View source for 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|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 $S$.



====Comments====
Cf. also [[#References|[a2]]], articles 803, 804 in volume III.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.P. Eisenhart,   "A treatise on the differential geometry of curves and surfaces" , Boston  (1909)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Darboux,   "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''1–4''' , Chelsea, reprint  (1972)</TD></TR></table>