Editing Appendix/Ramblings/ToroidalCoordinates (section)

====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,
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<math>~\xi_1|_\mathrm{max} > \xi_1 > ~\xi_1|_\mathrm{min} \, ,</math>
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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,
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<table border="0" cellpadding="5" align="center">

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

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

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  <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>

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  <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>

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  <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>
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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,
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<table border="0" cellpadding="5" align="center">

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

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

<tr>
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<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>
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  <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>
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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>.