Page 252 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 21: New Spherical Indicatrices and their Characterizations

21.1 Preliminaries Notes

To meet the requirements in the next sections, here, the basic elements of the theory of curves in
the space E are briefly presented; a more complete elementary treatment can be found further.
3
The Euclidean 3-space E provided with the standard flat metric given by
3
2
2
2
,  dx  dx  dx ,
3
1
2
where (x , x , x ) is a rectangular coordinate system of E . Recall that, the norm of an arbitrary
3
1
2
3
3
vector a  E is given by a  a,a .  is called a unit speed curve if velocity vector v of 
3
satisfies v = 1. For vectors v, w  E it is said to be orthogonal if and only if v,w  0. Let
v = v(s) be a regular curve in E . If the tangent vector of this curve forms a constant angle with a
3
fixed constant vector U, then this curve is called a general helix or an inclined curve. The sphere
of radius r > 0 and with center in the origin in the space E is defined by
3
S = {p = (p ; p ; p ) E : p; p = r }.
2
3
2

3
1
2
Denote by {T, N, B} the moving Frenet-Serret frame along the curve  in the space E . For an
3
arbitrary curve  with first and second curvature, K and T in the space E , the following Frenet-
3
Serret formulae are written under matrix form
T
 T'   0 K 0   
     
N ,


 N'   K 0 T   
B
 B'   0  T 0   
     
where
T,T  N,N  B,B  1,
T,N = T,B = T,N = N,B = 0.
Here, curvature functions are defined by K = K(s) = T'(s) and T(s) = N,B .
Let u = (u , u , u ), v = (v , v , v ) and w = (w ,w ,w ) be vectors in E and e , e , e be positive
3
3
3
3
2
1
2
3
1
1
1
2
2
oriented natural basis of E . Cross product of u and v is defined by
3
e 1 e 2 e 3
u × v = u 1 u 2 u .
3
v 1 v 2 v 3
Mixed product of u; v and w is defined by the determinant
u 1 u 2 u 3
[u, v, w] = v 1 v 2 v .
3
w 1 w 2 w 3
Torsion of the curve  is given by the aid of the mixed product
 ', '', '''   
T  .
K 2
LOVELY PROFESSIONAL UNIVERSITY 245