Editing Appendix/Ramblings/EllipticCylinderCoordinates (section)

==MF53==

===Definition===
From [[User:Tohline/Appendix/References#MF53|MF53]]'s ''Table of Separable Coordinates in Three Dimensions'' (see their Chapter 5, beginning on p. 655), we find the following description of '''Elliptic Cylinder Coordinates''' (p. 657).

<table border="1" cellpadding="10" align="center" width="80%">
<tr><td align="center">
'''Elliptic Cylindrical Coordinates'''<br />([[User:Tohline/Appendix/References#MF53|MF53]] Primary Definition)</td></tr>
<tr><td align="left">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi_1 \xi_2 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2)  \biggr]^{1 / 2} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi_3 </math>
  </td>
</tr>
</table>
</td></tr>
<tr><td align="center">
'''Alternate Definition'''
</td></tr>
<tr><td align="left">
Making the substitutions, <math>~\xi_3 \rightarrow z</math>, <math>~\xi_2 \rightarrow \cos\nu</math>, and <math>~\xi_1 \rightarrow d\cosh\mu</math>, we equally well obtain:
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~d\cosh\mu \cdot \cos\nu </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~d \sinh\mu \cdot \sin\nu </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z </math>
  </td>
</tr>
</table>

</td></tr></table>


Notice that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{x^2}{d^2 \cosh^2\mu} + \frac{y^2}{d^2 \sinh^2\mu}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cos^2\nu + \sin^2\nu = 1 \, .</math>
  </td>
</tr>
</table>
Hence, as is pointed out in a related [https://en.wikipedia.org/wiki/Elliptic_cylindrical_coordinates#Basic_definition Wikipedia discussion], "&hellip; this shows that curves of constant <math>~\mu</math> form ellipses."  For a given choice of <math>~\mu</math> &#8212; say, <math>~\mu_0</math> &#8212; let's see how the shape of the resulting ellipse relates to the standard ellipses described in our [[#Background|background discussion]], above.  The semi-major axis of the selected ellipse must be,
<div align="center">
<math>a = d\cosh\mu_0 \, .</math>
</div>

And its eccentricity must be obtainable from the relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a^2 - c^2 = a^2(1 - e^2)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~d^2 \sinh^2\mu_0</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a^2 \tanh^2\mu_0 = a^2 \biggl(1 - \frac{1}{\cosh^2\mu_0} \biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ e^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\cosh^2\mu_0} \, .</math>
  </td>
</tr>
</table>
We note, as well, that the x-coordinate location of the focus of the selected ellipse is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~c^2 = a^2 e^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~d^2\, .</math>
  </td>
</tr>
</table>
This emphasizes a key property of the MF53 Elliptic Cylindrical Coordinate system, viz., the family of ellipses that result from selecting various values of <math>~\mu_0</math> is a family of ''confocal'' ellipses.

===Scale Factors===

====Primary====

Appreciating that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial y}{\partial \xi_1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+\biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2)  \biggr]^{- 1 / 2}\xi_1(1-\xi_2^2) \, ,
</math> &nbsp; &nbsp; &nbsp; and that,
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial y}{\partial \xi_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2)  \biggr]^{- 1 / 2}\xi_2(\xi_1^2 - d^2) \, ,
</math>
  </td>
</tr>
</table>

we find that the respective [[User:Tohline/Appendix/Ramblings/DirectionCosines#Scale_Factors|scale factors]] are given by the expressions,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ h_1^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{\partial x}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial y}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial z}{\partial\xi_1} \biggr)^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi_2^2 +\biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2)  \biggr]^{- 1 }\xi_1^2 (1-\xi_2^2)^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ (\xi_1^2 - d^2)^{- 1 } [  (\xi_1^2 - d^2)\xi_2^2 +\xi_1^2 (1-\xi_2^2) ]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ \frac{  \xi_1^2 - d^2 \xi_2^2 }{\xi_1^2 - d^2} \biggr] \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ h_2^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{\partial x}{\partial\xi_2} \biggr)^2 + \biggl(\frac{\partial y}{\partial\xi_2} \biggr)^2 + \biggl(\frac{\partial z}{\partial\xi_2} \biggr)^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi_1^2 + \biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2)  \biggr]^{- 1 }\xi_2^2(\xi_1^2 - d^2)^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1 - \xi_2^2)^{- 1 } [\xi_1^2(1 - \xi_2^2)  + \xi_2^2(\xi_1^2 - d^2) ]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{ \xi_1^2  - d^2 \xi_2^2  }{1 - \xi_2^2} \biggr] \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ h_3^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{\partial x}{\partial\xi_3} \biggr)^2 + \biggl(\frac{\partial y}{\partial\xi_3} \biggr)^2 + \biggl(\frac{\partial z}{\partial\xi_3} \biggr)^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 \, . </math>
  </td>
</tr>
</table>
These match the scale-factor expressions found in [[User:Tohline/Appendix/References#MF53|MF53]].

====Alternatively====

Alternatively, the [https://en.wikipedia.org/wiki/Elliptic_cylindrical_coordinates#Scale_factors Wikipedia discussion] gives,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~h_\mu = h_\nu</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~d\sqrt{ \sinh^2\mu + \sin^2\nu}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\nabla^2\Phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{d^2(\sinh^2\mu + \sin^2\nu)} \biggl[ \frac{\partial^2 \Phi}{\partial \mu^2} + \frac{\partial^2 \Phi}{\partial \nu^2} \biggr] + \frac{\partial^2 \Phi}{\partial z^2} \, .
</math>
  </td>
</tr>
</table>

===Inverting Coordinate Mapping===

Inverting the original coordinate mappings, we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\xi_1^2 - a^2)\biggl[ 1 - \biggl(\frac{x}{\xi_1}\biggr)^2 \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\xi_1^2 - a^2) ( \xi_1^2 - x^2 ) - \xi_1^2 y^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\xi_1^2 - a^2) \xi_1^2  - (\xi_1^2 - a^2) x^2  - \xi_1^2 y^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\xi_1^4  - \xi_1^2 (a^2 + x^2  + y^2) + a^2 x^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \xi_1^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl\{
(a^2 + x^2 + y^2) \pm \biggl[ (a^2 + x^2 + y^2)^2 - 4a^2 x^2 \biggr]^{1 / 2}
\biggr\}
</math>
  </td>
</tr>
</table>
Only the ''superior'' &#8212; that is, only the ''positive'' &#8212; sign will ensure positive values of <math>~\xi_1^2</math>, so in summary we have,
<table border="1" width="80%" align="center" cellpadding="10"><tr><td align="left">

<table border="0" cellpadding="5" align="center">
<tr><td align="center" colspan="3">'''Coordinate Transformation'''</td></tr>
<tr>
  <td align="right">
<math>~\xi_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\sqrt{2}}\biggl\{
\biggl[ (a^2 + x^2 + y^2)^2 - 4a^2 x^2\biggr]^{1 / 2} + (a^2 + x^2 + y^2) 
\biggr\}^{1 / 2} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\xi_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{x}{\xi_1} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\xi_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
z \, .
</math>
  </td>
</tr>
</table>

</td></tr></table>

===Alternative Wikipedia Definition===

This same MF53 coordinate system &#8212; with different variable notation &#8212; is referred to in a [https://en.wikipedia.org/wiki/Elliptic_cylindrical_coordinates#Alternative_definition Wikipedia discussion] as an "alternative and geometrically intuitive set of elliptic coordinates."  The relevant mapping is, <math>~(d\sigma, \tau, z)_\mathrm{Wikipedia} = (\xi_1, \xi_2, \xi_3)_\mathrm{MF53}</math>.  The identified mapping to Cartesian coordinates is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(d\sigma)\tau </math>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi_1 \xi_2 \, ;</math>
  </td>

</tr>

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~d \biggl[ (\sigma^2 - 1 )(1 - \tau^2)  \biggr]^{1 / 2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2)  \biggr]^{1 / 2} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi_3 \, .</math>
  </td>
</tr>
</table>

According to the Wikipedia discussion, the three scale factors are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~h_\sigma^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
d^2\biggl[\frac{\sigma^2 - \tau^2}{\sigma^2 - 1}  \biggr] \, ;
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~h_\tau^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
d^2\biggl[\frac{\sigma^2 - \tau^2}{1 - \tau^2}  \biggr] \, ;
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~h_z^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 \, .
</math>
  </td>
</tr>
</table>

Interestingly, the Wikipedia discussion also includes the following expression for the Laplacian in this elliptic cylindrical coordinate system:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla^2\Phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{d^2(\sigma^2 - \tau^2)}
\biggl[ \sqrt{\sigma^2 - 1} \frac{\partial}{\partial\sigma}\biggl( \sqrt{\sigma^2 - 1} \frac{\partial\Phi}{\partial\sigma} \biggr)
+
\sqrt{1 - \tau^2 } \frac{\partial}{\partial\tau}\biggl( \sqrt{1 - \tau^2} \frac{\partial\Phi}{\partial\tau} \biggr) \biggr] + \frac{\partial^2\Phi}{\partial z^2} \, .
</math>
  </td>
</tr>
</table>