Editing Appendix/Ramblings/ToroidalCoordinates (section)

==Do It Again==

The question that I've had in the back of my mind for quite some time is, "For what astrophysically interesting problem might we effectively use the toroidal coordinate system in order to derive a much cleaner ''analytic'' description of an axisymmetric potential?"  Originally, I thought that a suitable configuration might be a uniform-density torus that has a perfectly circular cross-section.  After all, the surface of such a torus can be perfectly described as a <math>~\xi_1 = </math>&nbsp; constant configuration.  In a [[#Multipole_Moment_in_Toroidal_Coordinates|subsection presented below]], I began investigating this problem, setting up a toroidal coordinate system to appropriately conform to the surface of such a torus, then calling upon WolframAlpha's online integration tool to complete the integral over the orthogonal coordinate, <math>~\xi_2</math>, analytically.  After giving the problem considerable more thought, however, I realized that, while I could legitimately move the mass-density outside of that first integral, it was not legitimate to move the <math>~\mu K(\mu)</math> factor outside of that integral.  While it is true that CT99 showed that the <math>~\mu K(\mu)</math> factor only depends on the first coordinate in a toroidal coordinate system, it is a ''different'' toroidal coordinate system from the one that conveniently aligns with the physical torus!  Let's set up the double integral again, but this time let's use the toroidal coordinate system that is defined within the CCGF discussion.  We begin by describing geometric relationships between pairs of off-center circles and deriving algebraic expressions that define the conditions under which such circles overlap and/or simply intersect.

===Overlap Between Two Off-Center Circles===

Figure 2 displays two off-center circles.  The solid pink circle represents a meridional cross-section through a uniform-density, axisymmetric torus whose center lies in the equatorial plane of a <math>~(\varpi, Z) </math>, cylindrical coordinate system; as depicted, <math>~\varpi_t</math>  is the size of the major radius of this torus and its cross-sectional radius is <math>~r_t</math>.  The other circle represents a single, <math>~\xi_1</math> = constant (toroidal) surface in toroidal coordinates; its major radius is, <math>~R_0</math>, and its cross-sectional radius is <math>~r_0</math>.  The center of this <math>~\xi_1</math> = constant circle lies in the equatorial system of the associated toroidal coordinate system, which is parallel to but, as depicted, lies a distance, <math>~Z_0</math>, above the equatorial plane of the <math>~(\varpi, Z) </math>, cylindrical coordinate system.

As drawn, the figure does not identify the precise location of the ''origin'' of the toroidal coordinate system.  But, in accordance with the properties of such coordinate systems, the origin must lie inside of the referenced circle and to the left of &#8212; that is, closer to the <math>~Z</math> (symmetry) axis than &#8212; the center of the circle, <math>~R_0</math>.  

<div align="center" id="Figure2">
<table border="1" cellpadding="8">

<tr>
<th align="center"><font size="+1">Figure 2</font></th>
</tr>

<tr>
<td align="center">
[[File:DiagramToroidalCoordinates.png|350px|Diagram of Torus and Toroidal Coordinates]]
</td>
</tr>
</table>
</div>

If the size of the <math>~\xi_1</math> = constant surface is varied while all the other key parameters <math>~(R_0, Z_0, \varpi_t, r_t)</math> are held fixed, what is the range of values of <math>~r_0</math> over which the two depicted circles overlap and/or simply intersect?  

====Initial Contact====
Geometrically we appreciate that, as <math>~r_0</math> is increased, the two circles will first touch at a point that lies along the (blue-dashed) line-segment that connects the centers of both circles.  More specifically, the initial interception will be at the point identified in Figure 2 by the solid blue dot lying on the surface of the pink torus.  The distance between the two centers &#8212; which we will denote as <math>~h</math> &#8212; is also the hypotenuse of a right triangle whose other two sides are of length (opposite the angle, <math>~\alpha</math>) <math>~\varpi_t - R_0</math> and (adjacent to the angle, <math>~\alpha</math>) <math>~Z_0</math>.  We see that the initial interception will occur when <math>~r_0 + r_t = h</math>, that is, when 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_0 = r_+</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~h - r_t</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[(\varpi_t - R_0)^2 + Z_0^2 ]^{1/2} - r_t </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\varpi_t - R_0)[1 + \Lambda^2 ]^{1/2} - r_t \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Lambda</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{Z_0}{\varpi_t - R_0} \, .</math>
  </td>
</tr>
</table>
</div>
For later reference, we note that the cylindrical coordinates associated with this initial point of contact &#8212; ''i.e.,'' the point identified in Figure 2 by the solid blue dot lying on the surface of the pink torus &#8212; are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi_+</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\varpi_t - r_t \sin\alpha</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\varpi_t - \frac{r_t (\varpi_t-R_0)}{[(\varpi_t - R_0)^2 + Z_0^2 ]^{1/2} }</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\varpi_t - \frac{r_t }{[1+\Lambda^2 ]^{1/2} } \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Z_+</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_t \cos\alpha</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{r_t Z_0}{[(\varpi_t - R_0)^2 + Z_0^2 ]^{1/2} }</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{r_t \Lambda}{[1+\Lambda^2 ]^{1/2} } \, .</math>
  </td>
</tr>
</table>
</div>

====Final Contact====
It is easy to see, geometrically, that if the (blue-dashed) line-of-centers and, in particular, if <math>~r_0</math> is increased beyond the "initial contact" length of <math>~r_+</math>, by exactly a length that equals the diameter of the pink torus, <math>~2r_t</math>, then the <math>~\xi_1</math> = constant circle will make its ''last'' contact with the circle that defines the surface of the equatorial-plane torus.  Associating the subscript "-" with this point of last contact, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_-</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~r_+ + 2r_t</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\varpi_t - R_0)[1 + \Lambda^2 ]^{1/2} + r_t \, ,</math>
  </td>
</tr>
</table>
</div>
and the associated coordinate-location of this ''last'' point of contact,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi_-</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\varpi_t + \frac{r_t }{[1+\Lambda^2 ]^{1/2} } \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Z_-</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\frac{r_t \Lambda}{[1+\Lambda^2 ]^{1/2} } \, .</math>
  </td>
</tr>
</table>
</div>

====Region of Overlap====
From the above discussion and derivations, we conclude that the <math>~\xi_1</math> = constant circle will overlap the pink torus and will, accordingly, intersect the surface of that torus in two places for all values of <math>~r_+ < r_0 < r_-</math>, that is, for,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\varpi_t - R_0)[1 + \Lambda^2 ]^{1/2} - r_t </math>
  </td>
  <td align="center">
<math>~< r_0 < </math>
  </td>
  <td align="left">
<math>~(\varpi_t - R_0)[1 + \Lambda^2 ]^{1/2} + r_t \, .</math>
  </td>
</tr>
</table>
</div>

====Reality Check One====
Let's see if these derived results make sense.  As a first example, let's assign values of various Figure 2 parameters as follows:

<div id="Example1A">
<table align="center" border="1" cellpadding="5">
<tr><th align="center" colspan="4">Example 1A</th></tr>
<tr>
  <td align="center" width="25%"><math>~\varpi_t</math></td>
  <td align="center" width="25%"><math>~r_t</math></td>
  <td align="center" width="25%"><math>~Z_0</math></td>
  <td align="center"><math>~\alpha</math></td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{3}{4}</math></td>
  <td align="center"><math>~\tfrac{1}{4}</math></td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~\tfrac{\pi}{6}</math></td>
</tr>
</table>
</div>

(Notice that the first pair of these parameter values aligns with the properties of the pink torus that was sketched in Figure 4 of [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Hur&eacute; &amp; Hersant (2012)] &#8212; as [[#THH12Figure4|reprinted immediately below]] &#8212; and that the chosen value of <math>~Z_0</math> aligns with the z-coordinate of their "Point B.")  

<div align="center" id="THH12Figure4">
<table border="1" cellpadding="8">

<tr><td align="center">
Figure 4 extracted without modification from p. 2640 of [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Hur&eacute; &amp; Hersant (2012)]<p></p>
"''The Potential of Discs from a 'Mean Green Function' ''"<p></p>
Monthly Notices of the Royal Astronomical Society, vol. 424, pp. 2635-2645 &copy; RAS
</td>
</tr>

<tr>
<td align="center">
[[File:Figure4THH2012.png|350px|Figure 4 from Trova, Hur&eacute; &amp; Hersant (2012)]]
</td>
</tr>
<tr>
  <td align="center">
<math>~\varpi_t = \tfrac{3}{4}\, ; ~ r_t = \tfrac{1}{4}</math><p></p>
Point A:  <math>~(\varpi, Z) = (\tfrac{3}{4}, 0)</math><p></p>
Point B:  <math>~(\varpi, Z) = (1, 1)</math><p></p>
Point C:  <math>~(\varpi, Z) = (10, 10)</math>
  </td>
</tr>
</table>
</div>



Taken together, this choice for the values of <math>~\alpha</math> and <math>~Z_0</math> implies: (1) That the hypotenuse of the blue right-triangle in [[#THH12Figure4|our Figure 2]] and, hence, the distance between the centers of the two circles, is 
<div align="center">
<math>~h = \frac{Z_0}{\cos\alpha} = \frac{2\sqrt{3}}{3} \, ;</math>
</div>

and, (2) that the side of the triangle that is opposite the angle, <math>~\alpha</math>, is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi_t - R_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~h \sin\alpha = \frac{\sqrt{3}}{3} \, ,</math>
  </td>
</tr>
</table>
</div>
which, taken together with the choice of <math>~\varpi_t</math>, gives,
<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} \approx 0.17265\, .</math>
  </td>
</tr>
</table>
</div>

With this set of parameters held fixed, it is clear that, in order for the <math>~\xi_1</math> = constant circle to make first/final contact with the pink torus, it will need to have a radius,
<div align="center">
<math>~r_\pm = h \mp r_t = \frac{2\sqrt{3}}{3} \mp \frac{1}{4} \, .</math>
</div>

Let's see if this expectation matches the result obtained via the expressions derived above.   Specifically, we find,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Lambda</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{Z_0}{\varpi_t - R_0} = \sqrt{3} \, ;</math>
  </td>
</tr>
</table>
</div>
hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_\pm</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\varpi_t - R_0)[1 + \Lambda^2 ]^{1/2} \mp r_t </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{\sqrt{3}}{3}\biggl[1 + (\sqrt{3})^2 \biggr]^{1/2} \mp \frac{1}{4} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{2\sqrt{3}}{3} \mp \frac{1}{4} \, .</math>
  </td>
</tr>
</table>
</div>
This precisely matches our expectation.

===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>


===Other===
Now, the surface of an equatorial-plane torus having major radius, <math>\varpi_t</math>, and cross-sectional radius, <math>~r_t</math>, is described by the expression,

So, as the vertical coordinate varies over the range, <math>-r_t \leq z \leq + r_t</math>, the horizontal coordinate varies over the range, <math>(\varpi_t - r_t ) \leq \varpi \leq (\varpi_t + r_t)</math>.  But, more importantly, for a ''given'' value of <math>~\varpi</math>, the corresponding value of the vertical coordinate is,
<div align="center">
<math>
~ z = \pm \biggl[ r_t^2 - (\varpi-\varpi_t)^2 \biggr]^{1/2}.
</math>
</div>