Editing ThreeDimensionalConfigurations/ChallengesPt2 (section)

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

Now we attempt to transform this last expression into the form of the above-defined equation for an ''<font color="maroon">Off-Center Ellipse</font>'', which we rewrite here as,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1 - \frac{(x')^2}{x^2_\mathrm{max}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{y^2_\mathrm{max}}\biggl[ (y')^2 - 2(y')y_0 + y_0^2 \biggr] \, .</math>
  </td>
</tr>
</table>

An initial rearrangement of the relevant "last" expression gives,

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<tr>
  <td align="right">
<math>~1 - \frac{z_0^2}{c^2} - \frac{(x')^2}{a^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{c^2 \cos^2\theta + b^2\sin^2\theta}{b^2c^2}   
\biggl[(y')^2 - 2(y') \underbrace{ \biggl( - \frac{z_0 b^2 \sin\theta}{c^2 \cos^2\theta + b^2\sin^2\theta}\biggr) }_{y_0} \biggr] 
\, ,</math>
  </td>
</tr>
</table>
<span id="Result1">which, as indicated,</span> allows us to identify the appropriate expression for the y-offset, <math>~y_0</math>.  
<table border="1" align="center" cellpadding="10" width="60%" bordercolor="orange">
<tr><td align="center" bgcolor="lightblue">'''RESULT 1'''<br />(compare with [[ThreeDimensionalConfigurations/ChallengesPt4#Result2|Result 2]])</td></tr>
<tr><td align="left">
<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>~
-\frac{b^2\sin\theta}{c^2\cos^2\theta + b^2\sin^2\theta}
=
-\frac{\sin\theta}{c^2\kappa^2}
</math>
  </td>
</tr>
</table>

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

Dividing through by the leading coefficient,
<div align="center">
<math>~\kappa^2 \equiv \frac{c^2 \cos^2\theta + b^2\sin^2\theta}{b^2c^2} \, ,</math>
</div>
then adding <math>~y_0^2</math> to both sides gives,

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

<tr>
  <td align="right">
<math>~  
(y')^2 - 2(y') y_0 + y_0^2
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\kappa^2}\biggl[ 1 - \frac{z_0^2}{c^2} - \frac{(x')^2}{a^2} \biggr] + y_0^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\underbrace{\biggl[  \frac{1}{\kappa^2} - \frac{z_0^2}{c^2\kappa^2} + y_0^2 \biggr]}_{y^2_\mathrm{max}} - \frac{(x')^2}{\kappa^2 a^2} \, ,
</math>
  </td>
</tr>
</table>
which gives us the appropriate expression for <math>~y_\mathrm{max}^2</math>.  Finally, dividing through by <math>~y_\mathrm{max}^2</math> gives,

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

<tr>
  <td align="right">
<math>~  
\frac{1}{y_\mathrm{max}^2} \biggl[ (y')^2 - 2(y') y_0 + y_0^2 \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 - (x')^2 \underbrace{ \biggl[ \frac{1}{y_\mathrm{max}^2 \kappa^2 a^2} \biggr]}_{1/x^2_\mathrm{max}} \, ,
</math>
  </td>
</tr>
</table>
<span id="OffCenter">which identifies the appropriate</span> expression for <math>~x^2_\mathrm{max}</math>.  As viewed from the "tipped plane" (primed) coordinate frame, then, the equation for the orbit of each Lagrangian fluid element is that of an &hellip;

<table border="1" align="center" cellpadding="8" width="90%"><tr><td align="left">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="center" colspan="3"><font color="maroon">'''Off-Center Ellipse'''</font></td>
</tr>
<tr>
  <td align="right">
<math>~1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{x'}{x_\mathrm{max}} \biggr)^2 + \biggl(\frac{y' - y_0}{y_\mathrm{max}} \biggr)^2 \, ,</math>
  </td>
</tr>
</table>
with,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y_0</math>
  </td>
  <td align="center">
<math>~\equiv</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>

<tr>
  <td align="right">
<math>~y^2_\mathrm{max}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\kappa^2}\biggl( 1 - \frac{z_0^2}{c^2}\biggr) + y_0^2
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
b^2 c^2 \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\}\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~x_\mathrm{max}^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~a^2 \kappa^2 y_\mathrm{max}^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a^2\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\}
\, .
</math>
  </td>
</tr>
</table>
Note that the ratio,

<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\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]
\frac{1}{b^2 c^2} \biggl[ \frac{(c^2 \cos^2\theta + b^2\sin^2\theta)^2}{( c^2 -z_0^2)\cos^2\theta + b^2\sin^2\theta   } \biggr]
</math>
  </td>
</tr>

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