Randomly generate points uniformly on an ellipsoid in general position

Question

I want to uniformly generate points at random on the boundary of an ellipsoid in any dimension.

The method given by @elhuhdron here is very nice and it can be straightforwardly generalized to any dimension, but it is restricted to an ellipsoid in standard position (axes-aligned). How to do for the general position? Say the ellipsoid is defined by $x'Ax = r^2$. One can get the radii and sample on the axes-aligned ellipsoid having these radii but then it remains to rotate the simulated points.

So what is the rotation between an ellipsoid in general position and its "standard version"? (same radii and axes-aligned). That is the question.

I found the rotation here: https://math.stackexchange.com/a/2407719/38217. — Stéphane Laurent, May 19 '23 at 17:38
FYI. I have doubts that the method given by @elhuhdron in the link you provided is legit. (Use at own risk!)The problem is that surface area does not scale nicely under affine maps, in contrast to volume. The surface area of an ellipse is a notoriously messy quantity that involves elliptic function theory. Jacobi struggled with the triaxial case I think. — MathFont, May 24 '23 at 16:38
@MathWonk Thanks. I checked elhuhdron's method by evaluating the length of an elliptical arc with the help of simulations and this seemed to match very well the theoretical length. — Stéphane Laurent, May 24 '23 at 19:33
@MathWonk I checked in 3D and I'm afraid you're right: this method is not legit. — Stéphane Laurent, May 24 '23 at 21:43

Randomly generate points uniformly on an ellipsoid in general position

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