Lines of Curvature on Quadric Hypersurfaces of ${\mathbb{R}}^4$

Here are described the geometric structures of the lines of principal curvature and the partially umbilic singularities of the tridimensional non compact generic quadric hypersurfaces of ${\mathbb{R}}^4$. This includes the ellipsoidal hyperboloids of one and two sheets and the toroidal hyperboloids. The present study complements the analysis of the compact ellipsoidal hypersurfaces carried out by D. Lopes, R. Garcia and J. Sotomayor.