Page 256 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 21: New Spherical Indicatrices and their Characterizations

Let  = (s ) be M Bishop spherical image of a regular curve  = (s). We follow the same Notes
1

procedure to investigate the relations among Bishop and Frenet-Serret invariants. Thus, we
differentiate
d ds

'      k T.
1
ds ds

First, we have
ds
T = T and    k . (4)
 ds 1
So, one can calculate

ds
'
T  T   ds   k M  k M 2
1
2
1

or
k
'
T   M  k 2 1 M .
1
2

Since, we express
2
 k 

K  T  1   2  (5)


 k 1 
and
M k
N   K  1  k K 2  M .
2

1
Cross product of T × N gives us the binormal vector field of M spherical image of  = (s)
  1
k 1
B  k K 2  M  K  M .
2
1

1
Using the formula of the torsion, we write
'
k 1   k  
2
T    k 1  . (6)
2

k  k 2 2
1
Considering equations (5) and (6) by the Theorem 1, we get:
Corollary 2. Let  = (s ) be the M Bishop spherical image of the curve  = (s). If  = (s) is a
1

B-slant helix, then, the M Bishop spherical indicatrix (s ) is a circle in the osculating plane.
1

Theorem 2. Let  = (s ) be the M Bishop spherical image of a regular curve  = (s). There exists
1

a relation among Frenet-Serret invariants (s ) and Bishop curvatures of  = (s) as follows:

k 2 s  K T ds . (7)
2
k 1   0   

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