# Bispherical coordinates

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Illustration of bispherical coordinates, which are obtained by rotating a two-dimensional bipolar coordinate system about the axis joining its two foci. The foci are located at distance 1 from the vertical z-axis. The red self-interecting torus is the σ=45° isosurface, the blue sphere is the τ=0.5 isosurface, and the yellow half-plane is the φ=60° isosurface. The green half-plane marks the x-z plane, from which φ is measured. The black point is located at the intersection of the red, blue and yellow isosurfaces, at Cartesian coordinates roughly (0.841, -1.456, 1.239).

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.

## Definition

The most common definition of bispherical coordinates $(\sigma, \tau, \phi)$ 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 $\sigma$ coordinate of a point $P$ equals the angle $F_{1} P F_{2}$ and the $\tau$ coordinate equals the natural logarithm of the ratio of the distances $d_{1}$ and $d_{2}$ to the foci

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

### Coordinate surfaces

Surfaces of constant $\sigma$ 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 $\tau$ 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-$\tau$ spheres lie along the $z$-axis, whereas the constant-$\sigma$ tori are centered in the $xy$ plane.

### Inverse formulae

The formulae for the inverse transformation are:

$\sigma = \arccos((R^2-a^2)/Q)$
$\tau = \operatorname{arsinh}(2 a z/Q)$
$\phi = \operatorname{atan}(y/x)$

where $R=\sqrt{x^2+y^2+z^2}$ and $Q=\sqrt{(R^2+a^2)^2-(2 a z)^2}.$

### Scale factors

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

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

whereas the azimuthal scale factor equals

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

Thus, the infinitesimal volume element equals

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

and the Laplacian is given by

\begin{align} \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) \right. \\[8pt] &{} \quad + \left. \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] \end{align}

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

## Applications

The classic applications of bispherical coordinates are in solving partial differential equations, e.g., Laplace's equation, for which bispherical coordinates allow a separation of variables. However, the Helmholtz equation is not separable in bispherical coordinates. A typical example would be the electric field surrounding two conducting spheres of different radii.

## Bibliography

• Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. pp. 665–666.
• Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 182. LCCN 5914456.
• Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. p. 113. ISBN 0-86720-293-9.
• Moon PH, Spencer DE (1988). "Bispherical Coordinates (η, θ, ψ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer Verlag. pp. 110–112 (Section IV, E4Rx). ISBN 0-387-02732-7.