Editing ThreeDimensionalConfigurations/RiemannTypeI (section)

====Example Implementation====

<span id="Tipped">Drawing from</span> the set of [[#Case_I|''Case I'' parameters listed above]],  we will set a = 1, b = 1.25, c = 0.4703, &zeta;<sub>2</sub> = -2.2794, and &zeta;<sub>3</sub> = -1.9637.  This means that,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tan\theta </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{\zeta_2 }{ \zeta_3 } \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]\frac{c^2}{b^2} = - 0.3448 ~~~\Rightarrow ~~~ \theta = -19.02^\circ
\, .</math>
  </td>
</tr>
</table>

We deduce as well that,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
\frac{y_0}{z_0}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \biggl\{ \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3  \cos\theta  - \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \sin\theta  \biggr\}^{-1}
= -1.8666 \cdot \{-0.7245 - 0.6084 \}^{-1} = 1.4004 
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\frac{x_\mathrm{max}}{ y_\mathrm{max} }   
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{
\biggl[ \biggl( \frac{a^2}{a^2 + b^2} \biggr) \zeta_3  \cos\theta  
- \biggl( \frac{a^2}{a^2 + c^2} \biggr) \zeta_2 \sin\theta  \biggr] \biggl[ \frac{a^2 + b^2}{b^2} \biggr] \frac{\cos\theta}{\zeta_3} \biggr\}^{1 / 2}
= \{ [-1.3329] \cdot (-0.7895) \}^{1 / 2} = \sqrt{1.0524} = 1.0259 
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\varphi
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ 
\biggl[ \biggl( \frac{a^2}{a^2 + b^2} \biggr) \zeta_3   
- \biggl( \frac{a^2}{a^2 + c^2} \biggr) \zeta_2 \tan\theta  \biggr] \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 \biggr\}^{1 / 2} = 1.2993
\, .
</math>
  </td>
</tr>
</table>

<table border="1" cellpadding="10" align="center" width="80%">
<tr><td align="left">
It is important to keep in mind that each Lagrangian fluid element will complete one full orbit in its "tipped" plane when  <math>~\varphi t = 2\pi</math>, where <math>~t</math> is in units of <math>~[\pi G \rho]^{-1 / 2}</math> and <math>~\varphi</math>  is in units of <math>~[\pi G \rho]^{+1 / 2}</math>.  Note, as well, that in our COLLADA animations, we have adopted the convention that <math>~t = 1</math> when <math>~\mathrm{TIME} = 4</math>.  Hence, one complete orbit will be completed when, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{4} \cdot \mathrm{TIME}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2\pi}{\varphi} \, .</math>
  </td>
</tr>
</table>
That is to say, in the specific example being used here, on complete orbit is concluded when TIME = 19.343.
</td>
</tr></table>

Referring back to our [[#TippedPlane|earlier geometric prescription of a "tipped plane"]] we insert this new slope having m = tan&theta; = -0.3448, and choose a particular plane by setting a value of z<sub>0</sub>.  Then we can map out a locus of points that show the intersection of the chosen plane with the surface of the ellipsoid by specifying various values of z, then calculating the corresponding values of y and x via the respective equations:

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\tan\theta} \biggl[ z - z_0 \biggr] \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{x}{a} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ 1 - \biggl(\frac{z_0}{b \tan\theta}\biggr)^2 \biggl( \frac{z}{z_0} - 1 \biggr)^2 - \biggl( \frac{z}{c} \biggr)^2 \biggr]^{1 / 2} \, .</math>
  </td>
</tr>
</table>

For each specified value of z<sub>0</sub>, the relevant range of z values is given by the pair of values for which x/a = 0.  For example, if we set z_0 = b tan&theta; = -0.4310, this pair is given by the roots of the quadratic expression,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{z}{z_0} - 1 \biggr)^2 + \biggl( \frac{z}{c} \biggr)^2  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z^2\biggl[ \frac{1}{z_0^2} + \frac{1}{c^2} \biggr] -\frac{2z}{z_0} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z \biggl\{ z - \biggl[ \frac{2 z_0 c^2}{(c^2 + z_0^2)} \biggr] \biggr\} \biggl[ \frac{(c^2 + z_0^2)}{c^2 z_0^2} \biggr] \, .</math>
  </td>
</tr>
</table>

The roots are, then, <math>~z_\mathrm{max} = 0</math>, and

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~z_\mathrm{min}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2 z_0 c^2}{(c^2 + z_0^2)} = -0.468514 \, .</math>
  </td>
</tr>
</table>

<span id="ExampleTrajectories">The middle three columns</span> (table cells having bgcolor="lightblue") in the following table list values of |x| and y that correspond to various values of z that lie between this pair of limiting values.
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="11"><math>~m = \tan\theta = -0.3448 ~~~ \Rightarrow ~~~ \theta = -19.02^\circ</math></th>
</tr>
<tr>
  <td align="center" colspan="3">z_0 = -0.2500</td>
  <th align="center" rowspan="18">&nbsp;</th>
  <td align="center" colspan="3">z_0 = b tan&theta; = -0.4310</td>
  <th align="center" rowspan="18">&nbsp;</th>
  <td align="center" colspan="3">z_0 = -0.6000</td>
</tr>
<tr>
  <th align="center">z</th>
  <th align="center">y</th>
  <th align="center">|x|</th>
  <th align="center">z</th>
  <th align="center">y</th>
  <th align="center">|x|</th>
  <th align="center">z</th>
  <th align="center">y</th>
  <th align="center">|x|</th>
</tr>
<tr>
  <td align="center">0.1564</td><td align="center">-1.1788</td><td align="center">0.0000</td>
  <td align="center" bgcolor="lightblue">0</td>  <td align="center" bgcolor="lightblue">-1.2500</td>  <td align="center" bgcolor="lightblue">0</td> 
  <td align="center">-0.2182</td><td align="center">-1.1073</td><td align="center">0.0000</td>
</tr>
<tr>
  <td align="center">0.1147</td><td align="center">-1.0577</td><td align="center">0.4739</td>
  <td align="center" bgcolor="lightblue">-0.02</td>  <td align="center" bgcolor="lightblue">- 1.1920</td>  <td align="center" bgcolor="lightblue">0.2981</td> 
  <td align="center">-0.2336</td><td align="center">-1.0626</td><td align="center">0.1749</td>
</tr>
<tr>
  <td align="center">0.0729</td><td align="center">-0.9366</td><td align="center">0.6439</td>
  <td align="center" bgcolor="lightblue">-0.05</td>  <td align="center" bgcolor="lightblue">-1.1050</td>  <td align="center" bgcolor="lightblue">0.4553</td>
  <td align="center">-0.2490</td><td align="center">-1.0179</td><td align="center">0.2377</td>
</tr>
<tr>
  <td align="center">0.0312</td><td align="center">-0.8155</td><td align="center">0.7550</td>
  <td align="center" bgcolor="lightblue">-0.08</td>  <td align="center" bgcolor="lightblue">-1.0180</td>  <td align="center" bgcolor="lightblue">0.5548</td>
  <td align="center">-0.2645</td><td align="center">-0.9732</td><td align="center">0.2787</td>
</tr>
<tr>
  <td align="center">-0.0106</td><td align="center">-0.6943</td><td align="center">0.8312</td>
  <td align="center" bgcolor="lightblue">-0.1</td>  <td align="center" bgcolor="lightblue">-0.9600</td>  <td align="center" bgcolor="lightblue">0.6042</td>
  <td align="center">-0.2799</td><td align="center">-0.9285</td><td align="center">0.3068</td>
</tr>
<tr>
  <td align="center">-0.0524</td><td align="center">-0.5732</td><td align="center">0.8817</td>
  <td align="center" bgcolor="lightblue">-0.125</td>  <td align="center" bgcolor="lightblue">-0.8875</td>  <td align="center" bgcolor="lightblue">0.6521</td> 
  <td align="center">-0.2953</td><td align="center">-0.8838</td><td align="center">0.3255</td>
</tr>
<tr>
  <td align="center">-0.0941</td><td align="center">-0.4521</td><td align="center">0.9106</td>
  <td align="center" bgcolor="lightblue">-0.15</td>  <td align="center" bgcolor="lightblue">-0.8150</td>  <td align="center" bgcolor="lightblue">0.6879</td> 
  <td align="center">-0.3107</td><td align="center">-0.8390</td><td align="center">0.3361</td>
</tr>
<tr>
  <td align="center">-0.1359</td><td align="center">-0.3310</td><td align="center">0.9200</td>
  <td align="center" bgcolor="lightblue">-0.175</td>  <td align="center" bgcolor="lightblue">-0.7424</td>  <td align="center" bgcolor="lightblue">0.7133</td> 
  <td align="center">-0.3261</td><td align="center">-0.7943</td><td align="center">0.3396</td>
</tr>
<tr>
  <td align="center">-0.1776</td><td align="center">-0.2099</td><td align="center">0.9106</td>
  <td align="center" bgcolor="lightblue">-0.2</td>  <td align="center" bgcolor="lightblue">- 0.6699</td>  <td align="center" bgcolor="lightblue">0.7293</td>
  <td align="center">-0.3415</td><td align="center">-0.7496</td><td align="center">0.3361</td>
</tr>
<tr>
  <td align="center">-0.2194</td><td align="center">-0.0887</td><td align="center">0.8817</td>
  <td align="center" bgcolor="lightblue">-0.25</td>  <td align="center" bgcolor="lightblue">- 0.5249</td>  <td align="center" bgcolor="lightblue">0.7356</td>
  <td align="center">-0.3570</td><td align="center">-0.7049</td><td align="center">0.3255</td>
</tr>
<tr>
  <td align="center">-0.2612</td><td align="center">0.0324</td><td align="center">0.8312</td>
  <td align="center" bgcolor="lightblue">-0.30</td>  <td align="center" bgcolor="lightblue">- 0.3799</td>  <td align="center" bgcolor="lightblue">0.7076</td> 
  <td align="center">-0.3724</td><td align="center">-0.6602</td><td align="center">0.3068</td>
</tr>
<tr>
  <td align="center">-0.3029</td><td align="center">0.1535</td><td align="center">0.7550</td>
  <td align="center" bgcolor="lightblue">-0.35</td>  <td align="center" bgcolor="lightblue">- 0.2349</td>  <td align="center" bgcolor="lightblue">0.6410</td> 
  <td align="center">-0.3878</td><td align="center">-0.6155</td><td align="center">0.2787</td>
</tr>
<tr>
  <td align="center">-0.3447</td><td align="center">0.2746</td><td align="center">0.6439</td>
  <td align="center" bgcolor="lightblue">-0.4</td>  <td align="center" bgcolor="lightblue">- 0.0899</td>  <td align="center" bgcolor="lightblue">0.5210</td>
  <td align="center">-0.4032</td><td align="center">-0.5708</td><td align="center">0.2377</td>
</tr>
<tr>
  <td align="center">-0.3865</td><td align="center">0.3957</td><td align="center">0.4739</td>
  <td align="center" bgcolor="lightblue">-0.43</td>  <td align="center" bgcolor="lightblue">- 0.0029</td>  <td align="center" bgcolor="lightblue">0.4050</td> 
  <td align="center">-0.4186</td><td align="center">-0.5261</td><td align="center">0.1749</td>
</tr>
<tr>
  <td align="center">-0.4282</td><td align="center">0.5169</td><td align="center">0.0</td>
  <td align="center" bgcolor="lightblue">-0.46</td>  <td align="center" bgcolor="lightblue">+0.0841</td>  <td align="center" bgcolor="lightblue">0.1950</td> 
  <td align="center">-0.4340</td><td align="center">-0.4814</td><td align="center">0.0000</td>
</tr>
<tr>
  <td align="center" colspan="3">&nbsp;</td>
  <td align="center" bgcolor="lightblue">-0.4685</td>  <td align="center" bgcolor="lightblue">+0.1088</td>  <td align="center" bgcolor="lightblue">0.0081</td> 
  <td align="center" colspan="3">&nbsp;</td>
</tr>
</table>


More generally, letting <math>~q \equiv z_0/(b\tan\theta)</math>, the pair of limiting values of z are given by the roots of the quadratic expression,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~q^2 \biggl( \frac{z}{z_0} - 1 \biggr)^2 + \biggl( \frac{z}{c} \biggr)^2  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ z^2 \biggl[ \frac{ q^2c^2 + z_0^2)}{z_0^2 q^2 c^2} \biggr]  - \frac{2z}{z_0}    + \biggl[ 1  - \frac{1}{q^2} \biggr] \, .</math>
  </td>
</tr>
</table>

That is,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~z </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ \frac{z_0^2 q^2 c^2}{ 2(q^2 c^2 + z_0^2)} \biggr]\biggl\{ \frac{2}{z_0}  \pm   \biggl[\frac{4}{z_0^2} -  \frac{ 4 (1  - q^{-2})(q^2 c^2 + z_0^2)}{z_0^2 q^2 c^2} \biggr]^{1 / 2} \biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ \frac{z_0 q^2 c^2}{ (q^2 c^2 + z_0^2)} \biggr]\biggl\{ 1  \pm   \biggl[1 -  \frac{ (1  - q^{-2})(q^2 c^2 + z_0^2)}{ q^2 c^2} \biggr]^{1 / 2} \biggr\} \, .</math>
  </td>
</tr>
</table>
And we see that the constraint set on z<sub>0</sub> is given by the condition,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1</math>
  </td>
  <td align="center">
<math>~\ge</math>
  </td>
  <td align="left">
<math>~\frac{ (1  - q^{-2})(q^2c^2 + z_0^2)}{q^2c^2} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ q^2 c^2</math>
  </td>
  <td align="center">
<math>~\ge</math>
  </td>
  <td align="left">
<math>~(1  - q^{-2})(q^2c^2 + z_0^2) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\ge</math>
  </td>
  <td align="left">
<math>~(q^2 - 1)(c^2 + b^2\tan^2\theta) </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 0</math>
  </td>
  <td align="center">
<math>~\ge</math>
  </td>
  <td align="left">
<math>~ (q^2 - 1)b^2\tan^2\theta - c^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\ge</math>
  </td>
  <td align="left">
<math>~ z_0^2 - c^2 - b^2\tan^2\theta</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ z_0^2</math>
  </td>
  <td align="center">
<math>~\le</math>
  </td>
  <td align="left">
<math>~ c^2 + b^2\tan^2\theta \, .</math>
  </td>
</tr>
</table>

Therefore, for this example Type I ellipsoidal configuration, z<sub>0</sub> must lie within the range,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~-0.650165</math>
  </td>
  <td align="center">
<math>~\le z_0 \le</math>
  </td>
  <td align="left">
<math>~+0.650165 \, .</math>
  </td>
</tr>
</table>
In the [[#ExampleTrajectories|above table]], we have detailed the loci of points along the surface trajectories that correspond to the values, z<sub>0</sub> = - 0.2500 (the leftmost three columns of the table) and z<sub>0</sub> = -0.6000 (the rightmost three columns of the table).  The limiting values of z for these to choices of z<sub>0</sub> are, respectively, [z<sub>min</sub> = -0.4283, z<sub>max</sub> = +0.1564] and  [z<sub>min</sub> = -0.4340, z<sub>max</sub> = -0.2182].

<span id="Figure3">Using COLLADA,</span> we have constructed an animated and interactive 3D scene that displays in purple the surface of our example Type I ellipsoid; panels a and b of Figure 3 show what this ellipsoid looks like when viewed from two different perspectives.  (As a reminder &#8212; see the [[#explanation| explanation accompanying Figure 2, above]] &#8212; the ellipsoid is tilted about the x-coordinate axis at an angle of 61.25&deg; to the equilibrium spin axis, which is shown in green.)  Yellow markers also have been placed in this 3D scene at each of the coordinate locations specified in the  [[#ExampleTrajectories|above table]].  From the perspective presented in Figure 3b, we can immediately identify three separate, nearly circular trajectories; the largest one corresponds to our choice of z<sub>0</sub> = -0.25, the smallest corresponds to our choice of z<sub>0</sub> = -0.60, and the one of intermediate size correspond to our choice of z<sub>0</sub> = -0.4310.  When viewed from the perspective presented in Figure 3a, we see that these three trajectories define three separate planes; each plane is tipped at an angle of &theta; = -19.02&deg; to the ''untilted'' equatorial, x-y plane of the purple ellipsoid.

<div align="center">
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center">Figure 3a</th>
  <th align="center">Figure 3b</th>
</tr>
<tr>
  <td align="left" bgcolor="lightgrey">
[[File:B125c470B.cropped.png|400px|EFE Model b1.25c470B]]
  </td>
  <td align="left" bgcolor="lightgrey">
[[File:B125c470A.cropped.png|400px|EFE Model b1.25c470A]]
  </td>
</tr>
<tr>
  <th align="center" colspan="2">Figure 3c</th>
</tr>
<tr>
  <td align="center" bgcolor="white" colspan="2">
[[File:ProjectedOrbitsFlipped2.png|600px|EFE Model b1.25c470]]
  </td>
</tr>
</table>
</div>

As we have created each orbit in the xml-based COLLADA file, we have first manually typed in the (TIME, x, y, z) coordinates of each orbit that has a ''negative'' value of z<sub>0</sub>.  Specifically, using the Figure 3c projection as a guide/reference, we typed in coordinates for the following orbits:
<ul>
  <li><b>FLUID ELEMENT 1:</b>&nbsp; <math>~z_0 = -0.6000</math></li>
</ul>

<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center">nstep</td>
  <td align="center">TIME</td>
  <td align="center">x</td>
  <td align="center">y</td>
  <td align="center">z</td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="center">0.000</td>
  <td align="center">0.000</td>
  <td align="center">-1.107</td>
  <td align="center">-0.218</td>
</tr>
<tr>
  <td align="center">26</td>
  <td align="center">4.836</td>
  <td align="center">-0.340</td>
  <td align="center">-0.794</td>
  <td align="center">-0.326</td>
</tr>
<tr>
  <td align="center">51</td>
  <td align="center">9.672</td>
  <td align="center">0.000</td>
  <td align="center">-0.481</td>
  <td align="center">-0.434</td>
</tr>
<tr>
  <td align="center">76</td>
  <td align="center">14.507</td>
  <td align="center">+0.340</td>
  <td align="center">-0.794</td>
  <td align="center">-0.326</td>
</tr>
<tr>
  <td align="center">101</td>
  <td align="center">19.343</td>
  <td align="center">0.000</td>
  <td align="center">-1.107</td>
  <td align="center">-0.218</td>
</tr>
</table>

<ul>
  <li><b>FLUID ELEMENT 2:</b>&nbsp; <math>~z_0 = -0.4300</math></li>
</ul>