1),
 1     
1 1 1 1 1

     
 =22( 1 )4( 1 )22( 1 )32( 1)4( 1)(1 1), (2.38)
2      

1 1 1 1 1 1

     

 =2( 1 )4( 1 )2 4( 1 )32( 1)4( 1)(2 1)
 3      

1 1 1 1 1 1
Corollary 2.5 Let  be involute of . The geodesic curvature belonging to
1
( W )sin  W ( W )cos
 (s)= T  N  B,  -Smarandache curve of
 
2 22 2 W 2 22 2 W 2 22 2 W 2 2
evolute curve is,
 1
 2 = (2  ) (2.39)
g 5 1 2 3
(22)2
where coefficients are
 =232,  =132 24 ,  = 2 24 . (2.40)
1 2 3
Definition 2.6. Let (N ) spherical curve be of  , N , T and N T be Sabban vectors of
N N
1 1 1 1
1 1
(N ). In the fact  -Smarandache curve can be identified by

1
3
1
 (s)= (N T N T ) (2.41)
 N N
3 1 1
3 1 1
Theorem 2.6 Let  be involute of . The geodesic curvature belonging to
1
1
 (s)= ((sin cos)T N (sin cos)B )  -Smarandache curve of involute
 
3 1 1 1 1 1 1 1
3 3
curve is,
' ' '
(2 1 1)(1 1 ) (2 1 )
1 2 3
 W W W

 3 = 1 1 1 (2.42)
g ' ' 5
4 2(1 1 ( 1 )2)2
W W
1 1
where coefficients are

' ' ' ' '
 =24( 1 )( 1 )2 2( 1 )3( 1 )(2 1 1)
 1
W W W W W
 1 1 1 1 1

' ' ' ' ' '
 =22( 1 )4( 1 )2 2( 1 )32( 1 )4( 1 )(1 1 ) (2.43)
2
 W W W W W W
 1 1 1 1 1 1
 ' ' ' ' ' '
 =2( 1 )4( 1 )2 4( 1 )32( 1 )4 ( 1 )(2 1 )
3
 W W W W W W
 1 1 1 1 1 1
Corollary 2.6. Let  be involute of . The geodesic curvature belonging to
1
( W )sin 2 W 2 cos  W ( W )cos 2 W 2sin
 (s)= T  N B
3 323 W 2 323 W 2 323 W 2
 -Smarandache curve of evolute curve is,

3
( W )cos 2  W 2 sin
 B
32 3 W 2
 (21) (1) (2)
 3 = 1 2 3 (2.44)
g 5
4 2(12)2
where coefficients are
 =2442 232(21),  =2242 2324 (1)
1 2 (2.45)
 =242 4324 (2)
3
Definition 2.7. Let (B ) spherical curve be of  , B and T be Sabban vectors of (B ). In the
B
1 1 1 1
1
fact  -Smarandache curve can be identified by

1
1
 (s)= (B T ) (2.46)
 B
2 1
1 1
1
Theorem 2.7. Let  be involute of . The geodesic curvature belonging to  (s)= (N B )
1 1 2 1 1
 -Smarandache curve of involute curve is,

1
 4

 1 = 1 ( 2) (2.47)
g 5 1 1 1 2 1 3
(22 2)2
1 1
where coefficients are
      
=2( 1)2 ( 1)( 1),  =23( 1)2 ( 1)4 ( 1)( 1)
1    2    
1 1 1 1 1 1 1 (2.48)
  
 =2( 1)( 1)3( 1)
3   
1 1 1
Corollary 2.7. Let  be involute of . The geodesic curvature belonging to
1
1
 (s)= ((cossin)T (cossin)B,  -Smarandache curve of evolute curve is,
 
2
1 1
 4  
 1 = (W   2), (2.49)
g 5 1 2 3
(22  W 2)2
where coefficients are
W W W W W W W
 =2( )2 ( )( ),  = 23( )2 ( )4 ( )( )
1 2
      
(2.50)
W W W
 =2( )( )3 2( ).
3
  
Definition 2.8. Let (B ) spherical curve be of  , T and B T be Sabban vectors of (B ). In
B B
1 1 1 1
1 1
the fact  -Smarandache curve can be identified by

2
1
 (s)= (T B T ) (2.51)
 B B
3 1
2 1 1
Theorem 2.8. Let  be involute of . The geodesic curvature belonging to
1
1
 (s)= (T N ),  -Smarandache curve of involute curve is,
 
2 1 1
2 2
 4

 2 = 1 (2  ), (2.52)
g 5 1 1 1 2 1 3
(2 22)2
1 1
where coefficients are
      
 = 1 2( 1)32( 1)( 1),  =13( 1)2 2( 1)4 ( 1),
1     2   
1 1 1 1 1 1 1 (2.53)
  
 =( 1)2 2( 1)4 ( 1)
3   
1 1 1
Corollary 2.8. Let  be involute of . The geodesic curvature curvature belonging to
1
1
 (s)= (cosT N sinB),  -Smarandache curve of evolute curve is,
 
2
2 2
 4
 2 = (2 W    ), (2.54)
g 5 1 2 3
(2 2 W 2)2
where coefficients are
W W W W W W W
 =( )( )3 2( )( ),  = 13( )2 2( )4 ( )
1     2   
(2.55)
W W W
 =( )2 2( )4 ( )
3   
Definition 2.9. Let (B ) spherical curve be of  , B , T and B T be Sabban vectors of (B )
B B
1 1 1 1 1
1 1
In the fact  -Smarandache curve can be identified by

3
1
 (s)= (B T B T ) (2.56)
 B B
3 1 1
3 1 1
Theorem 2.9. Let  be involute of . The geodesic curvature belonging to
1
1
 = (T N B ),  -Smarandache curve of involute curve is,
 
3 3 1 1 1 3
 4

 3 = 1 ((2 ) ( ) (2 )), (2.57)
g 5 1 1 1 1 1 2 1 1 3
4 2(2  2)2
1 1 1 1
where coefficients are
     
 =24( 1)( 1)2 2( 1)3( 1)(2 1 1)
 1     
1 1 1 1 1

     

 =22( 1)4( 1)2 2( 1)32( 1)4 ( 1)(1 1) (2.58)
2      

1 1 1 1 1 1

     

 =2( 1)4( 1)2 4( 1)32( 1)4 ( 1)(2 1)
 3      

1 1 1 1 1 1
Corollary 2.9. Let  be involute of . The geodesic curvature belonging to
1
1
 (s)= ((sincos)T N (cossin)B),  -Smarandache curve of evolute curve is,
 
3
3 3
 4
 3 = ((2 W ) ( W ) (2 W ) ) (2.59)
g 5 1 2 3
4 2(5  W  W 2)2
where coefficients are
 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 ) (2.60)
2      

 W W W W W W
 =2( )4( )2 4( )32( )4 ( )(2 )
 3      
Example. Let us consider the unit speed evolute curve and involute curve, respectively
2 1 2 1 4 
(s)= sin2t sin8t, cos2t cos8t, sin3t
 
5 40 5 40 15 
2 1 4 2
(s)= sin2s sin8s 1scos5s, ,

1 5 40 5 5
1 4 4 3 3 
cos2s cos8s 1ssin5s, sin3s  s.
40 5 15 5 5 
The Frenet vectors belonging to involute curve,  are found as follows;
1
4 4 3
T = cos(5s), sin(5s),
 
1 5 5 5
1 4  4 1 
N = cos8s cos2s sin3s sin2s sin8s cos3s,
   
1 5 5  5 5 
 4 1  4 1  
 sin2s sin8ssin3scos3s cos2s cos8s,0,
 5 5  5 5  
 4 1  4 1 
B = cos3s cos2s cos8s sin3s sin2s sin8s ,
    
1  5 5  5 5 
4 1  4 1  4
cos3s sin2s sin8ssin3s cos2s cos8s, .
5 5  5 5  5
According to definitions, we reach specific Smarandache curves belonging to Sabban frame of this
curve. ,  , ,  ,  ,  ,  ,  and  (see Figure 1,2,3).
1 2 3      
1 2 3 1 2 3

Figure 1: -curve  -curve -curve
1 2 3

Figure 2:  -curve  -curve  - curve
  
1 2 3

Figure 3:  -curve  -curve  - curve
  
1 2 3

3. Conclusion

In this article, we reviewed the well-known involute and evolute curves in the literature. We
have created the Sabban frames on the unit sphere of the involute and evolute curves. We got
Smarandache curves from the Sabban frame and calculated the geodesic curvature of these
curves. Finally, we have given an example and have driven their shapes in the Mapple program.