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
where
,
. The following identities show that curves of constant
and
are
Circles in
-space.
![\begin{displaymath}
x^2+(y-a\cot u)^2 = a^2\csc^2 u
\end{displaymath}](b_1468.gif) |
(3) |
![\begin{displaymath}
(x-a\coth v)^2+y^2 = a^2\mathop{\rm csch}\nolimits ^2 v.
\end{displaymath}](b_1469.gif) |
(4) |
The Scale Factors are
The Laplacian is
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
,
Combining (8) and (9) gives
![\begin{displaymath}
{r_1}^2-{r_2}^2=4cx.
\end{displaymath}](b_1484.gif) |
(10) |
Solving for Cartesian Coordinates
and
gives
Solving for Polar Coordinates gives
References
Lockwood, E. H. ``Bipolar Coordinates.'' Ch. 25 in A Book of Curves. Cambridge, England: Cambridge University Press,
pp. 186-190, 1967.
© 1996-9 Eric W. Weisstein
1999-05-26