Bianchi transformation

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The transition from one focal surface $S$ of a Bianchi congruence to the other focal surface $S'$ of the same congruence (cf. Bianchi congruence). If $S$ is a pseudo-sphere then $S'$ also is a pseudo-sphere. The pseudo-spheres $S'$ which are Bianchi transforms of $S$ are orthogonal trajectories of the following congruence of circles. They are situated in the tangent plane to $S$ and have the same radius as $S$.


Cf. also [a2], articles 803, 804 in volume III.


[a1] L.P. Eisenhart, "A treatise on the differential geometry of curves and surfaces" , Boston (1909)
[a2] G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1–4 , Chelsea, reprint (1972)
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Bianchi transformation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article