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Two numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e0354401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e0354402.png" /> connected with rectangular Cartesian coordinates by the formulas
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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/e/e035/e035440/e0354403.png" /></td> </tr></table>
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 +
{{TEX|done}}
  
<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/e/e035/e035440/e0354404.png" /></td> </tr></table>
+
Two numbers  $  \sigma $
 +
and  $  \tau $
 +
connected with rectangular Cartesian coordinates by the formulas
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e0354405.png" />.
+
$$
 +
x  ^ {2}  = 
 +
\frac{( \sigma + a  ^ {2} ) ( \tau + a  ^ {2} ) }{a  ^ {2} - b  ^ {2} }
 +
,
 +
$$
 +
 
 +
$$
 +
y  ^ {2}  = 
 +
\frac{( \sigma + b  ^ {2} ) ( \tau + b  ^ {2} ) }{b  ^ {2} - a  ^ {2} }
 +
,
 +
$$
 +
 
 +
where $  - a  ^ {2} < \tau < - b  ^ {2} < \sigma < \infty $.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/e035440a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/e035440a.gif" />
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Figure: e035440a
 
Figure: e035440a
  
The coordinate lines are (see Fig.): confocal ellipses (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e0354406.png" />) and hyperbolas (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e0354407.png" />) with foci (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e0354408.png" />) and (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e0354409.png" />). The system of elliptic coordinates is orthogonal. To every pair of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e03544010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e03544011.png" /> correspond four points, one in each quadrant of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e03544012.png" />-plane.
+
The coordinate lines are (see Fig.): confocal ellipses ( $  \sigma = \textrm{ const } $)  
 +
and hyperbolas ( $  \tau = \textrm{ const } $)  
 +
with foci ( $  - \sqrt {a  ^ {2} - b  ^ {2} } , 0 $)  
 +
and ( $  \sqrt {a  ^ {2} - b  ^ {2} } , 0 $).  
 +
The system of elliptic coordinates is orthogonal. To every pair of numbers $  \sigma $
 +
and $  \tau $
 +
correspond four points, one in each quadrant of the $  xy $-
 +
plane.
  
 
The [[Lamé coefficients|Lamé coefficients]] are
 
The [[Lamé coefficients|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/e/e035/e035440/e03544013.png" /></td> </tr></table>
+
$$
 +
L _  \sigma  =
 +
\frac{1}{2}
 +
\sqrt {
 +
 
 +
\frac{\sigma - \tau }{( \sigma + a  ^ {2} )
 +
( \tau + b  ^ {2} ) }
 +
} ,
 +
$$
  
<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/e/e035/e035440/e03544014.png" /></td> </tr></table>
+
$$
 +
L _  \tau  =
 +
\frac{1}{2}
 +
\sqrt {
 +
\frac{\tau - \sigma }{(
 +
\sigma - a  ^ {2} ) ( \tau + b  ^ {2} ) }
 +
} .
 +
$$
  
 
In elliptic coordinates the Laplace equation allows separation of variables.
 
In elliptic coordinates the Laplace equation allows separation of variables.
  
Degenerate elliptic coordinates are two numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e03544015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e03544016.png" /> connected with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e03544017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e03544018.png" /> by the formulas (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e03544019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e03544020.png" />):
+
Degenerate elliptic coordinates are two numbers $  \widetilde \sigma  $
 +
and $  \widetilde \tau  $
 +
connected with $  \sigma $
 +
and $  \tau $
 +
by the formulas (for $  a = 1 $,  
 +
$  b = 0 $):
  
<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/e/e035/e035440/e03544021.png" /></td> </tr></table>
+
$$
 +
\sigma  = \sinh  ^ {2}  \widetilde \sigma  ,\ \
 +
\tau  = - \sin  ^ {2}  \widetilde \tau  ,
 +
$$
  
and with Cartesian coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e03544022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e03544023.png" /> by
+
and with Cartesian coordinates $  x $
 +
and $  y $
 +
by
  
<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/e/e035/e035440/e03544024.png" /></td> </tr></table>
+
$$
 +
= \cosh  \widetilde \sigma    \cos  \widetilde \tau  ,\ \
 +
= \sinh  \widetilde \sigma    \sin  \widetilde \tau  ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e03544025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035440/e03544026.png" />. Occasionally these coordinates are also called elliptic.
+
where $  0 \leq  \widetilde \sigma  < \infty $
 +
and  $  0 \leq  \widetilde \tau  < 2 \pi $.  
 +
Occasionally these coordinates are also called elliptic.
  
 
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/e/e035/e035440/e03544027.png" /></td> </tr></table>
+
$$
 +
L _ {\widetilde \sigma  }  = L _ {\widetilde \tau  }  = \
 +
\sqrt {\cosh  ^ {2}  \widetilde \sigma  -
 +
\cos  ^ {2}  \widetilde \tau  } .
 +
$$
  
 
The area element is:
 
The area element 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/e/e035/e035440/e03544028.png" /></td> </tr></table>
+
$$
 +
d s  = ( \cosh  ^ {2}  \widetilde \sigma  -
 +
\cos  ^ {2}  \widetilde \tau  )  d \widetilde \sigma    d \widetilde \tau  .
 +
$$
  
 
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/e/e035/e035440/e03544029.png" /></td> </tr></table>
+
$$
 +
\Delta \phi  =
 +
\frac{1}{\cosh  ^ {2}  \widetilde \sigma  -
 +
\cos  ^ {2}  \widetilde \tau  }
 +
\left (
  
 +
\frac{\partial  ^ {2} \phi }{\partial  \widetilde \sigma    ^ {2} }
 +
+
  
 +
\frac{\partial  ^ {2} \phi }{\partial  \widetilde \tau    ^ {2} }
 +
\right ) .
 +
$$
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Darboux,  "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''1''' , Gauthier-Villars  (1887)  pp. 1–18</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Darboux,  "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''1''' , Gauthier-Villars  (1887)  pp. 1–18</TD></TR></table>

Latest revision as of 19:37, 5 June 2020


Two numbers $ \sigma $ and $ \tau $ connected with rectangular Cartesian coordinates by the formulas

$$ x ^ {2} = \frac{( \sigma + a ^ {2} ) ( \tau + a ^ {2} ) }{a ^ {2} - b ^ {2} } , $$

$$ y ^ {2} = \frac{( \sigma + b ^ {2} ) ( \tau + b ^ {2} ) }{b ^ {2} - a ^ {2} } , $$

where $ - a ^ {2} < \tau < - b ^ {2} < \sigma < \infty $.

Figure: e035440a

The coordinate lines are (see Fig.): confocal ellipses ( $ \sigma = \textrm{ const } $) and hyperbolas ( $ \tau = \textrm{ const } $) with foci ( $ - \sqrt {a ^ {2} - b ^ {2} } , 0 $) and ( $ \sqrt {a ^ {2} - b ^ {2} } , 0 $). The system of elliptic coordinates is orthogonal. To every pair of numbers $ \sigma $ and $ \tau $ correspond four points, one in each quadrant of the $ xy $- plane.

The Lamé coefficients are

$$ L _ \sigma = \frac{1}{2} \sqrt { \frac{\sigma - \tau }{( \sigma + a ^ {2} ) ( \tau + b ^ {2} ) } } , $$

$$ L _ \tau = \frac{1}{2} \sqrt { \frac{\tau - \sigma }{( \sigma - a ^ {2} ) ( \tau + b ^ {2} ) } } . $$

In elliptic coordinates the Laplace equation allows separation of variables.

Degenerate elliptic coordinates are two numbers $ \widetilde \sigma $ and $ \widetilde \tau $ connected with $ \sigma $ and $ \tau $ by the formulas (for $ a = 1 $, $ b = 0 $):

$$ \sigma = \sinh ^ {2} \widetilde \sigma ,\ \ \tau = - \sin ^ {2} \widetilde \tau , $$

and with Cartesian coordinates $ x $ and $ y $ by

$$ x = \cosh \widetilde \sigma \cos \widetilde \tau ,\ \ y = \sinh \widetilde \sigma \sin \widetilde \tau , $$

where $ 0 \leq \widetilde \sigma < \infty $ and $ 0 \leq \widetilde \tau < 2 \pi $. Occasionally these coordinates are also called elliptic.

The Lamé coefficients are:

$$ L _ {\widetilde \sigma } = L _ {\widetilde \tau } = \ \sqrt {\cosh ^ {2} \widetilde \sigma - \cos ^ {2} \widetilde \tau } . $$

The area element is:

$$ d s = ( \cosh ^ {2} \widetilde \sigma - \cos ^ {2} \widetilde \tau ) d \widetilde \sigma d \widetilde \tau . $$

The Laplace operator is:

$$ \Delta \phi = \frac{1}{\cosh ^ {2} \widetilde \sigma - \cos ^ {2} \widetilde \tau } \left ( \frac{\partial ^ {2} \phi }{\partial \widetilde \sigma ^ {2} } + \frac{\partial ^ {2} \phi }{\partial \widetilde \tau ^ {2} } \right ) . $$

Comments

References

[a1] G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1 , Gauthier-Villars (1887) pp. 1–18
How to Cite This Entry:
Elliptic coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elliptic_coordinates&oldid=19184
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article