Bispherical coordinates

Definition

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

${\displaystyle \begin{array}{cc}x& =a\frac{\sin \sigma }{\cosh \tau -\cos \sigma }\cos \varphi ,\\ y& =a\frac{\sin \sigma }{\cosh \tau -\cos \sigma }\sin \varphi ,\\ z& =a\frac{\sinh \tau }{\cosh \tau -\cos \sigma },\end{array}}$

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 ranges are −∞ < ${\displaystyle \tau }$ < ∞, 0 ≤ ${\displaystyle \sigma }$ ≤ ${\displaystyle \pi }$ and 0 ≤ ${\displaystyle \varphi }$ ≤ 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:

${\displaystyle \begin{array}{cc}\sigma & =\arccos \left({\displaystyle \frac{R^2-a^2}{Q}}\right),\\ \tau & =\mathrm{arsinh}\left({\displaystyle \frac{2az}{Q}}\right),\\ \varphi & =\arctan \left({\displaystyle \frac{y}{x}}\right),\end{array}}$

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

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

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

Thus, the infinitesimal volume element equals

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

and the Laplacian is given by

${\displaystyle \begin{array}{cc}{\mathrm{\nabla }}^2\mathrm{\Phi }=\frac{{\left(\cosh \tau -\cos \sigma \right)}^3}{a^2\sin \sigma }& [\frac{\mathrm{\partial }}{\mathrm{\partial }\sigma }\left(\frac{\sin \sigma }{\cosh \tau -\cos \sigma }\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }\sigma }\right)\\ & +\sin \sigma \frac{\mathrm{\partial }}{\mathrm{\partial }\tau }\left(\frac{1}{\cosh \tau -\cos \sigma }\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }\tau }\right)+\frac{1}{\sin \sigma \left(\cosh \tau -\cos \sigma \right)}\frac{{\mathrm{\partial }}^2\mathrm{\Phi }}{\mathrm{\partial }{\varphi }^2}]\end{array}}$

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