View source for Ribaucour congruence

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A [[Congruence of lines|congruence of lines]] whose developable surfaces cut its mean surface by a [[Conjugate net|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====
<table><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">  S.P. Finikov,   "Theorie der Kongruenzen" , Akademie Verlag  (1959)  (Translated from Russian)</TD></TR></table>