Editing 2DStructure/ToroidalCoordinateIntegrationLimits (section)

==Integration Limits==
We define the following terms that are functions only of the four principal model parameters, <math>~(a, Z_0,  \varpi_t, r_t)</math>, and therefore can be treated as constants while carrying out the pair of nested integrals that determine <math>~q_0</math>.

===Zone I:===
<div id="ZoneI">
<table align="center" border="1" cellpadding="8">
<tr>
  <th align="center" colspan="2"><font size="+1">Figure 4:  </font>&nbsp;Zone I</th>
</tr>
<tr>
  <td align="center">Definition</td>
  <td align="center">Schematic Example</td>
</tr>
<tr>
  <td align="center">
<math>~Z_0 > r_t</math><p></p>for any <math>~a</math>
  </td>
  <td align="center">
[[File:Apollonian_myway5B.png|300px|Apollonian Circles]]
  </td>
</tr>
</table>
</div>

In an [[2DStructure/ToroidalCoordinates#Identifying_Limits_of_Integration|accompanying discussion]], we have derived the following integration limits; numerical values are given for the specific case, <math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{1}{3}, \tfrac{3}{4}, \tfrac{3}{4}, \tfrac{1}{4})</math>:

<div align="center" id="LambdaLimits">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\Lambda_1 = \xi_1|_\mathrm{max} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2  \biggr]^{-1/2}
\approx 1.1927843</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_1 = \xi_1|_\mathrm{min} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_-} \biggr)^2  \biggr]^{-1/2}
\approx 1.0449467</math>
  </td>
</tr>
</table>

</td></tr>
</table>
</div>

where,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\kappa</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
Z_0^2 + a^2  - (\varpi_t^2 - r_t^2) 
</math>
  </td>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="left">
<math>~
\frac{5^2}{2^4\cdot 3^2} \approx 0.17361111
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~C</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~1 + \biggl( \frac{2Z_0}{\kappa}\biggr)^2 ( \varpi_t^2 - r_t^2)
</math>
  </td>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="left">
<math>~\frac{17 \cdot 1409}{5^4} \approx 38.3248 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\beta_\pm</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
- \frac{\kappa}{2} \biggl[ \frac{\varpi_t \mp r_t \sqrt{C}}{(\varpi_t + r_t)(\varpi_t - r_t)} \biggr] 
</math>
  </td>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="left">
<math>~
-~\frac{5^2}{2^6\cdot 3}\biggl[ 1\mp \sqrt{ \frac{17\cdot 1409}{3^2\cdot 5^4}} \biggr] 
</math>
  </td>
</tr>
</table>

Notice that we have specified these integration limits such that, in going from the lower limit <math>~(\lambda_1)</math> to the upper limit <math>~(\Lambda_1)</math>, the value of <math>~\xi_1</math> monotonically increases.  <font color="red"><b>CAUTION:</b></font> This statement is often not true.  The quantity, <math>~\kappa</math>, changes signs, depending on whether <math>~(a^2 + Z_0^2) \gtrless (\varpi_t^2 -r_t^2)</math>. When <math>~\kappa</math> changes signs, the two quantities, <math>~\xi_1|_\mathrm{max}</math> and <math>~\xi_1|_\mathrm{min}</math> switch roles; specifically, <math>~\xi_1|_\mathrm{max}</math> becomes less than <math>~\xi_1|_\mathrm{min}</math> (or ''visa versa'').


Also,
<div align="center" id="Parameters">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\Gamma_1 = \xi_2\biggr|_+</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>

<tr>
  <td align="right">
<math>~\gamma_1 = \xi_2\biggr|_-</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>

</td></tr>
</table>
</div>


where, in addition to the quantities defined above,

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

<tr>
  <td align="right">
<math>~\biggl(\frac{\varpi_i}{a}\biggr)_\pm</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\kappa}{2a^2}\cdot \frac{\mathrm{B}}{\mathrm{A}} 
\biggl[1 \pm \sqrt{1-\frac{AC}{B^2}}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathrm{A}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{Z_0}{a}\biggr)^2 + \biggl[\frac{\varpi_t}{a} - \frac{\xi_1}{(\xi_1^2-1)^{1/2}}\biggr]^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathrm{B}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{2\varpi_t Z_0^2}{a\kappa}\biggr) - \biggl[\frac{\varpi_t}{a} - \frac{\xi_1}{(\xi_1^2-1)^{1/2}}\biggr] 
</math>
  </td>
</tr>
</table>

Here, our desire also is to specify the integration limits such that <math>~\xi_2</math> monotonically increases in going from the lower limit <math>~(\gamma_1)</math> to the upper limit <math>~(\Gamma_1)</math>.  In order to check to see if this is the case, let's test the limiting values of  <math>~\xi_2</math> when we are considering a radial-coordinate value roughly midway between its limits, say, when <math>~\xi_1 = 1.1</math>.  For this specific case, we find,
<div align="center">
<math>~\frac{\xi_1}{(\xi_1^2 - 1)^{1/2}} = 2.400397 \, ;</math><p></p>
<math>~\frac{\varpi_t}{a} - \frac{\xi_1}{(\xi_1^2-1)^{1/2}} = - 0.150397 \, ;</math><p></p>
<math>~A = 5.085119 \, ;</math><p></p>
<math>~B = 14.73040 \, ;</math><p></p>
<math>~\biggl(\frac{\varpi_i}{a}\biggr)_\pm = 2.263098\biggl[1 \pm \sqrt{1-0.898157}  \biggr] \, ;</math><p></p>
<math>~\Rightarrow ~~~ \biggl(\frac{\varpi_i}{a}\biggr)_+ = 2.985318 \, ;</math><p></p>
<math>~\Rightarrow ~~~ \biggl(\frac{\varpi_i}{a}\biggr)_- = 1.540878 \, ;</math><p></p>
<math>~\Rightarrow ~~~ \xi_2\biggr|_+ = 0.946496 \, ;</math><p></p>
<math>~\Rightarrow ~~~ \xi_2\biggr|_- = 0.802600 \, .</math>
</div>
Hence, our ordering of the limits appears to be the one desired.

===Zone II:===
<div id="ZoneII">
<table align="center" border="1" cellpadding="8">
<tr>
  <th align="center" colspan="3"><font size="+1">Figure 5:  </font>&nbsp;Zone II</th>
</tr>
<tr>
  <td align="center">Definition</td>
  <td align="center">Schematic Example</td>
  <td align="center">Quantitative Example: Partial Volumes Identified</td>
</tr>
<tr>
  <td align="center">
<math>~r_t > Z_0 > 0</math><p></p>and<p></p><math>~a < \varpi_t - \sqrt{r_t^2 - Z_0^2}</math>
  </td>
  <td align="center">
[[File:Apollonian_myway7B.png|240px|Apollonian Circles]]
  </td>
  <td align="center">
[[File:Zone_II08.png|240px|Zone II Partial Volumes]]
  </td>
</tr>
</table>
</div>

In an [[2DStructure/ToroidalCoordinates#Green_Cropped-Top_Volume|accompanying discussion]], we have derived the following integration limits; example numerical values of various parameters are provided for the specific case where, <math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{1}{3}, \tfrac{3}{20}, \tfrac{3}{4}, \tfrac{1}{4})</math>.  Note that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\kappa</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1319}{2^4\cdot 3^2 \cdot 5^2} \approx -0.36639 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~C</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \frac{2^5\cdot 3^6 \cdot 5^2}{(1319)^2} = \frac{2322961}{(1319)^2} \approx 1.33522 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\beta_+</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2^6\cdot 3^2\cdot 5^2} \biggl[3\cdot 1319 - \sqrt{2322961} \biggr] \approx 0.16895 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\beta_-</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2^6\cdot 3^2\cdot 5^2} \biggl[3\cdot 1319 + \sqrt{2322961} \biggr] \approx 0.38063 \, .</math>
  </td>
</tr>
</table>
</div>

<!-- OLDER EXAMPLE
In an [[2DStructure/ToroidalCoordinates#Green_Cropped-Top_Volume|accompanying discussion]], we have derived the following integration limits; example numerical values of various parameters are provided for the specific case where, <math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{1}{3}, \tfrac{1}{5}, \tfrac{3}{4}, \tfrac{1}{4})</math>.  Note that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\kappa</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{157}{2\cdot 3^2\cdot 5^2} \approx -0.34889 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~C</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{40849}{(157)^2} \approx 1.65723 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\beta_+</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2^3\cdot 3^2\cdot 5^2} \biggl[3\cdot 157 - \sqrt{40849} \biggr] \approx 0.14938 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\beta_-</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2^3\cdot 3^2\cdot 5^2} \biggl[3\cdot 157 + \sqrt{40849} \biggr] \approx 0.37395 \, .</math>
  </td>
</tr>
</table>
</div>
OLDER EXAMPLE -->

====Partial Volume #II-1====
This is the green cropped-top sub-volume identified as Partial Volume #1 (PV#1) in the right-hand panel of [[#ZoneII|Figure 5]], and [[2DStructure/ToroidalCoordinates#Green_Cropped-Top_Volume|discussed in detail elsewhere]].
<div align="center">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\Lambda_1 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{[\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2 + a^2}{[\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2-a^2} ~\rightarrow~\frac{1489}{13\cdot 53}
\approx 2.16110</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_1 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2 + a^2}{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2-a^2} ~\rightarrow~ \frac{41\cdot 89}{7\cdot 11\cdot 37} 
\approx 1.28080</math>
  </td>
</tr>
</table>

</td>
<td align="center">

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

<tr>
  <td align="right">
<math>~\Gamma_1 = \xi_2\biggr|_+</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>

<tr>
  <td align="right">
<math>~\gamma_1 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1</math>
  </td>
</tr>
</table>
</td>
<!-- OMIT Diagram
<td align="center">
[[File:CropTopB.png|175px|right|Diagram of "Cropped Top" Torus]]
</td>
-->
</tr>
</table>
</div>

====Partial Volume #II-2====

This is the sub-volume that is painted yellow and identified as Partial Volume #2 (PV#2) in the right-hand panel of [[#ZoneII|Figure 5]]. 
<div align="center">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\Lambda_2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2 + a^2}{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2-a^2} ~\rightarrow~ \frac{41\cdot 89}{7\cdot 11\cdot 37} 
\approx 1.28080</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi_1|_\mathrm{max} = \biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2  \biggr]^{-1/2}
\approx 1.22088</math>
  </td>
</tr>
</table>

</td>
<td align="center">

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

<tr>
  <td align="right">
<math>~\Gamma_2 = \xi_2\biggr|_+</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>

<tr>
  <td align="right">
<math>~\gamma_2 = \xi_2\biggr|_-</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>

</td></tr>
</table>
</div>

====Partial Volume #II-3====

This is the sub-volume that is painted orange and identified as Partial Volume #3 (PV#3) in the right-hand panel of [[#ZoneII|Figure 5]]. 

<div align="center">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\Lambda_3 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{[\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2 + a^2}{[\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2-a^2} ~\rightarrow~\frac{1489}{13\cdot 53}
\approx 2.16110</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_3 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2 + a^2}{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2-a^2} ~\rightarrow~ \frac{41\cdot 89}{7\cdot 11\cdot 37} 
\approx 1.28080</math>
  </td>
</tr>
</table>

</td>
<td align="center">

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

<tr>
  <td align="right">
<math>~\Gamma_3 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_3 = \xi_2\biggr|_-</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>

</td></tr>
</table>
</div>

====Partial Volume #II-4====

This is the sub-volume that is painted blue and identified as Partial Volume #4 (PV#4) in the right-hand panel of [[#ZoneII|Figure 5]]. 

<div align="center">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\Lambda_4 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi_1|_\mathrm{min} = \biggl[1-\biggl( \frac{a}{\varpi_t-\beta_-} \biggr)^2  \biggr]^{-1/2}
\approx 2.32125</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_4 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{[\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2 + a^2}{[\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2-a^2} ~\rightarrow~\frac{1489}{13\cdot 53}
\approx 2.16110</math>
  </td>
</tr>
</table>

</td>
<td align="center">

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

<tr>
  <td align="right">
<math>~\Gamma_4 = \xi_2\biggr|_+</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>

<tr>
  <td align="right">
<math>~\gamma_4 = \xi_2\biggr|_-</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>

</td></tr>
</table>
</div>


====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>
<!-- TEMPORARY ...
<tr>
  <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>
  <td align="center"><b><font color="#FFDD00">START</font></b><p></p><b><font color="lightgreen">END</font></b><p></p><b><font color="orange">END</font></b></td>
  <td align="center">
<b><font color="blue">END</font></b><p></p><b><font color="lightgreen">START</font></b><p></p><b><font color="ORANGE">START</font></b>
  </td>
  <td align="center">&nbsp;</td>
</tr>
END TEMPORARY -->
<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>

====Special Case of Z<sub>0</sub> = 0====

As has been recorded in an [[Appendix/Ramblings/ToroidalCoordinates#Tohline.27s_Ramblings|accompanying "Ramblings" appendix]], my inversion of the toroidal-coordinate definitions &#8212; which, by default, incorporate the assumption that <math>~Z_0 = 0</math> &#8212; 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>

It seems clear that, in this special case, the first radial coordinate that encounters the ''pink'' torus <math>~(\xi_1|_\mathrm{max})</math> should be given by the cylindrical-coordinate values, <math>~[\varpi,z] = [(\varpi_t - r_t),0]</math>; likewise the ''last'' radial coordinate <math>~(\xi_1|_\mathrm{min})</math> should be given by the cylindrical-coordinate values, <math>~[\varpi,z] = [(\varpi_t + r_t),0]</math>.  Hence, in both limits, <math>~\zeta = 0 ~~ \Rightarrow ~~
 r = \chi</math>, which further implies,
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
~\xi_1|_\mathrm{limits}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
\frac{\chi(\chi^2 + 1)}  {[\chi^2(\chi^2-1)^2 ]^{1/2}} = \frac{\chi^2 + 1}  {\chi^2-1} \, .
</math>
  </td>
</tr>
</table>
Separately, then, we expect,
<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
~\xi_1|_\mathrm{min}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
\frac{(\varpi_t - r_t)^2 + a^2}  {(\varpi_t - r_t)^2-a^2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\xi_1|_\mathrm{max}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
\frac{(\varpi_t + r_t)^2 + a^2}  {(\varpi_t + r_t)^2-a^2} \, .
</math>
  </td>
</tr>
</table>

By comparison, the [[#LambdaLimits|expressions that we have provided, above, for these two limits]] gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \xi_1|_\mathrm{max}\biggr]^{-2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[ \xi_1|_\mathrm{min} \biggr]^{-2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1-\biggl( \frac{a}{\varpi_t-\beta_-} \biggr)^2 \, ,
</math>
  </td>
</tr>
</table>
where, in the special case of <math>~Z_0 = 0</math>,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\kappa</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a^2  - (\varpi_t^2 - r_t^2) \, ,
</math>
</tr>

<tr>
  <td align="right">
<math>~\beta_\pm</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{\kappa}{2} \biggl[ \frac{\varpi_t \mp r_t }{(\varpi_t + r_t)(\varpi_t - r_t)} \biggr] 
</math>
  </td>
</tr>

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

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

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

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{a}{2} 
\biggl[\frac{a}{(\varpi_t \pm r_t )} + \frac{(\varpi_t \pm r_t)}{a} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl( \frac{\varpi_t - \beta_\pm}{a}\biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl(\frac{\Upsilon_\pm^2 + a^2}{2a\Upsilon_\pm} \biggr) \, ,
</math>
  </td>
</tr>
</table>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Upsilon_\pm</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\varpi_t \pm r_t \, .</math>
  </td>
</tr>
</table>
</div>
This also means that,

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{(\Upsilon_\pm^2 + a^2)^2 - 4a^2\Upsilon_\pm^2}{(\Upsilon_\pm^2 + a^2)^2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl[ \frac{\Upsilon_\pm^2 - a^2}{\Upsilon_\pm^2 + a^2}  \biggr]^2 \, .
</math>
  </td>
</tr>
</table>
</div>
That is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \xi_1|_\mathrm{max}\biggr]^{-2} = \biggl[ \frac{\Upsilon_+^2 - a^2}{\Upsilon_+^2 + a^2}  \biggr]^2</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\xi_1|_\mathrm{max} = 
\frac{\Upsilon_+^2 + a^2}{\Upsilon_+^2 - a^2}    \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[ \xi_1|_\mathrm{min} \biggr]^{-2} = \biggl[ \frac{\Upsilon_-^2 - a^2}{\Upsilon_-^2 + a^2}  \biggr]^2 </math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\xi_1|_\mathrm{min} =
\frac{\Upsilon_-^2 + a^2}{\Upsilon_-^2 - a^2}   \, .
</math>
  </td>
</tr>
</table>
Excellent!  This matches the earlier supposition.

Also, given that <math>~C = 1</math> and <math>~A = B^2</math>, the pair of values, <math>~(\varpi_i/a)_\pm</math>, is degenerate and given by the expression,

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

<tr>
  <td align="right">
<math>~\biggl(\frac{\varpi_i}{a}\biggr)_\pm</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\kappa}{2a^2 \mathrm{B}} 
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
<math>~\xi_2\biggr|_+</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>

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

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

===Zone III===

<div align="center" id="Figure7">
<table align="center" border="1" cellpadding="8">
<tr>
  <th align="center" colspan="3"><font size="+1">Figure 7:  </font>&nbsp;Zone III</th>
</tr>
<tr>
  <td align="center">Definition</td>
  <td align="center">Schematic Example</td>
  <td align="center">Quantitative Example: Partial Volumes Identified</td>
</tr>
<tr>
  <td align="center">
<math>~r_t > Z_0 > 0</math><p></p>and<p></p>
<math>~\varpi_t - \sqrt{r_t^2 - Z_0^2} < a < \varpi_t + \sqrt{r_t^2 - Z_0^2}</math><p></p>
<math>~\biggl[\frac{11}{20} < a < \frac{19}{20}\biggr]</math>
  </td>
  <td align="center">
[[File:Apollonian_myway8B.png|240px|Apollonian Circles]]
  </td>
  <td align="center">
[[File:LimitsOnTorus2ColoredSmall01.png|240px|Zone II Partial Volumes]]
  </td>
</tr>
</table>
</div>

Here, numerical values will be given for the specific case, <math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{4}{5}, \tfrac{3}{20}, \tfrac{3}{4}, \tfrac{1}{4})</math>:

<div align="center">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\xi_1|_\mathrm{max} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2  \biggr]^{-1/2}
\approx 8.9243615</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\xi_1|_\mathrm{min} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_-} \biggr)^2  \biggr]^{-1/2}
\approx 1.9116174</math>
  </td>
</tr>
</table>

</td></tr>
</table>
</div>

where,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\kappa</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
Z_0^2 + a^2  - (\varpi_t^2 - r_t^2) 
</math>
  </td>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="left">
<math>~\frac{13}{2^4\cdot 5} = 0.1625
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~C</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~1 + \biggl( \frac{2Z_0}{\kappa}\biggr)^2 ( \varpi_t^2 - r_t^2)
</math>
  </td>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="left">
<math>\frac{457}{13^2}  \approx 2.7041420 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\beta_\pm</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
- \frac{\kappa}{2} \biggl[ \frac{\varpi_t \mp r_t \sqrt{C}}{(\varpi_t + r_t)(\varpi_t - r_t)} \biggr] 
</math>
  </td>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="left">
<math>~
-~\frac{1}{2^6\cdot 5 }\biggl[ 39 \mp \sqrt{457} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~\beta_+</math>
  </td>
<td align="center">&nbsp;  <math>\approx</math>  &nbsp;</td>
  <td align="left">
<math>~
-~0.055070130
</math>
  </td>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
</tr>

<tr>
  <td align="right">
<math>~\beta_-</math>
  </td>
<td align="center">&nbsp;  <math>\approx</math>  &nbsp;</td>
  <td align="left">
<math>~
-~0.188679870
</math>
  </td>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
</tr>
</table>

Notice that we have specified these integration limits such that, in going from the lower limit <math>~(\lambda_1)</math> to the upper limit <math>~(\Lambda_1)</math>, the value of <math>~\xi_1</math> monotonically increases.

====Partial Volume #III-1====

This is the sub-volume that is painted blue and identified as Partial Volume #1 (PV#1) in the right-hand panel of [[#ZoneIII|Figure 7]]. 
<div align="center">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\Lambda_1 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a^2 + [\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2}{a^2 - [\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2} 
~\rightarrow~ \frac{2^8+11^2}{2^8-11^2} 
\approx 2.79259</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_1 </math>
  </td>
  <td align="center">
<math>~= </math>
  </td>
  <td align="left">
<math>~\xi_1|_\mathrm{min} =
\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_-} \biggr)^2  \biggr]^{-1/2}
\approx 1.9116174</math>
  </td>
</tr>
</table>

</td>
<td align="center">

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

<tr>
  <td align="right">
<math>~\Gamma_1 = \xi_2\biggr|_+</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>

<tr>
  <td align="right">
<math>~\gamma_1 = \xi_2\biggr|_- </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>

</td></tr>
</table>
</div>

====Partial Volume #III-2====

This is the sub-volume that is painted green and identified as Partial Volume #2 (PV#2) in the right-hand panel of [[#ZoneIII|Figure 7]]. 
<div align="center">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\Lambda_2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2+a^2}{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2-a^2} 
~\rightarrow~ \frac{19^2 + 2^8}{19^2 - 2^8} 
\approx 5.87619</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a^2 + [\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2}{a^2 - [\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2} 
~\rightarrow~ \frac{2^8+11^2}{2^8-11^2} 
\approx 2.79259</math>
  </td>
</tr>
</table>

</td>
<td align="center">

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

<tr>
  <td align="right">
<math>~\Gamma_2 = \xi_2\biggr|_+</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>

<tr>
  <td align="right">
<math>~\gamma_2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-1
</math>
  </td>
</tr>
</table>

</td></tr>
</table>
</div>

====Partial Volume #III-3====

This is the sub-volume that is painted orange and identified as Partial Volume #3 (PV#3) in the right-hand panel of [[#ZoneIII|Figure 7]]. 
<div align="center">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\Lambda_3 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\infty</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_3 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2+a^2}{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2-a^2} 
~\rightarrow~ \frac{19^2 + 2^8}{19^2 - 2^8} 
\approx 5.87619</math>
  </td>
</tr>
</table>

</td>
<td align="center">

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

<tr>
  <td align="right">
<math>~\Gamma_3 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+1
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_3 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-1
</math>
  </td>
</tr>
</table>

</td></tr>
</table>
</div>


'''Alternative determination of this partial volume:'''  &nbsp; Notice that this "orange" meridional-plane segment is a (semi)circle whose cross-sectional radius is (see [[Appendix/Ramblings/ToroidalCoordinates#Walk_Through_Step-By-Step|accompanying discussion]]),
<div align="center">
<math>~r_\mathrm{orange} = \frac{a}{\sqrt{\lambda_3^2 - 1}} \, ,</math>
</div>
and it is associated with a circular torus whose major radius is,
<div align="center">
<math>~R_\mathrm{orange} = \lambda_3 r_\mathrm{orange} \, .</math>
</div>
Hence, using the [[2DStructure/ToroidalCoordinates#Total_Mass|familiar expression for the volume of a circular torus]], we know that the volume associated with this "orange" partial volume is,

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

<tr>
  <td align="right">
<math>~V_\mathrm{orange}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\biggl[ 2\pi^2 R_\mathrm{orange} r_\mathrm{orange}^2 \biggr] = \pi^2 \lambda_3 r_\mathrm{orange}^3
= \frac{\pi^2 a^3 \lambda_3}{(\lambda_3^2-1)^{3/2}} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~\frac{V_\mathrm{orange}}{V_\mathrm{torus}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{a^3}{2\varpi_t r_t^2}\biggr) \frac{\lambda_3}{(\lambda_3^2-1)^{3/2}}  
\approx 0.165291952 \, .</math>
  </td>
</tr>
</table>
</div>

====Partial Volume #III-4====

This is the sub-volume that is painted pink and identified as Partial Volume #4a (PV#4a) in the right-hand panel of [[#ZoneIII|Figure 7]]. 
<div align="center">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\Lambda_4 </math>
  </td>
  <td align="center">
<math>~= </math>
  </td>
  <td align="left">
<math>~\xi_1|_\mathrm{max} =
\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2  \biggr]^{-1/2}
\approx 8.9243615</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\lambda_4 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a^2 + [\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2}{a^2 - [\varpi_t - \sqrt{r_t^2 - Z_0^2}]^2} 
~\rightarrow~ \frac{2^8+11^2}{2^8-11^2} 
\approx 2.79259</math>
  </td>
</tr>
</table>

</td>
<td align="center">

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

<tr>
  <td align="right">
<math>~\Gamma_{4a} = \xi_2\biggr|_-</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>

<tr>
  <td align="right">
<math>~\gamma_{4a} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-1
</math>
  </td>
</tr>
</table>

</td></tr>
</table>
</div>

This is the sub-volume that is painted red and identified as Partial Volume #4b (PV#4b) in the right-hand panel of [[#ZoneIII|Figure 7]]. 
<div align="center">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\Lambda_4 </math>
  </td>
  <td align="center">
<math>~= </math>
  </td>
  <td align="left">
<math>~\xi_1|_\mathrm{max} =
\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2  \biggr]^{-1/2}
\approx 8.9243615</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\lambda_4 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2+a^2}{[\varpi_t + \sqrt{r_t^2 - Z_0^2}]^2-a^2} 
~\rightarrow~ \frac{19^2 + 2^8}{19^2 - 2^8} 
\approx 5.87619</math>
  </td>
</tr>
</table>

</td>
<td align="center">

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

<tr>
  <td align="right">
<math>~\Gamma_{4b} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+1
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{4b} = \xi_2\biggr|_+</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>

</td></tr>
</table>
</div>

====Partial Volume #III-5====

This is the sub-volume that is painted black and identified as Partial Volume #5 (PV#5) in the right-hand panel of [[#ZoneIII|Figure 7]]. 
<div align="center">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\Lambda_5 </math>
  </td>
  <td align="center">
<math>~= </math>
  </td>
  <td align="left">
<math>~\infty</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_5 </math>
  </td>
  <td align="center">
<math>~= </math>
  </td>
  <td align="left">
<math>~\xi_1|_\mathrm{max} =
\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2  \biggr]^{-1/2}
\approx 8.9243615</math>
  </td>
</tr>
</table>

</td>
<td align="center">

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

<tr>
  <td align="right">
<math>~\Gamma_{5} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+1
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{5} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-1
</math>
  </td>
</tr>
</table>

</td></tr>
</table>
</div>


<!--  HIDE SEGMENT
Also,
<div align="center" id="Parameters">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\Gamma_1 = \xi_2\biggr|_+</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>

<tr>
  <td align="right">
<math>~\gamma_1 = \xi_2\biggr|_-</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>

</td></tr>
</table>
</div>


where, in addition to the quantities defined above,

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

<tr>
  <td align="right">
<math>~\biggl(\frac{\varpi_i}{a}\biggr)_\pm</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\kappa}{2a^2}\cdot \frac{\mathrm{B}}{\mathrm{A}} 
\biggl[1 \pm \sqrt{1-\frac{AC}{B^2}}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathrm{A}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{Z_0}{a}\biggr)^2 + \biggl[\frac{\varpi_t}{a} - \frac{\xi_1}{(\xi_1^2-1)^{1/2}}\biggr]^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathrm{B}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{2\varpi_t Z_0^2}{a\kappa}\biggr) - \biggl[\frac{\varpi_t}{a} - \frac{\xi_1}{(\xi_1^2-1)^{1/2}}\biggr] 
</math>
  </td>
</tr>
</table>

END HIDE SEGMENT -->

'''Alternative determination of this partial volume:'''  &nbsp; Notice that this "black" meridional-plane segment is a (semi)circle whose cross-sectional radius is (see [[Appendix/Ramblings/ToroidalCoordinates#Walk_Through_Step-By-Step|accompanying discussion]]),
<div align="center">
<math>~r_\mathrm{black} = \frac{a}{\sqrt{\lambda_5^2 - 1}} \, ,</math>
</div>
and it is associated with a circular torus whose major radius is,
<div align="center">
<math>~R_\mathrm{black} = \lambda_5 r_\mathrm{black} \, .</math>
</div>
Hence, using the [[2DStructure/ToroidalCoordinates#Total_Mass|familiar expression for the volume of a circular torus]], we know that the volume associated with this "black" partial volume is,

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

<tr>
  <td align="right">
<math>~V_\mathrm{black}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\biggl[ 2\pi^2 R_\mathrm{black} r_\mathrm{black}^2 \biggr] = \pi^2 \lambda_5 r_\mathrm{black}^3
= \frac{\pi^2 a^3 \lambda_5}{(\lambda_3^2-1)^{3/2}} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~\frac{V_\mathrm{black}}{V_\mathrm{torus}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{a^3}{2\varpi_t r_t^2}\biggr) \frac{\lambda_5}{(\lambda_5^2-1)^{3/2}}  
\approx 0.06988365 \, .</math>
  </td>
</tr>
</table>
</div>

====Summary (Zone III)====

<div align="center">
<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="7"><font size="+1">Table 2a:</font>&nbsp;&nbsp;Validate Pattern III-A</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:TorusTest000Sm.png|250px|Torus Test 000]]</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">8.92436</td>
  <td align="center">1.91162</td>
  <td align="center">5.87619</td>
  <td align="center">2.79259</td>
</tr>
<!-- TEMPORARY  <tr>
  <td align="center">Temporary</td>
  <td align="center"><b><font color="pink">END</font></b><p></p><b><font color="red">END</font></b></td>
  <td align="center"><b><font color="blue">START</font></b></td>
  <td align="center"><b><font color="green">END</font></b><p></p><b><font color="red">START</font></b></td>
  <td align="center">
<b><font color="blue">END</font></b><p></p><b><font color="green">START</font></b><p></p><b><font color="pink">START</font></b>
  </td>
  <td align="center">&nbsp;</td>
</tr>
END TEMPORARY-->
<tr>
  <td align="center"><math>~\xi_2|_+</math></td>
  <td align="center"><b><font color="red">START</font></b></td>
  <td align="center"><b><font color="blue">END</font></b></td>
  <td align="center">&nbsp;</td>
  <td align="center"><b><font color="green">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">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center"><b><font color="blue">START</font></b><p></p><b><font color="pink">END</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="red">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"><b><font color="pink">START</font></b></td>
  <td align="center">&nbsp;</td>
  <td align="center"><b><font color="green">START</font></b></td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
</tr>
</table>
</div>

<!-- BEGIN OMIT
<table border="1" cellpadding="8" align="center">
<tr><th align="center" colspan="13">
<font size="+1">Table 2:</font> Zone III Partial Volumes &amp; Integration Limits on <math>~\xi_1</math><p></p>
for model parameters <math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{4}{5}, \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<p></p>(blue)</td>
  <td align="center" colspan="2">PV #2<p></p>(green)</td>
  <td align="center" colspan="2">PV #3<p></p>(orange)</td>
  <td align="center" colspan="4">PV #4<p></p>(pink)&nbsp;&nbsp;&nbsp;&nbsp;and&nbsp;&nbsp;&nbsp;&nbsp;(red)</td>
  <td align="center" colspan="2">PV #5<p></p>(black)</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_{4a}</math></td>
  <td align="center"><math>~\Lambda_{4a}</math></td>
  <td align="center"><math>~\lambda_{4b}</math></td>
  <td align="center"><math>~\Lambda_{4b}</math></td>
  <td align="center"><math>~\lambda_5</math></td>
  <td align="center"><math>~\Lambda_5</math></td>
</tr>
<tr>
  <td align="center"><math>2.792593</math></td>
  <td align="center"><math>1.911617</math></td>
  <td align="center"><math>5.876190</math></td>
  <td align="center"><math>2.792593</math></td>
  <td align="center"><math>\infty</math></td>
  <td align="center"><math>5.876190</math></td>
  <td align="center"><math>8.924361</math></td>
  <td align="center"><math>2.792593</math></td>
  <td align="center"><math>8.924361</math></td>
  <td align="center"><math>5.876190</math></td>
  <td align="center"><math>\infty</math></td>
  <td align="center"><math>8.924361</math></td>
</tr>
<tr>
  <td align="center" rowspan="1" colspan="1">Volume<p></p>Fraction</td>
  <td align="center" colspan="2"><math>0.25309351</math></td>
  <td align="center" colspan="2"><math>0.43923615</math></td>
  <td align="center" colspan="2"><math>0.165291952</math></td>
  <td align="center" colspan="2"><math>0.059240412</math></td>
  <td align="center" colspan="2"><math>0.013254438</math></td>
  <td align="center" colspan="2"><math>0.0698836475</math></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="12">1.000000110 &nbsp; &nbsp; &nbsp; <math>~\Rightarrow~</math>&nbsp; &nbsp; &nbsp; Error = -1.1E-7
  </td>
</tr>
</table>
END OMIT -->

<div align="center">
<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="7"><font size="+1">Table 2b:</font>&nbsp;&nbsp;Validate Pattern III-A</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{1}{8}, \tfrac{3}{4}, \tfrac{1}{4})</math> 
</th>
  <td align="center" rowspan="7">[[File:TorusTest001Sm.png|250px|Torus Test 000]]</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">7.19569</td>
  <td align="center">1.98820</td>
  <td align="center">5.35175</td>
  <td align="center">2.60173</td>
</tr>
<!-- TEMPORARY  <tr>
  <td align="center">Temporary</td>
  <td align="center"><b><font color="pink">END</font></b><p></p><b><font color="red">END</font></b></td>
  <td align="center"><b><font color="blue">START</font></b></td>
  <td align="center"><b><font color="green">END</font></b><p></p><b><font color="red">START</font></b></td>
  <td align="center">
<b><font color="blue">END</font></b><p></p><b><font color="green">START</font></b><p></p><b><font color="pink">START</font></b>
  </td>
  <td align="center">&nbsp;</td>
</tr>
END TEMPORARY-->
<tr>
  <td align="center"><math>~\xi_2|_+</math></td>
  <td align="center"><b><font color="red">START</font></b></td>
  <td align="center"><b><font color="blue">END</font></b></td>
  <td align="center">&nbsp;</td>
  <td align="center"><b><font color="green">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">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center"><b><font color="blue">START</font></b><p></p><b><font color="pink">END</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="red">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"><b><font color="pink">START</font></b></td>
  <td align="center">&nbsp;</td>
  <td align="center"><b><font color="green">START</font></b></td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
</tr>
</table>
</div>


<div align="center">
<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="7"><font size="+1">Table 2c:</font>&nbsp;&nbsp;Validate Pattern III-A</th>
</tr>
<tr><th align="center" colspan="6">
Integration Limits for model parameters <p></p><math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{9}{10}, \tfrac{3}{40}, \tfrac{3}{4}, \tfrac{1}{4})</math> 
</th>
  <td align="center" rowspan="7">[[File:TorusTest003Sm.png|250px|Torus Test 000]]</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">11.4106</td>
  <td align="center">1.84912</td>
  <td align="center">10.6947</td>
  <td align="center">1.95431</td>
</tr>
<!-- TEMPORARY  <tr>
  <td align="center">Temporary</td>
  <td align="center"><b><font color="pink">END</font></b><p></p><b><font color="red">END</font></b></td>
  <td align="center"><b><font color="blue">START</font></b></td>
  <td align="center"><b><font color="green">END</font></b><p></p><b><font color="red">START</font></b></td>
  <td align="center">
<b><font color="blue">END</font></b><p></p><b><font color="green">START</font></b><p></p><b><font color="pink">START</font></b>
  </td>
  <td align="center">&nbsp;</td>
</tr>
END TEMPORARY-->
<tr>
  <td align="center"><math>~\xi_2|_+</math></td>
  <td align="center"><b><font color="red">START</font></b></td>
  <td align="center"><b><font color="blue">END</font></b></td>
  <td align="center">&nbsp;</td>
  <td align="center"><b><font color="green">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">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center"><b><font color="blue">START</font></b><p></p><b><font color="pink">END</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="red">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"><b><font color="pink">START</font></b></td>
  <td align="center">&nbsp;</td>
  <td align="center"><b><font color="green">START</font></b></td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
</tr>
</table>
</div>


<div align="center">
<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="7"><font size="+1">Table 3a:</font>&nbsp;&nbsp;Validate Pattern III-B</th>
</tr>
<tr><th align="center" colspan="6">
Integration Limits for model parameters <p></p><math>~(a, Z_0, \varpi_t, r_t) = (\tfrac{13}{20}, \tfrac{1}{10}, \tfrac{3}{4}, \tfrac{1}{4})</math> 
</th>
  <td align="center" rowspan="7">[[File:TorusTest002BSm.png|250px|Torus Test 000]]</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">2.10621</td>
  <td align="center">5.71122</td>
  <td align="center">2.57592</td>
  <td align="center">4.58888</td>
</tr>
<tr>
  <td align="center">Temporary</td>
  <td align="center"><b><font color="BLUE">END</font></b></td>
  <td align="center"><b><font color="pink">START</font></b><p></p><b><font color="red">START</font></b></td>
  <td align="center"><b><font color="red">END</font></b><p></p><b><font color="GREEN">END</font></b><p></p><b><font color="BLUE">START</font></b></td>
  <td align="center">
<b><font color="pink">END</font></b><p></p><b><font color="green">START</font></b>
  </td>
  <td align="center">&nbsp;</td>
</tr>
<tr>
  <td align="center"><math>~\xi_2|_+</math></td>
  <td align="center"><b><font color="BLUE">END</font></b></td>
  <td align="center"><b><font color="RED">START</font></b></td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
</tr>
<tr>
  <td align="center"><math>~\xi_2|_-</math></td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center"><b><font color="BLUE">START</font></b></td>
  <td align="center">
<b><font color="PINK">END</font></b><p></p><b><font color="green">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="GREEN">END</font></b><p></p><b><font color="RED">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">
<b><font color="PINK">START</font></b>
</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
</tr>
</table>
</div>



<table border="1" cellpadding="8" align="center">
<tr><th align="center" colspan="11">
<font size="+1">Table 4:</font> Various Zone III Partial Volumes<p></p>
for model parameters <math>~(\varpi_t, r_t) = (\tfrac{3}{4}, \tfrac{1}{4})</math> 
</th></tr>
<tr>
  <td align="center"><math>~a</math></td>
  <td align="center"><math>~Z_0</math></td>
  <td align="center">nzones</td>
  <td align="center">PV#1<p></p>(blue)</td>
  <td align="center">PV#2<p></p>(green)</td>
  <td align="center">PV#3<p></p>(orange)</td>
  <td align="center">PV#4a<p></p>(pink)</td>
  <td align="center">PV#4b<p></p>(red)</td>
  <td align="center">PV#5<p></p>(black)</td>
  <td align="center">Error</td>
  <td align="center">Thumbnail</td>
</tr>
<tr>
  <td align="center">0.8</td>
  <td align="center">0.15</td>
  <td align="center">5000</td>
  <td align="center" colspan="6">(see Table 2 for details)</td>
  <td align="center">-1.1E-7</td>
  <td align="center">[[File:TorusTest000Sm.png|75px|Torus Test 000]]
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">0.10</td>
  <td align="center">5000</td>
  <td align="center">0.075401233</td>
  <td align="center">0.44164551</td>
  <td align="center">0.230637507</td>
  <td align="center">0.088778341</td>
  <td align="center">0.010790998</td>
  <td align="center">0.152746462</td>
  <td align="center">-4.8E-8</td>
  <td align="center">&nbsp;</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">0.125</td>
  <td align="center">5000</td>
  <td align="center">0.14811529</td>
  <td align="center">0.45526220</td>
  <td align="center">0.201121481</td>
  <td align="center">0.073678658</td>
  <td align="center">0.013215276</td>
  <td align="center">0.108607177</td>
  <td align="center">-8.7E-8</td>
  <td align="center">[[File:TorusTest001Sm.png|75px|Torus Test 001]]</td>
</tr>
<tr>
  <td align="center">0.65</td>
  <td align="center">0.1</td>
  <td align="center">5000</td>
  <td align="center">0.21505365</td>
  <td align="center">0.38298966</td>
  <td align="center">0.149641005</td>
  <td align="center">0.0028992093</td>
  <td align="center">0.15531546</td>
  <td align="center">0.0941011094</td>
  <td align="center">-9.6E-8</td>
  <td align="center">[[File:TorusTest002BSm.png|75px|Torus Test 002]]</td>
</tr>
<tr>
  <td align="center">0.9</td>
  <td align="center">0.075</td>
  <td align="center">5000</td>
  <td align="center">0.013247868</td>
  <td align="center">0.60594555</td>
  <td align="center">0.0688878257</td>
  <td align="center">0.25103223</td>
  <td align="center">0.00046944204</td>
  <td align="center">0.0604172907</td>
  <td align="center">-2.0E-7</td>
  <td align="center">[[File:TorusTest003Sm.png|75px|Torus Test 003]]</td>
</tr>
</table>