# Bispherical coordinates

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 $(\tau ,\sigma ,\phi )$ is
    ${\begin{aligned}x&=a\ {\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}\cos \phi ,\\y&=a\ {\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}\sin \phi ,\\z&=a\ {\frac {\sinh \tau }{\cosh \tau -\cos \sigma }},\end{aligned}}$
where the $\sigma$ coordinate of a point $P$ equals the angle $F_{1}PF_{2}$ 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 -∞ < $\tau$ < ∞, 0 ≤ $\sigma$ ≤ $\pi$ 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:
    ${\begin{aligned}\sigma &=\arccos \left({\dfrac {R^{2}-a^{2}}{Q}}\right),\\\tau &=\operatorname {arsinh} \left({\dfrac {2az}{Q}}\right),\\\phi &=\arctan \left({\dfrac {y}{x}}\right),\end{aligned}}$
where R=√(x$^2$+y$^2$+z$^2$){\textstyle R={\sqrt {x^{2}+y^{2}+z^{2}}}} and Q=√((R$^2$+a$^2$)$^2$−(2az)$^2$).{\textstyle Q={\sqrt {\left(R^{2}+a^{2}\right)^{2}-\left(2az\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
    ${\begin{aligned}\nabla ^{2}\Phi ={\frac {\left(\cosh \tau -\cos \sigma \right)^{3}}{a^{2}\sin \sigma }}&\left[{\frac {\partial }{\partial \sigma }}\left({\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}{\frac {\partial \Phi }{\partial \sigma }}\right)\right.\\[8pt]&\quad +\left.\sin \sigma {\frac {\partial }{\partial \tau }}\left({\frac {1}{\cosh \tau -\cos \sigma }}{\frac {\partial \Phi }{\partial \tau }}\right)+{\frac {1}{\sin \sigma \left(\cosh \tau -\cos \sigma \right)}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}\right]\end{aligned}}$
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.