[R] random uniform sample of points on an ellipsoid (e.g. WG

[(Ted Harding)]

On 01-Mar-07 R Heberto Ghezzo, Dr wrote:
> 
> Hello,
> I do not know if I am completely out of it but . . .
> if x,y,z is a point in a sphere and [u,v,w]'A[u,v,w] = 1 is the
> equation of an ellipsoid and A = T'T (cholesky) then
> T.[x,y,z] should be a point in the ellipsoid ? isn't it?
> Heberto Ghezzo
> McGill University
> Montreal Canada

Yes, but the issue in this thread is to generate a sample of
points uniformly distributed over the surface of the ellipsoid.

It is easy to get points uniformly distributed over the surface
of a sphere; but if your transformation (or equivalent) is
applied to these points, the resulting points on the surface
of the ellipsoid are not uniformly distributed!

For example, if the transformation squashes the sphere vertically
(so that its vertical axis is shorter than its horizontal axis),
then the resulting points near the "equator" of the ellipsoid will
have a higher density per unit area than the points near the "poles".

Best wishes,
Ted.

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Date: 01-Mar-07                                       Time: 14:16:59
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