View source for Laplace transformation (in geometry)

{{TEX|done}}
The transition from one [[Focal net of a congruence|focal net of a congruence]] to another focal net of the same congruence. The concept of the Laplace transformation of a net was introduced by G. Darboux (1888), who discovered that an analytic transformation of the solutions of the Laplace equation

$$\frac{\partial^2\theta}{\partial u\partial v}=a\frac{\partial\theta}{\partial u}+b\frac{\partial\theta}{\partial v}+c\theta,$$

where $a$, $b$, $c$ are known functions of the variables $u$, $v$, can be interpreted geometrically as transition from one focal net of a congruence to another focal net of it. The Laplace transformation of nets establishes a correspondence between the theory of conjugate nets (cf. [[Conjugate net|Conjugate net]]) and line geometry. There are various generalizations of the Laplace transformation of a net.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Tzitzeica,   "Géométrie différentielle projective des réseaux" , Gauthier-Villars &amp; Acad. Roumaine  (1924)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.T. Bazylev,   "Multidimensional nets and their transformations"  ''Itogi Nauk. Geom. 1963''  (1965)  pp. 138–164  (In Russian)</TD></TR></table>



====Comments====
Cf. also [[Laplace sequence|Laplace sequence]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.P. Finikov,   "Theorie der Kongruenzen" , Akademie Verlag  (1959)  (Translated from Russian)</TD></TR></table>