Toroidal coordinates

The numbers $ \sigma $, $ \tau $ and $ \phi $ related to the Cartesian rectangular coordinates $ x $, $ y $ and $ z $ by the formulas:

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

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

where $ - \pi \leq \sigma \leq \pi $, $ 0 \leq \tau < \infty $, $ 0 \leq \phi < 2 \pi $. The coordinate surfaces $ \sigma = \textrm{ const } $ are spheres with centre $ ( 0, 0, a \mathop{\rm cot} \sigma ) $ and radius $ a/| \sin \sigma | $; the surfaces $ \tau = \textrm{ const } $ are tori with axial circle in the $ Oxy $- plane, centre at the origin and radius $ a \mathop{\rm coth} \tau $, while the circle of the transverse cross section has radius $ a/ \sinh \tau $; the surfaces $ \phi = \textrm{ const } $ are the half-planes $ y/x = \mathop{\rm tan} \phi $. The system of toroidal coordinates is orthogonal.

The Lamé coefficients are:

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

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

The Laplace operator is:

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

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