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:

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

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

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

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

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.