# Ellipsoidal coordinates

*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

Laplace's equation expressed in ellipsoidal coordinates is separable (cf Separation of variables, method of), and leads to Lamé functions.

#### 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 |

[a2] | Harold Jeffreys, Bertha Jeffreys, Methods of Mathematical Physics, 3rd edition, Cambridge University Press (1972) Zbl 0238.00004 |

**How to Cite This Entry:**

Ellipsoidal coordinates.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Ellipsoidal_coordinates&oldid=34027