Editing Appendix/Ramblings/ToroidalCoordinates (section)

===Off-center Circle===
The curves drawn in the above figure labeled "Toroidal Coordinate System" resemble circles whose centers are positioned a distance <math>~\chi_0</math> away from the origin.  Let's examine whether this is the case by drawing on the familiar expression for such a configuration, [[Appendix/Ramblings/ToroidalCoordinates#Off-center_Circle|as presented above]].  If this is the case, then the circle as illustrated in the figure will have <math>~z_0 = 0</math> and a radius,

<table align="center" border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
~\alpha_c \equiv \frac{r_c}{a}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\chi_\mathrm{outer} - \chi_0 
</math>
  </td>
</tr>

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

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

and the algebraic expression describing the circle will take the form,

<div align="center">
<math>
~(\chi - \chi_0)^2 + \zeta^2 = \alpha_c^2 = (\xi_1^2 - 1)^{-1} .
</math>
</div>

Let's evaluate the left-hand-side of this expression to see if it indeed reduces to <math>(\xi_1^2 - 1)^{-1}</math>.

<table align="center" border="0" cellpadding="10">
<tr>
  <td align="right">
<math>~\mathrm{LHS}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl\{ \biggl[ \frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2}
 \biggr] - \frac{\xi_1}{(\xi_1^2 - 1)^{1/2}} \biggr\}^2 + \biggl[ \frac{(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2}  \biggr]^2
</math>
  </td>
</tr>

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

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

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

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

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

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

</table>

Yes!  So this means that the <math>~\xi_1 = \mathrm{constant}</math> toroidal contours can be described by the off-center circle expression,

<div align="center">
<math>
~(\chi - \chi_0)^2 + \zeta^2 = (\chi_\mathrm{outer} - \chi_0)^2 \, ,
</math>
</div>
or,

<div align="center">
<math>
~\biggl[ \chi - \frac{\xi_1}{(\xi_1^2 - 1)^{1/2}} \biggr]^2 + \zeta^2 = \frac{1}{(\xi_1^2 - 1)} \, .
</math>
</div>

It also means that, while <math>~\xi_1</math> is the official ''radial'' coordinate of MF53's toroidal coordinate system, the actual dimensionless radius of the relevant cross-sectional circle is,

<table align="center" border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
~\alpha_c
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{1}{(\xi_1^2 - 1)^{1/2}} .
</math>
  </td>
</tr>
</table>