Page 254 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 21: New Spherical Indicatrices and their Characterizations

It is well-known that for a unit speed curve with non-vanishing curvatures the following Notes
propositions hold:
Proposition 1. Let  = (s) be a regular curve with curvatures K and T. The curve  lies on the
surface of a sphere if and only if

'
T  1 1  ' 0.


K       
T K   
 
Proposition 2. Let  = (s) be a regular curve with curvatures K and T.  is a general helix if and
only if
K constant.
T 
Proposition 3. Let  = (s) be a regular curve with curvatures K and T.  is a slant helix if and only
if

 k 2 
' 
 (s)   3   T   constant.

  K  
2
2 2
  (K  T )  
21.2 Tangent Bishop Spherical Images of a Regular Curve
Definition 2. Let  = (s) be a regular curve in E . If we translate of the first (tangent) vector field
3
of Bishop frame to the center O of the unit sphere S , we obtain a spherical image  = (s ). This
2

curve is called tangent Bishop spherical image or indicatrix of the curve  = (s).
Let  = (s ) be tangent Bishop spherical image of a regular curve  = (s). One can differentiate of

 with respect to s :

 '= d ds   k M  K M .
1
2
1
ds ds
2

Here, we shall denote differentiation according to s by a dash, and differentiation according to
s by a dot. In terms of Bishop frame vector fields, we have the tangent vector of the spherical

image as follows:
k M  k M
T  1 1 2 2 ,

2
k  k 2 2
1
where
ds   k  k  k(s).
2
2
ds 1 2
In order to determine the first curvature of , we write
k 3  k  ' k 3  k  '
T

T    2 2 2  1  M  1 2 2  2  M .
2
1

2
2
(k  k )  k 2  (k  k )  k 1 
2
1
2
1
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