What is a basic way to simulate points on the "level surfaces" of a multivariate normal distribution? That is, for a mean vector $\mu$ and covariance matrix $\Sigma$, how do we generate $X\sim N(\mu,\Sigma)$ such that the PDF $f_{\mu,\Sigma}(X)$ is equal to a constant $c$?
Given the form of $f_{\mu,\Sigma}$ this boils down to generating points $X\sim N(\mu,\Sigma)$ that satisfy $(X-\mu)^T\Sigma^{-1}(X-\mu)=c$.
What if we instead have a multivariate $t$-distribution? I imagine the algorithm would be similar because the level surfaces of a multivariate $t$-distribution are also defined by $(X-\mu)^T\Sigma^{-1}(X-\mu)=c$.