Editing 3Dconfigurations/RiemannEllipsoids (section)

== Step 1==
At the beginning of his discussion, Riemann denotes by <math>(x, y, z)</math> <font color="orange">"the [inertial-frame] coordinates of an element of the fluid body at time <math>t</math>,"</font> and he denotes by <math>(\xi, \eta, \zeta)</math> <font color="orange">"the coordinates of the point <math>(x, y, z)</math> with respect to a moving coordinate system, whose axes coincide at each instant with the principal axes of the ellipsoid."</font> Drawing from our [[Appendix/Mathematics/EulerAngles#Euler_Angles|accompanying discussion of Euler angles]], it seems appropriate to associate <math>\vec{A}_{XYZ}</math> with the vector that points from the origin to the <math>(x, y, z)</math> location of the fluid element as viewed from the inertial reference frame, and to associate <math>\vec{A}_\mathrm{body}</math> with the vector that points from the origin to the same location of the fluid element, but as viewed from Riemann's specified rotating frame.  With this in mind, we have the following notation mappings:

<table border="0" align="center" cellpadding="3">
<tr>
  <td align="right">
<math>\vec{A}_\mathrm{XYZ}(A_X, A_Y, A_Z)</math>
  </td>
  <td align="center">&nbsp; <math>\rightarrow</math> &nbsp;</td>
  <td align="left">
<math>\vec{A}_\mathrm{inertial}(x, y, z) = \vec{e}_X(x) + \vec{e}_Y(y) + \vec{e}_Z(z) \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>\vec{A}_\mathrm{body}(A_1, A_2, A_3)</math>
  </td>
  <td align="center">&nbsp; <math>\rightarrow</math> &nbsp;</td>
  <td align="left">
<math>\vec{A}_\mathrm{body}(\xi, \eta, \zeta) = \vec{e}_1(\xi) + \vec{e}_2(\eta) + \vec{e}_3(\zeta) \, .</math>
  </td>
</tr>
</table>
Again drawing from [[#FormMatrix|our accompanying Euler angles discussion]], quite generally these two coordinate representations  &#8212; of the ''same'' vector, <math>\vec{A}</math> &#8212; are related to one another via the matrix expression,

<div align="center">
<math>\vec{A}_\mathrm{body} = \hat{R} \cdot \vec{A}_\mathrm{inertial} \, ,</math>
</div>

that is,

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right">
<math>\begin{bmatrix}\xi \\ \eta \\ \zeta \end{bmatrix}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\begin{bmatrix}
\vec{e}_1 \cdot\vec{e}_X & \vec{e}_1 \cdot\vec{e}_Y & \vec{e}_1 \cdot\vec{e}_Z  \\
\vec{e}_2 \cdot\vec{e}_X & \vec{e}_2 \cdot\vec{e}_Y & \vec{e}_2 \cdot\vec{e}_Z \\
\vec{e}_3 \cdot\vec{e}_X & \vec{e}_3 \cdot\vec{e}_Y & \vec{e}_3 \cdot\vec{e}_Z 
\end{bmatrix}
\cdot
\begin{bmatrix}x \\ y \\ z \end{bmatrix}
= \hat{R} \cdot
\begin{bmatrix}x \\ y \\ z \end{bmatrix} \, .
</math>
  </td>
</tr>
</table>

[[Appendix/Mathematics/EulerAngles#RotationMatrix|In terms of the trio of Euler angles]], the rotation matrix is,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>\hat{R}(\phi, \theta, \psi)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\begin{bmatrix}
(\cos\psi \cos\phi -  \sin\phi \sin\psi \cos\theta) 
& ( \cos\psi \sin\phi + \cos\phi \sin\psi \cos\theta ) 
& (\sin\psi\sin\theta )  \\

(- \sin\psi \cos\phi - \sin\phi \cos\psi \cos\theta ) 
& (- \sin\phi \sin\psi + \cos\phi \cos\psi \cos\theta ) 
& (\cos\psi \sin\theta ) \\

(\sin\theta \sin\phi ) 
& ( -\sin\theta \cos\phi) 
& ( \cos\theta ) 
\end{bmatrix} \, .
</math>
  </td>
</tr>
</table>

This means that &#8212; see [[Appendix/Mathematics/EulerAngles#Switching_Coordinate_Representations_of_a_Vector|our accompanying discussion]] &#8212; in terms of the trio of Euler angles, the following three coordinate mappings hold:

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

<tr>
  <td align="right">
<math>\xi</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
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) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\eta</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
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) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\zeta</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
x (\sin\theta\sin\phi)
+ y (-\sin\theta \cos\phi)
+ z (\cos\theta) \, .
</math>
  </td>
</tr>
</table>

Notice the correspondence between this set of coordinate relations and the set marked by Riemann (1861) as equation (2) of &sect;1.

<table border="1" align="center" width="80%" cellpadding="5">
<tr><td align="center">From &sect;1 of '''Riemann (1861)'''</td></tr>
<tr><td align="left">
<font color="orange">Then <math>\xi</math>, <math>\eta</math>, <math>\zeta</math> are known &hellip; to be linear expressions in <math>x</math>, <math>y</math>, <math>z</math>,</font>

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

<tr>
  <td align="right">
<math>\xi</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\alpha x + \beta y + \gamma z \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\eta</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\alpha' x + \beta' y + \gamma' z \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\zeta</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\alpha'' x + \beta'' y + \gamma'' z \, .
</math>
  </td>
</tr>
</table>
<font color="orange">The coefficients are the cosines of the angles that the axes of one system form with the axes of the other &hellip;</font>  For example, <math>\alpha = \vec{e}_1 \cdot \vec{e}_X</math>, <math>\beta = \vec{e}_1 \cdot \vec{e}_Y</math>, and so on.
</td></tr></table>

Alternatively, given that,
<div align="center">
<math>\vec{A}_\mathrm{inertial} = \hat{R}^{-1} \cdot \vec{A}_\mathrm{body} \, ,</math>
</div>

the following additional three mapping relations also must hold:

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

<tr>
  <td align="right">
<math>x</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\xi(\cos\psi \cos\phi - \sin\psi \sin\phi \cos\theta)
+ \eta(-\sin\psi \cos\phi - \sin\phi \cos\theta \cos\psi)
+ \zeta(\sin\theta\sin\phi) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>y</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\xi(\sin\phi \cos\psi + \sin\psi \cos\theta \cos\phi)
+ \eta( - \sin\psi \sin\phi + \cos\psi \cos\theta\cos\phi )
+ \zeta (-\sin\theta \cos\phi) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>z</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>
\xi (\sin\psi \sin\theta)
+ \eta (\sin\theta \cos\psi)
+ \zeta (\cos\theta) \, .
</math>
  </td>
</tr>
</table>