Editing Appendix/Ramblings/EllipticCylinderCoordinates (section)

==Introduction==

As has been made clear in our above review of the Elliptic Cylinder Coordinate system <math>~(\xi_1, \xi_2, \xi_3) = (d\cosh\mu, \cos\nu, z)</math>, individual curves within a family of ''confocal'' ellipses are identified by one's choice of the "radial" coordinate parameter, <math>~\mu</math>, or, alternatively, <math>~\xi_1</math>.  Specifically, while the two foci of every ellipse are positioned along the x-axis at the same points &#8212; namely, <math>~(x, y) = (\pm~d, 0)</math> &#8212; the length of the semi-major axis is given by, <math>~a = \xi_1 = d\cosh\mu</math>.

In a [[User:Tohline/Appendix/Ramblings/T3Integrals|separate chapter]] we have introduced a different orthogonal curvilinear coordinate system that we refer to as, "[[User:Tohline/Appendix/Ramblings/T3Integrals|T3 Coordinates]]."  In this coordinate system, <math>~(\lambda_1, \lambda_2, \lambda_3)</math>, individual surfaces within a family of ''concentric'' spheroids are identified by one's choice of a different "radial" coordinate parameter, <math>~\lambda_1</math>.  Here we will adopt essentially this same set of orthogonal coordinates, using <math>~\lambda_1</math> and <math>~\lambda_2</math> to describe a family of ''concentric'' ellipses that is independent of the vertical-coordinate.  We will refer to it as the &hellip;

<table border="1" cellpadding="10" align="center" width="80%">
<tr><td align="center">
'''T5 Coordinate System'''</td></tr>
<tr><td align="left">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\lambda_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~x \cosh \zeta </math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(x^2 + q^2 y^2)^{1 / 2} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~x (\sinh\zeta)^{1/(1-q^2)} </math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{x^{q^2}}{qy} \biggr)^{1/(q^2-1)}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z </math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\zeta</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\sinh^{-1}\biggl( \frac{qy}{x} \biggr) </math>
  </td>
</tr>
</table>
and, <math>~0 < q < \infty</math> is the (fixed) parameter used to specify the eccentricity, <math>~e = [(q^2-1)^{1 / 2}/q]</math>, of every <math>~\lambda_1 = </math> constant curve within the family of ''concentric'' ellipses.

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


Checking these expressions, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_2 \equiv x (\sinh\zeta)^{1/(1-q^2)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x \biggl( \frac{qy}{x} \biggr)^{1/(1-q^2)} = x \biggl( \frac{x}{qy} \biggr)^{1/(q^2-1)} 
= \biggl( \frac{x^{q^2}}{qy} \biggr)^{1/(q^2-1)} \, .</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\lambda_1 \equiv x\cosh\zeta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x\biggl[ 1 + \sinh^2\zeta\biggr]^{1 / 2} = x\biggl[ 1 + \biggl( \frac{qy}{x} \biggr)^2\biggr]^{1 / 2}
= ( x^2 + q^2 y^2 )^{1 / 2} \, .</math>
  </td>
</tr>
</table>
Comparing this last expression with the [[#Background|above background description of ellipses]], we see that <math>~\lambda_1 = </math> constant &#8212; for example, <math>~\lambda_0</math> &#8212; is synonymous with an ellipse having &hellip;
<ul>
<li>A semi-major axis of length, <math>~a = \lambda_0</math>;</li>
<li>An eccentricity, <math>~e \equiv (1 - b^2/a^2)^{1 / 2} = [(q^2-1)/q^2]^{1 / 2}</math>;</li>
<li>A pair of foci whose coordinate locations along the major axis are, <math>~(x, y) = (\pm~c, 0)</math>, where, <math>~c = ae</math>.</li>
</ul>