Bispherical coordinates

Definition

The most common definition of bispherical coordinates (\tau, \sigma, \phi) is

:\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 \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}}

The coordinates ranges are −∞ \tau \sigma ≤ \pi and 0 ≤ \phi ≤ 2\pi.

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:

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

References

Bibliography