ThreeDimensionalConfigurations/ChallengesPt3 - Revision history

Jet53man: /* See Also */

2022-04-15T17:39:52Z

Jet53man at 17:33, 15 April 2022

2022-04-15T17:33:27Z

← Older revision		Revision as of 13:33, 15 April 2022
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	=Challenges Constructing Ellipsoidal-Like Configurations (Pt. 3)=		=Challenges Constructing Ellipsoidal-Like Configurations (Pt. 3)=

	This chapter extends the accompanying chapters titled, [[~~User:Tohline/~~ThreeDimensionalConfigurations/Challenges\|''Construction Challenges (Pt. 1)'']] and [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt2\|''(Pt. 2)'']]. The focus here is on firming up our understanding of the relationships between various "tilted" Cartesian coordinate frames.		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==		==Various Coordinate Frames==
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	</tr>		</tr>
	<tr><td align="center">		<tr><td align="center">
	For our chosen [[~~User:Tohline/~~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>.		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>		</td>
	</tr>		</tr>
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	</table>		</table>

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

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	</table>		</table>

	As we have summarized in an [[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannTypeI#EFEvelocities\|accompanying discussion]] of Riemann Type 1 ellipsoids, [[~~User:Tohline/~~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>) — see his Eq. (154), Chapter 7, §51 (p. 156). As viewed from the rotating frame of reference, the three component expressions are,		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>) — see his Eq. (154), Chapter 7, §51 (p. 156). As viewed from the rotating frame of reference, the three component expressions are,

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

	For our chosen [[~~User:Tohline/~~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>.		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>
	<td align="center">		<td align="center">
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	====Summary====		====Summary====
	In a [[~~User:Tohline/~~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,		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">		<table border="0" cellpadding="5" align="center">
	<tr>		<tr>
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	</tr>		</tr>
	<tr><td align="center">		<tr><td align="center">
	Given that b/a = 1.25 and c/a = 0.4703 for our chosen [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt2#Example_Equilibrium_Model\|Example Type I Ellipsoid]], we find that, <math>~\theta = - 0.3320 ~\mathrm{rad} = -19.02^\circ</math>.		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>		</td>
	</tr>		</tr>
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	</table>		</table>

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

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	</tr>		</tr>
	</table>		</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 [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt2#OffCenter\|accompanying derivation's expression]] for <math>~y_0</math>.		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====		====Demonstration====
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	</table>		</table>

	Recognizing, [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt2#Tipped_Orbital_Plane\|as before]], that the relevant coordinate mapping is,		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="1" align="center" width="40%" cellpadding="8"><tr><td align="left">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* See Also */

2022-02-17T23:40:33Z

Jet53man: /* See Also */

2022-02-06T01:48:40Z

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__FORCETOC__

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

This chapter extends the accompanying chapters titled, [[User:Tohline/ThreeDimensionalConfigurations/Challenges|''Construction Challenges (Pt. 1)'']] and [[User:Tohline/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 [[User:Tohline/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 [[User:Tohline/Appendix/References#EFE|[<font color="red">EFE</font>] ]] — see, for example, the first paragraph of §51 (p. 156) — 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>    and,    
<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>    and,    
<math>~\delta = \tan^{-1}\biggl[ \frac{\Omega_2}{\Omega_3} \biggr] \, .</math>
</td>
</tr>
</table>

As we have summarized in an [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#EFEvelocities|accompanying discussion]] of Riemann Type 1 ellipsoids, [[User:Tohline/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>) — see his Eq. (154), Chapter 7, §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">      and,       </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">
 
</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">
 
</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 [[User:Tohline/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 [[User:Tohline/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 [[User:Tohline/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">      and,      
<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">      and,      
<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 [[User:Tohline/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 [[User:Tohline/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 — that is, <math>~\dot{z}' = 0</math> — 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">
 
</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">
 
</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, [[User:Tohline/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">
 
</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">
 
</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">
 
</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">
 
</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">
 
</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">
 
</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">
 
</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">
 
</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">
 
</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">
 
</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">
 
</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">
 
</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">
 
</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">
 
</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">
 
</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">
 
</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>

=See Also=
* [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids|Riemann Type 1 Ellipsoids]]
* [[User:Tohline/ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations|Construction Challenges (Pt. 1)]]
* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2|Construction Challenges (Pt. 2)]]
* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3|Construction Challenges (Pt. 3)]]
* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt4|Construction Challenges (Pt. 4)]]
* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt5|Construction Challenges (Pt. 5)]]
* Related discussions of models viewed from a rotating reference frame:
** [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame|PGE]]
** <font color="red"><b>NOTE to Eric Hirschmann & David Neilsen... </b></font>I have moved the earlier contents of this page to a new Wiki location called [[User:Tohline/Apps/RiemannEllipsoids_Compressible|Compressible Riemann Ellipsoids]].

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← Older revision		Revision as of 13:39, 15 April 2022
Line 1,004:		Line 1,004:
	* Related discussions of models viewed from a rotating reference frame:		* Related discussions of models viewed from a rotating reference frame:
	** [[PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]		** [[PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]
	** <font color="red"><b>NOTE to Eric Hirschmann & David Neilsen... </b></font>I have moved the earlier contents of this page to a new Wiki location called [[Apps/~~RiemannEllipsoids_Compressible~~\|Compressible Riemann Ellipsoids]].		** <font color="red"><b>NOTE to Eric Hirschmann & David Neilsen... </b></font>I have moved the earlier contents of this page to a new Wiki location called [[Apps/RiemannEllipsoidsCompressible\|Compressible Riemann Ellipsoids]].


	{{ SGFfooter }}		{{ SGFfooter }}

← Older revision		Revision as of 19:40, 17 February 2022
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	=See Also=		=See Also=
	* [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Riemann Type 1 Ellipsoids]]
			* [[ThreeDimensionalConfigurations/DescriptionOfRiemannTypeI\|Description of Riemann Type I Ellipsoids]]
			* [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Riemann Type 1 Ellipsoids]] (old introduction)
	* [[ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Construction Challenges (Pt. 1)]]		* [[ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Construction Challenges (Pt. 1)]]
	* [[ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]		* [[ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]
Line 998:		Line 1,000:
	* [[ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]		* [[ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]
	* [[ThreeDimensionalConfigurations/ChallengesPt5\|Construction Challenges (Pt. 5)]]		* [[ThreeDimensionalConfigurations/ChallengesPt5\|Construction Challenges (Pt. 5)]]
			* [[ThreeDimensionalConfigurations/ChallengesPt6\|Construction Challenges (Pt. 6)]]

	* Related discussions of models viewed from a rotating reference frame:		* Related discussions of models viewed from a rotating reference frame:
	** [[PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]		** [[PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]

← Older revision		Revision as of 21:48, 5 February 2022
Line 992:		Line 992:

	=See Also=		=See Also=
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Riemann Type 1 Ellipsoids]]		* [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Riemann Type 1 Ellipsoids]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Construction Challenges (Pt. 1)]]		* [[ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Construction Challenges (Pt. 1)]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]		* [[ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]		* [[ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]		* [[ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt5\|Construction Challenges (Pt. 5)]]		* [[ThreeDimensionalConfigurations/ChallengesPt5\|Construction Challenges (Pt. 5)]]
	* Related discussions of models viewed from a rotating reference frame:		* Related discussions of models viewed from a rotating reference frame:
	** [[~~User:Tohline/~~PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]		** [[PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]
	** <font color="red"><b>NOTE to Eric Hirschmann & David Neilsen... </b></font>I have moved the earlier contents of this page to a new Wiki location called [[~~User:Tohline/~~Apps/RiemannEllipsoids_Compressible\|Compressible Riemann Ellipsoids]].		** <font color="red"><b>NOTE to Eric Hirschmann & David Neilsen... </b></font>I have moved the earlier contents of this page to a new Wiki location called [[Apps/RiemannEllipsoids_Compressible\|Compressible Riemann Ellipsoids]].


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ThreeDimensionalConfigurations/ChallengesPt3 - Revision history

Jet53man: /* See Also */

Jet53man at 17:33, 15 April 2022

Jet53man: /* See Also */

Jet53man: /* See Also */

Jet53man: Created page with "__FORCETOC__ =Challenges Constructing Ellipsoidal-Like Configurations (Pt. 3)= T..."

Jet53man: Created page with "FORCETOC =Challenges Constructing Ellipsoidal-Like Configurations (Pt. 3)= T..."