View source for Darboux quadric

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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|Darboux tensor]] vanishes [[#References|[2]]].

====References====
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<TR><TD valign="top">[1]</TD> <TD valign="top">  S.P. Finikov,   "Projective-differential geometry" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Cartan,   "Leçons sur la théorie des espaces à connexion projective" , Gauthier-Villars  (1937)  pp. Part II, Chapt. VI §II</TD></TR>
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