Editing Appendix/Ramblings/RiemannB74C692 (section)

===Cube_quaternion===

Here we construct the wall-clock.  First we recognize that the time required for the ellipsoid to complete "5 spins" is, as calculated above, TIME = 20/|&Omega;<sub>EFE</sub>|.  We have also dictated that the "minute hand" on the clock will complete one full cycle &#8212; that is, it will spin through an angle that starts at 0&deg; and runs to &gamma; = 360&deg; &#8212; every 4 TIME units.  While watching the ellipsoid spin five times, the "minute hand" must complete M = 5/|&Omega;<sub>EFE</sub>| cycles.

We have chosen to model the motion of the "minute hand" by breaking each cycle (360&deg;) into 16 equal divisions, that is, we have chosen to set &Delta;&gamma; = 360&deg;/16 = 22.5&deg;.  This also means that the number of discrete time values will be INT(M &times; 16) + 1 = INT(80/|&Omega;<sub>EFE</sub>|) + 1.

<table border="0" align="center" cellpadding="10"><tr>
<td align="left">

<table border="1" align="center" cellpadding="8">
<tr>
 <td align="center" colspan="3">To be used in the<br />Quaternion Array</td>
</tr>
<tr>
  <td align="center">&gamma; (degrees)</td>
  <td align="center">sin &gamma;</td>
  <td align="center">cos &gamma;</td>
</tr>
<tr>
  <td align="right">-90.0</td>
  <td align="right">-1.0000</td>
  <td align="right">0.0000</td>
</tr>
<tr>
  <td align="right">-112.5</td>
  <td align="right">-0.9239</td>
  <td align="right">-0.3827</td>
</tr>
<tr>
  <td align="right">-135.0</td>
  <td align="right">-0.7071</td>
  <td align="right">-0.7071</td>
</tr>
<tr>
  <td align="right">-157.5</td>
  <td align="right">-0.3827</td>
  <td align="right">-0.9239</td>
</tr>
<tr>
  <td align="right">-180.0</td>
  <td align="right">0.0000</td>
  <td align="right">-1.0000</td>
</tr>
<tr>
  <td align="right">-202.5</td>
  <td align="right">0.3827</td>
  <td align="right">-0.9239</td>
</tr>
<tr>
  <td align="right">-225.0</td>
  <td align="right">0.7071</td>
  <td align="right">-0.7071</td>
</tr>
<tr>
  <td align="right">-247.5</td>
  <td align="right">0.9239</td>
  <td align="right">-0.3827</td>
</tr>
<tr>
  <td align="right">-270.0</td>
  <td align="right">1.0000</td>
  <td align="right">0.0000</td>
</tr>
<tr>
  <td align="right">-292.5</td>
  <td align="right">0.9239</td>
  <td align="right">0.3827</td>
</tr>
<tr>
  <td align="right">-315.0</td>
  <td align="right">0.7071</td>
  <td align="right">0.7071</td>
</tr>
<tr>
  <td align="right">-337.5</td>
  <td align="right">0.3827</td>
  <td align="right">0.9239</td>
</tr>
<tr>
  <td align="right">0.0</td>
  <td align="right">0.0000</td>
  <td align="right">1.0000</td>
</tr>
<tr>
  <td align="right">-22.5</td>
  <td align="right">-0.3827</td>
  <td align="right">0.9239</td>
</tr>
<tr>
  <td align="right">-45.0</td>
  <td align="right">-0.7071</td>
  <td align="right">0.7071</td>
</tr>
<tr>
  <td align="right">-67.5</td>
  <td align="right">-0.9239</td>
  <td align="right">0.3827</td>
</tr>
</table>

</td>
<td align="left">

<table border="1" align="center" cellpadding="8">
<tr>
<th align="center" colspan="4">''Clock''</th>
</tr>
<tr>
  <td align="center">Model</td>
  <td align="center">"Minute Hand"<br />Cycles</td>
  <td align="center">Discrete<br />Steps</td>
  <td align="center">N/A</td>
</tr>
<tr>
  <td align="center">b41c385</td>
  <td align="center">9.1261</td>
  <td align="center">147</td>
  <td align="center">---</td>
</tr>
<tr>
  <td align="center" bgcolor="lightblue">b74c692</td>
  <td align="center" bgcolor="lightblue">7.8278</td>
  <td align="center" bgcolor="lightblue">126</td>
  <td align="center" bgcolor="lightblue">---</td>
</tr>
<tr>
  <td align="center">b90c333</td>
  <td align="center">11.1817</td>
  <td align="center">179</td>
  <td align="center">---</td>
</tr>
<tr>
  <td align="center">b28c256</td>
  <td align="center">10.9487</td>
  <td align="center">176</td>
  <td align="center"></td>
</tr>
</table>
</td></tr></table>

As we have [[Appendix/Ramblings/RiemannMeetsOculus#Final_Touches|detailed in a parallel discussion]], at each discrete time step, the equivalent '''<matrix>''' instruction should be of the form, <table border="0" align="center" cellpadding="8"><tr><td align="center"><matrix>[ 0 ] &nbsp; [ 0 ] &nbsp; [ -S<sub>x</sub> ]  &nbsp; [ T<sub>x</sub> ] &nbsp; &nbsp; [ S<sub>y</sub> &middot; sin(&gamma;) ] &nbsp; [ S<sub>y</sub> &middot; cos(&gamma;) ] &nbsp; [ 0 ] &nbsp; [ T<sub>y</sub> ] &nbsp;  &nbsp; [ S<sub>z</sub> &middot; cos(&gamma;) ] &nbsp; [ -S<sub>z</sub> &middot; sin(&gamma;) ] &nbsp; [ 0] &nbsp; [ T<sub>z</sub> ] &nbsp; &nbsp; [ 0 ] &nbsp; [ 0 ] &nbsp; [ 0 ] &nbsp; [ 1 ]</matrix> .</td></tr></table>

Note that after the clock was originally built as a separate object for these visual scenes, I realized that I needed the starting angle for both hands to be &gamma; = +270&deg; = -90&deg; in order for them to be properly registered at 0<sup>h</sup>0<sup>m</sup> at the start of each animation.  This is why the table of signs and cosines (immediately above) starts at -90&deg;.