Editing Appendix/Ramblings/ToroidalCoordinates (section)

==Yet Again==
===Walk Through Step-By-Step===
Keep the scale length of the toroidal coordinate system, <math>~a</math>, fixed while varying the value of <math>~\xi_1</math> and, hence, the radius, 
<div align="center">
<math>~r_0 = \frac{a}{(\xi_1^2 - 1)^{1/2}} \, ,</math>
</div>
of the <math>~\xi_1</math> = constant circle (hereafter, <math>\xi_1</math>-circle).  The (cylindrical) coordinate location of the center of this circle will be, <math>~(R_0, Z_0)</math>, where,
<div align="center">
<math>~R_0 = \frac{a\xi_1}{(\xi_1^2 - 1)^{1/2}} \, .</math>
</div>
For the time being, we will assume that <math>~Z_0 > 0</math>, as illustrated in our [[#Figure2|Figure 2]]. Our initial aim is to determine the range of values of <math>~\xi_1</math> for which the <math>\xi_1</math>-circle touches or overlaps the equatorial-plane torus, whose position and size are as defined in our [[#Figure2|Figure 2]].  

====Lowest Point on Circle====
We will identify the (cylindrical) coordinates of the lowest point on the <math>\xi_1</math>-circle as <math>~(R_0, Z_\mathrm{min})</math>, where,
<div align="center">
<math>~Z_\mathrm{min} = Z_0 - r_0 = Z_0 - \frac{a}{(\xi_1^2 - 1)^{1/2}} \, .</math>
</div>
The <math>\xi_1</math>-circle cannot possibly touch the equatorial-plane torus until <math>~\xi_1</math> drops to a value such that <math>~Z_\mathrm{min}</math> is less than or equal to the radius of the torus, <math>~r_t</math>.  This means that touching/overlap cannot occur unless,
<div align="center">
<math>~\xi_1 \leq \xi_\mathrm{max} \equiv \biggl[1 + \biggl(\frac{a}{Z_0 - r_t} \biggr)^2\biggr]^{1/2} \, .</math>
</div>

====A Critical Value of the Scale Length====
Now, the two circles will come into contact at this limiting value, <math>~\xi_\mathrm{max}</math>, only if the corresponding "radial" coordinate location of the center of the <math>~\xi_1</math> circle exactly equals <math>~\varpi_t</math>, that is, only if
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{a\xi_\mathrm{max}}{(\xi_\mathrm{max}^2 - 1)^{1/2}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\varpi_t</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
a^2\xi_\mathrm{max}^2
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\varpi_t^2(\xi_\mathrm{max}^2 - 1)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
\xi_\mathrm{max}^2
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\varpi_t^2}{(\varpi_t^2 - a^2)}</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
(\varpi_t^2 - a^2)(Z_0 - r_t)^2 + a^2(\varpi_t^2 - a^2)
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\varpi_t^2 (Z_0 - r_t)^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a^2 (Z_0 - r_t)^2 - a^2(\varpi_t^2 - a^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a^2 [(Z_0 - r_t)^2 - \varpi_t^2] + a^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ a = a_\mathrm{crit} </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
[\varpi_t^2 - (Z_0 - r_t)^2 ]^{1/2} \, .
</math>
  </td>
</tr>
</table>
</div>

====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>

====Limiting Values====
All other parameters <math>~(a, Z_0,  \varpi_t, r_t)</math> being held fixed, as the coordinate, <math>~\xi_1</math>, is varied, there will be a maximum value, <math>~\xi_1|_\mathrm{max}</math>, at which the <math>\xi_1</math>-circle will first make contact with the (pink) equatorial-plane torus, and there will be a minimum value, <math>~\xi_1|_\mathrm{min}</math>, at which it will have its final contact.  At all values within the parameter range,
<div align="center">
<math>~\xi_1|_\mathrm{max} > \xi_1 > ~\xi_1|_\mathrm{min} \, ,</math>
</div>
the <math>\xi_1</math>-circle will intersect the surface of the torus in two locations, defined by two different values of the associated angular coordinate, <math>~\xi_2</math> &#8212; see, for example, the coordinates listed in the table associated with [[#Example2|example 2, below]] &#8212; but ''at'' the first and final points of contact, the two values of <math>~\xi_2</math> will be degenerate.  Let's derive the mathematical relations that give the values of <math>~\xi_1|_\mathrm{max}</math> and <math>~\xi_1|_\mathrm{min}</math>.

The expression [[#IntersectionVarpi|derived above]] for the "radial" coordinate of the points of intersection, <math>~\varpi_i</math>, gives two physically viable, real numbers as long as the composite parameter, <math>~1 > \ell \geq 0</math>.  But only one real value is obtained when <math>~\ell = 1</math>, and that occurs when,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~( 2Z_0^2 \varpi_t-\Kappa \beta  )^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(Z_0^2 + \beta^2 )[\Kappa^2 +4Z_0^2 (\varpi_t^2- r_t^2)]   \, .</math>
  </td>
</tr>
</table>
</div>
In this expression, <math>~\beta</math> is the only parameter that depends on <math>~\xi_1</math>.  So, temporarily using the shorthand notation,
<div align="center">
<math>~\Lambda \equiv [\Kappa^2 +4Z_0^2 (\varpi_t^2- r_t^2)] \, </math>
</div>
let's solve for the "critical" value(s), <math>~\beta_\mathrm{crit}</math>.  We have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~4 Z_0^4 \varpi_t^2 - 4Z_0^2 \varpi_t \Kappa\beta + \Kappa^2\beta^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~Z_0^2\Lambda + \Lambda \beta^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~(\Kappa^2 - \Lambda)\beta^2 - (4Z_0^2 \varpi_t \Kappa)\beta + (4 Z_0^4 \varpi_t^2 -Z_0^2\Lambda)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>
The roots of this quadratic equation give,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\beta_\mathrm{crit}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{ 2Z_0^2 \varpi_t \Kappa}{ (\Kappa^2 - \Lambda)} \biggl[1\mp
\sqrt{1-\frac{(\Kappa^2 - \Lambda) (4 Z_0^4 \varpi_t^2 -Z_0^2\Lambda)}{4Z_0^4 \varpi_t^2 \Kappa^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}{ 4Z_0^2 (\varpi_t^2- r_t^2)} \biggl[1\mp
\sqrt{1+\frac{4Z_0^2 (\varpi_t^2- r_t^2)(4 Z_0^4 \varpi_t^2 -Z_0^2\Lambda)}{4Z_0^4 \varpi_t^2 \Kappa^2 }  }  \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{ \varpi_t \Kappa}{ 2 (\varpi_t^2- r_t^2)} \biggl[1\mp
\sqrt{1+\frac{(\varpi_t^2- r_t^2)(4 Z_0^2 r_t^2 -\Kappa^2)}{\varpi_t^2 \Kappa^2 }  }  \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Notice that a ''single'' critical value of <math>~\ell</math> &#8212; specifically, <math>~\ell = 1</math> &#8212; translates nicely into a pair of values of <math>~\beta_\mathrm{crit}</math>; these presumably relate directly to the pair of limiting coordinate values, <math>~\xi_1|_\mathrm{max}</math> and <math>~\xi_1|_\mathrm{min}</math>, that we are seeking.  Via the [[#BetaDefinition|definition of <math>~\beta</math>]], we find,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\xi_1}{(\xi_1^2-1)^{1/2}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{\varpi_t - \beta}{a} \biggr)</math>
  </td>
</tr>

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

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

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

Upon evaluation of this expression in conjunction with the ''pair'' of <math>~\beta_\mathrm{crit}</math> values, the table, below, provides numerical values for the limiting values of  <math>~\xi_1|_\mathrm{max}</math> and <math>~\xi_1|_\mathrm{min}</math>, along with the respective values of their accompanying (degenerate) coordinate, <math>~\xi_2</math>.

===Examples===


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

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

For reference purposes, Figure 2 has been displayed here, again, in the lefthand panel of Figure 4; the animation sequence presented in the righthand panel illustrates how the <math>\xi_1</math>-circle (depicted by the locus of small black dots) intersects the surface of the (pink) equatorial-plane torus as the value of <math>~\xi_1</math> is varied over the parameter range,
<div align="center">
<math>~\xi_1|_\mathrm{max} \geq \xi_1 \geq ~\xi_1|_\mathrm{min} \, ,</math>
</div>
for a toroidal coordinate system whose origin (filled, red dot) remains fixed at the (cylindrical) coordinate location, <math>~(\varpi, z) = (a, Z_0) = (\tfrac{1}{3}, \tfrac{3}{4})</math>.  For a toroidal coordinate system with this specified origin and an equatorial-plane torus having <math>~\varpi_t = \tfrac{3}{4}</math> and <math>~r_t = \tfrac{1}{4}</math> &#8212; as recorded in the top row of numbers in the Table, below &#8212; the <math>\xi_1</math>-circle makes ''first contact'' with the torus when <math>~\xi_1 = \xi_1|_\mathrm{max} = 1.1927843</math> and it makes ''final contact'' when <math>~\xi_1 = \xi_1|_\mathrm{min} = 1.0449467</math>.  The animation sequence contains ten unique frames: The value of <math>~\xi_1</math> that is associated with the <math>\xi_1</math>-circle in each case appears near the bottom-right corner of the animation frame.  These parameter values have also been recorded in the first column of ten separate rows in the following table, along with other relevant parameter values.  For example, in each frame of the animation, the points of intersection between the surface of the torus and the <math>\xi_1</math>-circle are identified by filled, green diamonds; the (cylindrical) coordinates associated with these points of intersection, <math>~(\varpi_i, z_i)</math>, are listed in each table row, along with the corresponding value of the toroidal coordinate system's angular, <math>~\xi_2</math> coordinate.

<div id="Example2" style="width: 85%; height: 15em; overflow: auto;">
<table align="center" border="1" cellpadding="5">
<tr><th align="center" colspan="10">Example 2</th></tr>
<tr>
  <td align="center" colspan="2" width="25%"><math>~\varpi_t</math></td>
  <td align="center" colspan="2" width="25%"><math>~r_t</math></td>
  <td align="center" colspan="2" width="25%"><math>~Z_0</math></td>
  <td align="center" colspan="2"><math>~a</math></td>
  <td align="center" colspan="2"><math>~\Kappa</math></td>
</tr>
<tr>
  <td align="center" colspan="2"><math>~\tfrac{3}{4}</math></td>
  <td align="center" colspan="2"><math>~\tfrac{1}{4}</math></td>
  <td align="center" colspan="2"><math>~\tfrac{3}{4}</math></td>
  <td align="center" colspan="2"><math>~\tfrac{1}{3}</math></td>
  <td align="center" colspan="2"><math>~(\tfrac{5}{12})^2</math></td>
</tr>
<tr>
  <th colspan="10" align="center">Torus Intersection Points</th>
</tr>
<tr>
  <td align="center" colspan="2" rowspan="2"><math>~\xi_1</math></td>
  <td align="center" colspan="1" rowspan="2"><math>~\beta</math></td>
  <td align="center" colspan="1" rowspan="2"><math>~\ell</math></td>
  <td align="center" colspan="3" bgcolor="yellow">Intersection #1 (''superior'' sign)</td>
  <td align="center" colspan="3" bgcolor="yellow">Intersection #2 (''inferior'' sign)</td>
</tr>
<tr>
  <td align="center"><math>~\xi_2</math>
  <td align="center"><math>~\varpi_i</math>
  <td align="center"><math>~z_i</math>
  <td align="center"><math>~\xi_2</math>
  <td align="center"><math>~\varpi_i</math>
  <td align="center"><math>~z_i</math>
</tr>
<tr>
  <td align="center" colspan="2"><math>~1.1927843</math></td>
  <td align="center" colspan="1"><math>~+0.138485</math></td>
  <td align="center" colspan="1"><math>~1.000000</math></td>
  <td align="center" colspan="1"><math>~0.885198</math></td>
  <td align="center" colspan="1"><math>~0.704606</math></td>
  <td align="center" colspan="1"><math>~0.245844</math></td>
  <td align="center" colspan="3">Degenerate Coordinate Values</td>
</tr>
<tr>
  <td align="center" colspan="2"><math>~1.176</math></td>
  <td align="center" colspan="1"><math>~+0.116568</math></td>
  <td align="center" colspan="1"><math>~0.981258</math></td>
  <td align="center" colspan="1"><math>~0.922142</math></td>
  <td align="center" colspan="1"><math>~0.812595</math></td>
  <td align="center" colspan="1"><math>~0.242037</math></td>
  <td align="center" colspan="1"><math>~0.841611</math></td>
  <td align="center" colspan="1"><math>~0.616896</math></td>
  <td align="center" colspan="1"><math>~0.211621</math></td>
</tr>
<tr>
  <td align="center" colspan="2"><math>~1.160</math></td>
  <td align="center" colspan="1"><math>~+0.092267</math></td>
  <td align="center" colspan="1"><math>~0.962725</math></td>
  <td align="center" colspan="1"><math>~0.933386</math></td>
  <td align="center" colspan="1"><math>~0.864726</math></td>
  <td align="center" colspan="1"><math>~0.222121</math></td>
  <td align="center" colspan="1"><math>~0.824945</math></td>
  <td align="center" colspan="1"><math>~0.584858</math></td>
  <td align="center" colspan="1"><math>~0.187691</math></td>
</tr>
<tr>
  <td align="center" colspan="2"><math>~1.144</math></td>
  <td align="center" colspan="1"><math>~+0.063705</math></td>
  <td align="center" colspan="1"><math>~0.943871</math></td>
  <td align="center" colspan="1"><math>~0.940238</math></td>
  <td align="center" colspan="1"><math>~0.908969</math></td>
  <td align="center" colspan="1"><math>~0.192948</math></td>
  <td align="center" colspan="1"><math>~0.813713</math></td>
  <td align="center" colspan="1"><math>~0.560766</math></td>
  <td align="center" colspan="1"><math>~0.163372</math></td>
</tr>
<tr>
  <td align="center" colspan="2"><math>~1.127</math></td>
  <td align="center" colspan="1"><math>~+0.027202</math></td>
  <td align="center" colspan="1"><math>~0.924221</math></td>
  <td align="center" colspan="1"><math>~0.944608</math></td>
  <td align="center" colspan="1"><math>~0.949856</math></td>
  <td align="center" colspan="1"><math>~0.150191</math></td>
  <td align="center" colspan="1"><math>~0.806047</math></td>
  <td align="center" colspan="1"><math>~0.539788</math></td>
  <td align="center" colspan="1"><math>~0.135318</math></td>
</tr>
<tr>
  <td align="center" colspan="2"><math>~1.111</math></td>
  <td align="center" colspan="1"><math>~-0.015045</math></td>
  <td align="center" colspan="1"><math>~0.907444</math></td>
  <td align="center" colspan="1"><math>~0.946487</math></td>
  <td align="center" colspan="1"><math>~0.980806</math></td>
  <td align="center" colspan="1"><math>~0.096065</math></td>
  <td align="center" colspan="1"><math>~0.802617</math></td>
  <td align="center" colspan="1"><math>~0.523232</math></td>
  <td align="center" colspan="1"><math>~0.105244</math></td>
</tr>
<tr>
  <td align="center" colspan="2"><math>~1.094</math></td>
  <td align="center" colspan="1"><math>~-0.071947</math></td>
  <td align="center" colspan="1"><math>~0.894425</math></td>
  <td align="center" colspan="1"><math>~0.945995</math></td>
  <td align="center" colspan="1"><math>~0.999208</math></td>
  <td align="center" colspan="1"><math>~0.019887</math></td>
  <td align="center" colspan="1"><math>~0.803522</math></td>
  <td align="center" colspan="1"><math>~0.509118</math></td>
  <td align="center" colspan="1"><math>~0.066901</math></td>
</tr>
<tr>
  <td align="center" colspan="2"><math>~1.078</math></td>
  <td align="center" colspan="1"><math>~-0.142539</math></td>
  <td align="center" colspan="1"><math>~0.892548</math></td>
  <td align="center" colspan="1"><math>~0.942353</math></td>
  <td align="center" colspan="1"><math>~0.989322</math></td>
  <td align="center" colspan="1"><math>~-0.072283</math></td>
  <td align="center" colspan="1"><math>~0.810056</math></td>
  <td align="center" colspan="1"><math>~0.500846</math></td>
  <td align="center" colspan="1"><math>~0.020554</math></td>
</tr>
<tr>
  <td align="center" colspan="2"><math>~1.061</math></td>
  <td align="center" colspan="1"><math>~-0.247448</math></td>
  <td align="center" colspan="1"><math>~0.916366</math></td>
  <td align="center" colspan="1"><math>~0.932024</math></td>
  <td align="center" colspan="1"><math>~0.916375</math></td>
  <td align="center" colspan="1"><math>~-0.186599</math></td>
  <td align="center" colspan="1"><math>~0.827074</math></td>
  <td align="center" colspan="1"><math>~0.505248</math></td>
  <td align="center" colspan="1"><math>~-0.050956</math></td>
</tr>
<tr>
  <td align="center" colspan="2"><math>~1.0449467</math></td>
  <td align="center" colspan="1"><math>~-0.398902</math></td>
  <td align="center" colspan="1"><math>~1.000000</math></td>
  <td align="center" colspan="1"><math>~0.885198</math></td>
  <td align="center" colspan="1"><math>~0.632605</math></td>
  <td align="center" colspan="1"><math>~-0.220722</math></td>
  <td align="center" colspan="3">Degenerate Coordinate Values</td>
</tr>
</table>
</div>

Notice in the animation that, while the origin of the selected toroidal coordinate system (the filled red dot) remains fixed, the ''center'' of the <math>\xi_1</math>-circle does not remain fixed.  In order to highlight this behavior, the location of the center of the <math>\xi_1</math>-circle has been marked by a filled, light-blue square and, in keeping with the earlier Figure 2 sketch, a vertical, light-blue line connects this center to the equatorial plane.


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