# Bispherical coordinates

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-intersecting 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 $\displaystyle{ F_{1} }$ and $\displaystyle{ F_{2} }$ in bipolar coordinates remain points (on the $\displaystyle{ z }$-axis, the axis of rotation) in the bispherical coordinate system.

## Definition

The most common definition of bispherical coordinates $\displaystyle{ (\tau, \sigma, \phi) }$ is

\displaystyle{ \begin{align} 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}, \end{align} }

where the $\displaystyle{ \sigma }$ coordinate of a point $\displaystyle{ P }$ equals the angle $\displaystyle{ F_{1} P F_{2} }$ and the $\displaystyle{ \tau }$ coordinate equals the natural logarithm of the ratio of the distances $\displaystyle{ d_{1} }$ and $\displaystyle{ d_{2} }$ to the foci

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

The coordinates range -∞ < $\displaystyle{ \tau }$ < ∞, 0 ≤ $\displaystyle{ \sigma }$$\displaystyle{ \pi }$, 0 ≤ $\displaystyle{ \phi }$ ≤ 2$\displaystyle{ \pi }$.

### Coordinate surfaces

Surfaces of constant $\displaystyle{ \sigma }$ correspond to intersecting tori of different radii

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

that all pass through the foci but are not concentric. The surfaces of constant $\displaystyle{ \tau }$ are non-intersecting spheres of different radii

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

that surround the foci. The centers of the constant-$\displaystyle{ \tau }$ spheres lie along the $\displaystyle{ z }$-axis, whereas the constant-$\displaystyle{ \sigma }$ tori are centered in the $\displaystyle{ xy }$ plane.

### Inverse formulae

The formulae for the inverse transformation are:

\displaystyle{ \begin{align} \sigma &= \arccos\left(\dfrac{R^2-a^2}{Q}\right), \\ \tau &= \operatorname{arsinh}\left(\dfrac{2az}{Q}\right), \\ \phi &= \arctan\left(\dfrac{y}{x}\right), \end{align} }

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

### Scale factors

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

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

whereas the azimuthal scale factor equals

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

Thus, the infinitesimal volume element equals

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

and the Laplacian is given by

\displaystyle{ \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 $\displaystyle{ \nabla \cdot \mathbf{F} }$ and $\displaystyle{ \nabla \times \mathbf{F} }$ can be expressed in the coordinates $\displaystyle{ (\sigma, \tau) }$ 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.
• 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.