I know how to project a 3d ellipsoid onto a plane (e.g: Projection of ellipsoid).
I was wondering if there was some way of projecting onto the coordinate planes, an $n$-dimensional hyperellipsoid of the form $z^TCz = k$, where $z$ is a vector of $n$ coordinates, $C$ is an $n$x$n$ symmetric positive definite matrix and $k$ is a positive constant.