Editing Appendix/Ramblings/ToroidalCoordinates (section)

===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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