Difference between revisions of "Elliptic coordinates"
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| + | $#C+1 = 29 : ~/encyclopedia/old_files/data/E035/E.0305440 Elliptic coordinates | ||
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| − | + | Two numbers $ \sigma $ | |
| + | and $ \tau $ | ||
| + | connected with rectangular Cartesian coordinates by the formulas | ||
| − | where < | + | $$ |
| + | x ^ {2} = | ||
| + | \frac{( \sigma + a ^ {2} ) ( \tau + a ^ {2} ) }{a ^ {2} - b ^ {2} } | ||
| + | , | ||
| + | $$ | ||
| + | |||
| + | $$ | ||
| + | y ^ {2} = | ||
| + | \frac{( \sigma + b ^ {2} ) ( \tau + b ^ {2} ) }{b ^ {2} - a ^ {2} } | ||
| + | , | ||
| + | $$ | ||
| + | |||
| + | 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" /> | ||
| Line 11: | Line 33: | ||
Figure: e035440a | Figure: e035440a | ||
| − | The coordinate lines are (see Fig.): confocal ellipses ( | + | 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 | ||
| − | + | $$ | |
| + | L _ \sigma = | ||
| + | \frac{1}{2} | ||
| + | \sqrt { | ||
| + | |||
| + | \frac{\sigma - \tau }{( \sigma + a ^ {2} ) | ||
| + | ( \tau + b ^ {2} ) } | ||
| + | } , | ||
| + | $$ | ||
| − | + | $$ | |
| + | L _ \tau = | ||
| + | \frac{1}{2} | ||
| + | \sqrt { | ||
| + | \frac{\tau - \sigma }{( | ||
| + | \sigma - a ^ {2} ) ( \tau + b ^ {2} ) } | ||
| + | } . | ||
| + | $$ | ||
In elliptic coordinates the Laplace equation allows separation of variables. | In elliptic coordinates the Laplace equation allows separation of variables. | ||
| − | Degenerate elliptic coordinates are two numbers | + | 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 | + | and with Cartesian coordinates $ x $ |
| + | and $ y $ | ||
| + | by | ||
| − | + | $$ | |
| + | x = \cosh \widetilde \sigma \cos \widetilde \tau ,\ \ | ||
| + | y = \sinh \widetilde \sigma \sin \widetilde \tau , | ||
| + | $$ | ||
| − | where < | + | 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: | ||
| − | + | $$ | |
| + | L _ {\widetilde \sigma } = L _ {\widetilde \tau } = \ | ||
| + | \sqrt {\cosh ^ {2} \widetilde \sigma - | ||
| + | \cos ^ {2} \widetilde \tau } . | ||
| + | $$ | ||
The area element is: | The area element is: | ||
| − | + | $$ | |
| + | d s = ( \cosh ^ {2} \widetilde \sigma - | ||
| + | \cos ^ {2} \widetilde \tau ) d \widetilde \sigma d \widetilde \tau . | ||
| + | $$ | ||
The Laplace operator is: | 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==== | ====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 |
Elliptic coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elliptic_coordinates&oldid=19184