Editing 3Dconfigurations/DescriptionOfRiemannTypeI (section)

====Off-Center Ellipse====

The yellow dots in Figures 2a and 2b trace three different, nearly circular, closed curves.  These curves each show what results from the intersection of the surface of the triaxial ellipsoid and a plane that is tilted with respect to the x = x' axis by the specially chosen angle, <math>\theta</math>; the different curves result from different choices of the intersection point, <math>z_0</math>.  Several additional such curves are displayed in Figure 2c.  Each of these curves necessarily also identifies the trajectory that is followed by a fluid element that sits on the surface of the ellipsoid.  

We have determined that the <math>y'(x')</math> function that defines each closed curve is describable analytically by the expression,

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

<tr>
  <td align="right">
<math>
\biggl[ \frac{y' - y'_0}{y'_\mathrm{max}} \biggr]^2
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 - \biggl( \frac{x'}{x'_\mathrm{max}} \biggr)^2 \, ,
</math>
  </td>
</tr>
</table>
where (see independent derivations with identical results from [[ThreeDimensionalConfigurations/ChallengesPt2#OffCenter|ChallengesPt2]] and [[ThreeDimensionalConfigurations/ChallengesPt6#Are_Orbits_Exact_Circles|ChallengesPt6]]),
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math> x'_\mathrm{max}  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
a\biggl[ 1 -\frac{z_0^2 \cos^2\theta}{(c^2  \cos^2\theta + b^2 \sin^2\theta )} \biggr]^{1 / 2}  
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>y'_\mathrm{max} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
bc \biggl[ (c^2  \cos^2\theta   + b^2 \sin^2\theta )  - z_0^2 \cos^2\theta   \biggr]^{1 / 2} ( c^2 \cos^2\theta + b^2 \sin^2\theta )^{-1}
 </math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>y'_0 </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-  \frac{z_0 b^2\sin\theta }{ ( c^2 \cos^2\theta + b^2 \sin^2\theta )} \, .
</math>
  </td>
</tr>

</table>

This is the equation that describes a closed ellipse with semi-axes, <math>(x'_\mathrm{max}, y'_\mathrm{max})</math>, that is offset from the z'-axis along the y'-axis by a distance, <math>y'_0</math>.  Notice that the degree of flattening,

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

<tr>
  <td align="right">
<math>\biggl[ \frac{x'}{y'} \biggr]_\mathrm{max} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{ a( c^2 \cos^2\theta + b^2 \sin^2\theta )^{1 / 2}}{bc} \, ,
</math>
  </td>
</tr>
</table>
is independent of <math>z_0</math>; that is to say, the degree of flattening of all of the elliptical trajectories is identical!  Notice, as well, that the y-offset, <math>y'_0</math>, is linearly proportional to <math>z_0</math>.

In a [[ThreeDimensionalConfigurations/ChallengesPt6#Plot_Off-Center,_Slightly_Flattened_Ellipse|separate discussion]], we have demonstrated that the [[#CompactFlowField|compact version of the ''tilted'' flow-field]] is everywhere orthogonal to the elliptical trajectory whose analytic definition is given by the off-set ellipse equation.