Ribaucour congruence

A congruence of lines whose developable surfaces cut its mean surface by a conjugate net of lines. Let $ S $ be the mean surface of a Ribaucour congruence. Then there is a family of surfaces corresponding to $ S $ by the orthogonality of the line elements, and having in each pair of corresponding points a normal parallel to a ray of the congruence. Conversely, if a pair of surfaces $ S $ and $ \widetilde{S} $ is given that correspond to each other by the orthogonality of the line elements, then the congruences formed by the rays passing through the points on $ S $ and collinear to the normals of $ \widetilde{S} $ at corresponding points are a Ribaucour congruence with mean surface $ S $. The surface $ \widetilde{S} $ is called the generating surface of the Ribaucour congruence. The curvature lines of $ \widetilde{S} $ correspond to those generating surfaces of the congruence whose lines of contraction intersect the ray in the centre. The developable surfaces of a Ribaucour congruence correspond to the asymptotic lines of the generating surface $ \widetilde{S} $. The generating surface of a normal Ribaucour congruence is a minimal surface. This type of congruence is formed by the normals of a surface with the isothermic spherical image of curvature lines.

Such congruences were examined for the first time by A. Ribaucour in 1881.

References

[1]	S.P. Finikov, "Projective-differential geometry" , Moscow-Leningrad (1937) (In Russian)
[2]	S.P. Finikov, "Theorie der Kongruenzen" , Akademie Verlag (1959) (Translated from Russian)