Ellipsoidal coordinates

spatial elliptic coordinates

The numbers $ \lambda $, $ \mu $ and $ \nu $ connected with Cartesian rectangular coordinates $ x $, $ y $ and $ z $ by the formulas

$$ x ^ {2} = \frac{( \lambda + a ^ {2} ) ( \mu + a ^ {2} ) ( \nu + a ^ {2} ) }{( b ^ {2} - a ^ {2} ) ( c ^ {2} - a ^ {2} ) } , $$

$$ y ^ {2} = \frac{( \lambda ^ {2} + b ^ {2} ) ( \mu ^ {2} + b ^ {2} ) ( \nu + b ^ {2} ) }{( a ^ {2} - b ^ {2} ) ( c ^ {2} - b ^ {2} ) } , $$

$$ z ^ {2} = \frac{( \lambda + c ^ {2} ) ( \mu + c ^ {2} ) ( \mu + c ^ {2} ) }{( a ^ {2} - c ^ {2} ) ( b ^ {2} - c ^ {2} ) } , $$

where $ - a ^ {2} < \nu < - b ^ {2} < \mu < - c ^ {2} < \lambda < \infty $. The coordinate surfaces are (see Fig.): ellipses $ ( \lambda = \textrm{ const } ) $, one-sheet hyperbolas ( $ \mu = \textrm{ const } $), and two-sheet hyperbolas ( $ \nu = \textrm{ const } $), with centres at the coordinate origin.

Figure: e035420a

The system of ellipsoidal coordinates is orthogonal. To every triple of numbers $ \lambda $, $ \mu $ and $ \nu $ correspond 8 points (one in each octant), which are symmetric to each other relative to the coordinate planes of the system $ O x y z $.

The Lamé coefficients are

$$ L _ \lambda = \frac{1}{2} \sqrt { \frac{( \lambda - \mu ) ( \mu - \nu ) }{( \lambda + a ^ {2} ) ( \lambda + b ^ {2} ) ( \lambda + c ^ {2} ) } } , $$

$$ L _ \mu = \frac{1}{2} \sqrt { \frac{( \lambda - \mu ) ( \nu - \mu ) }{( \mu + a ^ {2} ) ( \mu + b ^ {2} ) ( \mu + c ^ {2} ) } } , $$

$$ L _ \nu = \frac{1}{2} \sqrt { \frac{( \lambda - \nu ) ( \mu - \nu ) }{( \nu + a ^ {2} ) ( \nu + b ^ {2} ) ( \nu + c ^ {2} ) } } . $$

If one of the conditions $ a ^ {2} > b ^ {2} > c ^ {2} > 0 $ 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