Editing ThreeDimensionalConfigurations/ChallengesPt6 (section)

===Blue (x = 0) Ellipse===
By setting <math>x = 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</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{y}{b}\biggr)^2 ~~\Rightarrow ~~ y = y_\mathrm{max} = b \, .</math>
  </td>
</tr>
</table>

Similarly, by setting <math>x = y = 0</math>, we find the point where the z-axis intersects the surface of the ellipsoid, namely,
<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{z}{c}\biggr)^2 ~~\Rightarrow ~~ z = z_\mathrm{max} = c \, .</math>
  </td>
</tr>
</table>

If we only set, <math>x = 0</math>, this expression generates an ellipse in the y-z plane whose semi-axes are <math>(y_\mathrm{max}, z_\mathrm{max}) = (1.25, 0.4703)</math>.  The <math>(y, z)</math> coordinates of individual points along the ellipse can be determined by choosing values of <math>y</math> in the range,
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  <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,
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<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 - \frac{y^2}{y_\mathrm{max}^2}  \biggr]^{1 / 2}
\, .</math>
  </td>
</tr>
</table>

This ellipse is identified in Figure 2 by the dotted-blue curve.

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="2">Figure 2:  &nbsp; Y-Z plane(s) of Riemann Type I Ellipsoid[[File:DataFileButton02.png|right|60px|file = Dropbox/3Dviewers/RiemannModels/RiemannCalculations.xlsx --- worksheet = Feb22]]</td>
</tr>
<tr>
  <td align="center">[[File:YZplane3.png|400px|x = +0.70]]</td>
  <td align="center">[[File:YZplaneXm085.png|400px|x = -0.85]]
</tr>

<tr>
  <td align="center">Blue ellipse (x/a = 0.0); Green ellipse (x/a = + 0.70)</td>
  <td align="center">Blue ellipse (x/a = 0.0); Green ellipse (x/a = - 0.85)</td>
</tr>
</table>