Is there a simple way to decide if a hyperboloid is one-sheeted or two-sheeted, given the quadric equation? Let us say that we have a quadric equation, whose solution set lies in $\mathbb{R}^3$, and you know it's a hyperboloid. Is there a way to analytically decide through a criterion if the hyperboloid is one-sheeted or two sheeted? I know about the caveman method, namely finding the rotational axis of symmetry and deciding whether the conic section you revoluted was spinned around the transverse or conjugate axis. This is however a lot of work and for me only possible in some very nice cases where it's particularly easy, not every time. Is there any criterion that immediately tells me if it is one-sheeted or two-sheeted, like in $\mathbb{R}^2$, where you can immediately tell if the solution set of a quadric is an ellipse or a hyperboloid? Any help appreciated!

If you can put your equation in to this form

$\mathbf x^T A \mathbf x = K$ and since A is a symmetric matrix, you can diagonalize it, and not only diagonalize it, diagonalize it with ortho-normal P.

$\mathbf x^T P^T D P\mathbf x = K\\\ (P\mathbf x)^T D P\mathbf x = K$

The matrix $P$ represents a change to the coordinate system -- a basis change. And since $P$ is ortho-normal it is a basis change that is limited to reflections and rotations i.e. distance and volumes are preserved. It doesn't change the nature of the conic section.

$\mathbf y = P\mathbf x$ and $\mathbf y^T D \mathbf y = K$