Editing ThreeDimensionalConfigurations/ChallengesPt4 (section)

==Reconcile ''Results'' Differences==
<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>

===Body Frame===
As has been shown in [[ThreeDimensionalConfigurations/ChallengesPt2#Intersection_of_Tipped_Plane_With_Ellipsoid_Surface|our accompanying discussion]], we obtain the following,
<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>

<table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left">
Given the values of <math>~a, b, c, \theta, z_0</math>, we can immediately map this to the tipped plane to obtain the surface-intersection function, <math>~x'(y')</math>.  Specifically we find,
<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 - \cancelto{0}{z'} \sin\theta)^2 \biggl( \frac{c^2 + b^2\tan^2\theta}{b^2c^2} \biggr) + (y'\cos\theta - \cancelto{0}{z'} \sin\theta) 
\biggl( \frac{2z_0 \tan\theta}{c^2} \biggr) + \frac{z_0^2}{c^2} \biggr] \biggr\}^{1 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </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>

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

Along the ''tipped'' <math>~y'</math> axis, two points will mark the ends of the x'-y' orbit; they are identified by the roots of this ''Intersection Expression'' when x = 0.  That is, by the roots of,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y^2 \biggl[\frac{\kappa^2}{\cos^2\theta} \biggr] + y \biggl[ \frac{2z_0 \tan\theta}{c^2} \biggr] + \biggl[\frac{z_0^2}{c^2} -1 \biggr] \, ,</math>
  </td>
</tr>
</table>

where,
<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} \, .
</math>
  </td>
</tr>
</table>
The roots are &hellip;
<table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left">
Scratch notes:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y_\mathrm{edge}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{B}{2A} \biggl\{\pm \biggl[1 - \frac{4AC}{B^2}  \biggr]^{1 / 2} -1 \biggr\}</math>
  </td>
</tr>
</table>
where,
<div align="center">
<math>~
\frac{4AC}{B^2} = \frac{4\kappa^2}{\cos^2\theta} \biggl[\frac{z_0^2}{c^2} -1 \biggr] \biggl[ \frac{2z_0 \tan\theta}{c^2} \biggr]^{-2}
=
\frac{\kappa^2}{\cos^2\theta}  \biggl[ \frac{c^2(z_0^2 -  c^2)}{z_0^2 \tan^2\theta} \biggr]
=
\biggl[ \frac{c^2 \kappa^2 (z_0^2 -  c^2)}{z_0^2 \sin^2\theta} \biggr]
</math>
</div>
</td></tr></table>


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

<tr>
  <td align="right">
<math>~y_\mathrm{edge}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{z_0 \sin\theta \cos\theta}{c^2 \kappa^2} \biggr]\biggl\{\pm \biggl[1 - \frac{c^2 \kappa^2 (z_0^2 -  c^2)}{z_0^2 \sin^2\theta} \biggr]^{1 / 2} -1 \biggr\}</math>
  </td>
</tr>

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

Hereafter we will use <math>~y_p</math> to denote the "plus" root, and <math>~y_m</math> to denote the "minus" root; that is,

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

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

<tr>
  <td align="right">
<math>~y_m</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{\cos\theta}{c^2 \kappa^2} \biggr]\biggl\{\biggl[z_0^2 \sin^2\theta  + c^2 \kappa^2 (c^2 - z_0^2 ) \biggr]^{1 / 2} + z_0 \sin\theta  \biggr\} \, .</math>
  </td>
</tr>
</table>

<table border="1" align="center" cellpadding="8" width="80%">
<tr>
  <td align="left">

<ul>
  <li>If tan&theta; is positive &hellip;
    <ul>
<li>Quadrant 1 &#x21D2; both sin&theta; and cos&theta; are positive.</li>
<li>Quadrant 3 &rArr; both sin&theta; and cos&theta; are negative.</li>
    </ul>
  </li>
</ul>
<ul>
  <li>If tan&theta; is negative &hellip;
    <ul>
<li>Quadrant 2 &#x21D2; sin&theta; is positive but cos&theta; is negative.</li>
<li>Quadrant 4 &rArr; sin&theta; is negative while cos&theta; are positive.</li>
    </ul>
  </li>
</ul>

Taking numerical values from our [[#Riemann_Flow|chosen Example Model]] &#8212; and using z<sub>0</sub> = &plusmn; 0.25 as our 1<sup>st</sup> test cases &#8212; we have:
<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{\beta\Omega_2}{\gamma \Omega_3} = -0.344793 ~~~\Rightarrow~~~ \theta = -19.0238^\circ \, .
</math>
  </td>
</tr>

<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>

<tr>
  <td align="right">
<math>~\mathrm{TERM1}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~z_0^2 \sin^2\theta + c^2\kappa^2(c^2-z_0^2) 
=0.043577
</math>&nbsp; &nbsp; (for z<sub>0</sub> = &plusmn; 0.25).
  </td>
</tr>
</table>
z<sub>0</sub> = +0.25 and &theta; in QUADRANT #4 (sin&theta; is negative while cos&theta; is positive):

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

<tr>
  <td align="right">
<math>~y_p</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{\cos\theta}{c^2 \kappa^2} \biggr]\biggl\{\biggl[ \mathrm{TERM1} \biggr]^{1 / 2} - z_0 \sin\theta  \biggr\} 
=
1.1788
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y_m</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{\cos\theta}{c^2 \kappa^2} \biggr]\biggl\{\biggl[ \mathrm{TERM1} \biggr]^{1 / 2} + z_0 \sin\theta  \biggr\} 
=
- 0.5169
\, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z_p</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~-c \biggl[ 1 - \biggl( \frac{y_p}{b} \biggr)^2 \biggr]^{1 / 2} 
=
-0.15645
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z_m</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~c \biggl[ 1 - \biggl( \frac{y_m}{b} \biggr)^2 \biggr]^{1 / 2} 
=
0.42821
\, ,</math>
  </td>
</tr>
</table>

  </td>
  <td align="center">
<table border="1" align="center" cellpadding="5">
  <tr>
<td align="center" colspan="5"><math>~|z_0|_\mathrm{max} = 0.63792</math></td>
  </tr>
  <tr>
<td align="center"><math>~z_0</math></td>
<td align="center"><math>~y_p</math></td>
<td align="center"><math>~z_p</math></td>
<td align="center"><math>~y_m</math></td>
<td align="center"><math>~z_m</math></td>
  </tr>
  <tr>
<td align="right"><math>~+ 0.6379</math></td>
<td align="right"><math>~+ 0.8511</math></td>
<td align="right"><math>~+ 0.3444</math></td>
<td align="right"><math>~+ 0.8379</math></td>
<td align="right"><math>~+ 0.3490</math></td>
  </tr>
  <tr>
<td align="right"><math>~+ 0.60</math></td>
<td align="right"><math>~+ 1.1073</math></td>
<td align="right"><math>~+ 0.2182</math></td>
<td align="right"><math>~+ 0.4814</math></td>
<td align="right"><math>~+ 0.4340</math></td>
  </tr>
  <tr>
<td align="right"><math>~+ 0.55</math></td>
<td align="right"><math>~+ 1.1950</math></td>
<td align="right"><math>~+ 0.1380</math></td>
<td align="right"><math>~+ 0.2613</math></td>
<td align="right"><math>~+ 0.4599</math></td>
  </tr>
  <tr>
<td align="right"><math>~+ 0.50</math></td>
<td align="right"><math>~+ 1.2342</math></td>
<td align="right"><math>~+ 0.0744</math></td>
<td align="right"><math>~+ 0.0897</math></td>
<td align="right"><math>~+ 0.4691</math></td>
  </tr>
  <tr>
<td align="right"><math>~+ 0.45</math></td>
<td align="right"><math>~+ 1.2489</math></td>
<td align="right"><math>~- 0.0194</math></td>
<td align="right"><math>~- 0.0574</math></td>
<td align="right"><math>~+ 0.4698</math></td>
  </tr>
  <tr>
<td align="right"><math>~+ 0.40</math></td>
<td align="right"><math>~+ 1.2474</math></td>
<td align="right"><math>~- 0.0301</math></td>
<td align="right"><math>~- 0.1883</math></td>
<td align="right"><math>~+ 0.4649</math></td>
  </tr>
  <tr>
<td align="right"><math>~+ 0.35</math></td>
<td align="right"><math>~+ 1.2338</math></td>
<td align="right"><math>~- 0.0754</math></td>
<td align="right"><math>~- 0.3071</math></td>
<td align="right"><math>~+ 0.4559</math></td>
  </tr>
  <tr>
<td align="right"><math>~+ 0.30</math></td>
<td align="right"><math>~+ 1.2105</math></td>
<td align="right"><math>~- 0.1174</math></td>
<td align="right"><math>~- 0.4161</math></td>
<td align="right"><math>~+ 0.4435</math></td>
  </tr>
  <tr>
<td align="right" bgcolor="orange">+ 0.25</td>
<td align="right" bgcolor="orange">+ 1.1788</td>
<td align="right" bgcolor="orange">- 0.1564</td>
<td align="right" bgcolor="orange">- 0.5169</td>
<td align="right" bgcolor="orange">+ 0.4282</td>
  </tr>
  <tr>
<td align="right"><math>~+ 0.20</math></td>
<td align="right"><math>~+ 1.1399</math></td>
<td align="right"><math>~- 0.1930</math></td>
<td align="right"><math>~- 0.6103</math></td>
<td align="right"><math>~+ 0.4104</math></td>
  </tr>
  <tr>
<td align="right"><math>~+ 0.15</math></td>
<td align="right"><math>~+ 1.0943</math></td>
<td align="right"><math>~- 0.2273</math></td>
<td align="right"><math>~- 0.6971</math></td>
<td align="right"><math>~+ 0.3904</math></td>
  </tr>
  <tr>
<td align="right"><math>~+ 0.10</math></td>
<td align="right"><math>~+ 1.0426</math></td>
<td align="right"><math>~- 0.2595</math></td>
<td align="right"><math>~- 0.7778</math></td>
<td align="right"><math>~+ 0.3682</math></td>
  </tr>
  <tr>
<td align="right"><math>~+ 0.05</math></td>
<td align="right"><math>~+ 0.9849</math></td>
<td align="right"><math>~- 0.2896</math></td>
<td align="right"><math>~- 0.8525</math></td>
<td align="right"><math>~+ 0.3439</math></td>
  </tr>
  <tr>
<td align="right"><math>~+ 0.00</math></td>
<td align="right"><math>~+ 0.9216</math></td>
<td align="right"><math>~- 0.3177</math></td>
<td align="right"><math>~- 0.9216</math></td>
<td align="right"><math>~+ 0.3177</math></td>
  </tr>
  <tr>
<td align="right"><math>~- 0.05</math></td>
<td align="right"><math>~+ 0.8525</math></td>
<td align="right"><math>~- 0.3439</math></td>
<td align="right"><math>~- 0.9849</math></td>
<td align="right"><math>~+ 0.2895</math></td>
  </tr>
  <tr>
<td align="right"><math>~- 0.10</math></td>
<td align="right"><math>~+ 0.7778</math></td>
<td align="right"><math>~- 0.3682</math></td>
<td align="right"><math>~- 1.0426</math></td>
<td align="right"><math>~+ 0.2595</math></td>
  </tr>
  <tr>
<td align="center" colspan="5">''etc.''
  </tr>
  <tr>
<td align="right"><math>~- 0.6379</math></td>
<td align="right"><math>~- 0.8379</math></td>
<td align="right"><math>~- 0.3490</math></td>
<td align="right"><math>~- 0.8511</math></td>
<td align="right"><math>~- 0.3444</math></td>
  </tr>
</table>
  </td>
</tr>
</table>


The y-coordinate of the center of the orbit will lie halfway between these two "edges".  That is (numerical example given for the case of z<sub>0</sub> = + 0.25),
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y_\mathrm{center}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\biggl[ y_p + y_m \biggr] = +0.3310 \, .</math>
  </td>
</tr>
</table>
Correspondingly, the vertical (z) location of the orbit center in the ''body'' frame is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~z_\mathrm{center}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y_\mathrm{center}\tan\theta + z_0 = +0.1359 \, .</math>
  </td>
</tr>
</table>

The line that runs parallel to the x-axis &#8212; lies perpendicular to the y-z plane &#8212; and that passes through this "center" location intersects the surface of the ellipsoid at a value of x that is obtained by plugging <math>~y_\mathrm{center}</math> into the ''Intersection Expression''.  That is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{x_\mathrm{surf}}{a}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{
1 - y^2_\mathrm{center} \biggl[\frac{\kappa^2}{\cos^2\theta} \biggr] 
- y_\mathrm{center} \biggl[ \frac{2z_0 \tan\theta}{c^2} \biggr] 
- \frac{z_0^2}{c^2}
\biggr\}^{1 / 2} = 0.92001 \, ;</math>
  </td>
</tr>
</table>
along the relevant orbit, this is also the maximum value of x.

===Tipped Plane===

We use the expressions in the right panel of [[#Fig1|Figure 1, above]] to transform (x, y, z) ''body'' coordinates to (x', y', z') ''tipped-frame'' coordiantes.  Given that <math>~x' = x</math>, we appreciate that the x'-coordinate of the orbit center is zero, and the maximum value <math>~(x'_\mathrm{surf})</math> that is encountered along the relevant orbit is the same as the value of <math>~x_\mathrm{surf}</math> as determined in the ''body'' (unprimed) frame.

The z'-coordinate of the orbit center is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~z'_\mathrm{center}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(z_\mathrm{center} - z_0)\cos\theta - y_\mathrm{center}\sin\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(y_\mathrm{center}\tan\theta) \cos\theta - y_\mathrm{center}\sin\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0 \, .
</math>
  </td>
</tr>
</table>

Finally, in the ''tipped'' plane, the maximum (subscript_p) and minimum (subscript_m) values of y' are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y'_p</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(z_p - z_0)\sin\theta + y_p\cos\theta = 1.2469 \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y'_m</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(z_m - z_0)\sin\theta + y_m\cos\theta = -0.5467 \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ y'_\mathrm{center}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl[ y_p' + y_m'\biggr] = 0.3501\, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ y'_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
y'_p - y'_\mathrm{center} = 0.8968\, .
</math>
  </td>
</tr>
</table>
So, in the ''tipped'', x'-y' plane, the elliptical orbit that corresponds to z<sub>0</sub> = + 0.25 should be described by the 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{x'}{x_\mathrm{surf}} \biggr]^2 + \biggl[ \frac{(y' - y'_\mathrm{center} ) }{y'_\mathrm{surf}} \biggr]^2 \, .
</math>
  </td>
</tr>
</table>


NOTE:
*  <math>~y'_\mathrm{center}/z_0 = 0.350096/0.25 = 1.40038 \, .</math>
* <math>~x_\mathrm{surf}/y'_\mathrm{surf} = 0.92001/0.896819 = 1.0259 \, .</math>

At a minimum these numerical values should be compared to the Riemann model parameters [[ThreeDimensionalConfigurations/ChallengesPt2#Example_Equilibrium_Model|computed in our earlier (Pt. 2) examination]].