Bispherical coordinates

From Wikipedia, the free encyclopedia

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 $F 1$ and $F 2$ in bipolar coordinates remain points (on the $z$ -axis, the axis of rotation) in the bispherical coordinate system.

1 Basic definition
2 Scale factors
3 Applications
4 References

[edit] Basic definition

The most common definition of bispherical coordinates $(σ,τ,φ)$ is

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

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

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

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 the foci

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

Surfaces of constant $σ$ correspond to intersecting tori of different radii

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

that all pass through the foci but are not concentric. The surfaces of constant $τ$ are non-intersecting spheres of different radii

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

that surround the foci. The centers of the constant- $τ$ spheres lie along the $z$ -axis, whereas the constant- $σ$ tori are centered in the $x y$ plane.

[edit] Scale factors

The scale factors for the bispherical coordinates $σ$ and $τ$ are equal

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

whereas the azimuthal scale factor equals

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

Thus, the infinitesimal volume element equals

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

and the Laplacian is given by

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

Other differential operators such as $\nabla \cdot \mathbf{F}$ and $\nabla \times \mathbf{F}$ can be expressed in the coordinates $(σ,τ)$ 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.