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)

References

[a1]	O. Veblen, J.H.C. Whitehead, "The foundations of differential geometry" , Cambridge Univ. Press (1932)