Editing ThreeDimensionalConfigurations/ChallengesPt3 (section)

=Challenges Constructing Ellipsoidal-Like Configurations (Pt. 3)=

This chapter extends the accompanying chapters titled, [[ThreeDimensionalConfigurations/Challenges|''Construction Challenges (Pt. 1)'']] and [[ThreeDimensionalConfigurations/ChallengesPt2|''(Pt. 2)'']].  The focus here is on firming up our understanding of the relationships between various "tilted" Cartesian coordinate frames.

==Various Coordinate Frames==

===Riemann-Derived Expressions===
<table border="0" cellpadding="10" align="right" width="30%"><tr><td align="center">
<table border="1" align="center" cellpadding="8">
<tr><td align="center">
''Inertial Frame'' (green with subscript "0") <br />and ''Body Frame'' (black and unsubscripted).
</td>
</tr>
<tr>
  <td align="center">[[File:InertialAxes05.png|400px|Inertial and Body Frames]]</td>
</tr>
<tr><td align="center">
For our chosen [[ThreeDimensionalConfigurations/RiemannTypeI#Case_I|Example Type I Ellipsoid]], we have, <math>~\Omega_2 = 0.3639</math> and <math>~\Omega_3 = 0.6633</math>, in which case, <math>~\Omega_0 = 0.7566</math> and <math>~\delta = 0.5018 ~\mathrm{rad} = 28.75^\circ</math>.
</td>
</tr>
</table>
</td></tr></table>

The purple (ellipsoidal) configuration is spinning with frequency, <math>~\Omega_0</math> about the <math>~z_0</math>-axis of the "inertial frame," as illustrated; that is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol\Omega</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\boldsymbol{\hat{k}_0}\Omega_0 \, .</math>
  </td>
</tr>
</table>
Also as illustrated, the "body frame," which is attached to and aligned with the principal axes of the purple ellipsoid, is tilted at an angle, <math>~\delta</math>, with respect to the inertial frame.  Hence, as viewed from the ''body'' frame, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol\Omega</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \boldsymbol{\hat\jmath }\sin\delta + \boldsymbol{\hat{k} }\cos\delta \biggr]\Omega_0 \, .</math>
  </td>
</tr>
</table>

Now, adhering to the notation used by [[Appendix/References#EFE|[<font color="red">EFE</font>] ]] &#8212; see, for example, the first paragraph of &sect;51 (p. 156) &#8212; we should write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol\Omega</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\boldsymbol{\hat\jmath }\Omega_2 + \boldsymbol{\hat{k} }\Omega_3 
~~~~~\Rightarrow ~~~ \Omega_2 = \Omega_0\sin\delta
</math>&nbsp; &nbsp; and, &nbsp; &nbsp; 
<math>~\Omega_3 = \Omega_0\cos\delta \, .</math>
  </td>
</tr>
</table>
This means that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Omega_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\Omega_2^2 + \Omega_3^2 \biggr]^{1 / 2}
</math>&nbsp; &nbsp; and, &nbsp; &nbsp; 
<math>~\delta = \tan^{-1}\biggl[ \frac{\Omega_2}{\Omega_3} \biggr] \, .</math>
  </td>
</tr>
</table>

As we have summarized in an [[ThreeDimensionalConfigurations/RiemannTypeI#EFEvelocities|accompanying discussion]] of Riemann Type 1 ellipsoids, [[Appendix/References#EFE|[<font color="red">EFE</font>] ]]  provides an expression for the velocity vector of each fluid element, given its  instantaneous ''body''-coordinate position (x, y, z) = (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>) &#8212; see his Eq. (154), Chapter 7, &sect;51 (p. 156).  As viewed from the rotating frame of reference, the three component expressions are,

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

<tr>
  <td align="right">
<math>~\dot{x} = u_1 = \boldsymbol{\hat\imath} \cdot \boldsymbol{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 y - \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 z \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\dot{y} = u_2 = \boldsymbol{\hat\jmath} \cdot \boldsymbol{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \gamma \Omega_3 x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 x \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\dot{z} = u_3 = \boldsymbol{\hat{k}} \cdot \boldsymbol{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+ \beta \Omega_2 x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 x \, ,</math>
  </td>
</tr>
</table>
<span  id="betagamma">where,</span>

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

<tr>
  <td align="right">
<math>~\beta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \frac{\zeta_2}{\Omega_2}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\gamma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3}{\Omega_3} \, .
</math>
  </td>
</tr>
</table>

<table border="1" cellpadding="8" width="90%" align="center">
<tr><td align="left" colspan="2">
<div align="center">'''Rotating-Frame Vorticity'''</div>
</td>
</tr>
<tr>
<td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{\zeta} \equiv \boldsymbol{\nabla \times}\bold{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath} \biggl[ \frac{\partial \dot{z} }{\partial y} - \frac{\partial \dot{y}}{\partial z} \biggr]
+ \boldsymbol{\hat\jmath} \biggl[ \frac{\partial \dot{x}}{\partial z} - \frac{\partial \dot{z}}{\partial x} \biggr]
+ \bold{\hat{k}} \biggl[ \frac{\partial \dot{y}}{\partial x} - \frac{\partial \dot{x}}{\partial y} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\jmath} \biggl\{ 
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 + \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 
\biggr\}
+ \bold{\hat{k}} \biggl\{ 
\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 + \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\jmath} ~\zeta_2
+ \bold{\hat{k}} ~\zeta_3 \, .
</math>
  </td>
</tr>
</table>

For our chosen [[ThreeDimensionalConfigurations/RiemannTypeI#Case_I|Example Type I Ellipsoid]], we have, <math>~\zeta_2 = -2.2794</math> and <math>~\Omega_3 = -1.9637</math>, in which case, <math>~\zeta_\mathrm{rot} = (\zeta_2^2 + \zeta_3^2)^{1 / 2} = 2.2794</math> and <math>~\xi \equiv \tan^{-1}[\zeta_2/\zeta_3] = 4.0013 ~\mathrm{rad} = 229.26^\circ</math>.
  </td>
  <td align="center">
[[File:VorticityAxis04.png|350px|center|Vorticity Axis]]
  </td>
</tr>

</table>

===Tipped Orbit Planes===

====Summary====
In a [[ThreeDimensionalConfigurations/RiemannTypeI#Try_Again|separate discussion]], we have shown that, as viewed from a frame that "tumbles" with the (purple) body of a Type 1 Riemann ellipsoid, each Lagrangian fluid element moves along an elliptical path in a plane that is tipped by an angle <math>~\theta</math> about the x-axis of the body.  As viewed from the (primed) coordinates associated with this tipped plane, by definition, z' = constant and dz'/dt = 0, and the planar orbit is defined by the expression for an,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="center" colspan="3"><font color="maroon">'''Off-Center Ellipse'''</font></td>
</tr>
<tr>
  <td align="right">
<math>~1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{x'}{x_\mathrm{max}} \biggr]^2 + \biggl[\frac{y' - y_c(z')}{y_\mathrm{max}} \biggr]^2 \, .</math>
  </td>
</tr>
</table>

<table border="0" cellpadding="10" align="right" width="30%"><tr><td align="center">
<table border="1" align="center" cellpadding="8">
<tr><td align="center">
''Tipped Orbit Frame'' (yellow, primed) <br />
</td>
</tr>
<tr>
  <td align="center">[[File:TippedAxes03.png|350px|Tipped Orbital Planes]]</td>
</tr>
<tr><td align="center">
Given that b/a = 1.25 and c/a = 0.4703 for our chosen [[ThreeDimensionalConfigurations/ChallengesPt2#Example_Equilibrium_Model|Example Type I Ellipsoid]], we find that, <math>~\theta = - 0.3320 ~\mathrm{rad} = -19.02^\circ</math>.
</td>
</tr>
</table>
</td></tr></table>
Notice that the offset, <math>~y_c</math>, is a function of the tipped plane's vertical coordinate, <math>~z'</math>.  As a function of time, the x'-y' coordinates and associated velocity components of each Lagrangian fluid element are given by the expressions,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x_\mathrm{max}\cos(\dot\varphi t)</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>~y' - y_c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y_\mathrm{max}\sin(\dot\varphi t) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\dot{x}'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- x_\mathrm{max}~ \dot\varphi \cdot \sin(\dot\varphi t) = (y_c - y') \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] \dot\varphi </math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>~\dot{y}' </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y_\mathrm{max}~\dot\varphi \cdot \cos(\dot\varphi t) = x' \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr] \dot\varphi \, .</math>
  </td>
</tr>
</table>

As has been summarized in an [[ThreeDimensionalConfigurations/ChallengesPt2#Try_Tipped_Plane_Again|accompanying discussion]], we have determined that (numerical value given for our chosen example Type I ellipsoid),
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tan\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{\zeta_2}{\zeta_3} \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]  \frac{c^2}{b^2} 
=
- \frac{\beta \Omega_2}{\gamma \Omega_3}  
=
-0.34479\, ,
</math>
  </td>
</tr>
</table>
where, <math>~\beta</math> and <math>~\gamma</math> are as [[#betagamma|defined above]].  Also,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[\frac{x_\mathrm{max}}{y_\mathrm{max}}  \biggr]^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a^2}{b^2 c^2}  (c^2\cos^2\theta + b^2\sin^2\theta) 
= 1.05238  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~{\dot\varphi}^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\zeta_3^2\biggl[ \frac{b^2}{a^2 + b^2} \biggr]^2 
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]^2 \biggl[1 + \tan^2\theta \biggr] 
= 1.68818\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y_c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+ \frac{z' b^2 \tan\theta}{c^2 \cos^2\theta + b^2\sin^2\theta} 
=
+z' \tan\theta  \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}}  \biggr]^2 \frac{a^2}{c^2}
=
\biggl( \frac{z'}{ \cos\theta }\biggr)(-1.40038)
\, .</math>
  </td>
</tr>
</table>
Note that this last expression has been obtained by making the substitutions, <math>~y_0 \rightarrow y_c</math> and <math>~z_0 \rightarrow -z'/\cos\theta</math>, in the [[ThreeDimensionalConfigurations/ChallengesPt2#OffCenter|accompanying derivation's expression]] for <math>~y_0</math>.

====Demonstration====

In order to transform a vector from the "tipped orbit" frame (primed coordinates) to the "body" frame (unprimed), we use the following mappings of the three unit vectors:
<table border="1" align="center" width="40%" cellpadding="8"><tr><td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{\hat\imath'}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\boldsymbol{\hat\imath} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\boldsymbol{\hat\jmath'}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\boldsymbol{\hat\jmath}\cos\theta + \boldsymbol{\hat{k}}\sin\theta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\boldsymbol{\hat{k}'}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~-\boldsymbol{\hat\jmath}\sin\theta + \boldsymbol{\hat{k}}\cos\theta \, .</math>
  </td>
</tr>
</table>

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

Given that, by design in our "tipped orbit" frame, there is no vertical motion &#8212; that is, <math>~\dot{z}' = 0</math> &#8212; mapping the (primed coordinate) velocity to the body (unprimed) coordinate is particularly straightforward.  Specifically,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{u'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath'} \dot{x}'
+
\boldsymbol{\hat\jmath'} \dot{y}'
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~~~\rightarrow~~</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath} \dot{x}'
+
[\boldsymbol{\hat\jmath}\cos\theta + \boldsymbol{\hat{k}}\sin\theta] \dot{y}' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath} \biggl\{
(y_c - y') \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr) \dot\varphi
\biggr\}
+
[\boldsymbol{\hat\jmath}\cos\theta + \boldsymbol{\hat{k}}\sin\theta] \biggl\{
x' \biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr) \dot\varphi
\biggr\} \, .
</math>
  </td>
</tr>
</table>

Recognizing, [[ThreeDimensionalConfigurations/ChallengesPt2#Tipped_Orbital_Plane|as before]], that the relevant coordinate mapping is,
<table border="1" align="center" width="40%" cellpadding="8"><tr><td align="left">
<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">
<math>~x \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y'</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~y\cos\theta + z\sin\theta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z'</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~z\cos\theta - y\sin\theta \, ,</math>
  </td>
</tr>
</table>

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

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

<tr>
  <td align="right">
<math>~\boldsymbol{u'}</math>
  </td>
  <td align="center">
<math>~~~\rightarrow~~~</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath}  \dot\varphi \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)\biggl\{y_c - y\cos\theta - z\sin\theta\biggr\}
+
\boldsymbol{\hat\jmath}  \dot\varphi
\biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr)
\biggr\{ x\cos\theta \biggr\}
+
\boldsymbol{\hat{k}}  \dot\varphi
\biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr)
\biggr\{ x\sin\theta \biggr\} \, ,
</math>
  </td>
</tr>
</table>
where,

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

<tr>
  <td align="right">
<math>~y_c</math>
  </td>
  <td align="center">
<math>~~~\rightarrow~~~</math>
  </td>
  <td align="left">
<math>~
+[z\cos\theta - y\sin\theta] \tan\theta  \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}}  \biggr]^2 \frac{a^2}{c^2}
\, .</math>
  </td>
</tr>
</table>
Written in terms of the "body" frame coordinates, therefore, the 2<sup>nd</sup> and 3<sup>rd</sup> components of this velocity vector are, respectively:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{\hat\jmath}\cdot \boldsymbol{u'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x \dot\varphi
\biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr)
\cos\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x \biggl\{ \zeta_3^2\biggl[ \frac{b^2}{a^2 + b^2} \biggr]^2 
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]^2 \biggl[1 + \tan^2\theta \biggr] 
\biggr\}^{1 / 2}
\biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr)
\cos\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x \biggl\{ \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr] 
\biggr\} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\boldsymbol{\hat{k}}\cdot \boldsymbol{u'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x \dot\varphi
\biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr)
\sin\theta 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x \biggl\{ \zeta_3^2\biggl[ \frac{b^2}{a^2 + b^2} \biggr]^2 
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]^2 \biggl[1 + \tan^2\theta \biggr] 
\biggr\}^{1 / 2}
\biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr)
\sin\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x \biggl\{ \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] 
\biggr\}
\tan\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x \biggl\{ \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] 
\biggr\}
\biggl\{
- \frac{\zeta_2}{\zeta_3} \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]  \frac{c^2}{b^2}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-x \biggl\{ \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr] 
\biggr\} \, .
</math>
  </td>
</tr>
</table>
These expressions perfectly match the body-coordinate expressions derived by Riemann (see [[#Riemann-Derived_Expressions|above]]) for, respectively, <math>~\dot{y}</math> and <math>~\dot{z}</math>.  The 1<sup>st</sup> component is,

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

<tr>
  <td align="right">
<math>~\boldsymbol{\hat\imath}\cdot \boldsymbol{u'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\dot\varphi \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)\biggl\{y_c - y\cos\theta - z\sin\theta\biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \zeta_3^2\biggl[ \frac{b^2}{a^2 + b^2} \biggr]^2 
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]^2 \biggl[1 + \tan^2\theta \biggr] 
\biggr\}^{1 / 2}
\biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)
\biggl\{y_c
- y\cos\theta - z\sin\theta\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] 
\biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2
\biggl\{\frac{y_c}{\cos\theta}
- y - z\tan\theta\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 \biggl\{
\zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{y_c}{\cos\theta}
-~y\cdot \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] 
+~ z\cdot \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_2}{\zeta_3} \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]  \frac{c^2}{b^2}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 \biggl\{
\zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{y_c}{\cos\theta}
-~y\cdot \zeta_3\biggl[ \frac{a^2}{a^2 + b^2} \biggr] \frac{b^2}{a^2}
+~ z\cdot \zeta_2\biggl[ \frac{a^2}{a^2 + c^2} \biggr]   \frac{c^2}{a^2}
\biggr\} \, .
</math>
  </td>
</tr>
</table>
So, implementing the mapping of <math>~y_c</math>, the first term inside the curly braces becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{y_c}{\cos\theta}</math>
  </td>
  <td align="center">
<math>~~~\rightarrow~~~</math>
  </td>
  <td align="left">
<math>~
\frac{\zeta_3}{\cos\theta}\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggl\{
+[z\cos\theta - y\sin\theta] \tan\theta  \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}}  \biggr]^2 \frac{a^2}{c^2}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}}  \biggr]^2 \frac{a^2}{c^2} \biggl\{ -y\tan^2\theta  \biggr\}
+
\zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr]\tan\theta \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}}  \biggr]^2 \frac{a^2}{c^2}  \biggl\{ z \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~y \cdot \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}}  \biggr]^2 \frac{a^2}{c^2} \biggl\{ \tan^2\theta  \biggr\}
-
z \cdot \zeta_2  \biggl[ \frac{c^2 }{a^2 + c^2} \biggr]  \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}}  \biggr]^2 \frac{a^2}{c^2}  \biggl\{ \tan^2\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl(\frac{x_\mathrm{max}}{y_\mathrm{max}}  \biggr)^2 \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{y_c}{\cos\theta}</math>
  </td>
  <td align="center">
<math>~~~\rightarrow~~~</math>
  </td>
  <td align="left">
<math>~
-~y \cdot \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr]  \frac{a^2}{c^2} \biggl\{ \tan^2\theta  \biggr\} -
z \cdot \zeta_2  \biggl[ \frac{c^2 }{a^2 + c^2} \biggr]  \frac{a^2}{c^2}  \biggl\{ \tan^2\theta \biggr\}
</math>
  </td>
</tr>
</table>

<div align="left">
<math>
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr]^2~=~\frac{a^2}{b^2c^2}  (c^2\cos^2\theta + b^2\sin^2\theta) 
</math>

<math>
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr]^2 \biggl[ 1 + \tan^2\theta \biggr]~=~\frac{a^2}{b^2c^2}  (c^2 + b^2\tan^2\theta) 
</math>
</div>

Therefore,

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

<tr>
  <td align="right">
<math>~\boldsymbol{\hat\imath}\cdot \boldsymbol{u'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~y \cdot \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr]  \frac{a^2}{c^2} \biggl\{ \tan^2\theta  \biggr\} -
z \cdot \zeta_2  \biggl[ \frac{c^2 }{a^2 + c^2} \biggr]  \frac{a^2}{c^2}  \biggl\{ \tan^2\theta \biggr\}
~+~ 
\biggl\{
z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr]   
-~y\cdot \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] 
\biggr\} \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~y \cdot \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr]  \frac{a^2}{c^2} \biggl\{ \tan^2\theta  \biggr\}
-~y\cdot \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2
~+~ 
z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr]   \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2
-~z \cdot \zeta_2  \biggl[ \frac{c^2 }{a^2 + c^2} \biggr]  \frac{a^2}{c^2}  \biggl\{ \tan^2\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~y \cdot \zeta_3 \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \biggl\{ \frac{b^2}{c^2} \cdot \tan^2\theta 
+\frac{b^2}{a^2} \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 \biggr\}
~+~ 
z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \biggl\{  \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2
-~ \frac{a^2}{c^2}  \cdot \tan^2\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~y \cdot \zeta_3 \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \biggl\{ \frac{b^2}{c^2} \cdot \tan^2\theta 
+ \frac{1}{c^2}  (c^2\cos^2\theta + b^2\sin^2\theta) \biggr\}
~+~ 
z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \biggl\{  \frac{a^2}{b^2c^2}  (c^2\cos^2\theta + b^2\sin^2\theta)
-~ \frac{a^2}{c^2}  \cdot \tan^2\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~y \cdot \zeta_3 \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \frac{1}{c^2\cos^2\theta}\biggl\{b^2 \sin^2\theta 
+ (c^2\cos^2\theta + b^2\sin^2\theta)\cos^2\theta \biggr\}
~+~ 
z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \biggl\{  \frac{a^2}{b^2c^2}  (c^2\cos^2\theta + b^2\sin^2\theta)
-~ \frac{a^2}{c^2}  \cdot \tan^2\theta \biggr\}
</math>
  </td>
</tr>
</table>