Editing Appendix/Ramblings/ToroidalCoordinates (section)

==Toroidal Coordinates==
===Presentation by MF53===
The orthogonal toroidal coordinate system <math>(\xi_1,\xi_2,\xi_3=\cos\varphi)</math> discussed by MF53 has the following properties:
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
~\frac{x}{a}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\biggl[ \frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \biggr]\cos\varphi \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{y}{a}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\biggl[ \frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \biggr]\sin\varphi \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\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>
~\frac{\varpi}{a} \equiv \biggl[ \biggl(\frac{x}{a}\biggr)^2 + \biggl(\frac{y}{a}\biggr)^2 \biggr]^{1/2}
</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>

<tr>
  <td align="right">
<math>
~\frac{r}{a} \equiv \biggl[ \biggl(\frac{x}{a}\biggr)^2 + \biggl(\frac{y}{a}\biggr)^2 + \biggl(\frac{z}{a}\biggr)^2\biggr]^{1/2}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\biggl[ \frac{\xi_1 + \xi_2}{\xi_1 - \xi_2} \biggr]^{1/2} \, .
</math>
  </td>
</tr>
</table>

<span id="ToroidalScaleFactors">According to MF53, the associated scale factors of this orthogonal coordinate system are:</span>
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
~\frac{h_1}{a}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{1}{(\xi_1 - \xi_2)(\xi_1^2 - 1)^{1/2}} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{h_2}{a}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{1}{(\xi_1 - \xi_2)(1-\xi_2^2)^{1/2}} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{h_3}{a}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\biggl[ \frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \biggr]\frac{1}{\sin\varphi} \, .
</math>
  </td>
</tr>
</table>

That means that, in the meridional plane, an area element should be,
<div align="center">
<math>
~d\sigma = (h_1 d\xi_1)(h_2 d\xi_2) = a^2 \biggl[ \frac{d\xi_1}{(\xi_1 - \xi_2)(\xi_1^2 - 1)^{1/2}} \biggr] 
\biggl[ \frac{d\xi_2}{(\xi_1 - \xi_2)(1-\xi_2^2)^{1/2}} \biggr]   \, .
</math>
</div>

===Tohline's Ramblings===
My inversion of these coordinate definitions has led to the following expressions:
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
~\xi_1
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
\frac{r(r^2 + 1)}  {[\chi^2(r^2-1)^2 + \zeta^2(r^2+1)^2]^{1/2}} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\xi_2
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
\frac{r(r^2 - 1)}{[\chi^2(r^2-1)^2 + \zeta^2(r^2+1)^2]^{1/2}} \, ,
</math>
  </td>
</tr>
</table>
where,
<div align="center">
<math>
\chi \equiv \frac{\varpi}{a} ~~~;~~~\zeta\equiv\frac{z}{a} ~~~\mathrm{and} ~~~ r=(\chi^2 + \zeta^2)^{1/2} .
</math>
</div>

Apparently the allowed ranges of the two meridional-plane coordinates are:
<div align="center">
<math>
+1 \leq \xi_1 \leq \infty ~~~\mathrm{and} ~~~ -1 \leq \xi_2 \leq +1 .
</math>
</div>

===Example Toroidal Surfaces===
In the accompanying figure labeled "Toroidal Coordinate System," we've outlined three different <math>~\xi_1 = \mathrm{constant}</math> meridional contours for the MF53 toroidal coordinate system. The illustrated values are,
<table align="center" border="0" cellpadding="4">
<tr>
  <td align="right">
<math>
~\xi_1
</math>
  </td>
  <td align="center>
<math>~=</math>
  </td>
  <td align="left">
<math>~1.1</math>
  </td>
  <td align="left" width="25%">
&nbsp;
  </td>
  <td align="left">
<math>~\mathrm{(blue)} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\xi_1
</math>
  </td>
  <td align="center>
<math>~=</math>
  </td>
  <td align="left">
<math>~1.2</math>
  </td>
  <td align="left" width="25%">
&nbsp;
  </td>
  <td align="left">
<math>~\mathrm{(red)} \, ;</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
~\xi_1
</math>
  </td>
  <td align="center>
<math>~=</math>
  </td>
  <td align="left">
<math>~1.5</math>
  </td>
  <td align="left" width="25%">
&nbsp;
  </td>
  <td align="left">
<math>\mathrm{(gold)} \, .</math>
  </td>
</tr>
</table>

The inner and outer edges of the toroidal surface in the equatorial plane should be 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>
~\chi_\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>
~\chi_\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>
The equatorial-plane location of the "center" of each torus is,
<div align="center">
<math>
\chi_0 = \frac{1}{2} (\chi_\mathrm{outer} + \chi_\mathrm{inner}) = \frac{\xi_1}{(\xi_1^2 - 1)^{1/2}} ,
</math>
</div>
and the so-called distortion parameter,
<div align="center">
<math>
\delta \equiv \frac{\chi_\mathrm{outer}-\chi_\mathrm{inner}}{\chi_0}= \frac{2}{\xi_1} .
</math>
</div>


<table align="center" border="1" cellpadding="8">
<tr>
  <th align="center" colspan="6">
<font color="maroon">
Properties of <math>\xi_1 = \mathrm{constant}</math> Toroidal Surfaces
</font>
  </th>
</tr>
<tr>
  <td align="center">
Curve in<br />Figure
  </td>
  <td align="center">
<math>\xi_1</math>
  </td>
  <td align="center">
<math>\chi_\mathrm{inner}</math>
  </td>
  <td align="center">
<math>\chi_\mathrm{outer}</math>
  </td>
  <td align="center">
<math>\chi_0</math>
  </td>
  <td align="center">
<math>\delta</math>
  </td>
</tr>

<tr>
  <td align="center">
Blue
  </td>
  <td align="center">
1.1
  </td>
  <td align="center">
0.218
  </td>
  <td align="center">
4.583
  </td>
  <td align="center">
2.400
  </td>
  <td align="center">
1.818
  </td>
</tr>

<tr>
  <td align="center">
Red
  </td>
  <td align="center">
1.2
  </td>
  <td align="center">
0.302
  </td>
  <td align="center">
3.317
  </td>
  <td align="center">
1.809
  </td>
  <td align="center">
1.667
  </td>
</tr>

<tr>
  <td align="center">
Gold
  </td>
  <td align="center">
1.5
  </td>
  <td align="center">
0.447
  </td>
  <td align="center">
2.236
  </td>
  <td align="center">
1.342
  </td>
  <td align="center">
1.333
  </td>
</tr>
</table>


What function <math>~\zeta(\varpi)</math> coincides with these <math>~\xi_1 = \mathrm{constant}</math> surfaces? (To be answered!)


<div align="center">
[[File:LSU_CombinedTori.jpg|none|800px|Meridional contours of constant <math>\xi_1</math>.]]  
</div>

===Off-center Circle===
The curves drawn in the above figure labeled "Toroidal Coordinate System" resemble circles whose centers are positioned a distance <math>~\chi_0</math> away from the origin.  Let's examine whether this is the case by drawing on the familiar expression for such a configuration, [[Appendix/Ramblings/ToroidalCoordinates#Off-center_Circle|as presented above]].  If this is the case, then the circle as illustrated in the figure will have <math>~z_0 = 0</math> and a radius,

<table align="center" border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
~\alpha_c \equiv \frac{r_c}{a}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\chi_\mathrm{outer} - \chi_0 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\biggl[\frac{(\xi_1 + 1)}{(\xi_1 - 1)} \biggr]^{1/2} -  \frac{\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>
~\frac{1}{(\xi_1^2 - 1)^{1/2}} \, ,
</math>
  </td>
</tr>
</table>

and the algebraic expression describing the circle will take the form,

<div align="center">
<math>
~(\chi - \chi_0)^2 + \zeta^2 = \alpha_c^2 = (\xi_1^2 - 1)^{-1} .
</math>
</div>

Let's evaluate the left-hand-side of this expression to see if it indeed reduces to <math>(\xi_1^2 - 1)^{-1}</math>.

<table align="center" border="0" cellpadding="10">
<tr>
  <td align="right">
<math>~\mathrm{LHS}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl\{ \biggl[ \frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2}
 \biggr] - \frac{\xi_1}{(\xi_1^2 - 1)^{1/2}} \biggr\}^2 + \biggl[ \frac{(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2}  \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{1}{(\xi_1 - \xi_2)^2 (\xi_1^2 - 1)} \biggl\{ (\xi_1^2 - 1) - \xi_1 (\xi_1-\xi_2) \biggr\}^2 +  \frac{(1-\xi_2^2)}{(\xi_1 - \xi_2)^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{(\xi_1 \xi_2 - 1)^2}{(\xi_1 - \xi_2)^2 (\xi_1^2 - 1)}  +  \frac{(1-\xi_2^2)}{(\xi_1 - \xi_2)^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{1}{(\xi_1 - \xi_2)^2 (\xi_1^2 - 1)} \biggl[(\xi_1 \xi_2 - 1)^2  +  (\xi_1^2 - 1)(1-\xi_2^2) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{1}{(\xi_1 - \xi_2)^2 (\xi_1^2 - 1)} \biggl[ \xi_1^2 \xi_2^2 - 2\xi_1 \xi_2 + 1 )  +  (\xi_1^2 - 1 -\xi_1^2 \xi_2^2 + \xi_2^2) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{\xi_1^2 - 2\xi_1 \xi_2  + \xi_2^2}{(\xi_1 - \xi_2)^2 (\xi_1^2 - 1)}
</math>
  </td>
</tr>

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

</table>

Yes!  So this means that the <math>~\xi_1 = \mathrm{constant}</math> toroidal contours can be described by the off-center circle expression,

<div align="center">
<math>
~(\chi - \chi_0)^2 + \zeta^2 = (\chi_\mathrm{outer} - \chi_0)^2 \, ,
</math>
</div>
or,

<div align="center">
<math>
~\biggl[ \chi - \frac{\xi_1}{(\xi_1^2 - 1)^{1/2}} \biggr]^2 + \zeta^2 = \frac{1}{(\xi_1^2 - 1)} \, .
</math>
</div>

It also means that, while <math>~\xi_1</math> is the official ''radial'' coordinate of MF53's toroidal coordinate system, the actual dimensionless radius of the relevant cross-sectional circle is,

<table align="center" border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
~\alpha_c
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{1}{(\xi_1^2 - 1)^{1/2}} .
</math>
  </td>
</tr>
</table>