Quadrics are classified up to isometries of (translations and orthogonal transformations about the origin). When is invertible, completing the square with gives

so becomes . Diagonalising along the principal axes, the eigenvalues of together with determine the shape:

• all eigenvalues and : an ellipsoid;
• eigenvalues of both signs and : a hyperboloid;
• one or more zero eigenvalues: linear terms remain and require separate analysis.

For in :

• If , then defines an ellipsoid.
• If , the eigenvalues are , with orthonormal eigenvectors , , . The equation , i.e. , defines a two-sheeted hyperboloid, and a one-sheeted hyperboloid.