Separating two finite sets of points in a Euclidean space is a fundamental problem in classification. Customarily linear separation is used, but nonlinear separators such as spheres have been shown to have better performances in some tasks, such as edge detection in images. We exploit the relationships between the more general version of the spherical separation, where we use general ellipsoids, and the SVM model with quadratic kernel to propose a new classification approach. The implementation basically boils down to adding a SDP constraint to the standard SVM model in order to ensure that the chosen hyperplane in the feature space represents a non-degenerate ellipsoid in the input space; albeit being somewhat more costly than the original formulation, this still allows to exploit many of the techniques developed for SVR in combination with SDP approaches. We test our approach on several classification tasks, among which the edge detection problem for gray-scale images, proving that the approach is competitive with both the spherical classification one and the quadratic-kernel SVM one without the ellipsoidal restriction.