Editing Appendix/Ramblings/VirtualReality (section)

==Step-By-Step Production==

<ol type="I">
  <li>Choose an equilibrium Riemann S-type ellipsoidal model:</li>
  <ol type="A">
    <li>From EFE, or [https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract Ou (2006)], or Table 2 in [[ThreeDimensionalConfigurations/RiemannStype#Our_Parameter_Determinations|our separate discussion of Ou's work]], pick the desired pair of axis ratios, b/a and c/a; for example, b/a = 0.41 and c/a = 0.385.</li>
    <li>Find fortran code at, for example, philip.hpc.lsu.edu:/home/tohline/numRecipes/EllipticIntegrals/Riemann/Riemann01.for</li>
    <li>If necessary, compile this code then link it to the doubleELib.o library</li>
    <li>Upon execution, type in values of b/a and c/a (separated by a space), and save the output in a file; see example results in table, immediately below.</li>
  </ol>
</ol>

<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center" colspan="8">Example Riemann S-Type Ellipsoid  <br /><math>~b/a = 0.41</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>~c/a = 0.385</math></th>
</tr>
<tr>
  <td align="center" colspan="4"><font size="+1">Direct</font><br />all frequencies in units of <math>~\sqrt{G \rho}</math></td>
  <td align="center" colspan="4"><font size="+1">Adjoint</font><br />all frequencies in units of <math>~\sqrt{G \rho}</math></td>
</tr>
<tr>
  <td align="center"><math>~\Omega</math></td>
  <td align="center"><math>~\lambda</math></td>
  <td align="center"><math>~\zeta</math></td>
  <td align="center"><math>~f</math></td>
  <td align="center"><math>~\Omega</math></td>
  <td align="center"><math>~\lambda</math></td>
  <td align="center"><math>~\zeta</math></td>
  <td align="center"><math>~f</math></td>
</tr>
<tr>
  <td align="center">0.971082</td>
  <td align="center">0.141594</td>
  <td align="center">-0.403405</td>
  <td align="center">-0.415418</td>
  <td align="center">-0.141594</td>
  <td align="center">-0.971082</td>
  <td align="center">2.766637</td>
  <td align="center">-19.53923</td>
</tr>
<tr>
  <td align="left" colspan="8">All the frequency values should be divided by <math>~\sqrt{\pi}</math> in order to get them in EFE units of <math>~\sqrt{\pi G\rho}</math>.</td>
</tr>
</table>

<ol type="I" start="2">
  <li>Use associated Fortran code to generate surface of desired ellipsoid, etc.</li>
  <ol type="A">
    <li>Find code at, for example, philip.hpc.lsu.edu:/home/tohline/fortran/RiemannModels/Ou_b41c385_xx.for</li>
    <li>Near the top of the code, type in desired values of b/a, c/a, &Omega;<sub>D</sub> and &lambda;<sub>D</sub> for example, drawing from the table immediately above, we have &hellip; <br />(b/a, c/a, &Omega;<sub>D</sub>, &lambda;<sub>D</sub>) = (0.41, 0.385, 0.971082/&radic;&pi; = 0.547874, 0.141594/&radic;&pi; = 0.079886).</li>
    <li>Compile, Link, and Execute the code.</li>
  </ol>
  <li>Quantities needed for Animations</li>
    <ol type="A">
<li>'''Clock''' (Cube_rotation_euler_Z):  Here we ''choose'' a clock unit such that one complete clock cycle (360&deg;) &#8212; equivalent to <math>~[\pi G \rho]^{-1 / 2}</math> &#8212; takes ''four'' game-controller time units, that is, approximately 4 seconds of real time.  And we recognize that, by physics convention, a ''clockwise'' spin of the clock hand corresponds to a ''negative'' angle.  <font color="red">Hence, a ''TIME'' of 1 corresponds to an ''ANGLE'' of -90&deg;.</font>  So if at time zero the clock hand is positioned at -90&deg;, we can, for example, make the association &hellip;</li>
      <ol type="1">
        <li>''input'' (TIME) = 0, 4, 8, 12, 16</li>
        <li>''output'' (ANGLE in degrees) = -90, -450, -810, -1170, -1530</li>
      </ol>
      <li>'''Ellipsoid Octant1''' (Ellipsoid_rotation_euler_Z):  Given that, for the ''Direct'' configuration, &Omega;<sub>D</sub> = +0.547874, we should associate <font color="red">TIME = 4 with ANGLE = 360&deg; &times; &Omega;<sub>D</sub> = +197.23&deg;</font>.  Synchronizing with the '''Clock''' therefore means &hellip;
      <ol type="1">
        <li>''input'' (TIME) = 0, 4, 8, 12, 16</li>
        <li>''output'' (ANGLE in degrees) = 0, +197.23, +394.47, +591.70, +788.94</li>
      </ol>
    Alternatively, given that for the ''Adjoint'' configuration, &Omega;<sub>A</sub> = -0.079886, we should associate <font color="red">TIME = 4 with ANGLE = 360&deg; &times; &Omega;<sub>A</sub> = -28.76&deg;</font>.  That is &hellip;</li>
      <ol type="1">
        <li>''input'' (TIME) = 0, 4, 8, 12, 16</li>
        <li>''output'' (ANGLE in degrees) = 0, -28.76, -57.52, -86.28, -115.04</li>
      </ol>
    </li>
      <li>'''Ellipsoid Octant2''' (EllipFlip_rotation_euler_Z):  ''Octant2'' is the mirror image of ''Octant1'' with respect to the z = 0 equatorial plane of the ellipsoid.  Hence everything is the same except ''Octant2'' must be spun (about the z-axis) in the direction clockwise/counter-clockwise that is ''opposite'' to the spin direction of ''Octant1''.  Here, for example, for ''Octant2'' &hellip;
      <ol type="1">
        <li>''input'' (TIME) = 0, 4, 8, 12, 16</li>
        <li>''output'' (ANGLE in degrees) = 0, -197.23, -394.47, -591.70, -788.94</li>
      </ol>
    Alternatively, for the ''Adjoint'' configuration &hellip;</li>
      <ol type="1">
        <li>''input'' (TIME) = 0, 4, 8, 12, 16</li>
        <li>''output'' (ANGLE in degrees) = 0, +28.76, +57.52, +86.28, +115.04</li>
      </ol>
      <li>'''FLUID1''' (FluidElement1_location):  For the ''Direct'' configuration, each Lagrangian fluid element moves along an elliptical path (defined by the x and y coordinate positions given below) in a direction and at a rate specified by the value of &lambda;<sub>D</sub>.  Given that &lambda;<sub>D</sub> = 0.079886 in our example, we find that, P<sub>orb</sub> = 50.071.  If we choose to divide this into 50 equal time-steps, we have, &Delta;t = 1.0014, and &hellip;</li>
      <ol type="1">
        <li>''input'' (TIME) = 0, 1.0014, 2.0029, 3.0043, 4.0057 &hellip; 48.068, 49.070, 50.071</li>
        <li>''output'' (ANGLE in degrees) = 0, -7.2, -14.4, -21.6, -28.8 &hellip; -345.6, -352.8, -360.0</li>
        <li>''output'' (location_X) = 1.0000, 0.9921, 0.9686, 0.9298, 0.8763 &hellip; 0.9686, 0.9921, 1.0000</li>
        <li>''output'' (location_Y) = 0.0000, -0.05139, -0.1020, -0.1509, -0.1975 &hellip; 0.1020, 0.05139, 0.0000</li>
      </ol>
    </ol>
  <li>Concatenate and Edit/Blend</li>
  <ol>
    <li type="A">Copy the output file from philip.hpc.lsu.edu to the desktop Mac, then "cat" (concatenate) it to the end of the previously developed COLLADA vislualization file, for example &hellip;<br /> "cat MultiFluid22.PERFECT.dae outOu_b41c385_01 > Ou_b41c835_01.dae"</li>
  </ol>
</ol>


<table border="1" cellpadding="8" align="center" width="80%">
<tr><td align="left">
<table border="0" align="center"><tr><td align="center">
'''Motion of an Individual Lagrangian Fluid Element'''
</td></tr></table>

''Direct'' configuration &hellip;
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x_\mathrm{max} \cdot \cos\biggl[ - \lambda_D \biggl(\frac{\pi}{2}\biggr)\times \mathrm{TIME} \biggr]
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x_\mathrm{max} \biggl(\frac{b}{a}\biggr)\cdot \sin\biggl[ - \lambda_D \biggl(\frac{\pi}{2}\biggr)\times \mathrm{TIME} \biggr]
</math>
  </td>
</tr>
</table>
This means that, in the accompanying discussion titled [[ThreeDimensionalConfigurations/RiemannStype#Feeding_a_3D_Animation|''Feeding a 3D Animation'']], <math>~\varphi \rightarrow -2\pi \lambda_D</math> while <math>~t \rightarrow \mathrm{TIME}/4</math>.  Note that <math>~x</math> goes from <math>~+x_\mathrm{max}</math> to  <math>~-x_\mathrm{max}</math> over a ''TIME'' interval, <math>~T_\pi</math>, given by when the absolute value of the argument of the trigonometric functions goes to <math>~\pi</math>.  That is, as viewed from a frame that is rotating with the ellipsoidal figure, one complete elliptical orbital ''period'' for every individual Lagrangian fluid element will be, 
<div align="center">
<math>~P_\mathrm{orb} \equiv 2T_\pi =\frac{4}{|\lambda_D|} \, .</math>
</div>
<p><br /></p>
----
<p><br /></p>
''Adjoint'' configuration &hellip; just replace <math>~\lambda_D</math> with <math>~\lambda_A = -\Omega_D </math>.
</td></tr></table>