# Darboux surfaces

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wreath of

Surfaces associated with an infinitesimal deformation of one of them; discovered by G. Darboux . Darboux surfaces form a "wreath" of 12 surfaces, with radius vectors $\mathbf x _ {1} \dots \mathbf x _ {6} , \mathbf z _ {1} \dots \mathbf z _ {6}$ 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.

How to Cite This Entry:
Darboux surfaces. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux_surfaces&oldid=18810
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article