Darboux surfaces
From Encyclopedia of Mathematics
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wreath of
Surfaces associated with an infinitesimal deformation of one of them; discovered by G. Darboux [1]. Darboux surfaces form a "wreath" of 12 surfaces, with radius vectors 
satisfying the equations
 |
 |
 |
where 
and 
are in Peterson correspondence, 
and 
are in polar correspondence, while 
and 
are poles of a 
-congruence. A similar "wreath" is formed by pairs of isometric surfaces of an elliptic space.
References
| [1] | G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 4 , Gauthier-Villars (1896) |
| [2] | V.I. Shulikovskii, "Classical differential geometry in a tensor setting" , Moscow (1963) (In Russian) |
Comments
For the notion of a 
-congruence cf. Congruence of lines.
References
| [a1] | G. Fubini, E. Čech, "Introduction á la géométrie projective différentielle des surfaces" , Gauthier-Villars (1931) |
| [a2] | G. Bol, "Projective Differentialgeometrie" , Vandenhoeck & Ruprecht (1954) |
How to Cite This Entry:
Darboux surfaces. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux_surfaces&oldid=46581
Darboux surfaces. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux_surfaces&oldid=46581
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article