Editing 2DStructure/ToroidalCoordinates (section)

====Green Cropped-Top Volume====
Now, if we set <math>~Z_0 = h</math> with <math>~0 < h < r_t</math>, then the horizontal plane defined by <math>~z = Z_0</math> will cut through the circular torus, splitting it into two hemispheres &#8212; a lower, pink sub-volume and an upper, green sub-volume [[#Volume_with_Cropped_Top|as depicted in  the above diagram]].  While using toroidal coordinates to perform the volume integral, we recognize that this horizontal plane is also identified by setting the angular coordinate to <math>~\xi_2 = +1</math> [if <math>~a \leq (\varpi_t-b)</math>] or to <math>~\xi_2 = -1</math> [if <math>~a \geq (\varpi_t+b)</math>].   Then, using the former case as an example, for each value of <math>~\xi_1</math> (corresponding to a specific <math>\xi_1</math>- circle) the integral over the angular, <math>~\xi_2</math> coordinate should naturally break into two segments:  The segment falling within the green sub-volume should have integration limits, <math>~\xi_2|_+ \rightarrow 1</math>; and the segment falling within the pink sub-volume should have integration limits, <math>~1 \rightarrow \xi_2|_-</math>.  Let's see if specification of these limits allows us to derive an analytic expression for the green sub-volume that matches the expression for <math>~V_\mathrm{green}</math> [[#Volume_with_Cropped_Top|as derived above]] using cylindrical coordinates.


Notice that, for the green sub-volume, the limits on <math>~\xi_1</math> should correspond to <math>~\xi_2 = +1</math> and <math>~\varpi = (\varpi_t \pm b)</math>.  Because, in general,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a(\xi_1^2 -1)^{1/2}}{\xi_1-\xi_2} \, ,</math>
  </td>
</tr>
</table>
</div>
this means that the limits on <math>~\xi_1</math> are (valid only for <math>0 < Z_0 < r_t</math>),
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \varpi_t \pm \sqrt{r_t^2 - Z_0^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a(\xi_1 +1)^{1/2}(\xi_1 -1)^{1/2}}{\xi_1-1} </math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ (\xi_1 - 1)\biggl[\varpi_t \pm \sqrt{r_t^2 - Z_0^2}\biggr]^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a^2(\xi_1 +1) </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \xi_1\biggl\{\biggl[\varpi_t \pm \sqrt{r_t^2 - Z_0^2}\biggr]^2-a^2\biggr\} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\varpi_t \pm \sqrt{r_t^2 - Z_0^2}\biggr]^2 + a^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \xi_1\biggr|_\pm  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{[\varpi_t \pm \sqrt{r_t^2 - Z_0^2}]^2 + a^2}{[\varpi_t \pm \sqrt{r_t^2 - Z_0^2}]^2-a^2} \, .</math>
  </td>
</tr>
</table>
</div>

Hence, according to our [[#GeneralVolumeIntegration|just-derived volume integral]], we have,

<div align="center" id="GreenAnalytic">
<table border="1" cellpadding="8"><tr><th align="center" colspan="2">
Toroidal-Coordinate Integral Expression for Green Volume</th></tr>
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~V_\mathrm{green}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi a^3 \int\limits_{\xi_1|_-}^{\xi_1|_+}  d\xi_1
\biggl\{
\frac{(1-\xi_2^2)^{1/2} [ 4\xi_1^2 - 3\xi_1 \xi_2 - 1]}{(\xi_1^2-1)^2 (\xi_1 - \xi_2)^2}
+ \biggl[ \frac{(2\xi_1^2 + 1)}{(\xi_1^2-1)^{5/2}}\biggr] \cos^{-1}\biggl[ \frac{(\xi_1\xi_2 - 1 )}{(\xi_1- \xi_2)} \biggr] 
\biggr\}_{1}^{\xi_2|_+} \, .
</math>
  </td>
</tr>

</table>

</td>
<td align="center">
[[File:CropTopB.png|150px|right|Diagram of "Cropped Top" Torus]]
</td></tr>
</table>
</div>

Note from J. E. Tohline:  On 4 November 2015, a fortran subroutine was used to numerically perform this 1D integration and thereby determine the volume of the green segment of a "cropped top" torus.  The name of this fortran code was &hellip;
<pre>philip.hpc.lsu.edu:/home/tohline/fortran/Toroidal/testI3.for</pre>
The following table presents results of tests run with different sets of physical parameters and different numbers of 1D integration steps (nzones); note that the analytic expression for the ''angular'' integration limit, <math>~\xi_2|_+</math>, is given in the [[#Parameters|above table of parameter expressions]].  In the following table, values of <math>~V_\mathrm{green}</math> obtained by numerical integration are compared with values obtained from the [[#GreenAnalytic|analytic expression derived above via a cylindrical-coordinate formulation]].

<table border="1" cellpadding="5" align="center">
<tr><th align="center" colspan="11">
Comparison of "Cropped-Top" Volume Determinations
</th></tr>
<tr>
  <td align="center" rowspan="3"><math>~\varpi_t</math></td>
  <td align="center" rowspan="3"><math>~r_t</math></td>
  <td align="center" rowspan="3"><math>~a</math></td>
  <td align="center" rowspan="3"><math>~Z_0</math></td>
  <td align="center" colspan="5"><math>~V_\mathrm{green}/V_\mathrm{torus}</math></td>
</tr>
<tr>
  <td align="center" rowspan="2">Analytic</td>
  <td align="center" colspan="2">nzones = 5000</td>
  <td align="center" colspan="2">nzones = 500</td>
</tr>
<tr>
  <td align="center">1D Integration</td>
  <td align="center">Error</td>
  <td align="center">1D Integration</td>
  <td align="center">Error</td>
</tr>
<tr>
  <td align="center"><math>\tfrac{3}{4}</math></td>
  <td align="center"><math>\tfrac{1}{4}</math></td>
  <td align="center"><math>\tfrac{1}{3}</math></td>
  <td align="center">0.24</td>
  <td align="center">4.7727731D-03</td>
  <td align="center">4.7727731D-03</td>
  <td align="center">-1.66D-08</td>
  <td align="center">4.7727865D-03</td>
  <td align="center">-2.8D-06</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.23</td>
  <td align="center">1.3417064D-02</td>
  <td align="center">1.3417065D-02</td>
  <td align="center">-2.6D-08</td>
  <td align="center">1.3417115D-02</td>
  <td align="center">-3.8D-06</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.20</td>
  <td align="center">5.2044018D-02</td>
  <td align="center">5.2044021D-02</td>
  <td align="center">-6.3D-08</td>
  <td align="center">5.2044404D-02</td>
  <td align="center">-7.4D-06</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.15</td>
  <td align="center">1.4237849D-01</td>
  <td align="center">1.4237851D-01</td>
  <td align="center">-1.6D-07</td>
  <td align="center">1.4238095D-01</td>
  <td align="center">-1.7D-05</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.125</td>
  <td align="center">1.9550110D-01</td>
  <td align="center">1.9550115D-01</td>
  <td align="center">-2.4D-07</td>
  <td align="center">1.9550595D-01</td>
  <td align="center">-2.5D-05</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.10</td>
  <td align="center">2.5231578D-01</td>
  <td align="center">2.5231587D-01</td>
  <td align="center">-3.4D-07</td>
  <td align="center">2.5232459D-01</td>
  <td align="center">-3.5D-05</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.05</td>
  <td align="center">3.7353003D-01</td>
  <td align="center">3.7353031D-01</td>
  <td align="center">-7.4D-07</td>
  <td align="center">3.7355809D-01</td>
  <td align="center">-7.5D-05</td>
</tr>
<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.01</td>
  <td align="center">4.7454199D-01</td>
  <td align="center">4.7454347D-01</td>
  <td align="center">-3.3D-06</td>
  <td align="center">4.7465374D-01</td>
  <td align="center">-2.4D-04</td>
</tr>
</table>