Editing Appendix/Ramblings/ToroidalCoordinates (section)

====Revise Overlap Discussion====
Let's reassess the conclusions drawn in our [[#Overlap_Between_Two_Off-Center_Circles|overlap discussion, above]].  Rather than varying <math>~r_0</math> while holding <math>~R_0</math> fixed, let's consider varying <math>~\xi_1</math> while fixing the coordinate location of the origin of the toroidal coordinate system, <math>~(a, Z_0)</math>.  This is the approach that is appropriately aligned with integration over the (pink) toroidal mass distribution.

Re-expressed,  the pair of boundaries of the "region of overlap," <math>~r_\pm</math>, give:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(r_0 \pm r_t)^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\varpi_t - R_0)^2 + Z_0^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \biggl[ \frac{a}{(\xi_1^2 - 1)^{1/2}}  \pm r_t \biggr]^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \varpi_t - \frac{a\xi_1}{(\xi_1^2 - 1)^{1/2}} \biggr]^2 + Z_0^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \biggl[ a \pm r_t (\xi_1^2 - 1)^{1/2}\biggr]^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \varpi_t (\xi_1^2 - 1)^{1/2} - a\xi_1 \biggr]^2 + Z_0^2 (\xi_1^2 - 1)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
a^2 \pm 2a r_t (\xi_1^2 - 1)^{1/2} + r_t^2 (\xi_1^2 - 1)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\varpi_t^2 (\xi_1^2 - 1) - 2a \varpi_t \xi_1 (\xi_1^2 - 1)^{1/2} + a^2\xi_1^2  + Z_0^2 (\xi_1^2 - 1)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math> ~\Rightarrow~~~~(\xi_1^2 - 1)^{1/2}[2a \varpi_t \xi_1 \pm 2a r_t  ]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[\varpi_t^2 +a^2 + Z_0^2](\xi_1^2 - 1) </math>
  </td>
</tr>

<tr>
  <td align="right">
<math> ~\Rightarrow~~~~(\xi_1^2 - 1)^{1/2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2a(\varpi_t \xi_1 \pm r_t  )}{(\varpi_t^2 +a^2 + Z_0^2)} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\ell} \biggl[ \xi_1 \pm \frac{r_t}{\varpi_t}  \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>\ell \equiv \frac{1}{2}\biggl[ \frac{a^2 + \varpi_t^2 + Z_0^2}{a\varpi_t} \biggr] \, .</math>
</div>

After squaring both sides of this equation, we find that the values of <math>~\xi_1</math> corresponding to the limits of overlap can be obtained from the roots of the following quadratic equation:
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \xi_1^2 \pm \xi_1\biggl(\frac{2r_t}{\varpi_t}\biggr) +  \biggl(\frac{r_t}{\varpi_t}\biggr)^2 \, ,</math>
  </td>
</tr>
</table>
</div>
that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1-\ell^2) \xi_1^2 \pm \xi_1\biggl(\frac{2r_t}{\varpi_t}\biggr) +  \biggl[ \biggl(\frac{r_t}{\varpi_t}\biggr)^2 +\ell^2\biggr] \, .</math>
  </td>
</tr>
</table>
</div>

<!-- COMMENT OUT determination of quadratic roots because result is likely irrelevant
The roots are:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\xi_1\biggr|_\pm</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2(1-\ell^2)} \biggl\{ 
\mp \biggl(\frac{2r_t}{\varpi_t}\biggr) \pm  \sqrt{\biggl( \frac{2r_t}{\varpi_t}\biggr)^2 - 4(1-\ell^2) \biggl[ \biggl(\frac{r_t}{\varpi_t}\biggr)^2 +\ell^2\biggr]}
\biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(1-\ell^2)} \biggl(\frac{r_t}{\varpi_t}\biggr)\biggl\{ 
\mp 1 \pm  \sqrt{1 - (1-\ell^2) \biggl[ 1 +\biggl(\frac{\ell \varpi_t}{r_t}\biggr)^2\biggr]} 
\biggr\} \, .</math>
  </td>
</tr>
</table>
</div>

-->

After setting up this expression, it dawned on me that the "plus or minus" generalization is not appropriate in this situation.  While either result &#8212; say, the "plus" result &#8212; can be shifted from a <math>~r_0 - R_0</math> specification to a <math>~a - \xi_1</math> specification, the pair of results generally will not share the same value of the scale length, <math>~a</math>.  Hence the pair of solutions will be unrelated when viewed from the perspective of the toroidal coordinate system.  Instead, let's determine the value of <math>~a</math> from the "first contact" solution &#8212; the ''superior'' sign in the expression &#8212; then figure out what the "final contact" solution will be if this scale length is held fixed.  The solution to the quadratic equation is:

<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{1}{2(1-\ell^2)} \biggl\{
- \biggl(\frac{2r_t}{\varpi_t}\biggr) \pm \sqrt{\biggl(\frac{2r_t}{\varpi_t}\biggr)^2 -4(1-\ell^2)
\biggl[ \biggl(\frac{r_t}{\varpi_t}\biggr)^2 +\ell^2\biggr]}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{r_t}{\varpi_t(1-\ell^2)} \biggl\{
- 1 \pm \sqrt{1 -(1-\ell^2)
\biggl[ 1 +\biggl(\frac{\ell \varpi_t}{r_t}\biggr)^2\biggr]} 
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>
Given that the allowed range of values for the "radial" toroidal coordinate is, <math>~1 \leq \xi_1 \leq \infty</math>, the relevant root is, 

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

<tr>
  <td align="right">
<math>~\xi_1\biggr|_\mathrm{first}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{r_t}{\varpi_t(1-\ell^2)} \biggl\{
\sqrt{1 -(1-\ell^2)
\biggl[ 1 +\biggl(\frac{\ell \varpi_t}{r_t}\biggr)^2\biggr]} -1
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>