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Bipolar Coordinates

Bipolar coordinates are a 2-D system of coordinates. There are two commonly defined types of bipolar coordinates, the first of which is 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)

where $u \in [0, 2\pi)$, $v \in (-\infty, \infty)$. 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} (3)

(x-a\coth v)^2+y^2 = a^2\mathop{\rm csch}\nolimits ^2 v.
\end{displaymath} (4)

The Scale Factors are
$\displaystyle h_u$ $\textstyle =$ $\displaystyle {a\over\cosh v-\cos u}$ (5)
$\displaystyle h_v$ $\textstyle =$ $\displaystyle {a\over\cosh v-\cos u}.$ (6)

The Laplacian is
$\displaystyle \nabla^2$ $\textstyle =$ $\displaystyle {(\cosh v-\cos u)^2\over a^2} \left({{\partial^2\over\partial u^2} + {\partial^2\over\partial v^2}}\right).$ (7)

Laplace's Equation is separable.

Two-center bipolar coordinates are two coordinates giving the distances from two fixed centers $r_1$ and $r_2$, sometimes denoted $r$ and $r'$. For two-center bipolar coordinates with centers at $(\pm c, 0)$,

$\displaystyle {r_1}^2$ $\textstyle =$ $\displaystyle (x+c)^2+y^2$ (8)
$\displaystyle {r_2}^2$ $\textstyle =$ $\displaystyle (x-c)^2+y^2.$ (9)

Combining (8) and (9) gives
\end{displaymath} (10)

Solving for Cartesian Coordinates $x$ and $y$ gives
$\displaystyle x$ $\textstyle =$ $\displaystyle {{r_1}^2-{r_2}^2\over 4c}$ (11)
$\displaystyle y$ $\textstyle =$ $\displaystyle \pm {1\over 4c}\sqrt{16c^2{r_1}^2-({r_1}^2-{r_2}^2+4c^2)}.$ (12)

Solving for Polar Coordinates gives
$\displaystyle r$ $\textstyle =$ $\displaystyle \sqrt{{r_1}^2+{r_2}^2-2c^2\over 2}$ (13)
$\displaystyle \theta$ $\textstyle =$ $\displaystyle \tan^{-1}\left[{\sqrt{{8c^2({r_1}^2+{r_2}^2-2c^2)\over {r_1}^2-{r_2}^2}-1}\,}\right].$ (14)


Lockwood, E. H. ``Bipolar Coordinates.'' Ch. 25 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 186-190, 1967.

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© 1996-9 Eric W. Weisstein