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

$$x = {a\sinh v\over\cosh v-\cos u}$$ (1)

$$y = {a\sin u\over\cosh v-\cos u},$$ (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.

$$x^2+(y-a\cot u)^2 = a^2\csc^2 u$$ (3)

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

The Scale Factors are

$$h_u = {a\over\cosh v-\cos u}$$ (5)

$$h_v = {a\over\cosh v-\cos u}.$$ (6)

The Laplacian is

$$\nabla^2 = {(\cosh v-\cos u)^2\over a^2} \left({{\partial^2\over\partial u^2} + {\partial^2\over\partial v^2}}\right).$$ (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 $(\pm c, 0)$,

$${r_1}^2 = (x+c)^2+y^2$$ (8)

$${r_2}^2 = (x-c)^2+y^2.$$ (9)

Combining (8) and (9) gives

$${r_1}^2-{r_2}^2=4cx.$$ (10)

Solving for Cartesian Coordinates $x$ and $y$ gives

$$x = {{r_1}^2-{r_2}^2\over 4c}$$ (11)

$$y = \pm {1\over 4c}\sqrt{16c^2{r_1}^2-({r_1}^2-{r_2}^2+4c^2)}.$$ (12)

Solving for Polar Coordinates gives

$$r = \sqrt{{r_1}^2+{r_2}^2-2c^2\over 2}$$ (13)

$$\theta = \tan^{-1}\left[{\sqrt{{8c^2({r_1}^2+{r_2}^2-2c^2)\over {r_1}^2-{r_2}^2}-1}\,}\right].$$ (14)

References

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