Bipolar coordinates are a two-dimensional system of coordinates. There are two commonly defined types of bipolar coordinates, the first of which is defined by
(1)
| |||
(2)
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where ,
.
The following identities show that curves of constant
and
are circles in
-space.
(3)
|
(4)
|
The scale factors are
(5)
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(6)
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The Laplacian is
(7)
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Laplace's equation is separable.
Two-center bipolar coordinates are two coordinates giving the distances from two fixed centers
and
,
sometimes denoted
and
.
For two-center bipolar coordinates with centers at
,
(8)
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(9)
|
(10)
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Solving for Cartesian coordinates and
gives
(11)
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(12)
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Solving for polar coordinates gives
(13)
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(14)
|