Darboux surfaces
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) |
Darboux surfaces. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux_surfaces&oldid=46581