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''spatial elliptic coordinates''
 
''spatial elliptic coordinates''
  
The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035420/e0354201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035420/e0354202.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035420/e0354203.png" /> connected with Cartesian rectangular coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035420/e0354204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035420/e0354205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035420/e0354206.png" /> by the formulas
+
The numbers $  \lambda $,  
 +
$  \mu $
 +
and $  \nu $
 +
connected with Cartesian rectangular 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/e/e035/e035420/e0354207.png" /></td> </tr></table>
+
$$
 +
x  ^ {2}  =
 +
\frac{( \lambda + a  ^ {2} ) ( \mu + a  ^ {2} )
 +
( \nu + a  ^ {2} ) }{( b  ^ {2} - a  ^ {2} ) ( c  ^ {2} - a  ^ {2} ) }
 +
,
 +
$$
  
<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/e035420/e0354208.png" /></td> </tr></table>
+
$$
 +
y  ^ {2}  =
 +
\frac{( \lambda  ^ {2} + b  ^ {2} ) ( \mu  ^ {2} + b  ^ {2} ) ( \nu + b
 +
^ {2} ) }{( a  ^ {2} - b  ^ {2} ) ( c  ^ {2} - b  ^ {2} ) }
 +
,
 +
$$
  
<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/e035420/e0354209.png" /></td> </tr></table>
+
$$
 +
z  ^ {2}  =
 +
\frac{( \lambda + c  ^ {2} ) ( \mu + c  ^ {2} ) ( \mu + c  ^ {2}
 +
) }{( a  ^ {2} - c  ^ {2} ) ( b  ^ {2} - c  ^ {2} ) }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035420/e03542010.png" />. The coordinate surfaces are (see Fig.): ellipses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035420/e03542011.png" />, one-sheet hyperbolas (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035420/e03542012.png" />), and two-sheet hyperbolas (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035420/e03542013.png" />), with centres at the coordinate origin.
+
where $  - a  ^ {2} < \nu < - b  ^ {2} < \mu < - c  ^ {2} < \lambda < \infty $.  
 +
The coordinate surfaces are (see Fig.): ellipses $  ( \lambda = \textrm{ const } ) $,  
 +
one-sheet hyperbolas ( $  \mu = \textrm{ const } $),  
 +
and two-sheet hyperbolas ( $  \nu = \textrm{ const } $),  
 +
with centres at the coordinate origin.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/e035420a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/e035420a.gif" />
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Figure: e035420a
 
Figure: e035420a
  
The system of ellipsoidal coordinates is orthogonal. To every triple of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035420/e03542014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035420/e03542015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035420/e03542016.png" /> correspond 8 points (one in each octant), which are symmetric to each other relative to the coordinate planes of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035420/e03542017.png" />.
+
The system of ellipsoidal coordinates is orthogonal. To every triple of numbers $  \lambda $,  
 +
$  \mu $
 +
and $  \nu $
 +
correspond 8 points (one in each octant), which are symmetric to each other relative to the coordinate planes of the system $  O x y z $.
  
 
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/e035420/e03542018.png" /></td> </tr></table>
+
$$
 
+
L _  \lambda  =
<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/e035420/e03542019.png" /></td> </tr></table>
+
\frac{1}{2}
 +
\sqrt {
  
<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/e035420/e03542020.png" /></td> </tr></table>
+
\frac{( \lambda - \mu ) ( \mu - \nu ) }{( \lambda + a  ^ {2} ) ( \lambda + b  ^ {2} ) ( \lambda + c  ^ {2} ) }
 +
} ,
 +
$$
  
If one of the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035420/e03542021.png" /> in the definition of ellipsoidal coordinates is replaced by an equality, then degenerate ellipsoidal coordinate systems are obtained.
+
$$
 +
L _  \mu  =
 +
\frac{1}{2}
 +
\sqrt {
 +
\frac{( \lambda - \mu ) ( \nu - \mu ) }{(
 +
\mu + a  ^ {2} ) ( \mu + b  ^ {2} ) ( \mu + c  ^ {2} ) }
 +
} ,
 +
$$
  
 +
$$
 +
L _  \nu  = 
 +
\frac{1}{2}
 +
\sqrt {
 +
\frac{( \lambda - \nu ) ( \mu - \nu ) }{(
 +
\nu + a  ^ {2} ) ( \nu + b  ^ {2} ) ( \nu + c  ^ {2} ) }
 +
} .
 +
$$
  
 +
If one of the conditions  $  a  ^ {2} > b  ^ {2} > c  ^ {2} > 0 $
 +
in the definition of ellipsoidal coordinates is replaced by an equality, then degenerate ellipsoidal coordinate systems are obtained.
  
 
====Comments====
 
====Comments====
 
+
[[Laplace equation|Laplace's equation]] expressed in ellipsoidal coordinates is separable (cf [[Separation of variables, method of]]), and leads to [[Lamé function]]s.
  
 
====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>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  Harold Jeffreys, Bertha Jeffreys, ''Methods of Mathematical Physics'', 3rd edition, Cambridge University Press (1972) Zbl 0238.00004</TD></TR>
 +
</table>

Latest revision as of 19:37, 5 June 2020


spatial elliptic coordinates

The numbers $ \lambda $, $ \mu $ and $ \nu $ connected with Cartesian rectangular coordinates $ x $, $ y $ and $ z $ by the formulas

$$ x ^ {2} = \frac{( \lambda + a ^ {2} ) ( \mu + a ^ {2} ) ( \nu + a ^ {2} ) }{( b ^ {2} - a ^ {2} ) ( c ^ {2} - a ^ {2} ) } , $$

$$ y ^ {2} = \frac{( \lambda ^ {2} + b ^ {2} ) ( \mu ^ {2} + b ^ {2} ) ( \nu + b ^ {2} ) }{( a ^ {2} - b ^ {2} ) ( c ^ {2} - b ^ {2} ) } , $$

$$ z ^ {2} = \frac{( \lambda + c ^ {2} ) ( \mu + c ^ {2} ) ( \mu + c ^ {2} ) }{( a ^ {2} - c ^ {2} ) ( b ^ {2} - c ^ {2} ) } , $$

where $ - a ^ {2} < \nu < - b ^ {2} < \mu < - c ^ {2} < \lambda < \infty $. The coordinate surfaces are (see Fig.): ellipses $ ( \lambda = \textrm{ const } ) $, one-sheet hyperbolas ( $ \mu = \textrm{ const } $), and two-sheet hyperbolas ( $ \nu = \textrm{ const } $), with centres at the coordinate origin.

Figure: e035420a

The system of ellipsoidal coordinates is orthogonal. To every triple of numbers $ \lambda $, $ \mu $ and $ \nu $ correspond 8 points (one in each octant), which are symmetric to each other relative to the coordinate planes of the system $ O x y z $.

The Lamé coefficients are

$$ L _ \lambda = \frac{1}{2} \sqrt { \frac{( \lambda - \mu ) ( \mu - \nu ) }{( \lambda + a ^ {2} ) ( \lambda + b ^ {2} ) ( \lambda + c ^ {2} ) } } , $$

$$ L _ \mu = \frac{1}{2} \sqrt { \frac{( \lambda - \mu ) ( \nu - \mu ) }{( \mu + a ^ {2} ) ( \mu + b ^ {2} ) ( \mu + c ^ {2} ) } } , $$

$$ L _ \nu = \frac{1}{2} \sqrt { \frac{( \lambda - \nu ) ( \mu - \nu ) }{( \nu + a ^ {2} ) ( \nu + b ^ {2} ) ( \nu + c ^ {2} ) } } . $$

If one of the conditions $ a ^ {2} > b ^ {2} > c ^ {2} > 0 $ in the definition of ellipsoidal coordinates is replaced by an equality, then degenerate ellipsoidal coordinate systems are obtained.

Comments

Laplace's equation expressed in ellipsoidal coordinates is separable (cf Separation of variables, method of), and leads to Lamé functions.

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
[a2] Harold Jeffreys, Bertha Jeffreys, Methods of Mathematical Physics, 3rd edition, Cambridge University Press (1972) Zbl 0238.00004
How to Cite This Entry:
Ellipsoidal coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ellipsoidal_coordinates&oldid=14024
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article