View source for Laplace transformation (in geometry)

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057550/l0575501.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057550/l0575502.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057550/l0575503.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057550/l0575504.png" /> are known functions of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057550/l0575505.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057550/l0575506.png" />, 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>