Editing 2DStructure/ToroidalCoordinates (section)

===Properties===
Here we highlight certain properties and features of the [[Appendix/References#MF53|MF53]] toroidal coordinate system; more details can be found in a [[Appendix/Ramblings/ToroidalCoordinates#Toroidal_Coordinates|related set of our online notes]].  Most importantly in the context of our discussion, if (at all azimuthal angles) the origin of the toroidal coordinate system is placed at the ''cylindrical-coordinate'' location, <math>~(a, Z_0), </math> the pair of orthogonal coordinates, <math>~(\xi_1, \xi_2)</math>, is related to the cylindrical coordinate pair, <math>~(\varpi, Z)</math>, via the expressions,
<table align="center" border="0" cellpadding="5">
<tr><th align="center" colspan="1">&nbsp;</th>
       <th align="center" colspan="2">[<font color="red"><b>[[Appendix/References#MF5|MF53]]</b></font>]</th>
       <th align="center" colspan="2">[https://en.wikipedia.org/wiki/Toroidal_coordinates Wikipedia]</th>
</tr>
<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>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{\sinh\tau}{\cosh\tau - \cos\theta} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{(Z_0 - Z)}{a}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\pm~\frac{(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2} 
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{\sin\theta}{\cosh\tau - \cos\theta}  \, .
</math>
  </td>
</tr>
</table>

An off-center circle &#8212; such as the white circle with purple perimeter depicted in our Figure 1 diagram &#8212; is generated if a value of the "radial" coordinate, <math>~\xi_1</math>, is chosen from within the range, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
       <th align="center" colspan="1">[<font color="red"><b>[[Appendix/References#MF5|MF53]]</b></font>]</th>
       <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
       <th align="center" colspan="1">[https://en.wikipedia.org/wiki/Toroidal_coordinates Wikipedia]</th>
</tr>
<tr>
  <td align="right">
<math>~+1 \leq \xi_1 < \infty</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; or, equivalently &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="left">
<math>~0 \leq \tau < \infty \, ,</math>
  </td>
</tr>
</table>
</div>
and held fixed while the "angular" coordinate, <math>~\xi_2</math>, is varied over the range, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
       <th align="center" colspan="1">[<font color="red"><b>[[Appendix/References#MF5|MF53]]</b></font>]</th>
       <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
       <th align="center" colspan="1">[https://en.wikipedia.org/wiki/Toroidal_coordinates Wikipedia]</th>
</tr>
<tr>
  <td align="right">
<math>~ -1 \leq \xi_2 \leq +1</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; or, more completely &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="left">
<math>~-\pi < \theta \leq \pi</math>
  </td>
</tr>
</table>
</div>
Hereafter, we will refer to this <math>~\xi_1</math> = constant circle as a "<math>\xi_1</math>-circle."  A <math>\xi_1</math>-circle of radius zero and, hence, the origin of the toroidal coordinate system is associated with the ''upper'' limiting value of the radial coordinate, namely, <math>~\xi_1 = \infty</math>; as the value of <math>~\xi_1</math> is decreased monotonically, the radius of the circle (for example, the circle of radius, <math>~r_0</math>, in our Figure 1) steadily grows; and the radius of this circle becomes infinite at the radial coordinate's other limiting value, <math>~\xi_1 = 1</math>.

In the <math>~Z = Z_0</math> plane, the location of the inner and outer edges of the toroidal-coordinate surface are determined by setting <math>~\xi_2 = -1</math> (inner) and <math>~\xi_2 = +1</math> (outer).  Hence,
<table align="center" border="0" cellpadding="4">
<tr>
  <td align="right">
<math>
~\biggl(\frac{\varpi}{a}\biggr)_\mathrm{inner}
</math>
  </td>
  <td align="center>
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 +1} = \biggl[\frac{(\xi_1 - 1)}{(\xi_1 + 1)} \biggr]^{1/2} \, ,
</math>

  </td>
</tr>

<tr>
  <td align="right">
<math>
~\biggl(\frac{\varpi}{a}\biggr)_\mathrm{outer}
</math>
  </td>
  <td align="center>
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - 1} = \biggl[\frac{(\xi_1 + 1)}{(\xi_1 - 1)} \biggr]^{1/2} \, .
</math>

  </td>
</tr>
</table>
Hence, also, the (cylindrical) radial location of the "center" of each toroidal-coordinate surface &#8212; labeled <math>~R_0</math> in our Figure 1 &#8212; is given by the expression,
<div align="center">
<math>
R_0 = \frac{a}{2} \biggl[ \biggl(\frac{\varpi}{a}\biggr)_\mathrm{outer} 
+ \biggl(\frac{\varpi}{a}\biggr)_\mathrm{inner} \biggr] = \frac{a\xi_1}{(\xi_1^2 - 1)^{1/2}}  \, ,
</math>
</div>
and the surface's cross-sectional radius &#8212; labeled <math>~r_0</math> in our Figure 1 &#8212; is given by the expression,
<div align="center">
<math>
r_0 = \frac{a}{2} \biggl[ \biggl(\frac{\varpi}{a}\biggr)_\mathrm{outer} 
- \biggl(\frac{\varpi}{a}\biggr)_\mathrm{inner} \biggr] = \frac{a}{(\xi_1^2 - 1)^{1/2}} \, .
</math>
</div>

This last expression quantifies, and its simplicity reinforces, our earlier statement; that is, as the value of <math>~\xi_1</math> is decreased monotonically, the radius of the circle, <math>~r_0</math>, steadily grows.  The next-to-last expression makes it clear, as well, that <math>~R_0</math> grows larger and, therefore, the location of the center of a <math>\xi_1</math>-circle shifts farther away from the symmetry axis as the value of <math>~\xi_1</math> is decreased.  Notice that, for any off-center circle, the ratio of these to lengths gives the value of the toroidal-coordinate system's dimensionless "radial" coordinate, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{R_0}{r_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{a\xi_1}{(\xi_1^2 - 1)^{1/2}}\biggr] \biggl[ \frac{(\xi_1^2 - 1)^{1/2}}{a}\biggr]
= \xi_1 \, .</math>
  </td>
</tr>
</table>
</div>
Notice, furthermore, that there is a particular combination of these two lengths that is independent of <math>~\xi_1</math>, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_0 \biggl[\biggl( \frac{R_0}{r_0} \biggr)^2 - 1  \biggr]^{1/2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a}{(\xi_1^2 - 1)^{1/2}} \biggl[\xi_1^2 - 1  \biggr]^{1/2} = a \, .</math>
  </td>
</tr>
</table>
</div>
This is a manner in which one can determine the radial position, <math>~a</math>, of the origin of the toroidal coordinate system that could legitimately be associated with any particular off-center circle, such as the white circle with a purple perimeter drawn in our Figure 1.