Toroidal coordinates

Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci $F_{1}$ and $F_{2}$ in bipolar coordinates become a ring of radius $a$ in the $xy$ plane of the toroidal coordinate system; the $z$-axis is the axis of rotation.

Basic definition

The most common definition of toroidal coordinates $(\sigma ,\tau ,\phi )$ is

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

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

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

where the $\sigma$ coordinate of a point $P$ equals the angle $F_{1}PF_{2}$ and the $\tau$ coordinate equals the natural logarithm of the ratio of the distances $d_{1}$ and $d_{2}$ to opposite sides of the focal ring

$\tau =\ln {\frac {d_{1}}{d_{2}}}$

Surfaces of constant $\sigma$ correspond to spheres of different radii

$\left(x^{2}+y^{2}\right)+\left(z-a\cot \sigma \right)^{2}={\frac {a^{2}}{\sin ^{2}\sigma }}$

that all pass through the focal ring but are not concentric. The surfaces of constant $\tau$ are non-intersecting tori of different radii

$z^{2}+\left({\sqrt {x^{2}+y^{2}}}-a\coth \tau \right)^{2}={\frac {a^{2}}{\sinh ^{2}\tau }}$

that surround the focal ring. The centers of the constant-$\sigma$ spheres lie along the $z$-axis, whereas the constant-$\tau$ tori are centered in the $xy$ plane.

Scale factors

The scale factors for the toroidal coordinates $\sigma$ and $\tau$ are equal

$h_{\sigma }=h_{\tau }={\frac {a}{\cosh \tau -\cos \sigma }}$

whereas the azimuthal scale factor equals

$h_{\phi }={\frac {a\sinh \tau }{\cosh \tau -\cos \sigma }}$

Thus, the infinitesimal volume element equals

$dA={\frac {a^{3}\sinh \tau }{\left(\cosh \tau -\cos \sigma \right)^{3}}}d\sigma d\tau d\phi$

and the Laplacian is given by

$\nabla ^{2}\Phi ={\frac {\left(\cosh \tau -\cos \sigma \right)^{3}}{a^{2}\sinh \tau }}\left[\sinh \tau {\frac {\partial }{\partial \sigma }}\left({\frac {1}{\cosh \tau -\cos \sigma }}{\frac {\partial \Phi }{\partial \sigma }}\right)+{\frac {\partial }{\partial \tau }}\left({\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}{\frac {\partial \Phi }{\partial \tau }}\right)+{\frac {1}{\sinh \tau \left(\cosh \tau -\cos \sigma \right)}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}\right]$

Other differential operators such as $\nabla \cdot \mathbf {F}$ and $\nabla \times \mathbf {F}$ can be expressed in the coordinates $(\sigma ,\tau ,\phi )$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

Applications

The classic applications of toroidal coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which toroidal coordinates allow a separation of variables. A typical example would be the electric field surrounding a conducting ring.

References

• Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill.