I want to randomly sample points that have unit norm and lie on a n-dimensional quadratic form
$x^T Ax +b^T x +c =0 $.
$\|x\|=1$
Here, $x$ is a n-vector. $A \in R^{n \times n}$ is a real, symmetric matrix with both positive and negative real eigenvalues (in the general case). When $A$ only has positive eigenvalues, the quadratic form reduces to an n-dimensional ellipsoid. $b$ is a real n-vector and $c$ is a scalar. Of course in some modified coordinate system $x_1= V_e (x-x_0)$, where $V_e$ are the real eigenvectors of $A$, the above equation can be rewritten as
$x_1^T D x_1= d$
Where $D$ is a diagonal matrix with the eigenvalues (all real) of square, symmetric matrix $A$ on the diagonal and $x_0$ is a real vector ( the center of the quadratic form) .
Note that, now, the problem becomes sampling solutions $x_1$ to the quadratic form such that $\|x\|=1$.
I am familiar with sampling points from an n-sphere. Is it possible to somehow 'rotate ' points sampled from a n-sphere to this surface ?