# Elliptic coordinates

Two numbers $\sigma$ and $\tau$ connected with rectangular Cartesian coordinates by the formulas

$$x ^ {2} = \frac{( \sigma + a ^ {2} ) ( \tau + a ^ {2} ) }{a ^ {2} - b ^ {2} } ,$$

$$y ^ {2} = \frac{( \sigma + b ^ {2} ) ( \tau + b ^ {2} ) }{b ^ {2} - a ^ {2} } ,$$

where $- a ^ {2} < \tau < - b ^ {2} < \sigma < \infty$.

Figure: e035440a

The coordinate lines are (see Fig.): confocal ellipses ( $\sigma = \textrm{ const }$) and hyperbolas ( $\tau = \textrm{ const }$) with foci ( $- \sqrt {a ^ {2} - b ^ {2} } , 0$) and ( $\sqrt {a ^ {2} - b ^ {2} } , 0$). The system of elliptic coordinates is orthogonal. To every pair of numbers $\sigma$ and $\tau$ correspond four points, one in each quadrant of the $xy$- plane.

The Lamé coefficients are

$$L _ \sigma = \frac{1}{2} \sqrt { \frac{\sigma - \tau }{( \sigma + a ^ {2} ) ( \tau + b ^ {2} ) } } ,$$

$$L _ \tau = \frac{1}{2} \sqrt { \frac{\tau - \sigma }{( \sigma - a ^ {2} ) ( \tau + b ^ {2} ) } } .$$

In elliptic coordinates the Laplace equation allows separation of variables.

Degenerate elliptic coordinates are two numbers $\widetilde \sigma$ and $\widetilde \tau$ connected with $\sigma$ and $\tau$ by the formulas (for $a = 1$, $b = 0$):

$$\sigma = \sinh ^ {2} \widetilde \sigma ,\ \ \tau = - \sin ^ {2} \widetilde \tau ,$$

and with Cartesian coordinates $x$ and $y$ by

$$x = \cosh \widetilde \sigma \cos \widetilde \tau ,\ \ y = \sinh \widetilde \sigma \sin \widetilde \tau ,$$

where $0 \leq \widetilde \sigma < \infty$ and $0 \leq \widetilde \tau < 2 \pi$. Occasionally these coordinates are also called elliptic.

The Lamé coefficients are:

$$L _ {\widetilde \sigma } = L _ {\widetilde \tau } = \ \sqrt {\cosh ^ {2} \widetilde \sigma - \cos ^ {2} \widetilde \tau } .$$

The area element is:

$$d s = ( \cosh ^ {2} \widetilde \sigma - \cos ^ {2} \widetilde \tau ) d \widetilde \sigma d \widetilde \tau .$$

The Laplace operator is:

$$\Delta \phi = \frac{1}{\cosh ^ {2} \widetilde \sigma - \cos ^ {2} \widetilde \tau } \left ( \frac{\partial ^ {2} \phi }{\partial \widetilde \sigma ^ {2} } + \frac{\partial ^ {2} \phi }{\partial \widetilde \tau ^ {2} } \right ) .$$