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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
  
<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>
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$$\frac{\partial^2\theta}{\partial u\partial v}=a\frac{\partial\theta}{\partial u}+b\frac{\partial\theta}{\partial v}+c\theta,$$
  
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.
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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
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article