Editing ThreeDimensionalConfigurations/ChallengesPt5 (section)

==Tilted Plane Intersects Ellipsoid==

In a [[ThreeDimensionalConfigurations/RiemannTypeI#TippedPlane|an early subsection of the accompanying discussion]], we have pointed out that the intersection of each Lagrangian fluid element's tipped orbital plane with the surface of the (purple) ellipsoidal surface is given by the (unprimed) body-frame coordinates that simultaneously satisfy the expressions,
<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{x}{a}\biggr)^2 + \biggl( \frac{y}{b}\biggr)^2 + \biggl( \frac{z}{c}\biggr)^2 </math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y \tan\theta + z_0 \, ,</math>
  </td>
</tr>
</table>
where z<sub>0</sub> is the location where the tipped plane intersects the z-axis of the body frame.  Combining these two expressions, we see that an intersection between the tipped plane and the ellipsoidal surface will occur at (x, y)-coordinate pairs that satisfy what we will henceforth refer to as the,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3"><font color="maroon">'''Intersection Expression'''</font></td>
</tr>

<tr>
  <td align="right">
<math>~1 - \frac{x^2}{a^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y^2 \biggl[\frac{c^2 + b^2\tan^2\theta}{b^2c^2} \biggr] + y \biggl[ \frac{2z_0 \tan\theta}{c^2} \biggr] + \frac{z_0^2}{c^2} \, , </math>
  </td>
</tr>
</table>
as long as 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 \, .</math>
  </td>
</tr>
</table>
<span id="Fig1">Before calling upon any of Riemann's model parameters</span>, from geometric considerations alone we should be able to determine exactly what the expression is for any off-center ellipse that results from slicing &#8212; at a tipped angle &#8212; the chosen ellipsoid.
<table border="1" width="50%" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="3"><b>Figure 1</b></td>
</tr>

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

<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x' \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
y' \cos\theta - z'\sin\theta 
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~(z - z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
z' \cos\theta + y'\sin\theta 
\, .</math>
  </td>
</tr>
</table>
</td>

<td align="center">[[File:ExcelAxes02C.png|400px|Primed Coordinates]]</td>
<td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
y \cos\theta + (z - z_0) \sin\theta 
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(z-z_0) \cos\theta - y \sin\theta 
\, .</math>
  </td>
</tr>
</table>
</td>
</tr>
</table>

In the equatorial plane of the ''tipped'' coordinate system &#8212; that is, after mapping <math>~x \rightarrow x'</math> and <math>~y \rightarrow (y' \cos\theta - z'\sin\theta)</math>, then setting <math>~z' = 0</math> &#8212; this intersection expression becomes,
<table border="0" cellpadding="5" align="center">

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

The light-blue curve in the right-hand panel of the following animation is a plot of this <math>~x'(y')</math>function for various values of <math>~z_0</math> (as indicated by the light-blue numerical value in the upper-right corner of the figure's left-hand panel.

[[File:RiemannType1Tipped01.gif|center|600px|Animation of Intersection Curves]]

As it turns out &#8212; see our [[ThreeDimensionalConfigurations/ChallengesPt4#Tipped_Plane|accompanying discussion]] &#8212; this expression can be rewritten as,
<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{x'}{x_\mathrm{surf}} \biggr]^2 + \biggl[ \frac{(y' - y'_\mathrm{center} ) }{y'_\mathrm{surf}} \biggr]^2 \, ,
</math>
  </td>
</tr>
</table>
demonstrating that, as viewed from the x'-y' plane, the (light-blue) intersection curve is always an off-center ellipse.  See also our [[ThreeDimensionalConfigurations/ChallengesPt2#COLLADA-Based_Representation|COLLADA-based representation]] of these curves.