Bipolar Coordinates
Bipolar coordinates are a 2-D system of coordinates. There are two commonly defined types of bipolar coordinates, the first ofwhich is defined by
 |  |  | (1) |
 | |  | (2) |
where
, 
. The following identities show that curves of constant 
and 
areCircles in 
-space. | (3) |
 | (4) |
 | |  | (5) |
 | |  | (6) |
The Laplacian is
 | |  | (7) |
Laplace's Equation is separable.
Two-center bipolar coordinates are two coordinates giving the distances from two fixed centers 
and 
, sometimesdenoted 
and 
. For two-center bipolar coordinates with centers at 
,
 | |  | (8) |
 | |  | (9) |
Combining (8) and (9) gives
 | (10) |
and 
gives | |  | (11) |
| |  | (12) |
Solving for Polar Coordinates gives
 | |  | (13) |
 | |  | (14) |
References
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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