Editing Appendix/Ramblings/T6CoordinatesPt3 (section)

=COLLADA=
Here we try to use the 3D-visualization capabilities of COLLADA to test whether or not the three coordinates associated with the T6 Coordinate system are indeed orthogonal to one another.  We begin by making a copy of the '''Inertial17.dae''' text file, which we obtain from [[ThreeDimensionalConfigurations/MeetsCOLLADAandOculusRiftS#The_COLLADA_Code_.26_Initial_3D_Scene|an accompanying discussion]].  When viewed with the Mac's '''Preview''' application, this group of COLLADA-based instructions displays a purple ellipsoid with axis ratios, (b/a, c/a) = (0.41, 0.385).  This means that we are dealing with an ellipsoid for which,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>q \equiv \frac{a}{b}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2.44</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~p \equiv \frac{a}{c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2.60 \, .</math>
  </td>
</tr>
</table>

<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^2 y^2 + p^2 z^2)^{1 / 2} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\lambda_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{x y^{1/q^2}}{ z^{2/p^2}} \, ;
</math>
  </td>
</tr>

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

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

== First Trial==
<table border="1" align="center" width="80%" cellpadding="8">
<tr>
  <td align="center" colspan="6">'''First Trial'''<br />(specified variable values have bgcolor="pink")</td>
</tr>
<tr>
  <td align="center">x</td>
  <td align="center">y</td>
  <td align="center">z</td>
  <td align="center"><math>~\lambda_1</math></td>
  <td align="center"><math>~\ell_{3D}</math></td>
  <td align="center"><math>~\mathcal{D}</math></td>
</tr>
<tr>
  <td align="center" bgcolor="pink">0.5</td>
  <td align="center">0.35493</td>
  <td align="center" bgcolor="pink">0.00000</td>
  <td align="center" bgcolor="pink">1</td>
  <td align="center">0.46052</td>
  <td align="center">2.11310</td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\hat{e}_1 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath (x_0 \ell_{3D}) + \hat\jmath (q^2y_0 \ell_{3D}) + \hat{k} (p^2 z_0 \ell_{3D})
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat\imath (0.23026) + \hat\jmath (0.97313)  \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\hat{e}_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{x_0 q^2 y_0 p^2 z_0}{\mathcal{D}} \biggl\{
\hat{\imath} \biggl( \frac{1}{x_0} \biggr)
+ \hat{\jmath}  \biggl( \frac{1}{q^2 y_0} \biggr)
+ \hat{k} \biggl( -\frac{2}{p^2 z_0} \biggr)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\hat{k} ~\biggl(  \frac{2x_0 q^2 y_0  }{\mathcal{D}} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\hat{k} ~\biggl(  1 \biggr)
\ ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\hat{e}_3 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\ell_{3D}}{\mathcal{D}} 
\biggl\{
-\hat\imath \biggl[ x_0(2 q^4y_0^2 + p^4z_0^2 ) \biggl]
+ \hat\jmath \biggl[ q^2 y_0(p^4z_0^2 + 2x_0^2 )  \biggl]
+ \hat{k} \biggl[ p^2z_0( x_0^2 - q^4y_0^2 ) \biggl]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2q^2 x_0 y_0\ell_{3D}}{\mathcal{D}} 
\biggl\{
-\hat\imath ( q^2y_0 ) 
+ \hat\jmath  (x_0) 
\biggr\}
=
\biggl(1\biggr)\ell_{3D}\biggl\{ -\hat\imath ( q^2y_0 ) + \hat\jmath  (x_0) \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\hat\imath (0.97313 ) + \hat\jmath  (0.23026) \, .
</math>
  </td>
</tr>
</table>

===Tangent Plane===

From our [[Appendix/Ramblings/ConcentricEllipsoidalCoordinates#Other_Coordinate_Pair_in_the_Tangent_Plane|above derivation]], the plane that is tangent to the ellipsoid's surface at <math>~(x_0, y_0, z_0)</math> is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
x x_0  + q^2 y y_0  + p^2 z z_0 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(\lambda_1^2)_0 \, .
</math>
  </td>
</tr>
</table>
For this ''First Trial,'' we have (for all values of <math>~z</math>, given that <math>~z_0 = 0</math>) &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
(0.5)x   + (2.11310)y   
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ y   
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(1 - 0.5x)}{2.11310} \, .
</math>
  </td>
</tr>
</table>
So let's plot a segment of the tangent plane whose four corners are given by the coordinates,
<table border="1" cellpadding="5" align="center">
<tr>
  <td align="center">Corner</td>
  <td align="center">x</td>
  <td align="center">y</td>
  <td align="center">z</td>
</tr>
<tr>
  <td align="center">A</td>
  <td align="center" bgcolor="pink">x_0 - 0.25 = +0.25</td>
  <td align="center">0.41408</td>
  <td align="center" bgcolor="pink">-0.25</td>
</tr>
<tr>
  <td align="center">B</td>
  <td align="center" bgcolor="pink">x_0 + 0.25 = +0.75</td>
  <td align="center">0.29577</td>
  <td align="center" bgcolor="pink">-0.25</td>
</tr>
<tr>
  <td align="center">C</td>
  <td align="center" bgcolor="pink">x_0 - 0.25 = +0.25</td>
  <td align="center">0.41408</td>
  <td align="center" bgcolor="pink">+0.25</td>
</tr>
<tr>
  <td align="center">D</td>
  <td align="center" bgcolor="pink">x_0 + 0.25 = +0.75</td>
  <td align="center">0.29577</td>
  <td align="center" bgcolor="pink">+0.25</td>
</tr>
</table>
Now, in order to give some thickness to this tangent-plane, let's adjust the four corner locations by a distance of <math>~\pm 0.1</math> in the <math>~\hat{e}_1</math> direction.

===Eight Corners of Tangent Plane===

Corner 1:  Shift surface-point location <math>~(x_0, y_0, z_0)</math> by <math>~(+\Delta e_1)</math> in the <math>~\hat{e}_1</math> direction,  by <math>~(+\Delta e_2)</math> in the <math>~\hat{e}_2</math> direction, and by  by <math>~(+\Delta e_3)</math> in the <math>~\hat{e}_3</math> direction.  This gives &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x_0 + (\Delta e_1)0.23026 - (\Delta e_2)0.97313</math>
  </td>
</tr>
</table>

==Second Trial==
<table border="1" align="center" width="80%" cellpadding="8">
<tr>
  <td align="center" colspan="6">'''Second Trial''' &hellip; <math>~(q = 2.44, p = 2.60)</math><br />[specified variable values have bgcolor="pink"]</td>
</tr>
<tr>
  <td align="center">x_0</td>
  <td align="center">y_0</td>
  <td align="center">z_0</td>
  <td align="center"><math>~\lambda_1</math></td>
  <td align="center"><math>~\ell_{3D}</math></td>
  <td align="center"><math>~\mathcal{D}</math></td>
</tr>
<tr>
  <td align="center" bgcolor="pink">0.5</td>
  <td align="center">0.35493</td>
  <td align="center" bgcolor="pink">0.00000</td>
  <td align="center" bgcolor="pink">1</td>
  <td align="center">0.46052</td>
  <td align="center">2.11310</td>
</tr>
</table>

===Generic Unit Vector Expressions===

Let's adopt the notation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{e}_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\hat\imath ~[e_{ix}] + \hat\jmath ~[e_{iy}] + \hat{k} ~[e_{iz}]</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; for, &nbsp; &nbsp; &nbsp;</td>
  <td align="center"><math>~i = 1,3 \, .</math></td>
</tr>
</table>

Then, for the T6 Coordinate system, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~e_{1x}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~x_0 \ell_{3D} \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{1y}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~q^2y_0 \ell_{3D} \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{1z}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~p^2 z_0 \ell_{3D} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~e_{2x}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \frac{q^2 y_0 p^2 z_0}{\mathcal{D}}\, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{2y}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{x_0 p^2 z_0}{\mathcal{D}} \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{2z}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{2x_0 q^2 y_0}{\mathcal{D}}\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~e_{3x}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~-x_0(2 q^4y_0^2 + p^4z_0^2 )\frac{\ell_{3D}}{\mathcal{D}}  \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{3y}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~q^2 y_0(p^4z_0^2 + 2x_0^2 )\frac{\ell_{3D}}{\mathcal{D}}   \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{3z}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~p^2z_0( x_0^2 - q^4y_0^2 )\frac{\ell_{3D}}{\mathcal{D}} \, .</math>
  </td>
</tr>

</table>


<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="4">'''Second Trial'''</td>
</tr>
<tr>
  <td align="center"> &nbsp;</td>
  <td align="center"> x</td>
  <td align="center"> y</td>
  <td align="center"> z</td>
</tr>
<tr>
  <td align="center"> <math>~e_1</math></td>
  <td align="center">0.23026</td>
  <td align="center">0.97313</td>
  <td align="center"> 0.0</td>
</tr>
<tr>
  <td align="center"> <math>~e_2</math></td>
  <td align="center">0.0</td>
  <td align="center">0.0</td>
  <td align="center">-1.0</td>
</tr>
<tr>
  <td align="center"> <math>~e_3</math></td>
  <td align="center">- 0.97313</td>
  <td align="center">0.23026</td>
  <td align="center"> 0.0</td>
</tr>
</table>


What are the coordinates of the eight corners of a thin tangent-plane?  Let's say that we want the plane to extend &hellip;
<ul>
  <li>From <math>~(-\Delta_1)</math> to <math>~(+\Delta_1)</math> in the <math>~\hat{e}_1</math> direction &hellip; here we set <math>~\Delta_1 = 0.05</math>;</li>
  <li>From <math>~(-\Delta_2)</math> to <math>~(+\Delta_2)</math> in the <math>~\hat{e}_2</math> direction &hellip; here we set <math>~\Delta_2 = 0.25</math>;</li>
  <li>From <math>~(-\Delta_3)</math> to <math>~(+\Delta_3)</math> in the <math>~\hat{e}_3</math> direction &hellip; here we set <math>~\Delta_3 = 0.25</math>.</li>
</ul>

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

<tr>
  <td align="right">
<math>~\Delta_x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Delta_1 e_{1x} + \Delta_2 e_{2x} + \Delta_3 e_{3x} = -0.23177 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Delta_y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Delta_1 e_{1y} + \Delta_2 e_{2y} + \Delta_3 e_{3y} = +0.10622 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Delta_z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Delta_1 e_{1z} + \Delta_2 e_{2z} + \Delta_3 e_{3z} = -0.25000 \, .</math>
  </td>
</tr>
</table>

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="left">[[File:TangentPlaneSchematic.png|Tangent Plane Schematic]]</td>
  <td align="left">
<table border="1" cellpadding="5" align="center">
  <tr>
    <td align="center">vertex</td>
    <td align="center">x</td>
    <td align="center">y</td>
    <td align="center">z</td>
    <td align="center" rowspan="9" bgcolor="lightgray">&nbsp;</td>
    <td align="center">x</td>
    <td align="center">y</td>
    <td align="center">z</td>
  </tr>
  <tr>
    <td align="center">0<sup>&dagger;</sup></td>
    <td align="center"><math>~x_0 - |\Delta_x|</math></td>
    <td align="center"><math>~y_0 - |\Delta_y|</math></td>
    <td align="center"><math>~z_0 - |\Delta_z|</math></td>
    <td align="center">0.26823</td>
    <td align="center">0.24871</td>
    <td align="center">-0.25</td>
  </tr>
  <tr>
    <td align="center">1</td>
    <td align="center"><math>~x_0 - |\Delta_x|</math></td>
    <td align="center"><math>~y_0 + |\Delta_y|</math></td>
    <td align="center"><math>~z_0 - |\Delta_z|</math></td>
    <td align="center">0.26823</td>
    <td align="center">0.46115</td>
    <td align="center">-0.25</td>
  </tr>
  <tr>
    <td align="center">2</td>
    <td align="center"><math>~x_0 - |\Delta_x|</math></td>
    <td align="center"><math>~y_0 - |\Delta_y|</math></td>
    <td align="center"><math>~z_0 + |\Delta_z|</math></td>
    <td align="center">0.26823</td>
    <td align="center"> 0.24871 </td>
    <td align="center">0.25</td>
  </tr>
  <tr>
    <td align="center">3</td>
    <td align="center"><math>~x_0 - |\Delta_x|</math></td>
    <td align="center"><math>~y_0 + |\Delta_y|</math></td>
    <td align="center"><math>~z_0 + |\Delta_z|</math></td>
    <td align="center">0.26823</td>
    <td align="center"> 0.46115 </td>
    <td align="center">0.25</td>
  </tr>
  <tr>
    <td align="center">4</td>
    <td align="center"><math>~x_0 + |\Delta_x|</math></td>
    <td align="center"><math>~y_0 - |\Delta_y|</math></td>
    <td align="center"><math>~z_0 - |\Delta_z|</math></td>
    <td align="center">0.73177</td>
    <td align="center"> 0.24871 </td>
    <td align="center">-0.25</td>
  </tr>
  <tr>
    <td align="center">5</td>
    <td align="center"><math>~x_0 + |\Delta_x|</math></td>
    <td align="center"><math>~y_0 + |\Delta_y|</math></td>
    <td align="center"><math>~z_0 - |\Delta_z|</math></td>
    <td align="center"> 0.73177 </td>
    <td align="center"> 0.46115 </td>
    <td align="center">-0.25</td>
  </tr>
  <tr>
    <td align="center">6</td>
    <td align="center"><math>~x_0 + |\Delta_x|</math></td>
    <td align="center"><math>~y_0 - |\Delta_y|</math></td>
    <td align="center"><math>~z_0 + |\Delta_z|</math></td>
    <td align="center"> 0.73177 </td>
    <td align="center"> 0.24871 </td>
    <td align="center">0.25</td>
  </tr>
  <tr>
    <td align="center">7</td>
    <td align="center"><math>~x_0 + |\Delta_x|</math></td>
    <td align="center"><math>~y_0 + |\Delta_y|</math></td>
    <td align="center"><math>~z_0 + |\Delta_z|</math></td>
    <td align="center"> 0.73177 </td>
    <td align="center"> 0.46115 </td>
    <td align="center">0.25</td>
  </tr>
</table>
  </td>
</tr>
<tr>
  <td align="left" colspan="2">
<sup>&dagger;</sup>In the figure on the left, vertex 0 is hidden from view behind the (yellow) solid rectangle.
  </td>
</tr>
</table>

==Third Trial==
===GoodPlane01===
<table border="1" align="center" width="80%" cellpadding="8">
<tr>
  <td align="center" colspan="6">'''Third Trial''' &hellip; <math>~(q = 2.44, p = 2.60)</math><br />[specified variable values have bgcolor="pink"]</td>
</tr>
<tr>
  <td align="center">x_0</td>
  <td align="center">y_0</td>
  <td align="center">z_0</td>
  <td align="center"><math>~\lambda_1</math></td>
  <td align="center"><math>~\ell_{3D}</math></td>
  <td align="center"><math>~\mathcal{D}</math></td>
</tr>
<tr>
  <td align="center" bgcolor="pink">0.8</td>
  <td align="center">0.24600</td>
  <td align="center" bgcolor="pink">0.00000</td>
  <td align="center" bgcolor="pink">1</td>
  <td align="center">0.59959</td>
  <td align="center">2.34146</td>
</tr>
</table>

Again, for the T6 Coordinate system, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~e_{1x}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~x_0 \ell_{3D} \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{1y}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~q^2y_0 \ell_{3D} \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{1z}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~p^2 z_0 \ell_{3D} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~e_{2x}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \frac{q^2 y_0 p^2 z_0}{\mathcal{D}}\, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{2y}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{x_0 p^2 z_0}{\mathcal{D}} \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{2z}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{2x_0 q^2 y_0}{\mathcal{D}}\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~e_{3x}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~-x_0(2 q^4y_0^2 + p^4z_0^2 )\frac{\ell_{3D}}{\mathcal{D}}  \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{3y}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~q^2 y_0(p^4z_0^2 + 2x_0^2 )\frac{\ell_{3D}}{\mathcal{D}}   \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~e_{3z}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~p^2z_0( x_0^2 - q^4y_0^2 )\frac{\ell_{3D}}{\mathcal{D}} \, .</math>
  </td>
</tr>

</table>


<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="5">'''Third Trial'''</td>
</tr>
<tr>
  <td align="center"> &nbsp;</td>
  <td align="center"> x</td>
  <td align="center"> y</td>
  <td align="center"> z</td>
  <td align="center"> <math>~\Delta_\mathrm{TP}</math></td>
</tr>
<tr>
  <td align="center"> <math>~e_1</math></td>
  <td align="center">0.47967</td>
  <td align="center">0.87745</td>
  <td align="center"> 0.0</td>
  <td align="center"> 0.02</td>
</tr>
<tr>
  <td align="center"> <math>~e_2</math></td>
  <td align="center">0.0</td>
  <td align="center">0.0</td>
  <td align="center">-1.0</td>
  <td align="center"> 0.25</td>
</tr>
<tr>
  <td align="center"> <math>~e_3</math></td>
  <td align="center">- 0.87753</td>
  <td align="center"> 0.47952 </td>
  <td align="center">0.0</td>
  <td align="center"> 0.25</td>
</tr>
</table>

In constructing the Tangent-Plane (TP) for a 3D COLLADA display, we first move from the point that is on the surface of the ellipsoid, <math>~\vec{x}_0 = (x_0, y_0, z_0) = (0.8, 0.246, 0.0)</math>, to

<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" rowspan="2">vertex <br />"m"</td>
  <td align="center" rowspan="2"><math>~\vec{P}_m</math></td>
  <td align="center" colspan="3">Components</td>
</tr>
<tr>
  <td align="center"><math>~x_m = \hat\imath \cdot \vec{P}_m</math></td>
  <td align="center"><math>~y_m = \hat\jmath \cdot \vec{P}_m</math></td>
  <td align="center"><math>~z_m = \hat{k} \cdot \vec{P}_m</math></td>
</tr>

<tr>
  <td align="center">0</td>
  <td align="center"><math>~\vec{x}_0 - \Delta_1\hat{e}_1 - \Delta_2\hat{e}_2 - \Delta_3\hat{e}_3</math></td>
  <td align="center">
<math>~
x_0 - \Delta_1 e_{1x} - \Delta_2 e_{2x} - \Delta_3 e_{3x}
</math>
  </td>
  <td align="center">
<math>~
y_0 - \Delta_1 e_{1y} - \Delta_2 e_{2y} - \Delta_3 e_{3y}
</math>
  </td>
  <td align="center">
<math>~
z_0 - \Delta_1 e_{1z} - \Delta_2 e_{2z} - \Delta_3 e_{3z}
</math>
  </td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">
0.8 - 0.02 (0.47952)  - 0.25 (-0.87752) = 1.00979  
  </td>
  <td align="center">
0.24590 - 0.02 (0.87753)  - 0.25 (0.47952) = 0.10847
  </td>
  <td align="center">
- 0.25 (-1.0) = + 0.25
  </td>
</tr>

<tr>
  <td align="center">1</td>
  <td align="center"><math>~\vec{x}_0 - \Delta_1\hat{e}_1 + \Delta_2\hat{e}_2 - \Delta_3\hat{e}_3</math></td>
  <td align="center">
<math>~
x_0 - \Delta_1 e_{1x} + \Delta_2 e_{2x} - \Delta_3 e_{3x}
</math>
  </td>
  <td align="center">
<math>~
y_0 - \Delta_1 e_{1y} + \Delta_2 e_{2y} - \Delta_3 e_{3y}
</math>
  </td>
  <td align="center">
<math>~
z_0 - \Delta_1 e_{1z} + \Delta_2 e_{2z} - \Delta_3 e_{3z}
</math>
  </td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">
0.8 - 0.02 (0.47952)  - 0.25 (-0.87752) =  1.00979
  </td>
  <td align="center">
0.24590 - 0.02 (0.87753)  - 0.25 (0.47952) = 0.10847
  </td>
  <td align="center">
+ 0.25 (-1.0) = - 0.25
  </td>
</tr>

<tr>
  <td align="center">2</td>
  <td align="center"><math>~\vec{x}_0 - \Delta_1\hat{e}_1 - \Delta_2\hat{e}_2 + \Delta_3\hat{e}_3</math></td>
  <td align="center">
<math>~
x_0 - \Delta_1 e_{1x} - \Delta_2 e_{2x} + \Delta_3 e_{3x}
</math>
  </td>
  <td align="center">
<math>~
y_0 - \Delta_1 e_{1y} - \Delta_2 e_{2y} + \Delta_3 e_{3y}
</math>
  </td>
  <td align="center">
<math>~
z_0 - \Delta_1 e_{1z} - \Delta_2 e_{2z} + \Delta_3 e_{3z}
</math>
  </td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">
0.8 - 0.02 (0.47952)  + 0.25 (-0.87752) =   0.56307
  </td>
  <td align="center">
0.24590 - 0.02 (0.87753)  + 0.25 (0.47952) = 0.34823
  </td>
  <td align="center">
- 0.25 (-1.0) = + 0.25
  </td>
</tr>

<tr>
  <td align="center">3</td>
  <td align="center"><math>~\vec{x}_0 - \Delta_1\hat{e}_1 + \Delta_2\hat{e}_2 + \Delta_3\hat{e}_3</math></td>
  <td align="center">
<math>~
x_0 - \Delta_1 e_{1x} + \Delta_2 e_{2x} + \Delta_3 e_{3x}
</math>
  </td>
  <td align="center">
<math>~
y_0 - \Delta_1 e_{1y} + \Delta_2 e_{2y} + \Delta_3 e_{3y}
</math>
  </td>
  <td align="center">
<math>~
z_0 - \Delta_1 e_{1z} + \Delta_2 e_{2z} + \Delta_3 e_{3z}
</math>
  </td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="center">
0.8 - 0.02 (0.47952)  + 0.25 (-0.87752) = 0.57103
  </td>
  <td align="center">
0.24590 - 0.02 (0.87753)  + 0.25 (0.47952) = 0.34823
  </td>
  <td align="center">
+ 0.25 (-1.0) = - 0.25
  </td>
</tr>

<tr>
  <td align="center">4</td>
  <td align="center">&nbsp;</td>
  <td align="center">
0.8 + 0.02 (0.47952)  - 0.25 (-0.87752) =   1.0290
  </td>
  <td align="center">
0.24590 + 0.02 (0.87753)  - 0.25 (0.47952) = 0.1436
  </td>
  <td align="center">
- 0.25 (-1.0) = + 0.25
  </td>
</tr>

<tr>
  <td align="center">5</td>
  <td align="center">&nbsp;</td>
  <td align="center">
0.8 + 0.02 (0.47952)  - 0.25 (-0.87752) =  1.0290
  </td>
  <td align="center">
0.24590 + 0.02 (0.87753)  - 0.25 (0.47952) = 0.1436
  </td>
  <td align="center">
+ 0.25 (-1.0) = - 0.25
  </td>
</tr>

<tr>
  <td align="center">6</td>
  <td align="center">&nbsp;</td>
  <td align="center">
0.8 + 0.02 (0.47952)  + 0.25 (-0.87752) =   0.59021
  </td>
  <td align="center">
0.24590 + 0.02 (0.87753)  + 0.25 (0.47952) = 0.38333
  </td>
  <td align="center">
- 0.25 (-1.0) = + 0.25
  </td>
</tr>

<tr>
  <td align="center">7</td>
  <td align="center">&nbsp;</td>
  <td align="center">
0.8 + 0.02 (0.47952)  + 0.25 (-0.87752) = 0.59021
  </td>
  <td align="center">
0.24590 + 0.02 (0.87753)  + 0.25 (0.47952) = 0.38333
  </td>
  <td align="center">
+ 0.25 (-1.0) = - 0.25
  </td>
</tr>
</table>


<table border="1" align="center" cellpadding="8">
<tr>
  <td align="left">[[File:TangentPlaneSchematic.png|Tangent Plane Schematic]]</td>
  <td align="left">[[File:ExcelVertices080.png|Vertex Locations via Excel]]</td>
</tr>
<tr>
<td align="center" colspan="2"><math>~x_0 = 0.8, z_0 = 0.0, y_0 = 0.246, \lambda_1 = 1.0</math></td>
</tr>
<tr>
  <td align="center" colspan="2" bgcolor="lightgray">[[File:GoodPlane01.png|400px|Tangent Plane Schematic]]</td>
</tr>
<tr>
<td align="center" colspan="2"><math>~\Delta_1 = 0.02, \Delta_2 = 0.25, \Delta_3 = 0.25</math></td>
</tr>
</table>

===GoodPlane02===

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center"><math>~x_0 = 0.075, z_0 = 0.0, y_0 = 0.4089, \lambda_1 = 1.0</math></td>
</tr>
<tr>
  <td align="left" bgcolor="lightgray">[[File:GoodPlane02.png|400px|Tangent Plane Schematic]]</td>
</tr>
<tr>
<td align="center" colspan="2"><math>~\Delta_1 = 0.02, \Delta_2 = 0.25, \Delta_3 = 0.25</math></td>
</tr>
</table>

===GoodPlane03===

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="2"><math>~x_0 = 0.25, z_0 = 0.20, y_0 = 0.33501, \lambda_1 = 1.0</math></td>
</tr>
<tr>
  <td align="left" bgcolor="lightgray">[[File:GoodPlane03.png|400px|Tangent Plane Schematic]]</td>
  <td align="left" bgcolor="lightgray">[[File:GoodPlane03B.png|400px|Tangent Plane Schematic]]</td>
</tr>
<tr>
<td align="center" colspan="1"><math>~\Delta_1 = 0.02, \Delta_2 = 0.25, \Delta_3 = 0.25</math></td>
<td align="center" colspan="1"><math>~\Delta_1 = 0.02, \Delta_2 = 0.10, \Delta_3 = 0.25</math></td>
</tr>
<tr>
  <td align="left" colspan="2">
CAPTION: &nbsp; The image on the right differs from the image on the left in only one way &#8212; <math>~\Delta_2</math> = 0.1 instead of 0.25.  It illustrates more clearly that the <math>~\hat{e}_3</math> (longest) coordinate axis is not parallel to the z-axis when <math>~z_0 \ne 0.</math>
  </td>
</tr>
</table>

===GoodPlane04===

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="1"><math>~x_0 = 0.25, z_0 = 1/3, y_0 = 0.1777, \lambda_1 = 1.0</math></td>
</tr>
<tr>
  <td align="left" bgcolor="lightgray">[[File:GoodPlane04A.png|400px|Tangent Plane Schematic]]</td>
</tr>
<tr>
<td align="center" colspan="1"><math>~\Delta_1 = 0.02, \Delta_2 = 0.10, \Delta_3 = 0.25</math></td>
</tr>
</table>