Elliptic coordinates

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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:



[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. D.D. Sokolov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098