Sampling uniformly from the surface of an ellipsoid (in the sense of $\mu(dA) = \frac{1}{A}$) seems very nontrivial:

• How to sample uniformly from the surface of a hyper-ellipsoid (constant Mahalanobis distance)?
• How to generate points uniformly distributed on the surface of an ellipsoid?
• Random uniform points on the surface of (hyper) ellipsoid

with most suggesting rejection sampling.

On the other hand, a common approach for sampling on the $d$-sphere uses the spherical symmetry of the standard multivariate Gaussian to use $\frac{v}{||v||}$. More generally, for a zero-mean multivariate Gaussian with covariance $\Sigma$, the ellipsoids $$ v^\intercal \Sigma^{-1}v = c^2 $$ are sets of constant density. Yet, the following naive approach does not work (I assume due to inhomogeneous change in the area element):

$v \sim \mathcal{N}(0, \Sigma)$, construct samples $x = \frac{v}{\sqrt{v^\intercal \Sigma^{-1}v}}$

Yet, can we not use the level sets of a multivariate gaussian somehow to sample an ellipsoid uniformly without rejection?