Editing 2DStructure/ToroidalCoordinates (section)

===Move General Case===
<font color="red"><b>NOTE:</b></font>  A complete prescription of the toroidal-coordinate integration limits that are appropriate for a determination of the volume or the gravitational potential of a circular torus can be found in [[2DStructure/ToroidalCoordinateIntegrationLimits#Toroidal-Coordinate_Integration_Limits|an accompanying discussion]].

In the more general case, the expression for the volume integral should be the same; all we should have to do is incorporate the more general integration ''limits'' as specified in our above evaluation of the gravitational potential.  Hence, in the more general case we should have,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

</table>
</div>

Next, referencing various trigonometric relations, we note that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tan\biggl(\frac{\zeta}{2}\biggr) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \biggl[ \frac{1-\cos\zeta}{1+\cos\zeta} \biggr]^{1/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \biggl[ \frac{1-\xi_2}{1+\xi_2} \biggr]^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>
Hence, in our expression for the torus volume, the argument of the arctangent may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tan\biggl(\frac{\zeta}{2}\biggr) \cdot \frac{(\xi_1 + 1)^{1/2}}{(\xi_1 - 1)^{1/2}}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \biggl[ \frac{1-\xi_2}{1+\xi_2} \biggr]^{1/2} \cdot \frac{(\xi_1 + 1)^{1/2}}{(\xi_1 - 1)^{1/2}}  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \biggl[ \frac{(1-\xi_2)}{(1+\xi_2)} \cdot \frac{(\xi_1 + 1)}{(\xi_1 - 1)} \biggr]^{1/2}  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \biggl[ \frac{\xi_1+1-\xi_1\xi_2 - \xi_2}{\xi_1 - 1 +\xi_1\xi_2 - \xi_2}  \biggr]^{1/2}  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \biggl[ \frac{(\xi_1- \xi_2)-(\xi_1\xi_2-1) }{(\xi_1- \xi_2) +(\xi_1\xi_2- 1 ) }  \biggr]^{1/2}  </math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan\biggl(\frac{\alpha}{2}\biggr) \, , </math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\cos\alpha \equiv \frac{(\xi_1\xi_2- 1 )}{(\xi_1- \xi_2)} </math>
&nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math>&nbsp; &nbsp; &nbsp; 
<math>~\alpha \equiv \cos^{-1}\biggl[ \frac{(\xi_1\xi_2- 1 )}{(\xi_1- \xi_2)} \biggr]
= \cos^{-1}\biggl[ \frac{(\xi_1\cos\zeta - 1 )}{(\xi_1- \cos\zeta)} \biggr] \, .</math>
</div>
Hence, the volume integral may be written as,
<div align="center" id="GeneralVolumeIntegration">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pi a^3 \int\limits_{\xi_1|_\mathrm{min}}^{\xi_1|_\mathrm{max}}  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\}_{\xi_2|_-}^{\xi_2|_+} \, .
</math>
  </td>
</tr>

</table>
</div>

====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>

====Total Volume by Summing Four Zones====
Let's stick with a discussion of the situation where we set <math>~Z_0 = h</math> with <math>~0 < h < r_t</math>, and now determine the total volume by adding together four sub-volumes.  The green "cropped-top" region is the first of these sub-volume zones.  The remaining (pink) portion of the torus can be broken into three adjoining segments &#8212; left-to-right &#8212; whose two edge boundaries plus two internal interfaces are identified by the following four special values of the "radial" coordinate:
<div 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} \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\xi_1\biggr|_+  </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} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\xi_1\biggr|_-  </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} \, ,</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}\, .</math>
  </td>
</tr>
</table>
</div>
As presented in the following table, we have determined the values of these four boundary/interface radial coordinates for the same set of parameter values as in the previous table.  Notice that, as listed, they represent monotonically increasing values of the radial coordinate.  Between the first two boundaries and between the last two boundaries, the limits on the "angular" coordinate should be the normal, &nbsp;<math>~~~\xi_2|_-~\rightarrow ~\xi_2|_+ ~~~</math>.  But between the second and third boundaries, the integration limits on the angular coordinate should be, &nbsp;<math>~\xi_2|_- ~\rightarrow~ +1</math>.

<table border="1" cellpadding="5" align="center">
<tr><th align="center" colspan="10">
Various Boundary Values for <math>~\xi_1</math>
</th></tr>
<tr>
  <td align="center" rowspan="1"><math>~\varpi_t</math></td>
  <td align="center" rowspan="1"><math>~r_t</math></td>
  <td align="center" rowspan="1"><math>~a</math></td>
  <td align="center" rowspan="1"><math>~Z_0</math></td>
  <td align="center"><math>~\xi_1|_\mathrm{max}</math></td>
  <td align="center"><math>~\xi_1|_+</math></td>
  <td align="center"><math>~\xi_1|_-</math></td>
  <td align="center"><math>~\xi_1|_\mathrm{min}</math></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">1.1859</td>
  <td align="center">1.3959</td>
  <td align="center">1.6326</td>
  <td align="center">2.0312</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">1.1900</td>
  <td align="center">1.3655</td>
  <td align="center">1.7077</td>
  <td align="center">2.0630</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">1.2021</td>
  <td align="center">1.3180</td>
  <td align="center">1.8929</td>
  <td align="center">2.1603</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">1.2209</td>
  <td align="center">1.2808</td>
  <td align="center">2.1611</td>
  <td align="center">2.3213</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">1.2291</td>
  <td align="center">1.2700</td>
  <td align="center">2.2808</td>
  <td align="center">2.3963</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">1.2363</td>
  <td align="center">1.2622</td>
  <td align="center">2.3872</td>
  <td align="center">2.4637</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">1.2464</td>
  <td align="center">1.2529</td>
  <td align="center">2.5436</td>
  <td align="center">2.5638</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">1.2499</td>
  <td align="center">1.2501</td>
  <td align="center">2.5977</td>
  <td align="center">2.5985</td>
</tr>
</table>