Editing Appendix/Ramblings/AzimuthalDistortions (section)

====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">
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<b><font size="+1">Figure 7:</font></b> &nbsp;
Radial Eigenfunction Comparison
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  <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>
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[[File:PaperI_Fig16.png|300px|Figure 16 from HI11]]
</td><td align="center">[[File:PhaseMovieFrameFirst.png|250px|Figure 16 from HI11]]
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<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>
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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>.
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