Editing Appendix/Ramblings/AzimuthalDistortions (section)

===Constant Phase Loci===

Now let's work on the phase function, <math>~\phi_1(\varpi)</math>.  The phase function displayed in the right-hand panel of our Figure 2 &#8212; that is, the phase function that developed spontaneously from the linear stability analysis performed by [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H  HI11]  &#8212; appears to be fairly constant (''i.e.,'' the phase is independent of radius) in the innermost region of the torus and, then again, fairly constant in the outermost region of the torus with a smooth but fairly rapid phase shift of approximately <math>~\pi</math> radians between the two extremes.  This is the behavior exhibited by an arctangent (ATAN) function.  With this in mind, we have defined a new function, <math>~D(\varpi)</math>, ''in terms of'' our empirically derived radial eigenfunction, <math>~f_\ln(\varpi)</math>, as follows:

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

<tr>
  <td align="right">
<math>~D(\varpi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{f_\ln(\varpi) - [f_\ln]_\mathrm{min}}{[f_\ln]_\mathrm{max} - [f_\ln]_\mathrm{min}} \, .</math>
  </td>
</tr>
</table>
</div>

It has the following behavior:  
* At the inner edge of the torus <math>~(r_-)</math>, where <math>~f_\ln(\varpi) = [f_\ln]_\mathrm{max}</math>, <math>~D(\varpi) = 1</math>;
* At <math>~r_\mathrm{mid}</math>, where <math>~f_\ln(\varpi) = [f_\ln]_\mathrm{min}</math>, <math>~D(\varpi) = 0</math>;
* At the outer edge of the torus <math>~(r_+)</math>, where again <math>~f_\ln(\varpi) = [f_\ln]_\mathrm{max}</math>, <math>~D(\varpi) = 1</math>.
This function can therefore satisfactorily serve as an argument of the ATAN function, swinging the phase by <math>~\pi/2</math> over the inner (blue) region of the torus, then swinging the phase by an additional <math>~\pi/2</math> over the outer (green) region of the torus.  If we furthermore multiply the function by a variable coefficient &#8212; call it, <math>~\aleph</math> &#8212; before feeding it to the ATAN function, we can adjust the thickness of the radial domain over which the total phase transition occurs.  What appears to work well is the following:

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

<tr>
  <td align="center" bgcolor="blue">&nbsp;</td>
  <td align="right">
<math>~\phi_1(\varpi) + \frac{\pi}{2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+~\tan^{-1}[\aleph \cdot D(\varpi)]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_- < \varpi < r_\mathrm{mid} \, ;</math>
  </td>
</tr>
</table>
</div>

and

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

<tr>
  <td align="center" bgcolor="green">&nbsp;</td>
  <td align="right">
<math>~\phi_1(\varpi) + \frac{\pi}{2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\tan^{-1}[\aleph \cdot D(\varpi)] </math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_\mathrm{mid} < \varpi < r_+ \, .</math>
  </td>
</tr>
</table>
</div>

In the lefthand panel of Figure 4, the "constant phase loci" defined by this empirically constructed phase function have been mapped onto a polar-coordinate grid for ten different values of the leading coefficient in the range, <math>~1.0 \le \aleph \le 40.0</math>, as recorded in the bottom-right corner of the plot.  The constant phase locus created by setting <math>~\aleph = 8.0</math> has been singled out and displayed in the middle panel of Figure 4, because it closely resembles the "constant phase locus" published by HI11a (reprinted here as the righthand panel of Figure 4 to facilitate comparison).

<table border="1" cellpadding="8" align="center">
<tr><th align="center" colspan="3"><font size="+1">Figure 4</font></th></tr>
<tr>
  <td align="center" colspan="2">Constant Phase Loci Generated by Our Empirically Constructed Phase Function, <math>~\phi_1(\varpi)</math></td>
  <td align="center" rowspan="2">[http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11]'s Published Constant Phase Loci</td>
</tr>
<tr><td align="center">Ten Values of <math>~\aleph</math></td><td align="center">For <math>~\aleph = 8</math></td></tr>
<tr>
<td align="center">
[[File:ImamuraPhaseMovie.gif|230px|Movie Showing Empirically Constructed Phase]]
</td>
<td align="center">
[[File:a05.png|230px|Best Match]]
</td>
<td align="center">
[[File:ImamuraOriginalPhase.png|250px|HI11a constant phase loci]]
</td>
</tr>
</table>

<!--
Then we set,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\phi(\varpi) + \phi_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{\tan^{-1}[\aleph \cdot D(\varpi)] - \frac{\pi}{2} \biggr\} + \frac{\pi}{10} \, .</math>
  </td>
</tr>
</table>
</div>

Now &#8212; as stated earlier in the [[#Figure3|caption to Figure 3]] &#8212; for the specific case being graphically illustrated here, <math>~[f_\ln]_\mathrm{min} = -2.99448</math> and <math>~[f_\ln]_\mathrm{max} = 2.64665</math>.  Hence,

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

<tr>
  <td align="right">
<math>~\phi(\varpi) + \frac{\pi}{2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan^{-1}\biggl\{8\cdot \biggl[ 
\frac{f_\ln(\varpi) - [f_\ln]_\mathrm{min}}{[f_\ln]_\mathrm{max} - [f_\ln]_\mathrm{min}}  
\biggr]\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan^{-1}[a\cdot f(\varpi) + b]  \, ,
</math>
  </td>
</tr>
</table>
</div>
where, <math>~a = 1.41816</math> and <math>~b = 4.24664</math>.
-->