View source for Bipolar coordinates

<!--
b0164701.png
$#A+1 = 29 n = 0
$#C+1 = 29 : ~/encyclopedia/old_files/data/B016/B.0106470 Bipolar coordinates
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

The numbers  $  \tau $
and  $  \sigma $
which are connected with the Cartesian orthogonal coordinates  $  x $
and  $  y $
by the formulas

$$ 
x  =  
\frac{a  \sinh  \tau }{\cosh  \tau - \cos  \sigma }
 ,\ \ 
y  =  
\frac{a  \sin  \sigma }{\cosh  \tau - \cos  \sigma }
 ,
$$

where  $  0 \leq  \sigma < \pi , -\infty < \tau < \infty $. 
The coordinate lines are two families of circles  $  ( \tau = \textrm{ const } ) $
with poles  $  A $
and  $  B $
and the (half-c)ircles orthogonal with these  $  ( \sigma = \textrm{ const } ) $.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b016470a.gif" />

Figure: b016470a

The Lamé coefficients are:

$$ 
L _  \tau   =  L _  \sigma   = \ 

\frac{a  ^ {2} }{( \cosh  \tau - \cos  \sigma )  ^ {2} }
 .
$$

The Laplace operator is:

$$ 
\Delta f  =  
\frac{1}{a  ^ {2} }

( \cosh  \tau - \cos  \sigma )  ^ {2}
\left (

\frac{\partial   ^ {2} f }{\partial  \sigma  ^ {2} }
 +

\frac{\partial   ^ {2} f }{\partial  \tau  ^ {2} }
 \ 
\right ) .
$$

Bipolar coordinates in space (bispherical coordinates) are the numbers  $  \sigma , \tau $
and  $  \phi $, 
which are connected with the orthogonal Cartesian coordinates  $  x, y $
and  $  z $
by the formulas:

$$ 
x  =  
\frac{a  \sin  \sigma  \cos  \phi }{\cosh  \tau - \cos  \sigma }
 ,
$$

$$ 
y  =  
\frac{a  \sin  \sigma  \sin  \phi }{\cosh  \tau - \cos  \sigma }
 ,
$$

$$ 
z  =  
\frac{a  \sinh  \tau }{\cosh  \tau - \cos  \sigma }
 ,
$$

where  $  - \infty < \sigma < \infty , 0 \leq  \tau < \pi , 0 \leq  \phi < 2 \pi $. 
The coordinate surfaces are spheres ( $  \sigma = \textrm{ const } $), 
the surfaces obtained by rotation of arcs of circles ( $  \tau = \textrm{ const } $) 
and half-planes passing through the  $  Oz $-
axis. The system of bipolar coordinates in space is formed by rotating the system of bipolar coordinates on the plane  $  Oxy $
around the  $  Oz $-
axis.

The Lamé coefficients are:

$$ 
L _  \sigma   = \ 
L _  \tau   = \ 

\frac{a  ^ {2} }{( \cosh  \tau - \cos  \sigma )  ^ {2} }
 ,
$$

$$ 
L _  \phi   =  
\frac{a  ^ {2}  \sin  ^ {2}  \sigma }{( \cosh  \tau - \cos  \sigma )  ^ {2} }
 .
$$

The Laplace operator is:

$$ 
\Delta f  = \ 

\frac{( \cosh  \tau - \cosh  \sigma )  ^ {3} }{a  ^ {2}  \sin  \sigma }
 \left [

\frac \partial {\partial  \tau }
 \left (

\frac{\sin  \sigma }{\cosh  \tau - \cos  \sigma }

\frac{\partial  f }{\partial  \tau }
 \ 
\right )\right . +
$$

$$ 
+ \left .

\frac \partial {\partial  \sigma }
 \left ( 
\frac{\sin  \sigma
 }{\cosh  \tau - \cos  \sigma }
 
\frac{\partial  f }{\partial  \sigma }

 \right ) + 
\frac{1}{\sin  \sigma ( \cosh  \tau - \cos
 \sigma ) }
 
\frac{\partial   ^ {2} f }{\partial  \phi  ^ {2} }
  \right ] .
$$

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  E. Madelung,   "Die mathematischen Hilfsmittel des Physikers" , Springer  (1957)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  O. Veblen,   J.H.C. Whitehead,   "The foundations of differential geometry" , Cambridge Univ. Press  (1932)</TD></TR>
</table>

{{OldImage}}