Fundamental matrix estimation becomes ill-posed in degenerate two-view configurations. While practical tests exist for planar scenes and pure rotation, detecting more general critical surfaces—such as hyperboloids, cones, or cylinders containing the 3D points and camera centers—remains challenging. In this talk, I will present a new degeneracy test based on the geometry of homaloidal nets of conics. This viewpoint allows us to model the image-to-image relation induced by a critical surface as a quadratic transformation, which can be estimated linearly from point correspondences. Unlike previous methods, the test does not require a pre-computed fundamental matrix, making it more stable in precisely the cases where epipolar estimation is ill-posed.