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 ${\mathbf{x}}_1\dots {\mathbf{x}}_6,{\mathbf{z}}_1\dots {\mathbf{z}}_6$ 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.

References[edit]

Comments[edit]

For the notion of a $W$- congruence cf. Congruence of lines.

References[edit]