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The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b0164701.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b0164702.png" /> which are connected with the Cartesian orthogonal coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b0164703.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b0164704.png" /> by the formulas
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$#C+1 = 29 : ~/encyclopedia/old_files/data/B016/B.0106470 Bipolar coordinates
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b0164705.png" /></td> </tr></table>
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{{TEX|auto}}
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{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b0164706.png" />. The coordinate lines are two families of circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b0164707.png" /> with poles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b0164708.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b0164709.png" /> and the (half-c)ircles orthogonal with these <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647010.png" />.
+
The numbers  $  \tau $
 +
and  $  \sigma $
 +
which are connected with the Cartesian orthogonal coordinates  $  x $
 +
and  $  y $
 +
by the formulas
 +
 
 +
$$
 +
=
 +
\frac{a  \sinh  \tau }{\cosh  \tau - \cos  \sigma }
 +
,\ \
 +
=
 +
\frac{a  \sin  \sigma }{\cosh  \tau - \cos  \sigma }
 +
,
 +
$$
 +
 
 +
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" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b016470a.gif" />
Line 11: Line 38:
 
The Lamé coefficients are:
 
The Lamé coefficients are:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647011.png" /></td> </tr></table>
+
$$
 +
L _  \tau  = L _  \sigma  = \
 +
 
 +
\frac{a  ^ {2} }{( \cosh  \tau - \cos  \sigma )  ^ {2} }
 +
.
 +
$$
  
 
The Laplace operator is:
 
The Laplace operator is:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647012.png" /></td> </tr></table>
+
$$
 +
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647014.png" />, which are connected with the orthogonal Cartesian coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647016.png" /> by the formulas:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647017.png" /></td> </tr></table>
+
$$
 +
=
 +
\frac{a  \sin  \sigma  \cos  \phi }{\cosh  \tau - \cos  \sigma }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647018.png" /></td> </tr></table>
+
$$
 +
=
 +
\frac{a  \sin  \sigma  \sin  \phi }{\cosh  \tau - \cos  \sigma }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647019.png" /></td> </tr></table>
+
$$
 +
=
 +
\frac{a  \sinh  \tau }{\cosh  \tau - \cos  \sigma }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647020.png" />. The coordinate surfaces are spheres (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647021.png" />), the surfaces obtained by rotation of arcs of circles (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647022.png" />) and half-planes passing through the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647023.png" />-axis. The system of bipolar coordinates in space is formed by rotating the system of bipolar coordinates on the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647024.png" /> around the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647025.png" />-axis.
+
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:
 
The Lamé coefficients are:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647026.png" /></td> </tr></table>
+
$$
 +
L _  \sigma  = \
 +
L _  \tau  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647027.png" /></td> </tr></table>
+
\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:
 
The Laplace operator is:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647028.png" /></td> </tr></table>
+
$$
 +
\Delta f  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016470/b01647029.png" /></td> </tr></table>
+
\frac{( \cosh  \tau - \cosh  \sigma )  ^ {3} }{a  ^ {2}  \sin  \sigma }
 +
\left [
  
====References====
+
\frac \partial {\partial  \tau }
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Madelung,  "Die mathematischen Hilfsmittel des Physikers" , Springer (1957)</TD></TR></table>
+
  \left (
 +
 
 +
\frac{\sin \sigma }{\cosh  \tau - \cos  \sigma }
  
 +
\frac{\partial  f }{\partial  \tau }
 +
\
 +
\right )\right . +
 +
$$
  
 +
$$
 +
+ \left .
  
====Comments====
+
\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====
 
====References====
<table><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>
+
<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}}

Latest revision as of 08:18, 26 March 2023


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 } ) $.

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

[1] E. Madelung, "Die mathematischen Hilfsmittel des Physikers" , Springer (1957)
[a1] O. Veblen, J.H.C. Whitehead, "The foundations of differential geometry" , Cambridge Univ. Press (1932)


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How to Cite This Entry:
Bipolar coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bipolar_coordinates&oldid=11655
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article