Bispherical coordinates
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Bispherical coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that connects the two foci. Thus, the two foci <math>F_{1}</math> and <math>F_{2}</math> in bipolar coordinates remain points (on the <math>z</math>-axis, the axis of rotation) in the bispherical coordinate system.
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[edit] Basic definition
The most common definition of bispherical coordinates <math>(\sigma, \tau, \phi)</math> is
- <math>
x = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma} \cos \phi </math>
- <math>
y = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma} \sin \phi </math>
- <math>
z = a \ \frac{\sinh \tau}{\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 the foci
- <math>
\tau = \ln \frac{d_{1}}{d_{2}} </math>
Surfaces of constant <math>\sigma</math> correspond to intersecting tori of different radii
- <math>
z^{2} + \left( \sqrt{x^{2} + y^{2}} - a \cot \sigma \right)^{2} = \frac{a^{2}}{\sin^{2} \sigma} </math>
that all pass through the foci but are not concentric. The surfaces of constant <math>\tau</math> are non-intersecting spheres of different radii
- <math>
\left( x^{2} + y^{2} \right) + \left( z - a \coth \tau \right)^{2} = \frac{a^{2}}{\sinh^{2} \tau} </math>
that surround the foci. The centers of the constant-<math>\tau</math> spheres lie along the <math>z</math>-axis, whereas the constant-<math>\sigma</math> tori are centered in the <math>xy</math> plane.
[edit] Scale factors
The scale factors for the bispherical 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 \sin \sigma}{\cosh \tau - \cos\sigma} </math>
Thus, the infinitesimal volume element equals
- <math>
dA = \frac{a^{3}\sin \sigma}{\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}\sin \sigma} \left[ \frac{\partial}{\partial \sigma} \left( \frac{\sin \sigma}{\cosh \tau - \cos\sigma} \frac{\partial \Phi}{\partial \sigma} \right) + \sin \sigma \frac{\partial}{\partial \tau} \left( \frac{1}{\cosh \tau - \cos\sigma} \frac{\partial \Phi}{\partial \tau} \right) + \frac{1}{\sin \sigma \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)</math> by substituting the scale factors into the general formulae found in orthogonal coordinates.
[edit] Applications
The classic applications of bispherical coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which bispherical coordinates allow a separation of variables. A typical example would be the electric field surrounding two conducting spheres of different radii.
[edit] References
- Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill.
