Editing ThreeDimensionalConfigurations/ChallengesPt6 (section)

===Green (x/a = 0.7) Ellipse===

Next, let's examine the surface-intersection-ellipse that results from a y-z plane that slices through the ellipsoid at <math>x/a = 0.7</math>.  By setting <math>z = 0</math>, we find the point where the y-axis intersects the surface of the ellipsoid, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>1 - (0.7)^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{y}{b}\biggr)^2 ~~\Rightarrow ~~ y = y_\mathrm{max} = b [1 - (0.7)^2]^{1 / 2} = 0.89268\, .</math>
  </td>
</tr>
</table>

Similarly, by setting <math>y = 0</math>, we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>1 - (0.7)^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{z}{c}\biggr)^2 ~~\Rightarrow ~~ z = z_\mathrm{max} = c [1 - (0.7)^2]^{1 / 2} = 0.33586\, .</math>
  </td>
</tr>
</table>
The <math>(y, z)</math> coordinates of individual points along this ellipse can be determined, as before, by choosing values of <math>y</math> in the range,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>- y_\mathrm{max} \le y \le + y_\mathrm{max} \, ,</math>
  </td>
</tr>
</table>
then determining the corresponding pair of values of <math>z</math> via the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>z_\pm</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\pm ~z_\mathrm{max} \biggl[1 - (0.7)^2 - \frac{y^2}{y_\mathrm{max}^2}  \biggr]^{1 / 2}
\, .</math>
  </td>
</tr>
</table>

This ellipse is identified in Figure 2 by the dotted-green curve.  All four of the red arrows (velocity vectors, as  explained below) that are displayed in Figure 2 are anchored on this dotted-green curve; the <math>(x, y, z)_\mathrm{base}</math> coordinates of these anchor positions are listed in the yellow-colored elements of the following Table titled, "Red Arrows."  (There is nothing special about these four chosen anchor positions other than they lie on the dotted-green ellipse.)

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="12">'''Red Arrows'''<br />(Velocity Components in an y-z Plane)</td>
</tr>
<tr>
  <td align="center" rowspan="2">Number</td>
  <td align="center" colspan="4">Base of each Arrow</td>
  <td align="center" colspan="1" rowspan="6" bgcolor="lightgray">&nbsp;</td>
  <td align="center" colspan="2" rowspan="1">Velocity</td>
  <td align="center" colspan="1" rowspan="6" bgcolor="lightgray">&nbsp;</td>
  <td align="center" colspan="3">Arrow Tips</td>
</tr>
<tr>
  <td align="center"><math>x</math></td>
  <td align="center"><math>y</math></td>
  <td align="center"><math>z</math></td>
  <td align="center"><math>\biggl(\frac{x}{a}\biggr)^2 + \biggl(\frac{y}{b}\biggr)^2 + \biggl(\frac{z}{c}\biggr)^2</math></td>
  <td align="center" colspan="1" rowspan="1"><math>\dot{y} = u_2</math></td>
  <td align="center" colspan="1" rowspan="1"><math>\dot{z} = u_3</math></td>
  <td align="center"><math>x</math></td>
  <td align="center"><math>y</math></td>
  <td align="center"><math>z</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="right" bgcolor="yellow">0.7</td>
  <td align="right" bgcolor="yellow">-0.89268</td>
  <td align="right" bgcolor="yellow">0.00000</td>
  <td align="right">1.00000</td>
  <td align="right" rowspan="4">-1.19738 x</td>
  <td align="right" rowspan="4">+0.41285 x</td>
  <td align="right">0.7</td>
  <td align="right">-1.10222</td>
  <td align="right">+0.07225</td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="right" bgcolor="yellow">0.7</td>
  <td align="right" bgcolor="yellow">-0.16396</td>
  <td align="right" bgcolor="yellow">+0.33015</td>
  <td align="right">1.00001</td>
  <td align="right">0.7</td>
  <td align="right">-0.37350</td>
  <td align="right">+0.40240</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="right" bgcolor="yellow">0.7</td>
  <td align="right" bgcolor="yellow">+0.81981</td>
  <td align="right" bgcolor="yellow">+0.13291</td>
  <td align="right">1.00000</td>
  <td align="right">0.7</td>
  <td align="right">+0.61027</td>
  <td align="right">+0.20516</td>
</tr>
<tr>
  <td align="center">4</td>
  <td align="right" bgcolor="yellow">0.7</td>
  <td align="right" bgcolor="yellow">+0.38258</td>
  <td align="right" bgcolor="yellow">-0.30345</td>
  <td align="right">0.99999</td>
  <td align="right">0.7</td>
  <td align="right">+0.17304</td>
  <td align="right">-0.23121</td>
</tr>
</table>
As a check, we have also included in the "Red Arrows" table a column that tallies,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\biggl[ \biggl(\frac{x}{a}\biggr)^2 + 
\biggl(\frac{y}{b}\biggr)^2 + \biggl(\frac{z}{c}\biggr)^2
\biggr]_\mathrm{base} \, ,</math>
  </td>
</tr>
</table>
which in every case totals 1.000, as it should.