Page 255 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Since, we immediately arrive at

 k 3  k  '  2  k 3  k  '  2


K  T  1  2 2 2  1     1 2 2  2   . (2)
  2 k ) k 2 k ) k
  (k  2  2     (k  2  1  


1
1
Therefore, we have the principal normal
1   k 3  k  ' k 3  k  ' 

N    T  2 2 2  1  M  1 2 2  2  M 2  .
2
2
 K    (k  k )  k 2  1 (k  k )  k 1   
2
2
1
1
By the cross product of T × N , we obtain the binormal vector field
 
 k 4  k  ' k 4  k  '  k   k  
B = 1   1 5  2  – 2 5  1  T   2  M   1  M 2   .


2
2
2
 K  2 2 2  k 1  2 2 2  k 2    k  k  1  k  k 
2
 
   (k  k ) (k  k )    1 2   1 2   
1
2
1
2
By means of obtained equations, we express the torsion of the tangent Bishop spherical image
2
2
"
2
'
'
2
2
2
2
"
2
'
'
'
'

( k {3k (k k  k k ) (k  k )[k  k (k  k )]} k {3k (k k  k k ) (k  k )[k  k (k  k )}



1
2
1
1
2
2
2
1
2
1
2
1
1
1
2
2
2
1
2
1
1
1
2
2
T  2

 2 k  '  2 2 3

k  1  2    (k  k ) (3)
1
2
   k 1  

Consequently, we determined Frenet-Serret invariants of the tangent Bishop spherical indicatrix
according to Bishop invariants.
Corollary 1. Let  = (s ) be the tangent Bishop spherical image of a regular curve  = (s).

If  = (s) is a B-slant helix, then the tangent spherical indicatrix  is a circle in the osculating
plane.
Proof. Let  = (s ) be the tangent Bishop spherical image of a regular curve  = (s): If  = (s) is a

k
B-slant helix, then Theorem 1 holds. So, 1  constant. Substituting this to equations (2) and (3),
k 2
we have K = constant and T = 0, respectively. Therefore,  is a circle in the osculating plane.
 
s 
= T ds and using the transformation matrix, one can obtain the
Remark 1. Considering     
0
Bishop trihedra {T , M , M } of the curve  = (s ).
 1 2 
Here, one question may come to mind about the obtained tangent spherical image, since, Frenet-
Serret and Bishop frames have a common tangent vector field. Images of such tangent images
are the same as we shall demonstrate in subsequent section. But, here we are concerned with the
tangent Bishop spherical image’s Frenet-Serret apparatus according to Bishop invariants.
21.3 M Bishop Spherical Images of a Regular Curve
1
Definition 3. Let  = (s) be a regular curve in E . If we translate of the second vector field of
3
Bishop frame to the center O of the unit sphere S , we obtain a spherical image  = (s ). This curve
2

is called M Bishop spherical image or indicatrix of the curve  = (s).
1
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