Elliptic coordinates
From Encyclopedia of Mathematics
Two numbers 
and 
connected with rectangular Cartesian coordinates by the formulas
 |
 |
where 
.
Figure: e035440a
The coordinate lines are (see Fig.): confocal ellipses (
) and hyperbolas (
) with foci (
) and (
). The system of elliptic coordinates is orthogonal. To every pair of numbers 
and 
correspond four points, one in each quadrant of the 
-plane.
The Lamé coefficients are
 |
 |
In elliptic coordinates the Laplace equation allows separation of variables.
Degenerate elliptic coordinates are two numbers 
and 
connected with 
and 
by the formulas (for 
, 
):
 |
and with Cartesian coordinates 
and 
by
 |
where 
and 
. Occasionally these coordinates are also called elliptic.
The Lamé coefficients are:
 |
The area element is:
 |
The Laplace operator is:
 |
Comments
References
| [a1] | G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1 , Gauthier-Villars (1887) pp. 1–18 |
How to Cite This Entry:
Elliptic coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elliptic_coordinates&oldid=19184
Elliptic coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elliptic_coordinates&oldid=19184
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article