Editing Appendix/Ramblings/ToroidalCoordinates (section)

====Determine Overall Scale Length====
In order to fully tie our "region of overlap" discussion back to MF53's system of toroidal coordinates, we must identify the specific location of the origin of that coordinate system in, for example, the Figure 2 diagram.  [[#Presentation_by_MF53|As above]], we will place the origin of the coordinate system an, as yet unspecified, distance, <math>~a</math>, from the symmetry axis while, as illustrated in Figure 2, displacing it a distance, <math>~Z_0</math>, above the (cylindrical coordinate system's) equatorial plane.  Referring back to the properties of toroidal coordinate systems, as [[#Example_Toroidal_Surfaces|discussed above]], we know that in the <math>~Z = Z_0</math> plane, the inner and outer edges of a <math>~\xi_1</math> = constant torus/circle have radial locations,
<table align="center" border="0" cellpadding="4">
<tr>
  <td align="right">
<math>
~\frac{\varpi_\mathrm{inner}}{a} = \chi_\mathrm{inner}
</math>
  </td>
  <td align="center>
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl[\frac{(\xi_1 - 1)}{(\xi_1 + 1)} \biggr]^{1/2} \, ,
</math>
  </td>
</tr>

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

Hence, the major radius of the <math>~\xi_1</math> = constant toroidal surface is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} (\varpi_\mathrm{outer} + \varpi_\mathrm{inner})</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{a\xi_1}{(\xi_1^2 - 1)^{1/2}} \, ,
</math>
  </td>
</tr>
</table>
</div>
and its cross-sectional radius is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} (\varpi_\mathrm{outer} - \varpi_\mathrm{inner})</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{a}{(\xi_1^2 - 1)^{1/2}} \, .
</math>
  </td>
</tr>
</table>
</div>

This also means that, if <math>~r_0</math> and <math>~R_0</math> are specified, the associated values of <math>~\xi_1</math> and the scale length, <math>~a</math>, are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\xi_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{R_0}{r_0} \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_0 \biggl[\biggl( \frac{R_0}{r_0} \biggr)^2 - 1  \biggr]^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>