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Difference between revisions of "Bianchi transformation"

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The transition from one focal surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016040/b0160401.png" /> of a Bianchi congruence to the other focal surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016040/b0160402.png" /> of the same congruence (cf. [[Bianchi congruence|Bianchi congruence]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016040/b0160403.png" /> is a pseudo-sphere then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016040/b0160404.png" /> also is a pseudo-sphere. The pseudo-spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016040/b0160405.png" /> which are Bianchi transforms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016040/b0160406.png" /> are orthogonal trajectories of the following congruence of circles. They are situated in the tangent plane to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016040/b0160407.png" /> and have the same radius as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016040/b0160408.png" />.
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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 $S$.
  
  

Latest revision as of 17:28, 1 August 2014

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 $S$.


Comments

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

References

[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)
How to Cite This Entry:
Bianchi transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bianchi_transformation&oldid=17519
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article