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Bipolar Cylindrical Coordinates


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 $(u, v, z)$ is used in this work, Arfken (1970) prefers $(\eta, \xi, z)$. The following identities show that curves of constant $u$ and $v$ are Circles in $xy$-space.
x^2+(y-a\cot u)^2 = a^2\csc^2 u
\end{displaymath} (4)

(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.


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

© 1996-9 Eric W. Weisstein