 Bipolar coordinates

For a length scale $c>0$, planar bipolar coordinates may be defined byThe common scale factor is $h=c/(\cosh \tau -\cos \sigma )$, so Laplace's equation becomes ${\Phi }_{\sigma \sigma }+{\Phi }_{\tau \tau }=0$ away from coordinate singularities. For $\tau >0$, constant-$\tau$ curves are circlesTheir radii decrease as $\tau$ increases; for $0<a<b$ the $\tau =b$ circle lies inside the $\tau =a$ circle. The region $a<\tau <b$ is therefore an eccentric circular annulus. This ordering is useful when translating coordinate boundary data into geometric language.