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 <math>F_{1}</math> and <math>F_{2}</math> in bipolar coordinates become a ring of radius <math>a</math> in the <math>xy</math> plane of the toroidal coordinate system; the <math>z</math>-axis is the axis of rotation.

1 Basic definition
2 Scale factors
3 Toroidal Harmonics
4 Applications
5 References

[edit] Basic definition

The most common definition of toroidal coordinates <math>(\sigma, \tau, \phi)</math> is

<math>

x = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma} \cos \phi </math>

<math>

y = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma} \sin \phi </math>

<math>

z = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma} </math>

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

<math>

\tau = \ln \frac{d_{1}}{d_{2}} </math>

Surfaces of constant <math>\sigma</math> correspond to spheres of different radii

<math>

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

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

<math>

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

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

[edit] Scale factors

The scale factors for the toroidal coordinates <math>\sigma</math> and <math>\tau</math> are equal

<math>

h_{\sigma} = h_{\tau} = \frac{a}{\cosh \tau - \cos\sigma} </math>

whereas the azimuthal scale factor equals

<math>

h_{\phi} = \frac{a \sinh \tau}{\cosh \tau - \cos\sigma} </math>

Thus, the infinitesimal volume element equals

<math>

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

and the Laplacian is given by

<math>

\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] </math>

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

[edit] Toroidal Harmonics

The 3-variable Laplace equation

<math>\nabla^2\Psi=0</math>

admits solution via separation of variables in toroidal coordinates. The general solution is given in terms of complex exponentials in <math>\,\!\phi</math> and <math>\,\!\sigma</math>, and Legendre functions of the first and second kind with odd-half-integer degree, integer order, and argument real and greater than one, for <math>\,\!\tau</math>,

<math>\Psi=\sqrt{\cosh\tau-\cos\sigma}E(\tau)e^{i[\ell\sigma+m\phi]}</math>.

The associated Legendre functions <math>P_{n-\frac12}^m(\cosh\tau),\,Q_{n-\frac12}^m(\cosh\tau)</math> provide a basis of solutions for <math>\,\!E(\tau)</math>. These Legendre functions are often referred to as toroidal harmonics.

Toroidal harmonics have many interesting properties. If you make a variable substitution <math>\,\!1<z=\cosh\eta\,</math> then, for instance, with vanishing order (the convention is to not write the order when it vanishes) and <math>\,\!n=0</math>

<math>Q_{-\frac12}(z)=\sqrt{\frac{2}{1+z}}K\left(\sqrt{\frac{2}{1+z}}\right)</math>

and

<math>P_{-\frac12}(z)=\frac{2}{\pi}\sqrt{\frac{2}{1+z}}E \left( \sqrt{\frac{z-1}{z+1}} \right)</math>

where <math>\,\!K</math> and <math>\,\!E</math> are the complete elliptic integrals of the first and second kind respectively. The rest of the toroidal harmonics can be obtained, for instance, in terms of the complete elliptic integrals, by using recurrence relations for associated Legendre functions.

Toroidal harmonics show an interesting property under index interchange given by the Whipple transformations.

[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.