First I would like to say that geometry is far away from my domain.

I have encountered a problem that has a geometric formulation and I don't even know if this is a difficult or an easy problem.

So here it is, let us be given an n-dimensional ellipsoid such that it is centered at the origin, its algebraic equation is :

$\sum_{i=1}^{n}\big(\frac{x_i}{\lambda_i}\big)^2=1$ with $\sum_{i=1}^{n}\frac{1}{\lambda_i^2}=n$.

What I am looking for is an explicit parametrization (in any coordinate system) of the surface (a curve for n=3) defined by the intersection of this ellipsoid with the unit n-dimensional sphere.

Ultimately, my goal is to be able to sample uniformly a point on this surface using this parametrization. I believe this problem is so common that it must have been solved somewhere but I couldn't find any reference.