Editing Appendix/Ramblings/ConcentricEllipsoidalDaringAttack (section)

==Study the Functional Forms==

We know the functional forms of two of the desired curvilinear coordinates, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_1(x, y, z)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_3(x, y, z)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{y^{1/q^2}}{x} \, ,</math>
  </td>
</tr>
</table>
but we do not yet have a valid expression for the 2<sup>nd</sup> coordinate, <math>~\lambda_2(x, y, z)</math>.  Nevertheless, let's see if we can ''guess'' the functional forms for <math>~x_i(\lambda_1, \lambda_2, \lambda_3)</math>, by inverting the two known curvilinear-coordinate functions.  As a starting point, let's impose the following mappings:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~~\rightarrow~~</math>
  </td>
  <td align="left" colspan="3">
<math>~\frac{y^{1/q^2}}{\lambda_3} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~~\rightarrow~~</math>
  </td>
  <td align="left">
<math>~
\frac{1}{p}\biggl[
\lambda_1^2 - q^2y^2 - x^2
\biggr]^{1 / 2}
</math>
  </td>
  <td align="center">
<math>~~\rightarrow~~</math>
  </td>
  <td align="left">
<math>~
\frac{1}{p}\biggl[ \lambda_1^2 - q^2y^2 - \frac{y^{2/q^2}}{\lambda_3^2} \biggr]^{1 / 2} 
=
\frac{1}{p\lambda_3}\biggl[ \lambda_1^2 \lambda_3^2 - (qy\lambda_3)^2 - y^{2/q^2} \biggr]^{1 / 2} 
\, .
</math>
  </td>
</tr>
</table>
This means, for example, that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\ell_q^{-2}</math>
  </td>
  <td align="center">
<math>~~\rightarrow~~</math>
  </td>
  <td align="left">
<math>~
q^4y^2 + \frac{y^{2/q^2}}{\lambda_3^2} 
=
\lambda_3^{-2} \biggl[ q^2(qy\lambda_3)^2 + y^{2/q^2} \biggr]
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\ell_{3D}^{-2}</math>
  </td>
  <td align="center">
<math>~~\rightarrow~~</math>
  </td>
  <td align="left">
<math>~
q^4y^2 + \frac{y^{2/q^2}}{\lambda_3^2} +
p^2\biggl[
\lambda_1^2 - q^2y^2 - \frac{y^{2/q^2}}{\lambda_3^2}
\biggr]
=
\lambda_3^{-2} \biggl[
(q^2-p^2)(qy \lambda_3)^2 + (1-p^2)y^{2/q^2} + p^2\lambda_1^2\lambda_3^2
\biggr]
\, .
</math>
  </td>
</tr>
</table>

===Derivatives of x===
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial x}{\partial \lambda_1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x \lambda_1 \ell_{3D}^2 
=
y^{1/q^2} \lambda_1 \lambda_3 \biggl[
(q^2-p^2)(qy \lambda_3)^2 + (1-p^2)y^{2/q^2} + p^2\lambda_1^2\lambda_3^2
\biggr]^{-1} \, ,
 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial x}{\partial \lambda_3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- q^4 y^2 \ell_q^2 \biggl( \frac{x}{\lambda_3} \biggr)
=
- q^2 (qy)^2  \biggl( y^{1/q^2} \biggr) \lambda_3 \biggl[ q^2(qy\lambda_3)^2 + y^{2/q^2} \biggr]^{-1} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial x}{\partial \lambda_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_2 \ell_q \ell_{3D} (xp^2z)
=
\biggl[ h_2 \ell_q \ell_{3D}\biggr] \biggl(y^{1/q^2} \biggr)
\frac{p}{\lambda_3^2}\biggl[ \lambda_1^2 \lambda_3^2 - (qy\lambda_3)^2 - y^{2/q^2} \biggr]^{1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
h_2 \biggl( y^{1/q^2} \biggr) p \biggl[ \lambda_1^2 \lambda_3^2 - (qy \lambda_3)^2 - y^{2/q^2} \biggr]^{1 / 2}
\biggl[ q^2(qy\lambda_3)^2 + y^{2/q^2} \biggr]^{-1 / 2}
\biggl[
(q^2-p^2)(qy \lambda_3)^2 + (1-p^2)y^{2/q^2} + p^2\lambda_1^2\lambda_3^2
\biggr]^{-1 / 2}
</math>
  </td>
</tr>
</table>

===Struggling===

I have noticed that, in this last set of expressions, there are recurring terms of the form, <math>~(qy\lambda_3)</math> and <math>~(y^{2/q^2})</math>.  So, while keeping the same definition of the ccordinate, <math>~\lambda_1</math>, let's replace <math>~\lambda_2</math> and <math>~\lambda_3</math> with a pair of coordinates defined as follows:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_4 \equiv y\lambda_3 = \frac{y^{(q^2+1)/q^2}}{x} \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\lambda_5 \equiv y^{2/q^2} \, .</math>
  </td>
</tr>
</table>
This means that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda_5^{q^2/2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{y^{(q^2-1)/q^2}}{\lambda_4} 
=
\lambda_4^{-1} \lambda_5^{ (q^2-1)/2}
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{p^2} \biggl[\lambda_1^2 - x^2 - q^2y^2  \biggr]
=
\frac{1}{p^2 \lambda_4^2} \biggl[
\lambda_1^2 \lambda_4^2 - \lambda_5^{q^2-1} - q^2 \lambda_4^2\lambda_5^{q^2}
\biggr]
\, .</math>
  </td>
</tr>
</table>
Is this a set of orthogonal coordinates?  Well &hellip; &nbsp; &nbsp; &nbsp; <font color="red">No!</font>