Laplace transformation (in geometry)

The transition from one 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) and line geometry. There are various generalizations of the Laplace transformation of a net.

References

[1]	G. Tzitzeica, "Géométrie différentielle projective des réseaux" , Gauthier-Villars & Acad. Roumaine (1924)
[2]	V.T. Bazylev, "Multidimensional nets and their transformations" Itogi Nauk. Geom. 1963 (1965) pp. 138–164 (In Russian)

Comments

Cf. also Laplace sequence.

References

[a1]	S.P. Finikov, "Theorie der Kongruenzen" , Akademie Verlag (1959) (Translated from Russian)