Difference between revisions of "Laplace transformation (in geometry)"
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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 | 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 | + | 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==== | ====References==== | ||
Latest revision as of 09:35, 27 April 2014
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) |
How to Cite This Entry:
Laplace transformation (in geometry). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Laplace_transformation_(in_geometry)&oldid=31946
Laplace transformation (in geometry). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Laplace_transformation_(in_geometry)&oldid=31946
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article