Editing ThreeDimensionalConfigurations/MeetsCOLLADAandOculusRiftS (section)

===Animation===

When the "play" button is clicked on the Preview app's control panel (again, reference Figure 1b), the COLLADA-directed scene-animation begins.  Three different characteristic timescales are relevant:  
[[File:ClockImage01.png|175px|right|Clock Image]]<ol>
<li>
We use the motion of the hands on the wall-mounted clock to identify the natural oscillation frequency of the physical system, namely, <math>~\Omega_0 \equiv [\pi G \rho]^{1 / 2}</math>.  More specifically, by declaration, one oscillation period of the red "minute" hand &#8212; that is to say, one wall-clock "hour" &#8212; is <math>~P_0 = 2\pi[\pi G \rho]^{-1 / 2} = [4\pi/(G \rho)]^{1 / 2}</math>.  One full cycle of the wall-clock's black "hour" hand is therefore 12P<sub>0</sub>.  For pedagogical purposes, in all of our COLLADA-driven animations we have equated P<sub>0</sub> with a COLLADA TIME interval of 4.0, which means that during each animation, one wall-clock hour passes every (approximately) 4 seconds of real time.<sup>&Dagger;</sup></li>
<li>The purple ellipsoid spins in a prograde direction about its shortest axis &#8212; that is, about the Z<sub>0</sub>-axis &#8212; on a timescale set by the value of <math>~\Omega_\mathrm{EFE}</math>, which has already been normalized to &Omega;<sub>0</sub>.  Specifically, as viewed from the inertial frame of reference, the ellipsoid will complete one full spin in a time, <math>~P_\mathrm{spin} = |\Omega_\mathrm{EFE}|^{-1} = [0.5479]^{-1} = 1.825</math> wall-clock hours, which is approximately 7.3 seconds of real time.</li>
<li>As viewed from a frame of reference that is spinning with the ellipsoid, each Lagrangian fluid element inside as well as on the surface of the ellipsoid moves in a retro-grade direction along an elliptical trajectory that has the same ellipticity as does the equatorial surface of the 3D ellipsoid.  Each and every Lagrangian fluid element completes one full elliptical orbit in a time, <math>~P_\mathrm{Lagrange} = \lambda_\mathrm{EFE}^{-1} = [0.0799]^{-1} = 12.52</math> wall-clock hours, which is about 50 seconds of real time.
</ol>

In the COLLADA-based code (<b>Inertial17.dae</b>) that has been provided above in the scrollable box, we have instructed the animation sequence to follow the system's evolution through five complete spins of the purple ellipsoid.  (Unless the ''pause'' button is clicked, the animation will actually loop repeatedly, but the time on the wall-mounted clock will reset to 0<sup>h</sup>0<sup>m</sup> after every fifth spin period.)  In practice, this animation sequence lasts (5 &times; 1.825) &times; 4 = 36.5 seconds of real time.

Figure 2 displays a pair of still-frame images of our example (purple) ellipsoidal configuration after the ellipsoid has completed precisely five (counter-clockwise) spin cycles.  (The snapshots have been taken at the same point in time, but from two different camera viewing angles.)  In the left-hand image (Figure 2a),  the time on the clock appears to be about 9<sup>h</sup>8<sup>m</sup>.  This means that as the ellipsoid has completed five spin cycles, the "minute" hand of the clock has completed approximately [9 + 8/60] &#8776; 9.13 cycles.  Dividing this number by 5 gives a value of <math>~P_\mathrm{spin} \approx 1.827</math>, which is indeed the spin period we identified earlier.  This illustrates that the proper physical timescales governing oscillations of the wall-mounted clock and of the spinning ellipsoid have been implemented in a quantitatively accurate manner within the COLLADA code.


<div align="center">
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center">Figure 2a</th>
  <th align="center">Figure 2b</th>
</tr>
<tr>
  <td align="left" bgcolor="lightgrey">
[[File:COLLADA1stViewpoint.png|300px|EFE Model b41c385]]
  </td>
  <td align="left" bgcolor="lightgrey">
[[File:COLLADA4thViewpoint.png|300px|EFE Model b41c385]]
  </td>
</tr>
</table>
</div>


In Figure 1b, above, we have used a small red cube to mark the Lagrangian fluid element that is initially (TIME = 0.0) located at the end of the longest axis of the ellipsoid (in the +X direction).  In Figure 2b we can see where this fluid element is located after the ellipsoid has completed five complete spins.  After moving in a retrograde (clockwise) direction along its assigned elliptical trajectory while the ellipsoid completed five full spins, we see that the red-tagged Lagrangian fluid element has completed a little less than 3/4 of one full orbit; let's call it 73% of one orbit.  We therefore infer from the COLLADA-driven animation that <math>~0.73 P_\mathrm{Lagrange}\approx~5 P_\mathrm{spin}</math>, that is, <math>~P_\mathrm{Lagrange}\approx 6.849 P_\mathrm{spin}</math>.  This is indeed a proper reflection of the physical properties of our chosen example Riemann S-type ellipsoid, as its mathematically determined equilibrium frequency ratio is, <math>~|\Omega_\mathrm{EFE}|/|\lambda_\mathrm{EFE}| = 6.8582</math>.  This, too, illustrates that the proper physical timescales governing oscillations of our chosen example Riemann ellipsoid have been implemented in a quantitatively accurate manner within the COLLADA code.