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:

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

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

How to Cite This Entry:
Toroidal coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Toroidal_coordinates&oldid=48996
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article