Editing 3Dconfigurations/RiemannEllipsoids (section)

== Step 2==

Given that <math>(x, y, z)</math> and <math>(\xi, \eta, \zeta)</math> are components of the same position vector, <math>\vec{A}</math>, it must in general be the case that the square of the length of the vector is the same for both coordinate representations.  That is, as Riemann (1861) states in &sect;1 following equation (2), it must be the case that,
<div align="center">
<math> \xi^2 + \eta^2 + \zeta^2 = x^2 + y^2 + z^2 \, .</math>
</div>
Let's specifically assess whether or not this holds true when Euler angles are used to relate the components of the two position vector expressions.

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

<tr>
  <td align="right">
<math>\xi^2 + \eta^2 + \zeta^2</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\biggl[ x(\cos\psi \cos\phi - \sin\psi \sin\phi \cos\theta)
+ y(\sin\phi \cos\psi + \sin\psi \cos\theta \cos\phi)
+ z(\sin\psi \sin\theta) 
\biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+
\biggl[ x(-\sin\psi \cos\phi - \sin\phi \cos\theta \cos\psi)
+ y( - \sin\psi \sin\phi + \cos\psi \cos\theta\cos\phi )
+ z (\sin\theta \cos\psi) \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ \biggl[
x (\sin\theta\sin\phi)
+ y (-\sin\theta \cos\phi)
+ z (\cos\theta) \biggr]^2 \, .
</math>
  </td>
</tr>
</table>

As Riemann (1861) states, <font color="orange">between these coefficients, six equations hold &hellip;</font>. In particular, when they are written in terms of Euler angles, on the right-hand side, the coefficient of ''the six'' various terms is &hellip;

<table border="0" align="center" cellpadding="5" width="90%">

<tr>
  <td align="right" width="10%">
<math>x^2:</math>
  </td>
  <td align="left">
<math>
(\cos\psi \cos\phi - \sin\psi \sin\phi \cos\theta)^2 
+ (-\sin\psi \cos\phi - \sin\phi \cos\theta \cos\psi)^2
+ (\sin\theta\sin\phi)^2
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>=
\cos^2\psi \cos^2\phi - 2\cos\psi \cos\phi \sin\psi \sin\phi \cos\theta + \sin^2\psi \sin^2\phi \cos^2\theta 
+ \sin^2\psi \cos^2\phi 
+ 2\sin\psi \cos\phi\sin\phi \cos\theta \cos\psi + \sin^2\phi \cos^2\theta \cos^2\psi
+ \sin^2\theta \sin^2\phi
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>=
( 2\sin\psi \cos\phi\sin\phi \cos\theta \cos\psi - 2\cos\psi \cos\phi \sin\psi \sin\phi \cos\theta) 
+ (\sin^2\psi+ \cos^2\psi) \sin^2\phi \cos^2\theta  
+ (\sin^2 + \cos^2\psi )\cos^2\phi 
+ \sin^2\theta \sin^2\phi
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>=
\sin^2\phi \cos^2\theta  + \cos^2\phi + \sin^2\theta \sin^2\phi
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>= 1 \, .
</math>
  </td>
</tr>
</table>

<table border="0" align="center" cellpadding="5" width="90%">

<tr>
  <td align="right" width="10%">
<math>y^2:</math>
  </td>
  <td align="left">
<math>
(\sin\phi \cos\psi + \sin\psi \cos\theta \cos\phi )^2 
+ (- \sin\psi \sin\phi + \cos\psi \cos\theta\cos\phi  )^2
+ (-\sin\theta \cos\phi )^2
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>=
\sin^2\phi \cos^2\psi + 2\sin\phi \cos\psi \sin\psi \cos\theta \cos\phi + \sin^2\psi \cos^2\theta \cos^2\phi
+ \sin^2\psi \sin^2\phi - 2\sin\psi \sin\phi \cos\psi \cos\theta\cos\phi + \cos^2\psi \cos^2\theta \cos^2\phi 
+ \sin^2\theta \cos^2\phi
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>=
(2\sin\phi \cos\psi \sin\psi \cos\theta \cos\phi - 2\sin\psi \sin\phi \cos\psi \cos\theta\cos\phi ) 
+ (\cos^2\psi + \sin^2\psi )\sin^2\phi  
+ (\sin^2\psi  + \cos^2\psi) \cos^2\theta \cos^2\phi 
+ \sin^2\theta \cos^2\phi
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>=
\sin^2\phi  + \cos^2\theta \cos^2\phi
+ \sin^2\theta \cos^2\phi
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>= 1 \, .
</math>
  </td>
</tr>
</table>

<table border="0" align="center" cellpadding="5" width="90%">

<tr>
  <td align="right" width="10%">
<math>z^2:</math>
  </td>
  <td align="left">
<math>
(\sin\psi \sin\theta )^2 + (\sin\theta \cos\psi )^2 + ( \cos\theta)^2
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>= 1 \, .
</math>
  </td>
</tr>
</table>

<table border="0" align="center" cellpadding="5" width="90%">

<tr>
  <td align="right" width="10%">
<math>2xy:</math>
  </td>
  <td align="left">
<math>
(\cos\psi \cos\phi - \sin\psi \sin\phi \cos\theta)(\sin\phi \cos\psi + \sin\psi \cos\theta \cos\phi)
+ (-\sin\psi \cos\phi - \sin\phi \cos\theta \cos\psi)( - \sin\psi \sin\phi + \cos\psi \cos\theta\cos\phi )
+ (\sin\theta\sin\phi)(-\sin\theta \cos\phi)
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>=
\cos\psi \cos\phi (\sin\phi \cos\psi + \sin\psi \cos\theta \cos\phi)
- \sin\psi \sin\phi \cos\theta(\sin\phi \cos\psi + \sin\psi \cos\theta \cos\phi)
+ (\sin\psi \cos\phi + \sin\phi \cos\theta \cos\psi)( \sin\psi \sin\phi - \cos\psi \cos\theta\cos\phi )
- \sin^2\theta\sin\phi\cos\phi
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>=
\cos^2\psi (\sin\phi \cos\phi) + \sin\psi \cos\psi \cos\theta (-\sin^2\phi + \cos^2\phi)
- \sin^2\psi \cos^2\theta( \sin\phi\cos\phi)
+ \sin\psi \cos\phi ( \sin\psi \sin\phi )
- \sin\psi \cos\phi ( \cos\psi \cos\theta\cos\phi )
+ \sin\phi \cos\theta \cos\psi( \sin\psi \sin\phi )
- \sin\phi \cos\theta \cos\psi(\cos\psi \cos\theta\cos\phi )
- \sin^2\theta\sin\phi\cos\phi
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>=
[\sin\psi ( \cos\psi)\cos\theta (\sin^2\phi - \cos^2\phi)
+ \sin\psi \cos\psi \cos\theta (-\sin^2\phi + \cos^2\phi)]
+ \sin\phi \cos\phi[\sin^2\psi (1 - \cos^2\theta)
- \cos^2\theta \cos^2\psi- \sin^2\theta+\cos^2\psi] 
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>=
\sin\phi \cos\phi[\sin^2\psi (1 - \cos^2\theta)- \sin^2\theta \sin^2\psi ] 
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>=
\sin\phi \cos\phi\sin^2\psi[ 1 - \cos^2\theta - \sin^2\theta  ] 
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>= 0 \, .
</math>
  </td>
</tr>
</table>

<table border="0" align="center" cellpadding="5" width="90%">

<tr>
  <td align="right" width="10%">
<math>2xz:</math>
  </td>
  <td align="left">
<math>
(\cos\psi \cos\phi - \sin\psi \sin\phi \cos\theta)(\sin\psi \sin\theta)
+ (-\sin\psi \cos\phi - \sin\phi \cos\theta \cos\psi)(\sin\theta \cos\psi)
+ (\sin\theta\sin\phi)\cos\theta
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>=
\cos\psi \cos\phi (\sin\psi \sin\theta)
- \sin\psi \sin\phi \cos\theta(\sin\psi \sin\theta)
-\sin\psi \cos\phi (\sin\theta \cos\psi)
- \sin\phi \cos\theta \cos\psi(\sin\theta \cos\psi)
+ (\sin\theta\sin\phi)\cos\theta
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>=
\cos\psi \sin\psi [\cos\phi \sin\theta
- \cos\phi \sin\theta ]
- \sin^2\psi \sin\phi \cos\theta \sin\theta
+ \sin\theta\sin\phi \cos\theta [1
- \cos^2\psi]
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>= 0 \, .
</math>
  </td>
</tr>
</table>

<table border="0" align="center" cellpadding="5" width="90%">

<tr>
  <td align="right" width="10%">
<math>2yz:</math>
  </td>
  <td align="left">
<math>
(\sin\phi \cos\psi + \sin\psi \cos\theta \cos\phi)\sin\psi \sin\theta
+ ( - \sin\psi \sin\phi + \cos\psi \cos\theta\cos\phi )\sin\theta \cos\psi
+ (-\sin\theta \cos\phi)\cos\theta
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>=
[\sin\psi \cos\psi \sin\phi \sin\theta
- \sin\psi \cos\psi \sin\phi  \sin\theta ]
+ \sin^2\psi \cos\theta \cos\phi \sin\theta
+ (\cos^2\psi -1 )\cos\theta\cos\phi \sin\theta 
</math>
  </td>
</tr>

<tr>
  <td align="right" width="10%">
&nbsp;
  </td>
  <td align="left">
<math>= 0 \, .
</math>
  </td>
</tr>
</table>