Bipolar coordinates

The numbers $\tau$ and $\sigma$ which are connected with the Cartesian orthogonal coordinates $x$ and $y$ by the formulas

$$x = \frac{a \sinh \tau }{\cosh \tau - \cos \sigma } ,\ \ y = \frac{a \sin \sigma }{\cosh \tau - \cos \sigma } ,$$

where $0 \leq \sigma < \pi , -\infty < \tau < \infty$. The coordinate lines are two families of circles $( \tau = \textrm{ const } )$ with poles $A$ and $B$ and the (half-c)ircles orthogonal with these $( \sigma = \textrm{ const } )$.

Figure: b016470a

The Lamé coefficients are:

$$L _ \tau = L _ \sigma = \ \frac{a ^ {2} }{( \cosh \tau - \cos \sigma ) ^ {2} } .$$

The Laplace operator is:

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

Bipolar coordinates in space (bispherical coordinates) are the numbers $\sigma , \tau$ and $\phi$, which are connected with the orthogonal Cartesian coordinates $x, y$ and $z$ by the formulas:

$$x = \frac{a \sin \sigma \cos \phi }{\cosh \tau - \cos \sigma } ,$$

$$y = \frac{a \sin \sigma \sin \phi }{\cosh \tau - \cos \sigma } ,$$

$$z = \frac{a \sinh \tau }{\cosh \tau - \cos \sigma } ,$$

where $- \infty < \sigma < \infty , 0 \leq \tau < \pi , 0 \leq \phi < 2 \pi$. The coordinate surfaces are spheres ( $\sigma = \textrm{ const }$), the surfaces obtained by rotation of arcs of circles ( $\tau = \textrm{ const }$) and half-planes passing through the $Oz$- axis. The system of bipolar coordinates in space is formed by rotating the system of bipolar coordinates on the plane $Oxy$ around the $Oz$- axis.

The Lamé coefficients are:

$$L _ \sigma = \ L _ \tau = \ \frac{a ^ {2} }{( \cosh \tau - \cos \sigma ) ^ {2} } ,$$

$$L _ \phi = \frac{a ^ {2} \sin ^ {2} \sigma }{( \cosh \tau - \cos \sigma ) ^ {2} } .$$

The Laplace operator is:

$$\Delta f = \ \frac{( \cosh \tau - \cosh \sigma ) ^ {3} }{a ^ {2} \sin \sigma } \left [ \frac \partial {\partial \tau } \left ( \frac{\sin \sigma }{\cosh \tau - \cos \sigma } \frac{\partial f }{\partial \tau } \ \right )\right . +$$

$$+ \left . \frac \partial {\partial \sigma } \left ( \frac{\sin \sigma }{\cosh \tau - \cos \sigma } \frac{\partial f }{\partial \sigma } \right ) + \frac{1}{\sin \sigma ( \cosh \tau - \cos \sigma ) } \frac{\partial ^ {2} f }{\partial \phi ^ {2} } \right ] .$$

References

 [1] E. Madelung, "Die mathematischen Hilfsmittel des Physikers" , Springer (1957)