tetrahedral-cartesian transform

From: Timothy Golden (tttpppggg_at_yahoo.com)
Date: 12/04/04 

Date: 4 Dec 2004 15:35:22 -0800

There are four unit vectors S0, S1, S2, S3.
They are rays from the center of a tetrahedron to its corners.
This exists in 3D cartesian space( X0, X1, X2 ).
Aligning the center of the tetrahedron with the origin, and letting X0
be aligned with S0, then setting S1 in the X0-X1 plane toward X1, then
setting S2 closest to the X2 unit vector I get the following:

   X0 = S0 - (1/3)( S1 + S2 + S3 ).

   X1 = v( S1 - (1/2)(S2 + S3 )).

   X2 = uv( S2 - S3 ).

   where u = sqrt( 1 - 1/9 )
      and v = sqrt( 1 - 1/4 ).

Can you verify this or point out where I am wrong please?

This is important to me because I am seeing a phenomenon in polysigned
numbers where:
   |AB| = (|A|)(|B|)
does not hold past three signs. However I am not yet sure that it is
not my magnitude function, which first generates a cartesian vector
through the above puzzle then a pythagorean magnitude. Thanks in
advance for your help.

Tim Golden

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