Editing Appendix/Ramblings/ToroidalCoordinates (section)

===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.