Editing Appendix/Ramblings/ConcentricEllipsoidalDaringAttack (section)

===Setup===

The surface of an ellipsoid with semi-major axes (a, b, c) is defined 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}{a}\biggr)^2 + \biggl( \frac{y}{b}\biggr)^2 + \biggl( \frac{z}{c}\biggr)^2 \, .</math>
  </td>
</tr>
</table>
This is identical to our expression for <math>~\lambda_1</math> if we make the associations,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a = \lambda_1 \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; </td>
  <td align="center">
<math>~b = \frac{\lambda_1}{q} \ ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; </td>
  <td align="left">
<math>~c = \frac{\lambda_1}{p} \, .</math>
  </td>
</tr>
</table>

Now, given that <math>~\lambda_3</math> does not functionally depend on <math>~z</math>, let's consider that the choice of <math>~z</math> is tightly associated with the specification of the second coordinate, <math>~\lambda_2</math>.  Specifically, let's adopt the definition,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_2^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~1 - \biggl( \frac{z}{c}\biggr)^2 \, ,</math>
  </td>
</tr>
</table>
in which case, we see that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~z^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c^2(1-\lambda_2^2) = \frac{\lambda_1^2(1-\lambda_2^2)}{p^2} \, ,</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{x}{a}\biggr)^2 + \biggl( \frac{y}{b}\biggr)^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \lambda_2^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x^2 + q^2 y^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \lambda_1^2 \lambda_2^2 \, .</math>
  </td>
</tr>
</table>

[Note that <font color="red">in the case of spherical coordinates</font> (q<sup>2</sup> = p<sup>2</sup> = 1), <math>~\lambda_1 \rightarrow r</math>, and this "second" coordinate, <math>~\lambda_2</math>, becomes <math>~\sin\theta</math>.]  Combining this last expression with the <math>~x - y</math> relationship that is provided by the definition of <math>~\lambda_3</math>, gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_1^2 \lambda_2^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{y^{2/q^2}}{\lambda_3^2} + q^2y^2 \, .</math>
  </td>
</tr>
</table>

In general, the exponent of <math>~2q^{-2}</math> that appears in the first term on the right-hand side of this expression prevents us from being able to analytically prescribe the function, <math>~y(\lambda_1, \lambda_2, \lambda_3)</math>.  But a solution is obtainable for selected values of <math>~q^2 > 1</math>.