Editing 2DStructure/ToroidalCoordinateIntegrationLimits (section)

====Summary (Zone II)====

In summary, for a given set of the three model parameters <math>~(a, \varpi_t, r_t)</math> and the fourth, <math>~Z_0</math>, in the range, <math>~0 < Z_0 < r_t</math>,  the volume of the (pink) circular torus is determined by adding together four partial volumes &#8212; that is, adding together the results of four separate 1D integrations over the "radial" toroidal coordinate <math>~(\xi_1)</math>.  Although a total of eight radial integration limits (four lower limits and four upper limits) are required to fully determine the Zone II torus volume, only four unique limiting values need to be calculated because the partial volumes share <math>~\xi_1</math> boundaries.  This has been illustrated by the black vertical dashed and dot-dashed lines in the left-hand panel of Figure 6 &#8212; and, drawing from the above discussion, the numerical values of these limits have been recorded in Table 1 &#8212; for the ''specific'' case of <math>~Z_0 = \tfrac{3}{20}</math>.  

<table border="1" cellpadding="8" align="center">
<tr><th align="center" colspan="9">
<font size="+1">Table 1:</font> Zone II Partial Volumes &amp; Integration Limits on <math>~\xi_1</math><p></p>
for model parameters <math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{1}{3}, \tfrac{3}{20}, \tfrac{3}{4}, \tfrac{1}{4})</math> 
</th></tr>
<tr>
  <td align="center" colspan="1">&nbsp;</td>
  <td align="center" colspan="2">PV #1</td>
  <td align="center" colspan="2">PV #2</td>
  <td align="center" colspan="2">PV #3</td>
  <td align="center" colspan="2">PV #4</td>
</tr>
<tr>
  <td align="center" rowspan="2" colspan="1">Integration<p></p>Limits</td>
  <td align="center"><math>~\lambda_1</math></td>
  <td align="center"><math>~\Lambda_1</math></td>
  <td align="center"><math>~\lambda_2</math></td>
  <td align="center"><math>~\Lambda_2</math></td>
  <td align="center"><math>~\lambda_3</math></td>
  <td align="center"><math>~\Lambda_3</math></td>
  <td align="center"><math>~\lambda_4</math></td>
  <td align="center"><math>~\Lambda_4</math></td>
</tr>
<tr>
  <td align="center"><math>2.16110</math></td>
  <td align="center"><math>1.28080</math></td>
  <td align="center"><math>1.28080</math></td>
  <td align="center"><math>1.22088</math></td>
  <td align="center"><math>2.16110</math></td>
  <td align="center"><math>1.28080</math></td>
  <td align="center"><math>2.32125</math></td>
  <td align="center"><math>2.16110</math></td>
</tr>
<tr>
  <td align="center" rowspan="1" colspan="1">Volume<p></p>Fraction</td>
  <td align="center" colspan="2">0.14237851</td>
  <td align="center" colspan="2">0.21569718</td>
  <td align="center" colspan="2">0.63537958</td>
  <td align="center" colspan="2">0.0065448719</td>
</tr>
<tr>
  <td align="center" rowspan="1" colspan="1">Total Volume<p></p><font size="-1">(nzones = 5000)</font></td>
  <td align="center" colspan="8">1.00000014 &nbsp; &nbsp; &nbsp; <math>~\Rightarrow~</math>&nbsp; &nbsp; &nbsp; Error = -1.4E-7
  </td>
</tr>
</table>

While the [[#Volume_of_Circular_Torus|radial integrand expression for each partial volume]] is formally the same, it requires a specification of both limits, <math>~\gamma_i</math> and <math>~\Gamma_i</math>, for the "angular" coordinate integration which, as has also just been detailed, vary from one partial volume to the next and generally depend on the value of the radial coordinate.  This dependence of the angular coordinate integration limits on the specific value of the radial coordinate across the four separate partial volumes is quantitatively illustrated in Figure 6 for the specific set of model parameters, <math>~(a, \varpi_t, r_t) = (\tfrac{1}{3}, \tfrac{3}{4}, \tfrac{1}{4})</math>, and for twenty-four different values of <math>~Z_0</math>.


<table border="1" cellpadding="8" align="center">
<tr><th align="center" colspan="2">
<font size="+1">Figure 6:</font> Zone II Integration Limits on <math>~\xi_2</math><p></p>
for model parameters <math>~(a, \varpi_t, r_t) = (\tfrac{1}{3}, \tfrac{3}{4}, \tfrac{1}{4})</math> and various <math>~Z_0</math> 
</th></tr>
<tr>
<td align="center">
[[File:Torus_z15B.png|347px|Diagram of Torus and Toroidal Coordinates]]
</td>
<td align="center">
[[File:Xi2Movie04.gif|Animation of Torus and Toroidal Coordinates]]
</td>
</tr>
<tr>
  <td align="center" colspan="1" align="center">

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="4">Example <math>~\xi_2</math> Limits when <math>~Z_0 = \tfrac{3}{20} = 0.15</math></td>
</tr>
<tr>
  <td align="center" width="25%">
Partial<p></p>Volume (''i'') 
  </td>
  <td align="center" width="25%">
Example<p></p>
<math>~\xi_1</math>
  </td>
  <td align="center" width="25%">
<math>~\gamma_i</math>
  </td>
  <td align="center">
<math>~\Gamma_i</math>
  </td>
</tr>
<tr>
  <td align="center" bgcolor="green"><b>#1</b></td>
  <td align="center">2.00</td>
   <td align="center">1.00000</td>
   <td align="center">0.99625</td>
</tr>
<tr>
  <td align="center" bgcolor="yellow"><b>#2</b></td>
  <td align="center">1.25</td>
   <td align="center">0.91635</td>
   <td align="center">0.99789</td>
</tr>
<tr>
  <td align="center" bgcolor="orange"><b>#3</b></td>
  <td align="center">2.00</td>
   <td align="center">0.84538</td>
   <td align="center">1.00000</td>
</tr>
<tr>
  <td align="center" bgcolor="blue"><b>#4</b></td>
  <td align="center">2.30</td>
   <td align="center">0.94684</td>
   <td align="center">0.98886</td>
</tr>
</table>

  </td>
  <td align="center">&nbsp;</td>
</tr>
</table>

In both panels of Figure 6, at each of a variety of values of the "radial" coordinate, <math>~\xi_1</math> (horizontal axis), a pair of colored dots identify the values (vertical axis) of the two "angular" coordinate integration limits, <math>~\gamma_i</math> (lower value) and <math>~\Gamma_i</math> (upper value).  The left-hand panel has been constructed for the specific case, <math>~Z_0 = \tfrac{3}{20} = 0.15</math>; via an animation sequence, the right-hand panel shows how the limits vary with <math>~\xi_1</math> for twenty-four different values of <math>~Z_0</math> in the Zone II range <math>~(0 \leq Z_0 \leq r_t)</math>, as indicated in the lower right-hand corner of each frame.  The limits identified by yellow dots must be fed into the radial integrand expression when evaluating partial volume #2 and the blue dots provide the limits for partial volume #4.  Green dots identify the lower angular-coordinate integration limit across partial volume #1; orange dots identify the lower limit across partial volume #3; and the upper limit for both partial volume #1 and partial volume #3 is unity (black dots).  Four vertical black lines (two dashed and two dot-dashed) have been added to the plot displayed in the left-hand panel of Figure 6 in order to emphasize that the boundaries between the four partial volumes are defined by the radial-coordinate integration limits, <math>~\lambda_i</math> and <math>~\Lambda_i</math>; as has been detailed in Table 1 for the ''specific'' case <math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{1}{3}, \tfrac{3}{20}, \tfrac{3}{4}, \tfrac{1}{4})</math>, the boundaries occur at <math>~\xi_1 = 1.22088, 1.28080, 2.16110,</math> and <math>~2.32125</math>.


<div align="center">
<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="7"><font size="+1">Table 1b:</font>&nbsp;&nbsp;Validate Pattern II</th>
</tr>
<tr><th align="center" colspan="6">
Integration Limits for model parameters <p></p><math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{4}{5}, \tfrac{3}{20}, \tfrac{3}{4}, \tfrac{1}{4})</math> 
</th>
  <td align="center" rowspan="7">[[File:Zone_II08.png|250px|Zone II Partial Volumes]]</td>
</tr>
<tr>
  <td align="center" colspan="1" rowspan="2">&nbsp;</td>
  <td align="center"><math>~\xi_1|_\mathrm{max}</math></td>
  <td align="center"><math>~\xi_1|_\mathrm{min}</math></td>
  <td align="center"><math>~\xi_1|_+</math></td>
  <td align="center"><math>~\xi_1|_-</math></td>
  <td align="center" rowspan="2"><math>~\infty</math></td>
</tr>
<tr>
  <td align="center">1.22088</td>
  <td align="center">2.32125</td>
  <td align="center">1.28080</td>
  <td align="center">2.16110</td>
</tr>
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  <td align="center">Temporary</td>
  <td align="center"><b><font color="#FFDD00">END</font></b></td>
  <td align="center"><b><font color="blue">START</font></b></td>
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  <td align="center">&nbsp;</td>
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<tr>
  <td align="center"><math>~\xi_2|_+</math></td>
  <td align="center"><b><font color="yellow">END</font></b></td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center"><b><font color="lightgreen">START</font></b><p></p><b><font color="blue">END</font></b></td>
  <td align="center">&nbsp;</td>
</tr>
<tr>
  <td align="center"><math>~\xi_2|_-</math></td>
  <td align="center">&nbsp;</td>
  <td align="center"><b><font color="blue">START</font></b></td>
  <td align="center"><b><font color="yellow">START</font></b></td>
  <td align="center"><b><font color="ORANGE">START</font></b></td>
  <td align="center">&nbsp;</td>
</tr>
<tr>
  <td align="center"><math>~+1</math></td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center"><b><font color="lightgreen">END</font></b><p></p><b><font color="orange">END</font></b></td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
</tr>
<tr>
  <td align="center"><math>~-1</math></td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
</tr>
</table>
</div>