Bipolar coordinates
= Bipolar coordinates
{title2=$(\sigma,\tau)$}
For a length scale $c>0$, planar <bipolar coordinates> may be defined by
$$
x=\frac{c\sinh\tau}{\cosh\tau-\cos\sigma},\qquad y=\frac{c\sin\sigma}{\cosh\tau-\cos\sigma},\qquad0\leq\sigma<2\pi.
$$
The 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 circles
$$
(x-c\coth\tau)^2+y^2=c^2\operatorname{csch}^2\tau.
$$
Their 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.