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)|
Bianchi transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bianchi_transformation&oldid=32669