Page 263 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 2. If the tangent vector of this curve forms a constant angle with a fixed constant vector U,
then this curve is called a .................. or an inclined curve
3. Let  = (s) be a regular curve in E . If we translate of the first (tangent) vector field of
3
Bishop frame to the center O of the unit sphere S , we obtain a spherical image  = (s ). This
2

curve is called .................. or indicatrix of the curve  = (s).
4. Let  = (s) be a regular curve in E . If we translate of the second vector field of Bishop frame
3
to the center O of the unit sphere S , we obtain a spherical image  = (s ). This curve is
2

called .................. or indicatrix of the curve  = (s).
5. Let  = (s) be a regular curve in E . If we translate of the third vector field of Bishop frame
3
to the center O of the unit sphere S , we obtain a spherical image of  = (s ). This curve is
2

called the .................. or the indicatrix of the curve  = (s).
21.8 Review Questions

 T  1  0 0   T 
 




1. Find Relation Matrix N    0 cos (s) sin (s)   M .
 

1

   0  sin (s) cos (s)  M 
B


     2 
2. 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.
3. Next, let us consider the following unit speed curve (s) = ( ,  ,  ):
1
3
2
   9 sin16s  1 sin36s
 1 208 116

 9 1
    327 cos16s  117 cos36s
2

 6 sin10s
  
3
 65
Answers: Self Assessment
1. slant helix 2. general helix
3. tangent Bishop spherical image 4. M Bishop spherical image
1
5. M Bishop spherical image
2
21.9 Further Readings
Books Ahelfors, D.V. : Complex Analysis
Conway, J.B. : Function of one complex variable
Pati, T. : Functions of complex variable
Shanti Narain : Theory of function of a complex Variable
Tichmarsh, E.C. : The theory of functions
H.S. Kasana : Complex Variables theory and applications

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