Toroidal coordinates

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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 $x y$ plane of the toroidal coordinate system; the $z$ -axis is the axis of rotation.

1 Basic definition
2 Scale factors
3 Applications
4 References

[edit] Basic definition

The most common definition of toroidal coordinates $(σ,τ,φ)$ 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 $σ$ coordinate of a point $P$ equals the angle $F 1 P F 2$ and the $τ$ 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 $σ$ 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 $τ$ 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- $σ$ spheres lie along the $z$ -axis, whereas the constant- $τ$ tori are centered in the $x y$ plane.

[edit] Scale factors

The scale factors for the toroidal coordinates $σ$ and $τ$ 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 $(σ,τ,φ)$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

[edit] Applications

The classic applications of toroidal coordinates are in solving partial differential equations, e.g., Laplace's equation for which toroidal coordinates allow a separation of variables or the Helmholtz equation, for which toroidal coordinates does not allow a separation of variables. A typical example would be the electric field surrounding a conducting ring.