Editing Appendix/Ramblings/ToroidalCoordinates (section)

===Relate to Toroidal Coordinate System===

====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>



====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>

====Reality Check Two====
Let's examine the behavior of these expressions, given the structural parameters provided in [[#Example1A|Example 1A, as defined above]].  [[#Reality_Check_One|Earlier]], we deduced that "first contact" occurs when, 
<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{3}{4} - \frac{\sqrt{3}}{3} = \frac{9-4\sqrt{3}}{12} \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_0 = r_+</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{2\sqrt{3}}{3} - \frac{1}{4} = \frac{8\sqrt{3} - 3}{12} \, .</math>
  </td>
</tr>
</table>
</div>

Hence, we should find that,
<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_0}{r_0} = \frac{9-4\sqrt{3}}{8\sqrt{3} - 3} \approx 0.19084\, ,</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>