Editing Appendix/Ramblings/ToroidalCoordinates (section)

====Points of Intersection====
In all meridional planes, the surface of the equatorial-plane torus is defined by the off-center circle expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\varpi - \varpi_t)^2 + z^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_t^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~z^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_t^2 - (\varpi - \varpi_t)^2 \, .</math>
  </td>
</tr>
</table>
</div>

Independently, we know that the surface of the off-center, <math>\xi_1</math>-circle is defined by the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

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

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

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

When the two circles intersect, the (cylindrical) coordinates of the point(s) at which the intersection occurs,  <math>~(\varpi, z)=(\varpi_i, z_i)</math> must be shared by both circles.  Eliminating <math>~z</math> between these two off-center circle expressions allows us to solve for the "radial" coordinate, <math>~\varpi_i</math>, of the intersection point(s).  Specifically we find,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~2 [ r_t^2 - (\varpi - \varpi_t)^2 ]^{1/2} Z_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~Z_0^2 + a^2 + \varpi^2 + [ r_t^2 - (\varpi - \varpi_t)^2 ] - \frac{2a \varpi \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>~Z_0^2 + a^2 + \varpi^2 + r_t^2 - (\varpi^2  - 2 \varpi \varpi_t + \varpi_t^2)  - \varpi\biggl[ \frac{2a \xi_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>~\Kappa + 2 \varpi \biggl[ \varpi_t - \frac{a \xi_1}{(\xi_1^2 - 1)^{1/2}} \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>\Kappa \equiv Z_0^2 + a^2  - (\varpi_t^2 - r_t^2) \, .</math>
</div>

Squaring both sides of this expression gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~4Z_0^2 [ r_t^2 - (\varpi^2 - 2\varpi \varpi_t + \varpi_t^2) ] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Kappa^2 + 4 \varpi \Kappa\biggl[ \varpi_t - \frac{a \xi_1}{(\xi_1^2 - 1)^{1/2}} \biggr] 
+ 4 \varpi^2 \biggl[ \varpi_t - \frac{a \xi_1}{(\xi_1^2 - 1)^{1/2}} \biggr]^2</math>
  </td>
</tr>
</table>
</div>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[\Kappa^2 +4Z_0^2 (\varpi_t^2- r_t^2)] + \varpi [ 4\Kappa \beta - 8Z_0^2 \varpi_t ] 
+ 4\varpi^2  [Z_0^2 + \beta^2 ] \, ,</math>
  </td>
</tr>
</table>
</div>
<span id="BetaDefinition">where,</span>
<div align="center">
<math>\beta \equiv \varpi_t - \frac{a \xi_1}{(\xi_1^2 - 1)^{1/2}}  \, .</math>
</div>

<span id="IntersectionVarpi">The roots of this quadratic equation provide the sought-after coordinate(s), <math>~\varpi_i</math>, of the point(s) of intersection.</span>  Specifically,

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

<tr>
  <td align="right">
<math>~\varpi_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{8 [Z_0^2 + \beta^2 ]} \biggl\{
[ 8Z_0^2 \varpi_t- 4\Kappa \beta  ] \pm \sqrt{[ 8Z_0^2 \varpi_t- 4\Kappa \beta  ]^2 - 16[Z_0^2 + \beta^2 ]  [\Kappa^2 +4Z_0^2 (\varpi_t^2- r_t^2)]}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{[ 8Z_0^2 \varpi_t- 4\Kappa \beta  ]}{8 [Z_0^2 + \beta^2 ]} \biggl\{
1 \pm \sqrt{1 - \frac{16[Z_0^2 + \beta^2 ]  [\Kappa^2 +4Z_0^2 (\varpi_t^2- r_t^2)]}{[ 8Z_0^2 \varpi_t- 4\Kappa \beta  ]^2 }}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2Z_0^2 \varpi_t-\Kappa \beta  }{2 (Z_0^2 + \beta^2 )} \biggl\{
1 \pm \sqrt{1 - \ell } \biggr\}\, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>\ell \equiv \frac{(Z_0^2 + \beta^2 )[\Kappa^2 +4Z_0^2 (\varpi_t^2- r_t^2)]}{( 2Z_0^2 \varpi_t-\Kappa \beta  )^2 } \, .</math>
</div>

Now, from the [[#Toroidal_Coordinates|definition of Toroidal Coordinates, as provided above]], we know that the cylindrical coordinate, <math>~\varpi</math>, is related to the pair of meridional-plane toroidal coordinates via the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\varpi}{a}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1-\xi_2} \, .</math>
  </td>
</tr>
</table>
</div>

Therefore, once <math>~\varpi_i</math> has been determined for a given choice of <math>~\xi_1</math>, the corresponding value of <math>~\xi_2</math> at the intersection point is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\xi_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi_1 - \frac{(\xi_1^2 - 1)^{1/2}}{(\varpi_i/a)} \, .</math>
  </td>
</tr>
</table>
</div>
Finally, given the pair of coordinate values, <math>~(\xi_1, \xi_2)_i</math>, the value of the (cylindrical) z-coordinate at the intersection point can be obtained via the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{Z_0}{a} - \frac{z}{a}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(1-\xi_2^2 )^{1/2}}{\xi_1-\xi_2} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~Z_0 - \frac{a(1-\xi_2^2 )^{1/2}}{\xi_1-\xi_2} \, .</math>
  </td>
</tr>
</table>
</div>