Editing ThreeDimensionalConfigurations/ChallengesPt6 (section)

==Graphically Compare Velocities==

Using Excel, choose a particular elliptical orbit trajectory in the ''tipped'' plane, and plot the ratio of the Cartesian components of the fluid velocity as specified: (a) by EFE; and (b) by the tangent to the elliptical trajectory.

Adopting the axis-ratios, <math>b/a = 1.25</math> and <math>c/a = 0.4703</math>, we can refer to the table labeled "Example Type I Ellipsoid" immediately above (under <font color="red">STEP #5</font>) to obtain and/or check values of <math>\Omega_2, \Omega_3, \zeta_2, \zeta_3, \beta^+, \gamma^+</math>.  From these tabulated values, we determine that the ''tip'' angle, <math>\tan\theta = -0.344793 ~\Rightarrow~ \theta = -0.33203</math>.

We then know that z<sub>0</sub> lies within the range,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~z_0^2</math>
  </td>
  <td align="center">
<math>~\le</math>
  </td>
  <td align="left">
<math>~c^2 + b^2\tan^2\theta = 0.40694</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
<math>\Rightarrow ~~~~- 0.63792 \le z_0 \le + 0.63792 \, .</math>
  </td>
</tr>
</table>
As our hand-determined example, let's choose, <math>z_0 = -0.4310</math>, which corresponds to the solid green ellipse that is displayed in [[ThreeDimensionalConfigurations/RiemannTypeI#Figure3|Figure 3c of an accompanying discussion]] of this problem.  <font color="red">NOTE: &nbsp; Just above the referenced "Figure 3," we have stated that the limits are, z<sub>0</sub> = &pm; 0.650165; not sure why that is!</font>

Next, we find that,  
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\kappa^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{c^2 \cos^2\theta + b^2 \sin^2\theta}{b^2c^2} = 1.05238 \, .
</math>
  </td>
</tr>
</table>

<span id="FixedMistake">So we can evaluate a number of terms that define the relevant elliptical orbit.</span>
<table border="1" cellpadding="8" align="center" width="60%">
<tr><td align="left">
<font color="red">CORRECTION:</font>

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

<tr>
  <td align="right"><math>y_\mathrm{max}^2</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{1}{\kappa^2} - \frac{z_0^2}{c^2 \kappa^2} + y_c^2</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{1}{\kappa^2}\biggl[1 - \frac{z_0^2}{c^2} + \frac{z_0^2 \sin^2\theta}{c^4 \kappa^2}\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{1}{c^4\kappa^4}\biggl[c^4 \kappa^2 - z_0^2 c^2\kappa^2 + z_0^2 \sin^2\theta \biggr] = 0.51646</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ y_\mathrm{max}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>0.71865 \, .</math>
  </td>
</tr>
</table>
</td></tr>

<tr><td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{y_c}{z_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{\sin\theta}{c^2 \kappa^2} = +1.40038~ \Rightarrow ~ y_c = -0.60356\, ,</math>
  </td>
</tr>
</table>
Note as well that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a^2\kappa^2 = \frac{a^2}{b^2 c^2} \biggl[ c^2 \cos^2\theta + b^2 \sin^2\theta \biggr] = 1.05238 
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\frac{x_\mathrm{max}}{y_\mathrm{max}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>1.02585
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~x_\mathrm{max}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>0.73723
\, .</math>
  </td>
</tr>
</table>

</td></tr></table>

Select points along the elliptical orbit by first specifying various values of <math>y'</math> over the range,
<div align="center">
<math> 
(y_c - y_\mathrm{max}) \le y'\le (y_c + y_\mathrm{max})
~\Rightarrow~ -1.32221 \le y'\le 0.11509
</math>
</div>
Then, for each specified value of <math>y'</math>, calculate the corresponding value(s) of <math>x'</math> via the relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[\frac{x'}{x_\mathrm{max}} \biggr]^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>1 - \biggl[\frac{y' - y_c}{y_\mathrm{max}} \biggr]^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x'</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> \pm x_\mathrm{max} \biggl\{1 - \biggl[\frac{y' - y_c}{y_\mathrm{max}} \biggr]^2 \biggr\}^{1 / 2} \, . </math>
  </td>
</tr>
</table>