# 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 ${\displaystyle F_{1}}$ 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 ${\displaystyle (\sigma ,\tau ,\phi )}$ is

${\displaystyle x=a\ {\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}\cos \phi }$
${\displaystyle y=a\ {\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}\sin \phi }$
${\displaystyle z=a\ {\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}}$

where the ${\displaystyle \sigma }$ coordinate of a point ${\displaystyle P}$ equals the angle ${\displaystyle F_{1}PF_{2}}$ 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

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

### 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:

${\displaystyle \sigma =\arccos \left({\dfrac {R^{2}-a^{2}}{Q}}\right)}$
${\displaystyle \tau =\operatorname {arsinh} \left({\dfrac {2az}{Q}}\right)}$
${\displaystyle \phi =\operatorname {atan} \left({\dfrac {y}{x}}\right)}$

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

### 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{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 ${\displaystyle \nabla \cdot \mathbf {F} }$ 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. LCCN 59014456.
• 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.