View source for Ribaucour congruence

A [[Congruence of lines|congruence of lines]] whose developable surfaces cut its mean surface by a [[Conjugate net|conjugate net]] of lines. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081750/r0817501.png" /> be the mean surface of a Ribaucour congruence. Then there is a family of surfaces corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081750/r0817502.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081750/r0817503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081750/r0817504.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081750/r0817505.png" /> and collinear to the normals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081750/r0817506.png" /> at corresponding points are a Ribaucour congruence with mean surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081750/r0817507.png" />. The surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081750/r0817508.png" /> is called the generating surface of the Ribaucour congruence. The curvature lines of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081750/r0817509.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081750/r08175010.png" />. 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>