Namespaces
Variants
Actions

Darboux surfaces

From Encyclopedia of Mathematics
Jump to: navigation, search


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

d \mathbf z _ {i} = [ \mathbf z _ {i + 1 } , d \mathbf x _ {i} ] ,\ \ d \mathbf x _ {i} = [ \mathbf x _ {i - 1 } , d \mathbf z _ {i} ] ,

\mathbf z _ {i} - \mathbf x _ {i + 1 } = [ \mathbf z _ {i+ 1 } , \mathbf x _ {i} ],\ i = 1 \dots 6 ,

\mathbf x _ {i + 6 } = \mathbf x _ {i} ,\ \mathbf z _ {i + 6 } = \mathbf z _ {i} ;

where \mathbf z _ {i+} 1 and \mathbf x _ {i} are in Peterson correspondence, \mathbf z _ {i+} 1 and \mathbf x _ {i-} 1 are in polar correspondence, while \mathbf z _ {i} and \mathbf x _ {i+} 1 are poles of a W - 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 W - 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
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article