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Ellipsoidal coordinates

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spatial elliptic coordinates

The numbers , and connected with Cartesian rectangular coordinates , and by the formulas

where . The coordinate surfaces are (see Fig.): ellipses , one-sheet hyperbolas (), and two-sheet hyperbolas (), with centres at the coordinate origin.

Figure: e035420a

The system of ellipsoidal coordinates is orthogonal. To every triple of numbers , and correspond 8 points (one in each octant), which are symmetric to each other relative to the coordinate planes of the system .

The Lamé coefficients are

If one of the conditions in the definition of ellipsoidal coordinates is replaced by an equality, then degenerate ellipsoidal coordinate systems are obtained.


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:
Ellipsoidal coordinates. D.D. Sokolov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Ellipsoidal_coordinates&oldid=14024
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098