Editing Appendix/Ramblings/AzimuthalDistortions (section)

===Playing with the Radial Eigenfunction===
Up to this point, we've only considered radial eigenfunctions composed of two components (a "blue" inner component and a "green" outer component) that do not overlap.  Here we'll allow the two components to overlap by assigning different values of <math>~r_\mathrm{mid}</math> to the two separate components &#8212; more specifically, we'll allow <math>~r_\mathrm{mid}|_\mathrm{green} \le r_\mathrm{mid}|_\mathrm{blue}</math> &#8212; then add the two functions over the region of overlap.  Let's consider components of the following form:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center" bgcolor="blue">&nbsp;</td>
  <td align="right">
<math>~f_\mathrm{blue}(\varpi) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{r_\mathrm{blue} - \varpi}{\varpi - r_-} \biggr)^{p}
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_- < \varpi < r_\mathrm{blue} \, ,</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>~f_\mathrm{green}(\varpi) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{r_\mathrm{green} - \varpi}{\varpi - r_+} \biggr)^{p}
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_\mathrm{green} < \varpi < r_+ \, ,</math>
  </td>
</tr>
</table>
</div>
where, <math>~p</math> is a exponent yet to be specified.  So, for fixed values of the inner and outer radii of the torus, <math>~r_-</math> and <math>~r_+</math>, this two-component function has three adjustable variables.  They are, <math>~r_\mathrm{blue}</math>, <math>~r_\mathrm{green}</math>, and <math>~p</math>.

====Experimenting====
In Figure 5, <math>~r_\mathrm{blue}</math> and <math>~r_\mathrm{green}</math> are fixed, and <math>~p</math> is varied.

<div align="center">
<table border="1" align="center" cellpadding="5" width="80%">
<tr>
  <th align="center"><font size="+1">Figure 5:</font>&nbsp; Variable exponent, <math>~p</math></th>
</tr>
<tr><td align="center">
<math>~r_- = 0.5</math>, <math>~r_+ = 1.5</math>
&nbsp; &nbsp; &nbsp; &nbsp; &hellip; &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~r_\mathrm{blue} = 1.25</math>, <math>~r_\mathrm{green}= 1.1</math>
</td></tr>
<tr><td align="center">
[[File:MontageAbrief.png|500px|Playing with radial eigenfunction]]
</td></tr>
<tr><td align="left">
In this example, the exponent, <math>~p</math>, is varied over the range, <math>~0.25 \le p \le 1.2</math>, as indicated by the numerical values shown in the upper-lefthand corner of each panel.
</td></tr>
</table>
</div>
Based on the [[#SquareRoot|above discussion]], I expected that the best match to the eigenfunctions found in [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11] would be <math>~p=0.5</math>, that is, a square-root.  However, as illustrated in Figure 5, this and other fractional exponents less than unity generate noncontinuous derivatives at the overlapping edges of our two-piece function.  Instead, a value of <math>~p = 1.2</math> seems to exhibit a more desired, smooth behavior.


In Figure 6, <math>~r_\mathrm{green}</math> and <math>~p</math> are fixed, and <math>~r_\mathrm{blue}</math> is varied.

<div align="center">
<table border="1" align="center" cellpadding="5" width="80%">
<tr>
  <th align="center"><font size="+1">Figure 6:</font>&nbsp; Variable <math>~r_\mathrm{blue}</math></th>
</tr>
<tr><td align="center">
<math>~r_- = 0.5</math>, <math>~r_+ = 1.5</math>
&nbsp; &nbsp; &nbsp; &nbsp; &hellip; &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~p = 1.2</math>, <math>~r_\mathrm{green}= 0.9</math>
</td></tr>
<tr><td align="center">
[[File:MontageBbrief.png|500px|Playing with radial eigenfunction]]
</td></tr>
<tr><td align="left">
In this example, the "blue" edge is varied over the range, <math>~0.91 \le r_\mathrm{blue} \le 1.25</math>, as indicated by the numerical values shown in the upper-lefthand corner of each panel.
</td></tr>
</table>
</div>

The frames of Figure 6 illustrate the qualitative behavior we have been seeking.  Setting the exponent, <math>~p</math>, to a value greater than unity then varying one of the edges of the two-part eigenfunction provides a natural variation from "pointed" curves that look like adjoined arc-hyperbolic tangents to others that look more like a parabola.


====Trial Comparison with HI11====

In Figure 7, we show how a straightforward, smooth adjustment of one parameter &#8212; namely, <math>~r_\mathrm{blue}</math> &#8212; generates a series of eigenfunctions that nicely match the set of radial eigenfunctions that are displayed in Figure 16 of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11].

<div align="center" id="Figure7">
<table border="1" cellpadding="3" align="center" width="60%">
<tr>
  <td align="center" colspan="2">
<b><font size="+1">Figure 7:</font></b> &nbsp;
Radial Eigenfunction Comparison
  </td>
</tr>
<tr>
  <td align="center">'''(a)''' &nbsp; Extracted from Figure 16 of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11]</td>
  <td align="center">'''(b)''' &nbsp; Our Empirically Constructed Function</td>
</tr>
<tr><td align="center" rowspan="4">
[[File:PaperI_Fig16.png|300px|Figure 16 from HI11]]
</td><td align="center">[[File:PhaseMovieFrameFirst.png|250px|Figure 16 from HI11]]
</td></tr>
<tr><td align="center">[[File:EigenfunctionMovie01.gif|250px|Figure 16 from HI11]]</td></tr>
<tr><td align="center">[[File:PhaseMovieFrameLast.png|250px|Figure 16 from HI11]]</td></tr>
<tr>
  <td align="left" colspan="1">
Here we set <math>~r_- = 0.6</math>, <math>~r_+ = 1.5</math>, <math>~r_\mathrm{blue}= 1.15</math>, and
<math>~p = 1.2</math>, then let the "green" edge vary over the range, <math>~0.605 \le r_\mathrm{green} \le 1.144</math>.  Via an animation, the middle panel illustrates the behavior of our empirically constructed eigenfunction over this entire range of values as indicated by the numerical value shown in the bottom-righthand corner of the panel; the top and bottom panels display the shape of our eigenfunction at the two extreme values of <math>~r_\mathrm{green}</math>.
  </td>
</tr>
</table>
</div>