# 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.