Editing Appendix/Ramblings/T9Coordinates

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=T9 Coordinates=

==Establish 1<sup>st</sup> and 3<sup>rd</sup> Coordinates==

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

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

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

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

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

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

<tr>
  <td align="right">
<math>~h_3^{-2}</math>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \biggl( \frac{q^2}{1 - q^2 } \biggr) \frac{\lambda_3}{x} \biggr]^2
+
\biggl[ \biggl( \frac{1}{q^2 - 1} \biggr) \frac{\lambda_3}{y} \biggr]^2
=
\frac{\lambda_3^2}{(q^2-1)^2}\biggl[ \frac{x^2 + q^4y^2 }{x^2 y^2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ h_3</math>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(q^2-1)}{\lambda_3}\biggl[ \frac{x y}{(x^2 + q^4y^2)^{1 / 2} } \biggr] \, .
</math>
  </td>
</tr>
</table>
Hence, the three direction-cosines are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\gamma_{31} = h_3 \biggl(\frac{\partial \lambda_3}{\partial x} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(q^2-1)}{\lambda_3}\biggl[ \frac{x y}{(x^2 + q^4y^2)^{1 / 2} } \biggr]\biggl( \frac{q^2}{1-q^2} \biggr) \frac{\lambda_3}{x}
=
- \biggl[ \frac{q^2y}{(x^2 + q^4y^2)^{1 / 2} } \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{32} = h_3 \biggl(\frac{\partial \lambda_3}{\partial y} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(q^2-1)}{\lambda_3}\biggl[ \frac{x y}{(x^2 + q^4y^2)^{1 / 2} } \biggr]\biggl( \frac{1}{q^2 - 1} \biggr) \frac{\lambda_3}{y}
=
\biggl[ \frac{x }{(x^2 + q^4y^2)^{1 / 2} } \biggr] \, .
</math>
  </td>
</tr>
</table>

And the position vector is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathbf{\vec{x}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 (\gamma_{11} x + \gamma_{12} y + \gamma_{13} z)
+
\mathbf{\hat{e}}_2 (\gamma_{21} x + \gamma_{22} y + \gamma_{23} z)
+
\mathbf{\hat{e}}_3 (\gamma_{31} x + \gamma_{32} y + \gamma_{33} z) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 \biggl[ x^2\ell_{3D} + q^2 y^2 \ell_{3D} + p^2 z^2 \ell_{3D} \biggr]
+
\mathbf{\hat{e}}_2 (\gamma_{21} x + \gamma_{22} y + \gamma_{23} z)
+
\mathbf{\hat{e}}_3 \biggl[- \frac{q^2xy}{(x^2 + q^4y^2)^{1 / 2} } +  \frac{xy }{(x^2 + q^4y^2)^{1 / 2} }  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 \biggl[ \lambda_1^2 \ell_{3D} \biggr]
+
\mathbf{\hat{e}}_2 (\gamma_{21} x + \gamma_{22} y + \gamma_{23} z)
-
\mathbf{\hat{e}}_3 \biggl[\frac{(q^2-1) xy }{(x^2 + q^4y^2)^{1 / 2} }  \biggr] \, .
</math>
  </td>
</tr>
</table>

==Guess 2<sup>nd</sup> Coordinate==

===Unspecified Coefficients===

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

<tr>
  <td align="right">
<math>~\lambda_2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(ax^2 + by^2 + ez^2)^{-1 / 2} (ax^2 + by^2)^{1 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{(ax^2 + by^2)}{(ax^2 + by^2 + ez^2)}\biggr]^{1 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-1 / 2} \, .</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{1}{2}\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-3 / 2} \frac{\partial}{\partial x}\biggl[ez^2 (ax^2 + by^2)^{-1 } \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{ez^2 }{2}\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-3 / 2} \biggl( -1\biggr) \biggl[(ax^2 + by^2)^{-2} \biggr]2ax
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a x e z^2 }{(ax^2 + by^2)^{2}}\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-3 / 2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a x e z^2 ~\lambda_2^3}{(ax^2 + by^2)^{2}} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{1}{2}\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-3 / 2} \frac{\partial}{\partial y}\biggl[ez^2 (ax^2 + by^2)^{-1} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{ez^2 }{2}\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-3 / 2} \biggl( -1\biggr) \biggl[(ax^2 + by^2)^{-2} \biggr]2by
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{by e z^2 }{(ax^2 + by^2)^{2}}\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-3 / 2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{b y e z^2 ~\lambda_2^3}{(ax^2 + by^2)^{2}} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{1}{2}\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-3 / 2} \frac{\partial}{\partial z}\biggl[ez^2 (ax^2 + by^2)^{-1 } \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{ez ~ \lambda_2^3}{(ax^2 + by^2)}  \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~h_2^{-2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{a x e z^2 ~\lambda_2^3}{(ax^2 + by^2)^{2}}\biggr]^2
+
\biggl[ \frac{b y e z^2 ~\lambda_2^3}{(ax^2 + by^2)^{2}} \biggr]^2
+
\biggl[\frac{ez ~ \lambda_2^3}{(ax^2 + by^2)} \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{e z ~\lambda_2^3}{(ax^2 + by^2)^{2}}\biggr]^2
\biggl[
(a x z)^2
+
(b y z )^2
+
(ax^2 + by^2)^2
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ h_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{(ax^2 + by^2)^{2}}{e z ~\lambda_2^3}\biggr]
\biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2
\biggr]^{-1 / 2} 
\, .
</math>
  </td>
</tr>
</table>

The three direction-cosines are, then,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\gamma_{21} = h_2 \biggl(\frac{\partial\lambda_2}{\partial x} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a x e z^2 ~\lambda_2^3}{(ax^2 + by^2)^{2}}
\biggl[\frac{(ax^2 + by^2)^{2}}{e z ~\lambda_2^3}\biggr]
\biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2
\biggr]^{-1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
axz
\biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2
\biggr]^{-1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{22} = h_2 \biggl(\frac{\partial\lambda_2}{\partial y} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{b y e z^2 ~\lambda_2^3}{(ax^2 + by^2)^{2}}
\biggl[\frac{(ax^2 + by^2)^{2}}{e z ~\lambda_2^3}\biggr]
\biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2
\biggr]^{-1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
byz
\biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2
\biggr]^{-1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{23} = h_2 \biggl(\frac{\partial\lambda_2}{\partial z} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{ez ~ \lambda_2^3}{(ax^2 + by^2)}
\biggl[\frac{(ax^2 + by^2)^{2}}{e z ~\lambda_2^3}\biggr]
\biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2
\biggr]^{-1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-(ax^2 + by^2)
\biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2
\biggr]^{-1 / 2} \, .
</math>
  </td>
</tr>
</table>

<table border="1" cellpadding="9" align="center">
<tr>
  <td align="center" colspan="9">'''Direction Cosine Components for T9 Coordinates'''</td>
</tr>
<tr>
  <td align="center"><math>~n</math></td>
  <td align="center"><math>~\lambda_n</math></td>
  <td align="center"><math>~h_n</math></td>
  <td align="center"><math>~\frac{\partial \lambda_n}{\partial x}</math></td>
  <td align="center"><math>~\frac{\partial \lambda_n}{\partial y}</math></td>
  <td align="center"><math>~\frac{\partial \lambda_n}{\partial z}</math></td>
  <td align="center"><math>~\gamma_{n1}</math></td>
  <td align="center"><math>~\gamma_{n2}</math></td>
  <td align="center"><math>~\gamma_{n3}</math></td>
</tr>
<tr>
  <td align="center">(1)</td>
  <td align="center">(2)</td>
  <td align="center">(3)</td>
  <td align="center">(4)</td>
  <td align="center">(5)</td>
  <td align="center">(6)</td>
  <td align="center">(7)</td>
  <td align="center">(8)</td>
  <td align="center">(9)</td>
</tr>

<tr>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} </math></td>
  <td align="center"><math>~\lambda_1 \ell_{3D}</math></td>
  <td align="center"><math>~\frac{x}{\lambda_1}</math></td>
  <td align="center"><math>~\frac{q^2 y}{\lambda_1}</math></td>
  <td align="center"><math>~\frac{p^2 z}{\lambda_1}</math></td>
  <td align="center"><math>~(x) \ell_{3D}</math></td>
  <td align="center"><math>~(q^2 y)\ell_{3D}</math></td>
  <td align="center"><math>~(p^2z) \ell_{3D}</math></td>
</tr>

<tr>
  <td align="center"><math>~2A</math></td>
  <td align="center">---</td>
  <td align="center" bgcolor="pink"><math>~\ell_{3D}(x^2 + q^4 y^2)^{1 / 2}</math></td>
  <td align="center" bgcolor="white"><math>~\frac{xp^2z}{(x^2 + q^4y^2)} </math></td>
  <td align="center" bgcolor="white"><math>~\frac{q^2 y p^2z}{(x^2 + q^4y^2)}</math></td>
  <td align="center" bgcolor="white"><math>~-1</math></td>
  <td align="center"><math>~\frac{x p^2 z\ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} </math></td>
  <td align="center"><math>~\frac{q^2 y p^2 z\ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} </math></td>
  <td align="center"><math>~-\frac{(x^2 + q^4 y^2)\ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} </math></td>
</tr>

<tr>
  <td align="center"><math>~2B</math></td>
  <td align="center" bgcolor="yellow"><math>~\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-1 / 2}</math></td>
  <td align="center"><math>~\biggl[\frac{(ax^2 + by^2)^2}{ez~\lambda_2^3} \biggr]\mathfrak{J}_{2\mathrm{B}}</math></td>
  <td align="center"><math>~\frac{ax~ez^2 ~\lambda_2^3}{(ax^2 + by^2)^2} </math></td>
  <td align="center"><math>~\frac{by~ez^2 ~\lambda_2^3}{(ax^2 + by^2)^2} </math></td>
  <td align="center"><math>~- \frac{ez ~\lambda_2^3}{(ax^2 + by^2)} </math></td>
  <td align="center"><math>~axz ~\mathfrak{J}_{2\mathrm{B}}</math></td>
  <td align="center"><math>~byz ~\mathfrak{J}_{2\mathrm{B}}</math></td>
  <td align="center"><math>~-(ax^2 + by^2)~\mathfrak{J}_{2\mathrm{B}} </math></td>
</tr>

<tr>
  <td align="center"><math>~3</math></td>
  <td align="center"><math>~\biggl[ \frac{q^2y}{x^{q^2}} \biggr]^{1/(q^2-1)}</math></td>
  <td align="center"><math>~\frac{(q^2-1)x y }{\lambda_3(x^2 + q^4y^2)^{1 / 2}} </math></td>
  <td align="center"><math>~- \biggl( \frac{q^2}{q^2 - 1} \biggr) \frac{\lambda_3}{x}</math></td>
  <td align="center"><math>~\biggl( \frac{q^2}{q^2 - 1} \biggr) \frac{\lambda_3}{q^2y}</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~- \frac{q^2 y }{(x^2 + q^4y^2)^{1 / 2}} </math></td>
  <td align="center"><math>~\frac{x }{(x^2 + q^4y^2)^{1 / 2}} </math></td>
  <td align="center"><math>~0</math></td>
</tr>

<tr>
  <td align="left" colspan="9">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>~\ell_{3D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(x^2 + q^4y^2 + p^4 z^2 )^{- 1 / 2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{J}^{-2}_0 \equiv \frac{(x^2 + q^4 y^2)}{\ell_{3D}^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[(xp^2 z)^2 + (q^2 y p^2 z )^2 + (x^2 + q^4y^2)^2\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{J}_{2\mathrm{B}}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2\biggr]^{-1 / 2} \, .
</math>
  </td>
</tr>
</table>
  </td>
</tr>
</table>


This table titled, "<i>Direction Cosine Components for T9 Coordinates</i>," contains the following information:
<ol>
<li>
The (first) row labeled, <math>~n = 1</math>, correctly details the scale-factor, <math>~h_1</math>, and the unit vector expression, <math>~\hat{e}_1 = (\hat\imath \gamma_{11} + \hat\jmath \gamma_{12} + \hat{k} \gamma_{13})</math>, that result from the given specification of the <math>~\lambda_1</math> coordinate.  By design, the unit vector, <math>~\hat{e}_1</math>, is everywhere normal to the "surface" of the ellipsoid.
</li>
<li>
The (fourth) row labeled, <math>~n = 3</math>, correctly details the scale-factor, <math>~h_3</math>, and the unit vector expression, <math>~\hat{e}_3 = (\hat\imath \gamma_{31} + \hat\jmath \gamma_{32} + \hat{k} \gamma_{33})</math>, that result from the given specification of the <math>~\lambda_2</math> coordinate.  By design, this unit vector, <math>~\hat{e}_3</math>, has no vertical component &#8212; that is, <math>~\gamma_{33} = 0</math> &#8212; and, by design, it is everywhere perpendicular to the "surface-normal" unit vector, <math>~\hat{e}_1</math>. 
</li>
<li>
We desire a unit vector, <math>~\hat{e}_2</math>, that is mutually orthogonal to the other two unit vectors; this has been accomplished by examining their cross-product, namely, we have set <math>~\hat{e}_{2\mathrm{A}} = \hat{e}_3 \times \hat{e}_1</math>.  Determined in this manner, the expressions for the three direction-cosine components of <math>~\hat{e}_{2\mathrm{A}}</math> have been written in the last three columns of the (second) row labeled, <math>~n=2\mathrm{A}</math>.  While we are confident that the correct specification of <math>~\hat{e}_2</math> is, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_{2\mathrm{A}} = (\hat\imath \gamma_{21} + \hat\jmath \gamma_{22} + \hat{k} \gamma_{23})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath (xp^2 z)\mathfrak{J}_0 + \hat\jmath (q^2y p^2 z)\mathfrak{J}_0 - \hat{k} (x^2 + q^4y^2) \mathfrak{J}_0 \, ,
</math>
  </td>
</tr>
</table>
as yet (18 February 2021), we have been unable to determine an expression for the coordinate, <math>~\lambda_{2\mathrm{A}}(x, y, z)</math>, from which all three of these direction-cosine expressions can be simultaneously derived &#8212; hence, the dashes in the second column of row 2A.  The expression for <math>~h_{2\mathrm{A}}</math> that has been presented in the third column of row 2A (and framed in pink) is a ''guess'' which, when divided into each of the three direction cosines, gives respectively the three ''guessed'' partial-derivative expressions shown in columns 4, 5, and 6 of row 2A.
</li>
<li>
The second column of row 2B contains a ''guess'' (framed in yellow) for the coordinate expression, <math>~\lambda_{2\mathrm{B}}</math>; this expression contains three unspecified scalar coefficients, <math>~a</math>, <math>~b</math> and <math>~e</math>.  The remaining columns of this row contain the three partial derivatives, the associated scale factor, and the three direction cosines that result from this ''guessed'' coordinate expression.  If we can find values of the three scalar coefficients that give (row 2B) expressions for the three direction cosines that perfectly match the direction cosines written in row 2A, then we will be able to state that <math>~\lambda_{2\mathrm{B}}</math> is &#8212; at least one form of &#8212; our sought-after third coordinate expression.
</li>
</ol>

===Yellow-Framed Guess 2B===

Referencing the 2B table row, above, we are looking for coefficient values that map,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~axz \rightarrow xp^2 z \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="center">
<math>~byz \rightarrow q^2 yp^2 z \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="left">
<math>~(ax^2 + by^2) \rightarrow (x^2 + q^4y^2) \, ,</math>
  </td>
</tr>
</table>

and that map,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\mathfrak{J}_{2\mathrm{B}} \equiv \biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2 \biggr]^{-1 / 2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[z^2(a^2 x^2 + b^2 y^2) + (ax^2 + by^2)^2 \biggr]^{-1 / 2} \, ,
</math>
  </td>
</tr>
</table>

into the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{J}_0 \equiv \frac{\ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[
(x^2 + q^4y^2 + p^4 z^2)(x^2 + q^4 y^2)
\biggr]^{-1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[
p^4 z^2(x^2 + q^4 y^2)
+
(x^2 + q^4y^2 )^2
\biggr]^{-1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[
x^2 p^4 z^2
+
q^4 y^2 p^4 z^2
+
(x^2 + q^4y^2 )^2
\biggr]^{-1 / 2} \, .
</math>
  </td>
</tr>
</table>

A portion of these mappings are accomplished by setting <math>~a = p^2</math> and <math>~b = q^2p^2</math>, but this pair of specified coefficient values does not satisfy other mappings.  Alternatively, a separate subset of mappings &#8212; but, again, not all mappings &#8212; is satisfied by setting <math>~a = 1</math> and <math>~b = q^4</math>.  So the yellow-framed <font color="red">guess 2B does not provide a correct second-coordinate expression</font>.

===Blue-Framed Guess 2C===
Let's try,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_{2\mathrm{B}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(cz)^{2e} (ax^2 + by^2)^{-f} \, .</math>
  </td>
</tr>
</table>


<table border="1" cellpadding="9" align="center">
<tr>
  <td align="center" colspan="9">'''Direction Cosine Components for Additional T9 Coordinate Guesses'''</td>
</tr>
<tr>
  <td align="center"><math>~n</math><br />&nbsp; <br />(1)</td>
  <td align="center"><math>~\lambda_n</math><br />&nbsp; <br />(2)</td>
  <td align="center"><math>~h_n</math><br />&nbsp; <br />(3)</td>
  <td align="center"><math>~\partial \lambda_n/\partial x</math><br />&nbsp; <br />(4)</td>
  <td align="center"><math>~\partial \lambda_n/\partial y</math><br />&nbsp; <br />(5)</td>
  <td align="center"><math>~\partial \lambda_n/\partial z</math><br />&nbsp; <br />(6)</td>
  <td align="center"><math>~\gamma_{n1}</math><br />&nbsp; <br />(7)</td>
  <td align="center"><math>~\gamma_{n2}</math><br />&nbsp; <br />(8)</td>
  <td align="center"><math>~\gamma_{n3}</math><br />&nbsp; <br />(9)</td>
</tr>

<tr>
  <td align="center"><math>~2A</math></td>
  <td align="center">---</td>
  <td align="center" bgcolor="white">---</td>
  <td align="center" bgcolor="white">---</td>
  <td align="center" bgcolor="white">---</td>
  <td align="center" bgcolor="white">---</td>
  <td align="center"><math>~x p^2 z ~\mathfrak{J}_0 </math></td>
  <td align="center"><math>~q^2 y p^2 z~\mathfrak{J}_0 </math></td>
  <td align="center"><math>~-(x^2 + q^4 y^2)~\mathfrak{J}_0 </math></td>
</tr>

<tr>
  <td align="center"><math>~2B</math></td>
  <td align="center" bgcolor="yellow"><math>~\biggl[1 + \frac{ez^2}{(ax^2 + by^2)} \biggr]^{-1 / 2}</math></td>
  <td align="center"><math>~\biggl[\frac{(ax^2 + by^2)^2}{ez~\lambda_2^3} \biggr]\mathfrak{J}_{2\mathrm{B}}</math></td>
  <td align="center"><math>~\frac{ax~ez^2 ~\lambda_2^3}{(ax^2 + by^2)^2} </math></td>
  <td align="center"><math>~\frac{by~ez^2 ~\lambda_2^3}{(ax^2 + by^2)^2} </math></td>
  <td align="center"><math>~- \frac{ez ~\lambda_2^3}{(ax^2 + by^2)} </math></td>
  <td align="center"><math>~axz ~\mathfrak{J}_{2\mathrm{B}}</math></td>
  <td align="center"><math>~byz ~\mathfrak{J}_{2\mathrm{B}}</math></td>
  <td align="center"><math>~-(ax^2 + by^2)~\mathfrak{J}_{2\mathrm{B}} </math></td>
</tr>

<tr>
  <td align="center"><math>~2C</math></td>
  <td align="center" bgcolor="lightblue"><math>~(cz)^{2e} (ax^2 + by^2)^{-f}</math></td>
  <td align="center"><math>~\biggl[\frac{z(ax^2 + by^2)}{2e~\lambda_{2\mathrm{C}}} \biggr]\mathfrak{J}_{2\mathrm{C}}</math></td>
  <td align="center"><math>~- \frac{2afx~\lambda_{2\mathrm{C}} }{(ax^2 + by^2)}</math></td>
  <td align="center"><math>~- \frac{2bfy~\lambda_{2\mathrm{C}} }{(ax^2 + by^2)}</math></td>
  <td align="center"><math>~\frac{2e~\lambda_{2\mathrm{C}} }{z}</math></td>
  <td align="center"><math>~ -\biggl( \frac{af xz}{e} \biggr)~\mathfrak{J}_{2\mathrm{C}}</math></td>
  <td align="center"><math>~- \biggl( \frac{bf yz}{e} \biggr)~\mathfrak{J}_{2\mathrm{C}}</math></td>
  <td align="center"><math>~(ax^2 + by^2)~\mathfrak{J}_{2\mathrm{C}} </math></td>
</tr>

<tr>
  <td align="left" colspan="9">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>~\ell_{3D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(x^2 + q^4y^2 + p^4 z^2 )^{- 1 / 2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{J}^{-2}_0 \equiv \frac{(x^2 + q^4 y^2)}{\ell_{3D}^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[(xp^2 z)^2 + (q^2 y p^2 z )^2 + (x^2 + q^4y^2)^2\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{J}_{2\mathrm{B}}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2\biggr]^{-1 / 2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{J}_{2\mathrm{C}}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[(x z)^2 \biggl( \frac{fa}{e} \biggr)^2 + (y z )^2\biggl( \frac{fb}{e} \biggr)^2 + (ax^2 + by^2)^2\biggr]^{-1 / 2} \, .
</math>
  </td>
</tr>
</table>

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


Examining just the expression for <math>~\mathfrak{J}_{2\mathrm{C}}</math>, we see that we definitely need:  <math>~a = 1</math> and <math>~b=q^4</math>.  Also, we need,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{af}{e}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~p^2</math>
  </td>
<td align="center">&nbsp; &nbsp; <math>~\Rightarrow </math>&nbsp; &nbsp; 
  <td align="right">
<math>~\frac{f}{e}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~p^2 \, ;</math>
  </td>
</tr>
</table>
and, separately we need,

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

<tr>
  <td align="right">
<math>~\frac{fb}{e}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~q^2 p^2</math>
  </td>
<td align="center">&nbsp; &nbsp; <math>~\Rightarrow </math>&nbsp; &nbsp; 
  <td align="right">
<math>~\frac{f}{e}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{p^2}{q^2} \, .</math>
  </td>
</tr>
</table>
These cannot simultaneously be satisfied.    So the blue-framed <font color="red">guess 2C does not provide a correct second-coordinate expression</font>.  But we are very close!  We need one additional scalar coefficient degree of freedom.

===Next Thought===
More generally, this leads to an expression for the scale-factor of the form,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~h_2^{-2}</math>
  </td>
  <td align="center">
<math>~\sim</math>
  </td>
  <td align="left" colspan="9">
<math>~
\mathfrak{G}(x, y, z) 
\biggl[
x^2 p^4 z^2
+
q^4 y^2 p^4 z^2
+
(x^2 + q^4y^2 )^2
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{\partial \lambda_2}{\partial x}</math>
  </td>
  <td align="center">
<math>~\sim</math>
  </td>
  <td align="left">
<math>~
[\mathfrak{G}(x, y, z) ]^{1 / 2}
\biggl[
x p^2 z
\biggr] \, ;
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial y}</math>
  </td>
  <td align="center">
<math>~\sim</math>
  </td>
  <td align="left">
<math>~
[\mathfrak{G}(x, y, z) ]^{1 / 2}
\biggl[
q^2 y p^2 z
\biggr] \, ;
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{\partial \lambda_2}{\partial z}</math>
  </td>
  <td align="center">
<math>~\sim</math>
  </td>
  <td align="left">
<math>~
[\mathfrak{G}(x, y, z) ]^{1 / 2}
\biggl[
x^2 + q^4y^2
\biggr] \, .
</math>
  </td>
</tr>
</table>


----

Now, if we set, <math>~a = p^2</math> and <math>~b = q^2p^2</math>, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(a x z)^2 + (b y z )^2 + (ax^2 + by^2)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( x p^2 z)^2 + (q^2p^2 y z )^2 + (p^2 x^2 + q^2p^2 y^2)^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
p^4[ q^4 y^2 z^2 + x^2 z^2 +  (x^2 + q^2 y^2 )^2] \, ,
</math>
  </td>
</tr>
</table>

in which case,

=Complete Orthogonality Check=
If the set of unit vectors is indeed orthogonal, then we must find that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\gamma_{mn}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~M_{mn} \, ,</math>
  </td>
</tr>
</table>
where the quantity <math>~M_{mn}</math> is the minor of <math>~\gamma_{mn}</math> in the determinant, <math>~|\gamma_{mn}|</math>.  (Note:  This last expression is true only for right-handed coordinate systems.  If the coordinate system is left-handed, we should find, <math>~\gamma_{mn} = - M_{mn}</math>.)  More specifically, for any right-handed, orthogonal curvilinear coordinate system we should find:
<table  border="1" align="center" cellpadding="10">
<tr>
  <td align="left">
<table border="0" cellpadding="5" align="left">

<tr>
  <td align="right">
<math>~\gamma_{11}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{22}\gamma_{33} - \gamma_{23}\gamma_{32} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{12}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{23}\gamma_{31} - \gamma_{21}\gamma_{33} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{13}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{21}\gamma_{32} - \gamma_{22}\gamma_{31} \, ,</math>
  </td>
</tr>
</table>
  </td>
<td align="center">&nbsp;&nbsp;&nbsp;</td>
  <td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\gamma_{21}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{32}\gamma_{13} - \gamma_{33}\gamma_{12} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{22}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{33}\gamma_{11} - \gamma_{31}\gamma_{13} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{23}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{31}\gamma_{12} - \gamma_{32}\gamma_{11} \, ,</math>
  </td>
</tr>
</table>
  </td>
<td align="center">&nbsp;&nbsp;&nbsp;</td>
  <td align="right">
<table border="0" cellpadding="5" align="right">

<tr>
  <td align="right">
<math>~\gamma_{31}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{12}\gamma_{23} - \gamma_{13}\gamma_{22} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{32}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{13}\gamma_{21} - \gamma_{11}\gamma_{23} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{33}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_{11}\gamma_{22} - \gamma_{12}\gamma_{21} \, .</math>
  </td>
</tr>
</table>
</td></tr></table>

and the position vector is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathbf{\vec{x}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat\imath} x + 
\mathbf{\hat\jmath} y +
\mathbf{\hat{k}} z
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 (\gamma_{11} x + \gamma_{12} y + \gamma_{13} z)
+
\mathbf{\hat{e}}_2 (\gamma_{21} x + \gamma_{22} y + \gamma_{23} z)
+
\mathbf{\hat{e}}_3 (\gamma_{31} x + \gamma_{32} y + \gamma_{33} z) \, .
</math>
  </td>
</tr>
</table>

==For (&kappa;<sub>1</sub>, &kappa;<sub>2</sub>, &kappa;<sub>3</sub>)==

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

<tr>
  <td align="right">
<math>~\gamma_{22}\gamma_{33} - \gamma_{23}\gamma_{32}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{q^2y p^2z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
\biggl[ 0 \biggr]
+
\biggl[ \frac{( x^2 + q^4y^2 ) \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}}  \biggr]
\biggl[ \frac{x}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
=
x\ell_{3D}
=
\gamma_{11} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{23}\gamma_{31} - \gamma_{21}\gamma_{33}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\bigg[ (x^2 + q^4y^2)^{1 / 2} \ell_{3D} \biggr]
\biggl[ \frac{q^2y}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
-
\biggl[ \frac{xp^2 z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
\biggl[ 0 \biggr]
=
q^2 y \ell_{3D}
=
\gamma_{12} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{21}\gamma_{32} - \gamma_{22}\gamma_{31}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{xp^2 z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
\biggl[ \frac{x}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
+
\biggl[ \frac{q^2 y p^2 z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
\biggl[ \frac{q^2y}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{x^2p^2 z \ell_{3D}}{(x^2 + q^4y^2)} \biggr]
+
\biggl[ \frac{q^4 y^2 p^2 z \ell_{3D}}{(x^2 + q^4y^2)} \biggr]
=
p^2 z \ell_{3D} 
=
\gamma_{13} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\gamma_{32}\gamma_{13} - \gamma_{33}\gamma_{12}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{x}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
\biggl[ p^2z \ell_{3D} \biggr]
-
\biggl[ 0\biggr]
\biggl[ q^2y \ell_{3D} \biggr]
=
\biggl[ \frac{xp^2 z\ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
=
\gamma_{21} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{33}\gamma_{11} - \gamma_{31}\gamma_{13}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 0\biggr]
\biggl[ x\ell_{3D} \biggr]
+
\biggl[ \frac{q^2y}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
\biggl[ p^2z \ell_{3D} \biggr]
=
\biggl[ \frac{q^2y p^2z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
=
\gamma_{22} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{31}\gamma_{12} - \gamma_{32}\gamma_{11}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{q^2y}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
\biggl[ q^2y \ell_{3D} \biggr]
-
\biggl[ \frac{x}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
\biggl[ x\ell_{3D} \biggr]
=
-(x^2 + q^4y^2)^{1 / 2}\ell_{3D}
=
\gamma_{23} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\gamma_{12}\gamma_{23} - \gamma_{13}\gamma_{22}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ q^2y \ell_{3D} \biggr]
\biggl[ (x^2 + q^4y^2)^{1 / 2}\ell_{3D} \biggr]
-
\biggl[ p^2 z\ell_{3D} \biggr]
\biggl[ \frac{q^2y p^2 z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{ q^2 y \ell_{3D}^2 }{(x^2 + q^4y^2)^{1 / 2} } 
\biggl[ x^2 + q^4y^2 
+
p^4 z^2 \biggr] 
=
- \frac{ q^2 y }{(x^2 + q^4y^2)^{1 / 2} } 
=
\gamma_{31} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{13}\gamma_{21} - \gamma_{11}\gamma_{23}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ p^2z \ell_{3D} \biggr]
\biggl[ \frac{x p^2z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
+
\biggl[ x \ell_{3D} \biggr]
\biggl[ (x^2 + q^4y^2)^{1 / 2} \ell_{3D}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{x \ell_{3D}^2 }{ (x^2 + q^4y^2)^{1 / 2} } \biggl[
x^2 + q^4y^2 + p^4 z^2 
\biggr]
=
\frac{x }{ (x^2 + q^4y^2)^{1 / 2} } 
=
\gamma_{32} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{11}\gamma_{22} - \gamma_{12}\gamma_{21}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ x \ell_{3D} \biggr]
\biggl[ \frac{q^2y p^2z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
-
\biggl[ q^2y \ell_{3D} \biggr]
\biggl[ \frac{xp^2 z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}} \biggr]
=
0
=
\gamma_{33} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>
</table>

Given that the prescribed interrelationships between all nine direction cosines are satisfied, we conclude that the <math>~(\kappa_1, \kappa_2, \kappa_3)</math> coordinate system is an orthogonal one.  Accordingly, the position vector is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathbf{\vec{x}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 \biggl\{
x^2 \ell_{3D} 
+ q^2y^2 \ell_{3D} 
+ p^2z^2 \ell_{3D}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\mathbf{\hat{e}}_2 \biggl\{
x^2 + q^2y^2  - \frac{1}{p^2}(x^2 + q^4 y^2) 
\biggr\} \frac{p^2 z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~
\mathbf{\hat{e}}_3 \biggl\{ 
- \frac{xq^2y}{(x^2 + q^4y^2)^{1 / 2}} 
+ \frac{xy}{(x^2 + q^4 y^2)^{1 / 2}}
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 \biggl[ \kappa_1^2 \ell_{3D} \biggr]
~+~
\mathbf{\hat{e}}_2 \biggl[
x^2 + q^2y^2  - \frac{1}{p^2}(x^2 + q^4 y^2) 
\biggr] \frac{p^2 z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}}
~+~
\mathbf{\hat{e}}_3 
\biggl[ 
\frac{(1-q^2)xy}{(x^2 + q^4 y^2)^{1 / 2}} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 \biggl[ \kappa_1^2 \ell_{3D} \biggr]
~+~
\mathbf{\hat{e}}_2 \biggl[
x^2\biggl(1 - \frac{1}{p^2}\biggr) - q^2y^2 \biggl(q^2 - 1 \biggr) 
\biggr] \frac{p^2 z \ell_{3D}}{(x^2 + q^4y^2)^{1 / 2}}
~-~
\mathbf{\hat{e}}_3 
\biggl[ 
\frac{(q^2 - 1)xy}{(x^2 + q^4 y^2)^{1 / 2}} \biggr] \, .
</math>
  </td>
</tr>
</table>

==For (&kappa;<sub>1</sub>, &kappa;<sub>4</sub>, &kappa;<sub>5</sub>)==

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

<tr>
  <td align="right">
<math>~\gamma_{42}\gamma_{53} - \gamma_{43}\gamma_{52}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{xq^2y p^2z}{\mathcal{D}} \biggl( \frac{1}{q^2y} \biggr) \biggr]
\biggl[ \frac{p^2 z \ell_{3D}}{\mathcal{D}} \biggl(x^2 - q^4y^2 \biggr)  \biggr]
+
\biggl[ \frac{xq^2 y p^2 z}{\mathcal{D}} \biggl( \frac{2}{p^2z} \biggr) \biggr]
\biggl[ \frac{q^2y \ell_{3D}}{\mathcal{D}}\biggl( p^4 z^2 + 2x^2 \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{x\ell_{3D}}{\mathcal{D}^2} \biggl[
(x^2 - q^4y^2) p^4z^2 
+
( p^4 z^2 + 2x^2 )2q^4y^2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{x\ell_{3D}}{\mathcal{D}^2} \biggl[
x^2p^4z^2  
+
4x^2 q^4y^2 
+
q^4y^2 p^4 z^2 \biggr]
=
x\ell_{3D} = \gamma_{11} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{43}\gamma_{51} - \gamma_{41}\gamma_{53}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{xq^2y p^2z}{\mathcal{D}} \biggl( \frac{2}{p^2 z} \biggr) \biggr]
\biggl[ \frac{x \ell_{3D}}{\mathcal{D}} \biggl(2q^4y^2 +p^4z^2\biggr)  \biggr]
-
\biggl[ \frac{xq^2 y p^2 z}{\mathcal{D}} \biggl( \frac{1}{x} \biggr) \biggr]
\biggl[ \frac{p^2 z \ell_{3D}}{\mathcal{D}}\biggl( x^2 - q^4y^2 \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{q^2 y \ell_{3D}}{\mathcal{D}^2}  \biggl[
2 x^2 (2q^4y^2 +p^4z^2 ) 
-
p^4 z^2 ( x^2 - q^4y^2) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{q^2 y \ell_{3D}}{\mathcal{D}^2}  \biggl[
4x^2 q^4y^2 + x^2 p^4z^2 + q^4y^2 p^4 z^2 
\biggr]
=
q^2y \ell_{3D}
= \gamma_{12} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{41}\gamma_{52} - \gamma_{42}\gamma_{51}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{xq^2y p^2z}{\mathcal{D}} \biggl( \frac{1}{x} \biggr) \biggr]
\biggl[ \frac{q^2 y\ell_{3D}}{\mathcal{D}} \biggl(p^4z^2 + 2x^2\biggr)  \biggr]
+
\biggl[ \frac{xq^2 y p^2 z}{\mathcal{D}} \biggl( \frac{1}{q^2 y} \biggr) \biggr]
\biggl[ \frac{x \ell_{3D}}{\mathcal{D}}\biggl( 2q^4y^2 + p^4z^2 \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{p^2 z \ell_{3D}}{\mathcal{D}^2} \biggl[
q^4y^2 (p^4z^2 + 2x^2)
+
x^2 ( 2q^4y^2 + p^4z^2 ) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{p^2 z \ell_{3D}}{\mathcal{D}^2} \biggl[
q^4y^2 p^4z^2 + 4x^2 q^4 y^2
+ x^2p^4z^2  
\biggr]
=
p^2 z \ell_{3D}
=
\gamma_{13} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\gamma_{52}\gamma_{13} - \gamma_{53}\gamma_{12}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{q^2 y p^2 z\ell_{3D}^2}{\mathcal{D}} \biggl( p^4z^2 + 2x^2 \biggr) \biggr]
-
\biggl[ \frac{q^2 y p^2 z\ell_{3D}^2}{\mathcal{D}} \biggl( x^2 - q^4y^2 \biggr)  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{q^2 y p^2 z\ell_{3D}^2}{\mathcal{D}} \biggl[ x^2 + q^4y^2 + p^4z^2 \biggr]
=
\frac{q^2 y p^2 z }{\mathcal{D}} 
=
\gamma_{41} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{53}\gamma_{11} - \gamma_{51}\gamma_{13}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{xp^2 z\ell_{3D}^2}{\mathcal{D}} \biggl( x^2 - q^4y^2 \biggr)  \biggr]
+
\biggl[ \frac{xp^2 z\ell_{3D}^2}{\mathcal{D}} \biggl( 2q^4y^2 +p^4z^2\biggr)  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{xp^2 z\ell_{3D}^2}{\mathcal{D}} \biggl[ x^2 + q^4y^2 +p^4z^2  \biggr]
=
\frac{xp^2 z}{\mathcal{D}}
=
\gamma_{42} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{51}\gamma_{12} - \gamma_{52}\gamma_{11}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\biggl[ \frac{x q^2y \ell_{3D}^2}{\mathcal{D}} \biggl( 2q^4y^2 +p^4z^2\biggr)  \biggr]
-
\biggl[ \frac{x q^2y \ell_{3D}^2}{\mathcal{D}} \biggl( p^4z^2 + 2x^2\biggr)  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{2x q^2y \ell_{3D}^2}{\mathcal{D}}\biggl[ x^2 + q^4y^2 +p^4z^2  \biggr]
=
-\frac{2x q^2y }{\mathcal{D}}
=
\gamma_{43} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\gamma_{12}\gamma_{43} - \gamma_{13}\gamma_{42}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{2x q^4 y^2 \ell_{3D} }{\mathcal{D}} \biggr]
- \biggl[ \frac{x p^4 z^2 \ell_{3D} }{\mathcal{D}} \biggr]
=
- \biggl[ \frac{x \ell_{3D} }{\mathcal{D}} \biggr] (2q^4 y^2 + p^4z^2)
= \gamma_{51} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{13}\gamma_{41} - \gamma_{11}\gamma_{43}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{q^2 y p^4z^2 \ell_{3D} }{\mathcal{D}} \biggr]
+ 
\biggl[ \frac{2x^2 q^2 y \ell_{3D} }{\mathcal{D}} \biggr]
=
\biggl[ \frac{q^2 y  \ell_{3D} }{\mathcal{D}} \biggr](p^4z^2 + 2x^2)
=
\gamma_{52} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\gamma_{11}\gamma_{42} - \gamma_{12}\gamma_{41}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{x^2 p^2z \ell_{3D}}{\mathcal{D}} \biggr]
- 
\biggl[ \frac{q^4 y^2 p^2 z\ell_{3D} }{\mathcal{D}} \biggr]
=
\biggl[ \frac{p^2z \ell_{3D}}{\mathcal{D}} \biggr](x^2 - q^4y^2)
=
\gamma_{53} \, .
</math>
&nbsp; &nbsp; &nbsp; <font color="red">Yes!</font>
  </td>
</tr>
</table>

Given that the prescribed interrelationships between all nine direction cosines are satisfied, we conclude that the <math>~(\kappa_1, \kappa_4, \kappa_5)</math> coordinate system is an orthogonal one.  Accordingly, the position vector is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathbf{\vec{x}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 (\gamma_{11} x + \gamma_{12} y + \gamma_{13} z)
+
\mathbf{\hat{e}}_2 (\gamma_{41} x + \gamma_{42} y + \gamma_{43} z)
+
\mathbf{\hat{e}}_3 (\gamma_{51} x + \gamma_{52} y + \gamma_{53} z) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 \biggl\{
x^2\ell_{3D} 
+ q^2y^2 \ell_{3D} 
+ p^2z^2\ell_{3D}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\mathbf{\hat{e}}_2 \biggl\{
\frac{xq^2yp^2z}{\mathcal{D}} 
+ \frac{x y p^2z}{\mathcal{D}}
- \frac{2 xq^2y z}{\mathcal{D}}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\mathbf{\hat{e}}_3 \biggl\{
- \frac{\ell_{3D} x^2}{\mathcal{D}} (2q^4y^2 + p^4z^2)
+ \frac{\ell_{3D} q^2 y^2}{\mathcal{D}} (p^4z^2 + 2x^2) 
+ \frac{\ell_{3D} p^2z^2}{\mathcal{D}} (x^2 - q^4y^2) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 \biggl\{
\kappa_1^2 \ell_{3D}
\biggr\}
+
\mathbf{\hat{e}}_2 \biggl\{
q^2p^2 + p^2 - 2 q^2 
\biggr\} \frac{xyz}{\mathcal{D}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\mathbf{\hat{e}}_3 \biggl\{
- x^2 (2q^4y^2 + p^4z^2)
+ q^2 y^2 (p^4z^2 + 2x^2) 
+ p^2z^2 (x^2 - q^4y^2) 
\biggr\} \frac{\ell_{3D}}{\mathcal{D}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{\hat{e}}_1 \biggl\{
\kappa_1^2 \ell_{3D}
\biggr\}
~+~
\mathbf{\hat{e}}_2 \biggl\{
q^2p^2 + p^2 - 2 q^2 
\biggr\} \frac{xyz}{\mathcal{D}}
~-~
\mathbf{\hat{e}}_3 \biggl\{ 2 x^2 q^2y^2(q^2-1) + x^2 p^2z^2(p^2-1) + q^2 y^2 p^2 z^2(q^2 - p^2)
\biggr\} \frac{\ell_{3D}}{\mathcal{D}} \, .
</math>
  </td>
</tr>
</table>

=See Also=

<ul>
  <li>
[[User:Tohline/Appendix/Ramblings/DirectionCosines|Direction Cosines]]
  </li>
  <li>[[User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates#Background|Trials up through T7 Coordinates]]</li>
</ul>


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