Demoulin quadrilateral

A tetragon formed by two pairs $l_1,l_1'$ and $l_2,l_2'$ of rectilinear generators of the Lie quadric at a hyperbolic point $M$ of a surface in three-dimensional (projective) space. The straight lines $l_1,l_1',l_2,l_2'$ are called Demoulin straight lines. The canonical tetrahedron $T(M,M_1,M_2,M_3)$ associated with the Lie quadric is known as the Demoulin tetrahedron. The Demoulin tetrahedron is non-degenerate if and only if the third Fubini form is non-degenerate. Studied by A. Demoulin [1].

Comments

In general, the Demoulin tetrahedron is defined at all points of a surface, not only in the case of negative curvature (a hyperbolic point), see [a1]. The main characterization of the vertices of the Demoulin tetrahedron is as follows: There are four envelopes for the set of Lie quadrics of a surface different from the surface itself. These surfaces meet the Lie quadrics exactly at the vertices of the Demoulin tetrahedron.

References

[1]	A. Demoulin, "Sur la théorie des lignes asymptotiques, etc." C.R. Acad. Sci. Paris , 147 (1908) pp. 493–496
[a1]	G. Bol, "Projective Differentialgeometrie" , 2 , Vandenhoeck & Ruprecht (1954)
[a2]	E.P. Lane, "A treatise on projective differential geometry" , Univ. Chicago Press (1942)