Editing Appendix/Ramblings/ToroidalCoordinates (section)

===Example Toroidal Surfaces===
In the accompanying figure labeled "Toroidal Coordinate System," we've outlined three different <math>~\xi_1 = \mathrm{constant}</math> meridional contours for the MF53 toroidal coordinate system. The illustrated values are,
<table align="center" border="0" cellpadding="4">
<tr>
  <td align="right">
<math>
~\xi_1
</math>
  </td>
  <td align="center>
<math>~=</math>
  </td>
  <td align="left">
<math>~1.1</math>
  </td>
  <td align="left" width="25%">
&nbsp;
  </td>
  <td align="left">
<math>~\mathrm{(blue)} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\xi_1
</math>
  </td>
  <td align="center>
<math>~=</math>
  </td>
  <td align="left">
<math>~1.2</math>
  </td>
  <td align="left" width="25%">
&nbsp;
  </td>
  <td align="left">
<math>~\mathrm{(red)} \, ;</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
~\xi_1
</math>
  </td>
  <td align="center>
<math>~=</math>
  </td>
  <td align="left">
<math>~1.5</math>
  </td>
  <td align="left" width="25%">
&nbsp;
  </td>
  <td align="left">
<math>\mathrm{(gold)} \, .</math>
  </td>
</tr>
</table>

The inner and outer edges of the toroidal surface in the equatorial plane should be determined by setting <math>~\xi_2 = -1</math> (inner) and <math>~\xi_2 = +1</math> (outer).  Hence,
<table align="center" border="0" cellpadding="4">
<tr>
  <td align="right">
<math>
~\chi_\mathrm{inner}
</math>
  </td>
  <td align="center>
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 +1} = \biggl[\frac{(\xi_1 - 1)}{(\xi_1 + 1)} \biggr]^{1/2}
</math>

  </td>
</tr>

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

  </td>
</tr>
</table>
The equatorial-plane location of the "center" of each torus is,
<div align="center">
<math>
\chi_0 = \frac{1}{2} (\chi_\mathrm{outer} + \chi_\mathrm{inner}) = \frac{\xi_1}{(\xi_1^2 - 1)^{1/2}} ,
</math>
</div>
and the so-called distortion parameter,
<div align="center">
<math>
\delta \equiv \frac{\chi_\mathrm{outer}-\chi_\mathrm{inner}}{\chi_0}= \frac{2}{\xi_1} .
</math>
</div>


<table align="center" border="1" cellpadding="8">
<tr>
  <th align="center" colspan="6">
<font color="maroon">
Properties of <math>\xi_1 = \mathrm{constant}</math> Toroidal Surfaces
</font>
  </th>
</tr>
<tr>
  <td align="center">
Curve in<br />Figure
  </td>
  <td align="center">
<math>\xi_1</math>
  </td>
  <td align="center">
<math>\chi_\mathrm{inner}</math>
  </td>
  <td align="center">
<math>\chi_\mathrm{outer}</math>
  </td>
  <td align="center">
<math>\chi_0</math>
  </td>
  <td align="center">
<math>\delta</math>
  </td>
</tr>

<tr>
  <td align="center">
Blue
  </td>
  <td align="center">
1.1
  </td>
  <td align="center">
0.218
  </td>
  <td align="center">
4.583
  </td>
  <td align="center">
2.400
  </td>
  <td align="center">
1.818
  </td>
</tr>

<tr>
  <td align="center">
Red
  </td>
  <td align="center">
1.2
  </td>
  <td align="center">
0.302
  </td>
  <td align="center">
3.317
  </td>
  <td align="center">
1.809
  </td>
  <td align="center">
1.667
  </td>
</tr>

<tr>
  <td align="center">
Gold
  </td>
  <td align="center">
1.5
  </td>
  <td align="center">
0.447
  </td>
  <td align="center">
2.236
  </td>
  <td align="center">
1.342
  </td>
  <td align="center">
1.333
  </td>
</tr>
</table>


What function <math>~\zeta(\varpi)</math> coincides with these <math>~\xi_1 = \mathrm{constant}</math> surfaces? (To be answered!)


<div align="center">
[[File:LSU_CombinedTori.jpg|none|800px|Meridional contours of constant <math>\xi_1</math>.]]  
</div>