Editing ThreeDimensionalConfigurations/ChallengesPt2 (section)

=Challenges Constructing Ellipsoidal-Like Configurations (Pt. 2)=

This chapter extends an [[ThreeDimensionalConfigurations/Challenges|accompanying chapter titled, ''Construction Challenges (Pt. 1)'']].  The focus here is on an SCF technique that will incorporate specification of a Lagrangian flow-flied.
==Motivation==

===Where Are We Headed?===
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.  (See the yellow-dotted orbits in Figure panels 1a and 1b below).  As viewed from the (primed) coordinates associated with this tipped plane, by definition, z' = 0 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_0}{y_\mathrm{max}} \biggr)^2 \, .</math>
  </td>
</tr>
</table>
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_0</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_0 - 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>

Notice that this is a divergence-free flow-field:

<table border="1" cellpadding="8" width="90%" align="center"><tr><td align="left">
<div align="center">'''Divergence'''</div>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla \cdot \vec{v'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial \dot{x}'}{\partial x'} + \frac{\partial \dot{y}'}{\partial y'} + \frac{\partial \dot{z}'}{\partial z'}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial }{\partial x'} \biggl[ (y' - y_0) \dot\varphi \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr)  \biggr] 
+ 
\frac{\partial }{\partial y'} \biggl[ x'  \dot\varphi \biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr)  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0\, .
</math>
  </td>
</tr>
</table>

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


<span id="VorticitySetup">Also,</span> along the lines of our [[ThreeDimensionalConfigurations/Challenges#Riemann_S-type_Ellipsoids|accompanying discussion of Riemann S-Type Ellipoids]], it is useful to develop the expression for the fluid vorticity as viewed from the rotating- and tipped-reference frame.
<table border="1" cellpadding="8" width="90%" align="center"><tr><td align="left">
<div align="center">'''Vorticity'''</div>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{\zeta'} \equiv \boldsymbol{\nabla \times}\bold{v'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath'} \biggl[ \frac{\partial \cancelto{}{\dot{z}'} }{\partial y'} - \frac{\partial \dot{y}'}{\partial z'} \biggr]
+ \boldsymbol{\hat\jmath'} \biggl[ \frac{\partial \dot{x}'}{\partial z'} - \frac{\partial \cancelto{}{\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\imath'} (x' \dot\varphi )\frac{\partial }{\partial z'} \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max} }\biggr]
+ \boldsymbol{\hat\jmath'} \biggl\{
\dot\varphi (y_0 - y')\frac{\partial }{\partial z'}\biggl[  \frac{x_\mathrm{max}}{y_\mathrm{max} } \biggr]
+ \dot\varphi \biggl[  \frac{x_\mathrm{max}}{y_\mathrm{max} } \biggr] \frac{\partial y_0}{\partial z'}
\biggr\}
+ \bold{\hat{k}'} \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max} } - \frac{x_\mathrm{max} }{y_\mathrm{max} } \biggr] \dot\varphi \, .
</math>
  </td>
</tr>
</table>
Further evaluation is completed, [[#Vorticity_Determination|below]], after we determine how <math>~y_0</math> and <math>~[x_\mathrm{max}/y_\mathrm{max}]^{\pm 1}</math> depend on <math>~z_0</math>; and after appreciating that, in order to introduce the functional dependence on <math>~z' \ne 0</math> in every relevant expression, we need to make the replacement, <math>~z_0 \rightarrow (z_0 + z'\cos\theta)</math>.  <font color="red"><==&nbsp; &nbsp; Figure this out!</font>
</td></tr></table>


In the subsections of this chapter that follow, we provide analytic expressions for these various quantities &#8212; <math>~x_\mathrm{max}, y_\mathrm{max}, y_0, \dot\varphi</math> &#8212; in terms of the properties of any chosen Type 1 Riemann ellipsoid.

===Intersection of Tipped Plane With Ellipsoid Surface===

====Body Frame====

In a [[ThreeDimensionalConfigurations/RiemannTypeI#TippedPlane|an early subsection of the accompanying discussion]], we have pointed out that the intersection of each Lagrangian fluid element's tipped orbital plane with the surface of the (purple) ellipsoidal surface is given by the (unprimed) body-frame coordinates that simultaneously satisfy the expressions,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{x}{a}\biggr)^2 + \biggl( \frac{y}{b}\biggr)^2 + \biggl( \frac{z}{c}\biggr)^2 </math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y \tan\theta + z_0 \, ,</math>
  </td>
</tr>
</table>
where z<sub>0</sub> is the location where the tipped plane intersects the z-axis of the body frame.  Combining these two expressions, we see that an intersection between the tipped plane and the ellipsoidal surface will occur at (x, y)-coordinate pairs that satisfy what we will henceforth refer to as the,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3"><font color="maroon">'''Intersection Expression'''</font></td>
</tr>

<tr>
  <td align="right">
<math>~1 - \frac{x^2}{a^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{y^2}{b^2} + \biggl[ \frac{y\tan\theta + z_0}{c}\biggr]^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y^2 \biggl[\frac{c^2 + b^2\tan^2\theta}{b^2c^2} \biggr] + y \biggl[ \frac{2z_0 \tan\theta}{c^2} \biggr] + \frac{z_0^2}{c^2} \, , </math>
  </td>
</tr>
</table>
as long as z<sub>0</sub> lies within the range,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~z_0^2</math>
  </td>
  <td align="center">
<math>~\le</math>
  </td>
  <td align="left">
<math>~c^2 + b^2\tan^2\theta \, .</math>
  </td>
</tr>
</table>

====Tipped Orbital Plane====

A table provided in [[ThreeDimensionalConfigurations/RiemannTypeI#Try_Again|our accompanying discussion]] shows how to transform from the body-frame coordinates (unprimed) to the (primed) frame that aligns with the Lagrangian fluid element's orbit.  Specifically,
<table border="1" width="50%" cellpadding="8" align="center">
<tr>
<td align="left">
<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' \, ,</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~(z - z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
z' \cos\theta + y'\sin\theta 
\, .</math>
  </td>
</tr>
</table>
</td>

<td align="center">[[File:PrimedCoordinates3.png|250px|Primed Coordinates]]</td>
<td align="left">
<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 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
y \cos\theta + (z - z_0) \sin\theta 
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(z-z_0) \cos\theta - y \sin\theta 
\, .</math>
  </td>
</tr>
</table>
</td>
</tr>

<tr>
  <td align="center" colspan="3">NOTE:  &nbsp; &nbsp;<math>~z_0 \rightarrow z_0 + z'\cos\theta \, .</math><font color="red"> &nbsp; &nbsp;<== &nbsp; &nbsp; Figure this out!</font></td>
</tr>
</table>

Using the 2<sup>nd</sup> and 3<sup>rd</sup> of these relations, we see from the equation that defines the "tipped plane," that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~z_0 + z' \cos\theta + y'\sin\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[ y' \cos\theta - z'\sin\theta ]\tan\theta + z_0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~z' \biggl[ \cos\theta + \tan\theta \cdot \sin\theta \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y' [ \cos\theta \cdot \tan\theta - \sin\theta] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\frac{z'}{\cos\theta} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~z' </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
Hence, as viewed from the primed coordinate frame, all points of intersection between the tipped plane and the surface of the ellipsoid will be found in the <math>~z' = 0</math> plane, as desired.  Inserting the 1<sup>st</sup> and 2<sup>nd</sup> of these relations into the above-defined <font color="maroon">''Intersection Expression''</font>, we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1 - \frac{(x')^2}{a^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ y' \cos\theta - \cancelto{0}{z'} \sin\theta \biggr]^2 \biggl[\frac{c^2 + b^2\tan^2\theta}{b^2c^2} \biggr] 
+ \biggl[ y' \cos\theta - \cancelto{0}{z'} \sin\theta  \biggr] \biggl[ \frac{2z_0 \tan\theta}{c^2} \biggr] + \frac{z_0^2}{c^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(y')^2\biggl[\frac{c^2 \cos^2\theta + b^2\sin^2\theta}{b^2c^2} \biggr] 
- 2(y')\biggl[ - \frac{z_0 \sin\theta}{c^2} \biggr] + \frac{z_0^2}{c^2}
\, .</math>
  </td>
</tr>
</table>

====Off-Center Ellipse====

Now we attempt to transform this last expression into the form of the above-defined equation for an ''<font color="maroon">Off-Center Ellipse</font>'', which we rewrite here as,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1 - \frac{(x')^2}{x^2_\mathrm{max}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{y^2_\mathrm{max}}\biggl[ (y')^2 - 2(y')y_0 + y_0^2 \biggr] \, .</math>
  </td>
</tr>
</table>

An initial rearrangement of the relevant "last" expression gives,

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

<tr>
  <td align="right">
<math>~1 - \frac{z_0^2}{c^2} - \frac{(x')^2}{a^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{c^2 \cos^2\theta + b^2\sin^2\theta}{b^2c^2}   
\biggl[(y')^2 - 2(y') \underbrace{ \biggl( - \frac{z_0 b^2 \sin\theta}{c^2 \cos^2\theta + b^2\sin^2\theta}\biggr) }_{y_0} \biggr] 
\, ,</math>
  </td>
</tr>
</table>
<span id="Result1">which, as indicated,</span> allows us to identify the appropriate expression for the y-offset, <math>~y_0</math>.  
<table border="1" align="center" cellpadding="10" width="60%" bordercolor="orange">
<tr><td align="center" bgcolor="lightblue">'''RESULT 1'''<br />(compare with [[ThreeDimensionalConfigurations/ChallengesPt4#Result2|Result 2]])</td></tr>
<tr><td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{y_0}{z_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{b^2\sin\theta}{c^2\cos^2\theta + b^2\sin^2\theta}
=
-\frac{\sin\theta}{c^2\kappa^2}
</math>
  </td>
</tr>
</table>

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

Dividing through by the leading coefficient,
<div align="center">
<math>~\kappa^2 \equiv \frac{c^2 \cos^2\theta + b^2\sin^2\theta}{b^2c^2} \, ,</math>
</div>
then adding <math>~y_0^2</math> to both sides gives,

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

<tr>
  <td align="right">
<math>~  
(y')^2 - 2(y') y_0 + y_0^2
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\kappa^2}\biggl[ 1 - \frac{z_0^2}{c^2} - \frac{(x')^2}{a^2} \biggr] + y_0^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\underbrace{\biggl[  \frac{1}{\kappa^2} - \frac{z_0^2}{c^2\kappa^2} + y_0^2 \biggr]}_{y^2_\mathrm{max}} - \frac{(x')^2}{\kappa^2 a^2} \, ,
</math>
  </td>
</tr>
</table>
which gives us the appropriate expression for <math>~y_\mathrm{max}^2</math>.  Finally, dividing through by <math>~y_\mathrm{max}^2</math> gives,

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

<tr>
  <td align="right">
<math>~  
\frac{1}{y_\mathrm{max}^2} \biggl[ (y')^2 - 2(y') y_0 + y_0^2 \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 - (x')^2 \underbrace{ \biggl[ \frac{1}{y_\mathrm{max}^2 \kappa^2 a^2} \biggr]}_{1/x^2_\mathrm{max}} \, ,
</math>
  </td>
</tr>
</table>
<span id="OffCenter">which identifies the appropriate</span> expression for <math>~x^2_\mathrm{max}</math>.  As viewed from the "tipped plane" (primed) coordinate frame, then, the equation for the orbit of each Lagrangian fluid element is that of an &hellip;

<table border="1" align="center" cellpadding="8" width="90%"><tr><td align="left">
<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_0}{y_\mathrm{max}} \biggr)^2 \, ,</math>
  </td>
</tr>
</table>
with,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y_0</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{z_0 b^2 \sin\theta}{c^2 \cos^2\theta + b^2\sin^2\theta} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y^2_\mathrm{max}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\kappa^2}\biggl( 1 - \frac{z_0^2}{c^2}\biggr) + y_0^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{b^2(c^2 - z_0^2)}{c^2 \cos^2\theta + b^2\sin^2\theta} + \biggl[\frac{z_0 b^2 \sin\theta}{c^2 \cos^2\theta + b^2\sin^2\theta}\biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~b^2 \biggl\{
\frac{(c^2-z_0^2) ( c^2 \cos^2\theta + b^2\sin^2\theta ) + z_0^2 b^2 \sin^2\theta}{(c^2 \cos^2\theta + b^2\sin^2\theta)^2} \biggr\}\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
b^2 c^2 \biggl\{ \frac{( c^2 -z_0^2)\cos^2\theta + b^2\sin^2\theta   }{(c^2 \cos^2\theta + b^2\sin^2\theta)^2} \biggr\}\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~x_\mathrm{max}^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~a^2 \kappa^2 y_\mathrm{max}^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a^2\biggl\{ \frac{( c^2 -z_0^2)\cos^2\theta + b^2\sin^2\theta   }{c^2 \cos^2\theta + b^2\sin^2\theta} \biggr\}
\, .
</math>
  </td>
</tr>
</table>
Note that the ratio,

<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>~
a^2\biggl[ \frac{( c^2 -z_0^2)\cos^2\theta + b^2\sin^2\theta   }{c^2 \cos^2\theta + b^2\sin^2\theta} \biggr]
\frac{1}{b^2 c^2} \biggl[ \frac{(c^2 \cos^2\theta + b^2\sin^2\theta)^2}{( c^2 -z_0^2)\cos^2\theta + b^2\sin^2\theta   } \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a^2}{b^2c^2} \biggl[ (c^2 \cos^2\theta + b^2\sin^2\theta)\biggr] \, ,
</math>
  </td>
</tr>
</table>
which is independent of <math>~z_0</math>.
</td></tr></table>

===Velocities===

====Tipped Orbit Velocities====
From the generic expressions for [[#Motivation|(primed) velocities associated with an off-center elliptical orbit]], we expect,

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

<tr>
  <td align="right">
<math>~\frac{\dot{x}'}{\dot\varphi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-x_\mathrm{max} \sin(\dot\varphi t)
=
- (y' - y_0) \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] \, ,
</math>
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; 
  </td>
</tr>

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

====Body Frame Velocities====

From the already-referenced table provided in [[ThreeDimensionalConfigurations/RiemannTypeI#Try_Again|our accompanying discussion]], we can transform this pair of expressions for the velocity components in the "tipped orbit" frame &#8212; remember that the third component, <math>~\dot{z}' = 0</math> &#8212; into the (three-component) velocities of the body frame using the expressions,
<table border="1" width="75%" cellpadding="8" align="center">
<tr>
    <td align="left" width="50%">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\dot{x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\dot{x}' \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\dot{y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\dot{y}' \cos\theta
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\dot{z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\dot{y}' \sin\theta
\, .</math>
  </td>
</tr>
</table>

    </td>
    <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-z_0)\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-z_0)\cos\theta - y\sin\theta
\, .</math>
  </td>
</tr>
</table>

    </td>
</tr>
</table>
That is to say,

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

<tr>
  <td align="right">
<math>~\frac{\dot{x}}{\dot\varphi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- (y' - y_0) \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] 
=
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] \biggl\{ y_0 - [y\cos\theta + (z-z_0)\sin\theta]  \biggr\} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\dot{y}}{\dot\varphi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x' \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] \cos\theta
=
x \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] \cos\theta \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\dot{z}}{\dot\varphi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x' \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] \sin\theta
=
x \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] \sin\theta \, ,
</math>
  </td>
</tr>
</table>
where,
<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}\biggl[ c^2 \cos^2\theta + b^2\sin^2\theta \biggr] \, .
</math>
  </td>
</tr>
</table>
Notice that the all-important tipping angle, <math>~\theta</math>, is related to these body-frame velocity components via the simple relation,
<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>~\biggl( \frac{\dot{z}}{\dot{y}} \biggr) \, .</math>
  </td>
</tr>
</table>

====Vorticity Determination====
Given that the ratio, <math>~(x_\mathrm{max}/y_\mathrm{max})</math>, does not depend on <math>~z'</math>, and that, after mapping <math>~z_0 \rightarrow (z_0 + z'\cos\theta)</math>, we find,

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

<tr>
  <td align="right">
<math>~\frac{\partial y_0}{\partial z'} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial }{\partial z'}
\biggl[
- \frac{(z_0 + z'\cos\theta) b^2 \sin\theta}{c^2 \cos^2\theta + b^2\sin^2\theta}
\biggr]
=
- \frac{ b^2 \sin\theta\cos\theta}{c^2 \cos^2\theta + b^2\sin^2\theta}
=
- b^2 \sin\theta\cos\theta \biggl(\frac{a^2}{b^2c^2}\biggr) \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 \, ,
</math>
  </td>
</tr>
</table>
the [[#VorticitySetup|above vorticity expression]] becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{\zeta'} \equiv \boldsymbol{\nabla \times}\bold{v'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\boldsymbol{\hat\imath'} (x' \dot\varphi ) \cancelto{0}{\frac{\partial }{\partial z'} \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max} }\biggr]}
+ \boldsymbol{\hat\jmath'} \biggl\{
\dot\varphi (y_0 - y') \cancelto{0}{\frac{\partial }{\partial z'}\biggl[  \frac{x_\mathrm{max}}{y_\mathrm{max} } \biggr]}
+ \dot\varphi \biggl[  \frac{x_\mathrm{max}}{y_\mathrm{max} } \biggr] \frac{\partial y_0}{\partial z'}
\biggr\}
+ \bold{\hat{k}'} \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max} } - \frac{x_\mathrm{max} }{y_\mathrm{max} } \biggr] \dot\varphi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \boldsymbol{\hat\jmath'} 
\biggl[  \frac{x_\mathrm{max}}{y_\mathrm{max} } \biggr] b^2 \sin\theta\cos\theta \biggl(\frac{a^2}{b^2c^2}\biggr) \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2\dot\varphi 
+ \bold{\hat{k}'} \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max} } - \frac{x_\mathrm{max} }{y_\mathrm{max} } \biggr] \dot\varphi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[  \frac{y_\mathrm{max}}{x_\mathrm{max} } \biggr] \dot\varphi \biggl\{
- \boldsymbol{\hat\jmath'} 
~\sin\theta\cos\theta \biggl(\frac{a^2}{c^2}\biggr)  
+ \bold{\hat{k}'} \biggl[ 1 - \frac{x^2_\mathrm{max} }{y^2_\mathrm{max} } \biggr]  
\biggr\} \, .
</math>
  </td>
</tr>
</table>

Referring back to our [[#Tipped_Orbital_Plane|aboved-defined tipped plane]], we see that the unprimed Cartesian unit vectors are related to the primed unit vectors via the relations &hellip;
<table border="1" align="center" cellpadding="10" width="60%"><tr><td align="left">
<table border="0" cellpadding="5" align="center">

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

</td></tr></table>
Hence, from the perspective of the body frame, the expression for the vorticity becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{\zeta} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[  \frac{y_\mathrm{max}}{x_\mathrm{max} } \biggr] \dot\varphi \biggl\{
- \biggl[ \boldsymbol{\hat{\jmath}}\cos\theta  + \boldsymbol{\hat{k}}\sin\theta  \biggr]
~\sin\theta\cos\theta \biggl(\frac{a^2}{c^2}\biggr)  
+ \biggl[ -\boldsymbol{\hat{\jmath}}\sin\theta  + \boldsymbol{\hat{k}}\cos\theta \biggr] \biggl[ 1 - \frac{x^2_\mathrm{max} }{y^2_\mathrm{max} } \biggr]  
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\cos\theta}\biggl[ \frac{b^2 \zeta_3}{a^2 + b^2} \biggr] \biggl\{
- \boldsymbol{\hat{\jmath}} ~\sin\theta\cos^2\theta \biggl(\frac{a^2}{c^2}\biggr)  
- \boldsymbol{\hat{k}}~\sin^2\theta\cos\theta \biggl(\frac{a^2}{c^2}\biggr)  
-\boldsymbol{\hat{\jmath}}\sin\theta  \biggl[ 1 - \frac{x^2_\mathrm{max} }{y^2_\mathrm{max} } \biggr]  
+ 
\boldsymbol{\hat{k}}\cos\theta \biggl[ 1 - \frac{x^2_\mathrm{max} }{y^2_\mathrm{max} } \biggr]  
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{b^2 \zeta_3}{a^2 + b^2} \biggr] \biggl\{
- \boldsymbol{\hat{\jmath}} \biggl[  1 - \frac{x^2_\mathrm{max} }{y^2_\mathrm{max} }   
~+~\cos^2\theta \biggl(\frac{a^2}{c^2}\biggr) \biggr]\tan\theta 
+ 
\boldsymbol{\hat{k}} \biggl[ 1 - \frac{x^2_\mathrm{max} }{y^2_\mathrm{max} }  
~-~ \sin^2\theta \biggl(\frac{a^2}{c^2}\biggr) \biggr] 
\biggr\} \, .
</math>
  </td>
</tr>
</table>
Now, we appreciate that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1 - \frac{x^2_\mathrm{max}}{y^2_\mathrm{max}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 - \frac{a^2}{b^2 c^2}\biggl( c^2 \cos^2\theta + b^2\sin^2\theta \biggr)
=
1 - \biggl(\frac{a^2}{b^2}\biggr)\cos^2\theta - \biggl(\frac{a^2}{c^2}\biggr)\sin^2\theta \, .
</math>
  </td>
</tr>
</table>
Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{\zeta} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{b^2 \zeta_3}{a^2 + b^2} \biggr] \biggl\{
- \boldsymbol{\hat{\jmath}} \biggl[  1 - \biggl(\frac{a^2}{b^2}\biggr)\cos^2\theta - \biggl(\frac{a^2}{c^2}\biggr)\sin^2\theta   
~+~\cos^2\theta \biggl(\frac{a^2}{c^2}\biggr) \biggr]\tan\theta 
+ 
\boldsymbol{\hat{k}} \biggl[ 1 - \biggl(\frac{a^2}{b^2}\biggr)\cos^2\theta - \biggl(\frac{a^2}{c^2}\biggr)\sin^2\theta  
~-~ \sin^2\theta \biggl(\frac{a^2}{c^2}\biggr) \biggr] 
\biggr\} 
</math>
  </td>
</tr>

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

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

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

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

===Riemann-Derived Velocity Components===

====Inertial-Frame Expressions====
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</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</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</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>
where,

<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">
<div align="center">'''Rotating-Frame Vorticity'''</div>
<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>
</td></tr></table>

In the inertial frame, the velocity components are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\bold{u}^{(0)} = \bold{u} + \boldsymbol{\Omega \times}\bold{x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( \boldsymbol{\hat\imath} \dot{x}
+ \boldsymbol{\hat\jmath} \dot{y}
+ \bold{\hat{k}} \dot{z} )
+
(\boldsymbol{\hat\jmath}\Omega_2 + \boldsymbol{\hat{k}}\Omega_3) \boldsymbol\times (\boldsymbol{\hat\imath} x + \boldsymbol{\hat\jmath}y + \boldsymbol{\hat{k}}z)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( \boldsymbol{\hat\imath} \dot{x}
+ \boldsymbol{\hat\jmath} \dot{y}
+ \bold{\hat{k}} \dot{z} )
+
\Omega_2(\boldsymbol{\hat\imath}z -\boldsymbol{\hat{k}} x)
+ \Omega_3 (\boldsymbol{\hat\jmath}x - \boldsymbol{\hat\imath}y)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath}(\dot{x} + \Omega_2 z - \Omega_3y)
+
\boldsymbol{\hat\jmath}(\dot{y} +  \Omega_3x)
+
\boldsymbol{\hat{k}}(\dot{z} - \Omega_2 x)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath}\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 y - \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 z + \Omega_2 z - \Omega_3y\biggr]
+
\boldsymbol{\hat\jmath}\biggr[ - \gamma \Omega_3 x +  \Omega_3x \biggr]
+
\boldsymbol{\hat{k}}\biggl[ + \beta \Omega_2 x - \Omega_2 x \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath}\biggl\{ \biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma - 1 \biggr]\Omega_3y + \biggl[ 1 -  \biggl(\frac{a}{c}\biggr)^2 \beta\biggr] \Omega_2 z\biggr\}
+
\boldsymbol{\hat\jmath} ( 1- \gamma ) \Omega_3 x
+
\boldsymbol{\hat{k}} ( \beta -1 ) \Omega_2 x \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\boldsymbol{\zeta}^{(0)} = \boldsymbol{\nabla \times}\bold{u}^{(0)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath} \biggl[ 0\biggr]
+ \boldsymbol{\hat\jmath} \biggl\{ \biggl[ 1 -  \biggl(\frac{a}{c}\biggr)^2 \beta\biggr] \Omega_2 + (1-\beta)\Omega_2 \biggr\}
+ \bold{\hat{k}} \biggl\{ (1-\gamma)\Omega_3 +  \biggl[1 -  \biggl(\frac{a}{b}\biggr)^2 \gamma \biggr]\Omega_3 \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\jmath} \biggl[ 1 -  \biggl(\frac{a}{c}\biggr)^2 \beta  + (1-\beta) \biggr]\Omega_2
+ \bold{\hat{k}} \biggl[ (1-\gamma) +  1 -  \biggl(\frac{a}{b}\biggr)^2 \gamma  \biggr] \Omega_3 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\jmath} \biggl[ 2 -  \biggl( 1 + \frac{a^2}{c^2} \biggr)\beta   \biggr]\Omega_2
+ \bold{\hat{k}} \biggl[ 2 - \biggl(1 +  \frac{a^2}{b^2} \biggr) \gamma  \biggr] \Omega_3 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\jmath} \biggl[ 2 + \frac{\zeta_2}{\Omega_2}   \biggr]\Omega_2
+ \bold{\hat{k}} \biggl[ 2 + \frac{\zeta_3}{\Omega_3}  \biggr] \Omega_3 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\boldsymbol{\Omega} + \boldsymbol{\zeta} \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>

====Coefficient Expression in Tipped Plane ====
In order for our expressions for the body-frame velocity components to align with Riemann's velocity components, we see, first, that,
<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{\dot{z}}{\dot{y}}  
=
- \frac{\zeta_2}{\zeta_3} \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]  \frac{c^2}{b^2} \, .
</math>
  </td>
</tr>
</table>
As a result, we find that,
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ c^2 + \frac{b^2 \zeta_2^2}{\zeta_3^2} \biggl( \frac{a^2 + b^2}{a^2 + c^2} \biggr)^2 \frac{c^4}{b^4}  \biggr]
\biggl[1 + \frac{\zeta_2^2}{\zeta_3^2} \biggl( \frac{a^2 + b^2}{a^2 + c^2} \biggr)^2  \frac{c^4}{b^4}  \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl[\frac{x_\mathrm{max}}{y_\mathrm{max}}  \biggr]^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a^2
\biggl[ b^2 \zeta_3^2 (a^2 + c^2)^2 + c^2 \zeta_2^2 ( a^2 + b^2)^2  \biggr]
\biggl[b^4\zeta_3^2 (a^2 + c^2)^2 + c^4 \zeta_2^2 ( a^2 + b^2 )^2  \biggr]^{-1} 
\, .
</math>
  </td>
</tr>
</table>

Finally, setting the (square of the) two expressions for the <math>~\dot{y}</math> velocity component equal to one another gives,
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~{\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]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a^2\zeta_3^2\biggl[ \frac{ b^2}{a^2 + b^2} \biggr]^2 
\biggl[ b^2 \zeta_3^2 (a^2 + c^2)^2 + c^2 \zeta_2^2 ( a^2 + b^2)^2  \biggr]
\biggl[b^4\zeta_3^2 (a^2 + c^2)^2 + c^4 \zeta_2^2 ( a^2 + b^2 )^2  \biggr]^{-1} 
\biggl\{ 1 + \biggl[ - \frac{\zeta_2}{\zeta_3} \biggl( \frac{a^2 + b^2}{a^2 + c^2} \biggr)  \frac{c^2}{b^2} \biggr]^2 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{ a^2 b^4 \zeta_3^2}{(a^2 + b^2)^2} \biggr] 
\biggl[ b^2 \zeta_3^2 (a^2 + c^2)^2 + c^2 \zeta_2^2 ( a^2 + b^2)^2  \biggr]
\biggl[b^4\zeta_3^2 (a^2 + c^2)^2 + c^4 \zeta_2^2 ( a^2 + b^2 )^2  \biggr]^{-1} 
\biggl\{ \frac{b^4 \zeta_3^2 (a^2 + c^2)^2 + c^4 \zeta_2^2 (a^2 + b^2)^2}{b^4 \zeta_3^2 (a^2 + c^2)^2} \biggr\}
</math>
  </td>
</tr>

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

===Try Tipped Plane Again===

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

<tr>
  <td align="right">
<math>~\frac{\dot{x}'}{\dot\varphi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- (y' - y_0) \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] \, ,
</math>
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="right">
<math>~\frac{\dot{y}'}{\dot\varphi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x' \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr] \, .
</math>
  </td>
</tr>
</table>

<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}  \, ,
</math>
  </td>
</tr>
</table>
where,

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

<tr>
  <td align="right">
<math>~\biggl[\frac{x_\mathrm{max}}{y_\mathrm{max}}  \biggr]^2 (1+\tan^2\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a^2}{b^2 c^2}  (c^2 + b^2\tan^2\theta)  \, ,
</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<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] \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~y_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{(z_0 + z'\cos\theta) b^2 \sin\theta}{c^2 \cos^2\theta + b^2\sin^2\theta} \, .</math>
  </td>
</tr>
</table>


<table border="1" align="center" cellpadding="10" width="60%"><tr><td align="left">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~\Rightarrow~~~\boldsymbol{\Omega} = \boldsymbol{\hat\jmath} \Omega_2 + \boldsymbol{\hat{k}} \Omega_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Omega_2 ( \boldsymbol{\hat{\jmath}'}\cos\theta  - \boldsymbol{\hat{k}'}\sin\theta )
+
\Omega_3 ( \boldsymbol{\hat{\jmath}'}\sin\theta  + \boldsymbol{\hat{k}'}\cos\theta )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\jmath'} (\Omega_2 \cos\theta + \Omega_3\sin\theta) + \boldsymbol{\hat{k}'} (- \Omega_2\sin\theta + \Omega_3\cos\theta) \, .
</math>
  </td>
</tr>
</table>

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

In the inertial reference frame,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\bold{u'}^{(0)} = \bold{u'} + \boldsymbol{\Omega \times}\bold{x'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( \boldsymbol{\hat\imath'} \dot{x}'
+ \boldsymbol{\hat\jmath'} \dot{y}'
+ \boldsymbol{\hat{k}'} \cancelto{0}{\dot{z}' })
+
[  \boldsymbol{\hat\jmath'} (\Omega_2 \cos\theta + \Omega_3\sin\theta) + \boldsymbol{\hat{k}'} (- \Omega_2\sin\theta + \Omega_3\cos\theta) ]
\boldsymbol\times (\boldsymbol{\hat\imath'} x' + \boldsymbol{\hat\jmath'}y' + \boldsymbol{\hat{k'}}z')
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath'} \dot{x}'
+ \boldsymbol{\hat\jmath'} \dot{y}'
+
[  -\boldsymbol{\hat{k'}} (\Omega_2 \cos\theta + \Omega_3\sin\theta) x']
+
[  \boldsymbol{\hat\imath'} (\Omega_2 \cos\theta + \Omega_3\sin\theta) z']
+
[  \boldsymbol{\hat\jmath'} (- \Omega_2\sin\theta + \Omega_3\cos\theta)x' ]
+
[ - \boldsymbol{\hat\imath'} (- \Omega_2\sin\theta + \Omega_3\cos\theta) y']
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath'}  \biggl[ \dot{x}' + (\Omega_2\sin\theta - \Omega_3\cos\theta) y'  + (\Omega_2 \cos\theta + \Omega_3\sin\theta) z' \biggr]
+ \boldsymbol{\hat\jmath'} \biggl[ \dot{y}' +  (- \Omega_2\sin\theta + \Omega_3\cos\theta)x'  \biggr]
~-~\boldsymbol{\hat{k'}} \biggl[ (\Omega_2 \cos\theta + \Omega_3\sin\theta) x' \biggr] \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\boldsymbol{\zeta'}^{(0)} \equiv \boldsymbol{\nabla \times}\bold{u}^{(0)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath'} \biggl[ \frac{\partial u_z' }{\partial y'} - \frac{\partial u_y'}{\partial z'} \biggr]^{(0)}
+ \boldsymbol{\hat\jmath'} \biggl[ \frac{\partial u_x'}{\partial z'} - \frac{\partial u_z'}{\partial x'} \biggr]^{(0)}
+ \bold{\hat{k}'} \biggl[ \frac{\partial u_y'}{\partial x'} - \frac{\partial u_x'}{\partial y'} \biggr]^{(0)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath'} \biggl[ 0 \biggr]
+ \boldsymbol{\hat\jmath'} \biggl[\frac{\partial \dot{x}'}{\partial z'} +  (\Omega_2 \cos\theta + \Omega_3\sin\theta)  + (\Omega_2 \cos\theta + \Omega_3\sin\theta) \biggr]
+ \bold{\hat{k}'} \biggl[\frac{\partial \dot{y}'}{\partial x'} + (- \Omega_2\sin\theta + \Omega_3\cos\theta) - \frac{\partial \dot{x}'}{\partial y'} - (\Omega_2\sin\theta - \Omega_3\cos\theta) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\boldsymbol{\Omega} +
\boldsymbol{\hat\jmath'} \biggl[ \frac{\partial \dot{x}'}{\partial z'} \biggr]
+ \bold{\hat{k}'} \biggl[\frac{\partial \dot{y}'}{\partial x'} - \frac{\partial \dot{x}'}{\partial y'} \biggr] 
</math>
  </td>
</tr>
</table>

We appreciate that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \dot{x}'}{\partial z'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial }{\partial z'} \biggl[
(y_0 - y' ) \dot\varphi \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr)
\biggr]
=
\dot\varphi \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr)
\frac{\partial y_0}{\partial z'} 
=
-\dot\varphi \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr)
\biggl[\frac{b^2 \sin\theta \cos\theta }{c^2 \cos^2\theta + b^2\sin^2\theta} \biggr] \, .
</math>
  </td>
</tr>
</table>
Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{\zeta'}^{(0)} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\boldsymbol{\Omega} -
\boldsymbol{\hat\jmath'} \dot\varphi \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr)
\biggl[\frac{b^2 \sin\theta \cos\theta }{c^2 \cos^2\theta + b^2\sin^2\theta} \biggr] 
+ \bold{\hat{k}'} \biggl[ \biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr) + \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr) \biggr]\dot\varphi \, .
</math>
  </td>
</tr>
</table>

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

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

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

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

<tr>
  <td align="right">
<math>~\boldsymbol{\zeta'}^{(0)} - 2\boldsymbol{\Omega}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \dot\varphi \biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr) \biggl\{
-\boldsymbol{\hat\jmath'} \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr)^2
\biggl[\frac{b^2 \sin\theta \cos\theta }{c^2 \cos^2\theta + b^2\sin^2\theta} \biggr] 
+ \bold{\hat{k}'} \biggl[ 1+ \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{\zeta_3}{\cos\theta} \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggl\{
-\boldsymbol{\hat\jmath'} \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr)^2
\biggl[\frac{b^2 \sin\theta \cos\theta }{c^2 \cos^2\theta + b^2\sin^2\theta} \biggr] 
+ \bold{\hat{k}'} \biggl[ 1+ \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{\zeta_3}{\cos\theta} \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggl\{
-\boldsymbol{\hat\jmath'} \frac{a^2 }{b^2 c^2}  (c^2 \cos^2\theta + b^2\sin^2\theta)
\biggl[\frac{b^2 \sin\theta \cos\theta }{c^2 \cos^2\theta + b^2\sin^2\theta} \biggr] 
+ \bold{\hat{k}'} \biggl[ 1+ \frac{a^2 }{b^2 c^2}  (c^2 \cos^2\theta + b^2\sin^2\theta) \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\boldsymbol{\hat\jmath'} \frac{a^2 }{c^2}  \biggl[ \sin\theta \cos\theta\biggr]  \frac{\zeta_3}{\cos\theta} \biggl[ \frac{b^2}{a^2 + b^2} \biggr] 
+ \bold{\hat{k}'} \biggl[ 1+ \frac{a^2 }{b^2 c^2}  (c^2 \cos^2\theta + b^2\sin^2\theta) \biggr]  \frac{\zeta_3}{\cos\theta} \biggl[ \frac{b^2}{a^2 + b^2} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\jmath'}~ \biggl[ \frac{a^2 \zeta_2}{a^2 + c^2} \biggr]  \cos\theta
+ \bold{\hat{k}'} \biggl[ 1 \biggr]  \frac{\zeta_3}{\cos\theta} \biggl[ \frac{b^2}{a^2 + b^2} \biggr] 
+ \bold{\hat{k}'} \biggl[ \frac{a^2 }{b^2 c^2}  (c^2 \cos^2\theta ) \biggr]  \frac{\zeta_3}{\cos\theta} \biggl[ \frac{b^2}{a^2 + b^2} \biggr] 
+ \bold{\hat{k}'} \biggl[ \frac{a^2 }{b^2 c^2}  (b^2\sin^2\theta) \biggr]  \frac{\zeta_3}{\cos\theta} \biggl[ \frac{b^2}{a^2 + b^2} \biggr] 
</math>
  </td>
</tr>

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

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

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

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

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

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

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


----


<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}  \, ,
</math>
  </td>
</tr>
</table>

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

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

<tr>
  <td align="right">
<math>~\Rightarrow~~~\frac{1}{\cos\theta} \biggl[ \frac{\zeta_3b^2}{a^2 + b^2} \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\biggl[ \frac{c^2 \zeta_2}{a^2 + c^2} \biggr]  \frac{1}{\sin\theta}
</math>
  </td>
</tr>
</table>

===Example Equilibrium Model===

These key parameters have been drawn from [[Appendix/References#EFE|[<font color="red">EFE</font>] ]]Chapter 7, Table XIII (p. 170):
<table width="80%" align="center" cellpadding="8" border="0">
<tr><td align="left"><math>~a = a_1 = 1</math></td></tr>
<tr><td align="left"><math>~b = a_2 = 1.25</math></td></tr>
<tr><td align="left"><math>~c = a_3 = 0.4703</math></td></tr>
<tr><td align="left"><math>~\Omega_2 = 0.3639</math></td></tr>
<tr><td align="left"><math>~\Omega_3 = 0.6633</math></td></tr>
<tr><td align="left"><math>~\zeta_2 = - 2.2794</math></td></tr>
<tr><td align="left"><math>~\zeta_3 = - 1.9637</math></td></tr>
</table>

As a consequence &#8212; see [[ThreeDimensionalConfigurations/RiemannTypeI#Try_Again|an accompanying discussion]] for details &#8212; the values of other parameters are &hellip;

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="4">&nbsp;</td>
  <td align="center" rowspan="6" colspan="1" bgcolor="lightgrey">&nbsp; </td>
  <td align="center" colspan="2">'''Example Values'''</td>
</tr>

<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} = -0.344793</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~~ \theta =</math>
  </td>
  <td align="left">
<math>~- 19.0238^\circ</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\Lambda
</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3  \cos\theta    
-
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \sin\theta 
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\Lambda =</math>
  </td>
  <td align="left">
<math>~-1.332892 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\frac{y_0}{z_0}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \frac{\zeta_2}{\Lambda} 
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{y_0}{z_0} =</math>
  </td>
  <td align="left">
<math>~+ 1.400377</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\frac{x_\mathrm{max}}{ y_\mathrm{max} }   
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \Lambda \biggl[ \frac{a^2 + b^2}{b^2} \biggr] \frac{\cos\theta}{\zeta_3} \biggr\}^{1 / 2}
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
&nbsp; &nbsp; <math>~\frac{x_\mathrm{max}}{y_\mathrm{max}} =</math>
  </td>
  <td align="left">
<math>~+ 1.025854</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\dot\varphi
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \Lambda \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3 }{\cos\theta} \biggr\}^{1 / 2}
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\dot\varphi =</math>
  </td>
  <td align="left">
<math>~+1.299300</math>
  </td>
</tr>
</table>

===COLLADA-Based Representation===

As has been described in an [[ThreeDimensionalConfigurations/RiemannTypeI#Figure3|accompanying discussion of Riemann Type 1 ellipsoids]], we have used COLLADA to construct  an animated and interactive 3D scene that displays in purple the surface of an example Type I ellipsoid; panels a and b of Figure 1 show what this ellipsoid looks like when viewed from two different perspectives.  (As a reminder &#8212; see the [[#explanation| explanation accompanying Figure 2 of that accompanying discussion]] &#8212; the ellipsoid is tilted about the x-coordinate axis at an angle of 61.25&deg; to the equilibrium spin axis, which is shown in green.)  Yellow markers also have been placed in this 3D scene at each of the coordinate locations specified in the [[#ExampleTrajectories|table that accompanies that discussion]].  From the perspective presented in Figure 3b, we can immediately identify three separate, nearly circular trajectories; the largest one corresponds to our choice of z<sub>0</sub> = -0.25, the smallest corresponds to our choice of z<sub>0</sub> = -0.60, and the one of intermediate size correspond to our choice of z<sub>0</sub> = -0.4310.  When viewed from the perspective presented in Figure 1a, we see that these three trajectories define three separate planes; each plane is tipped at an angle of &theta; = -19.02&deg; to the ''untilted'' equatorial, x-y plane of the purple ellipsoid.

<div align="center">
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center">Figure 1a</th>
  <th align="center">Figure 1b</th>
</tr>
<tr>
  <td align="left" bgcolor="lightgrey">
[[File:B125c470B.cropped.png|500px|EFE Model b41c385]]
  </td>
  <td align="left" bgcolor="lightgrey">
[[File:B125c470A.cropped.png|500px|EFE Model b41c385]]
  </td>
</tr>
<tr>
  <td align="center" colspan="2" bgcolor="lightgrey">
[[File:DataFileButton02.png|75px|file = Dropbox/3Dviewers/AutoRiemann/TypeI/Lagrange/TL15.lagrange.dae]] <font size="+2">&#x21b2;</font>
  </td>
</tr>
<tr>
  <th align="center" colspan="2">Figure 1c</th>
</tr>
<tr>
  <td align="center" bgcolor="white" colspan="2">
[[File:ProjectedOrbitsFlipped2.png|600px|EFE Model b41c385]]<br />
  <div align="center">[[File:DataFileButton02.png|75px|file = Dropbox/3Dviewers/RiemannModels/RiemannCalculations.xlsx --- worksheet = TypeI_1b]] <font size="+2">&#x21b2;</font></div>
  </td>
</tr>
</table>
</div>