From: posse@masg1.epfl.ch (Christian Posse)
Newsgroups: sci.math
Subject: Repost : special param. of 2 unit orthogonal vectors in R^n
Date: 15 Apr 92 15:14:35 GMT
Organization: Ecole Polytechnique Federale de Lausanne

Sorry to repost this file. There was a small mistake in the description
of vector B.
...


Dear Mathematicians on RN,

Could anyone help me in the following problem.

I am interested in a  parametrization of two orthogonal unit 
vectors in R^n.

Presentation
============

I consider two-dimensional planes in R^n which are defined by two unit 
orthogonal vectors, let's say A and B. So

	    ||A|| = 1 , ||B|| = 1  and A.B = 0. 

One way to parametrize A and B is the well-known :


				/   \
				| 1 |
				| 0 |
				| 0 |
				| . |
				| . |
A = Q_{n-1} Q_{n-2} ... Q_2 Q_1 |   |
				|   |
				|   |
				|   |
				| 0 |
				\   /


     /					        \
     |		   cos(u_1)		        |
     |	       sin(u_1)cos(u_2)		        |
     |	   sin(u_1)sin(u_2)cos(u_3)             |
     |					        |
     |					        |
   = |					        | 
     |					        |
     |					        |
     |sin(u_1)sin(u_2)..sin(u_{n-2})cos(u_{n-1})|
     |sin(u_1)sin(u_2)..sin(u_{n-2})sin(u_{n-1})|
     \					        /



				/		   0			   \
				|		cos(w_1)		   |
				|	    sin(w_1)cos(w[M_2)		   |
				|	 sin(w_1)sin(w_2)cos(w_3)          |
				|					   |
				|					   |
B = Q_{n-1} Q_{n-2} ... Q_2 Q_1 |					   | 
				|					   |
				|					   |
				|sin(w_1)sin(w_2)..sin(w_{n-3})cos(w_{n-2})|
				|sin(w_1)sin(w_2)..sin(w_{n-3})sin(w_{n-2})|
				\					   /


where

	/							\	
	|1							|
	|   1							|
	|	..						|
	|	    1						|
	|		cos(u_i)    -sin(u_i)			|
	|		sin(u_i)     cos(u_1)			| i = 1,...,n-1	
Q_i =   |					1		|
	|					    1		|
	|						..	|
	|						    1	|
	\							/

(blanks are zeros).

With this parametrization, we can show some interesting properties, but 
also other ones that I must avoid in all cases. Properties are like the
following :

Let d denote the partial derivative operator, A = (A_1,A_2,...,A_n) and
B= (B_1,B_2,...,B_n).

\sum{k=1,n} dA_k  A_k = 0, \sum{k=1,n} dB_k  B_k = 0  \sum{k=1,n} dB_k  B_k = 0	
	    du_i		       du_i                       dw_j


\sum{k=1,n} dA_k  dA_k   =  delta_Kronecker(i,j)
            du_i  du_j

\sum{k=1,n} dB_k  dB_k   =  delta_Kronecker(i,j)
            dw_i  dw_j

BUT this diagonalization does not work for the other combinations

\sum{k=1,n} dA_k  dB_k , \sum{k=1,n} dA_k  dB_k, \sum{k=1,n} dB_k  dB_k
            du_i  dw_j		     du_i  du_j              du_i  dw_j

Question
========

Does it exist a parametrization of A and B such that the following
properties are verified :

\sum{k=1,n} dA_k  dA_k   =  delta_Kronecker(i,j)
            du_i  du_j

\sum{k=1,n} dB_k  dB_k   =  delta_Kronecker(i,j)
            dw_i  dw_j

and

\sum{k=1,n} dA_k  dB_k	 =  0	 \sum{k=1,n} dA_k  dB_k  = 0  (if there are such
            du_i  dw_j			     dw_i  du_j	       derivatives)


\sum{k=1,n} dB_k  dB_k	 =  0
            du_i  dw_j	


\sum{k=1,n} dA_k  dB_k  =  0 or  delta_Kronecker(i,j) 
            du_i  du_j


And at least, where can I find good references concerning parametrizations
on the unit Sphere S^n in R^n ?


Many thanks in advance for your help and your time.

Christian

-------------------------------------------------------------------------------
      ____/  __  /  ____/   /   Christian POSSE
     /      /   /  /       /    Swiss Federal Institute of Technology
    ____/  ____/   ____/  /     DMA-Ecublens                                   
   /      /       /      /      CH-1015 Lausanne (Switzerland)
 _____/ _/      _/     _____/   
   			        e-mail : posse@masg6.epfl.ch

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