Editing 2DStructure/ToroidalCoordinateIntegrationLimits (section)

===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>