Bipolar Cylindrical Coordinates

$\begin{figure}\begin{center}\BoxedEPSF{curv_coords_Bipolar.epsf}\end{center}\end{figure}$

A set of Curvilinear Coordinates defined by

$\displaystyle x$	$\textstyle =$	$\displaystyle {a\sinh v\over\cosh v-\cos u}$	(1)
$\displaystyle y$	$\textstyle =$	$\displaystyle {a\sin u\over \cosh v-\cos u}$	(2)
$\displaystyle z$	$\textstyle =$	$\displaystyle z,$	(3)

where $u \in [0, 2\pi)$ , $v \in (-\infty, \infty)$ , and $z \in (-\infty, \infty)$ . There are several notational conventions, and whereas

is used in this work, Arfken (1970) prefers $(\eta, \xi, z)$ . The following identities show that curves of constant

and

are Circles in

-space.

$\begin{displaymath} x^2+(y-a\cot u)^2 = a^2\csc^2 u \end{displaymath}$

(4)

$\begin{displaymath} (x-a\coth v)^2+y^2 = a^2\mathop{\rm csch}\nolimits ^2 v. \end{displaymath}$

(5)

The Scale Factors are

$\displaystyle h_u$	$\textstyle =$	$\displaystyle {a\over\cosh v-\cos u}$	(6)
$\displaystyle h_v$	$\textstyle =$	$\displaystyle {a\over\cosh v-\cos u}$	(7)
$\displaystyle h_z$	$\textstyle =$	$\displaystyle 1.$	(8)

The Laplacian is

$\displaystyle \nabla^2$

$\textstyle =$

$\displaystyle {(\cosh v-\cos u)^2\over a^2} \left({{\partial^2 \over \partial u^2} + {\partial^2 \over\partial v^2}}\right) + {\partial^2\over \partial z^2}.$

(9)

Laplace's Equation is not separable in Bipolar Cylindrical Coordinates, but it is in 2-D Bipolar Coordinates.

References

Arfken, G. ``Bipolar Coordinates ( $\xi$ , $\eta$ , ).'' §2.9 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 97-102, 1970.