Darboux quadric

A second-order surface with second-order contact with a surface $S$ in three-dimensional projective space $P_3$ at a point $x$, in which the line of intersection with the surface $S$ at the point $x$ has a special type of singularity. Out of the set of quadrics with second-order contact with $S$ at $x$ one can select the quadrics in which the line of intersection with $S$ has a singular point $x$ with three coincident tangents. On the surface $S$ there are three directions (Darboux directions) for these three coincident tangents. At $x\in S$ there exists a one-parameter family of Darboux quadrics — the Darboux pencil. A pencil of hyper-quadrics, which are in contact at a point $x$ with a hypersurface $S$ in projective space $P_{n+1}$, is an extension of the Darboux pencil. A (non-developable) hypersurface $S$ degenerates into a hyper-quadric if and only if its generalized Darboux tensor vanishes [2].

References

[1]	S.P. Finikov, "Projective-differential geometry" , Moscow-Leningrad (1937) (In Russian)
[2]	G.F. Laptev, "Differential geometry of imbedded manifolds. Group theoretical methods of differential geometric investigation" Trudy Moskov. Mat. Obshch. , 2 (1953) pp. 275–382 (In Russian)
[a1]	E. Cartan, "Leçons sur la théorie des espaces à connexion projective" , Gauthier-Villars (1937) pp. Part II, Chapt. VI §II