User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3 - Revision history

Tohline: /* See Also */

2021-05-11T21:51:04Z

Tohline: /* See Also */

2021-04-30T16:23:48Z

Tohline at 16:51, 29 April 2021

2021-04-29T16:51:33Z

← Older revision		Revision as of 16:51, 29 April 2021
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	__FORCETOC__<!-- will force the creation of a Table of Contents -->		__FORCETOC__<!-- will force the creation of a Table of Contents -->
	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
	=Challenges Constructing Ellipsoidal-Like Configurations (Pt. 2)=		=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, [[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.

Tohline: /* Demonstration */

2021-04-28T20:58:06Z

Demonstration

← Older revision		Revision as of 20:58, 28 April 2021
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	~+~		~+~
	z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \biggl\{ \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2		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\}		-~ \frac{a^2}{c^2} \cdot \tan^2\theta \biggr\}
	</math>		</math>

Tohline: /* Tipped Orbit Planes */

2021-04-28T20:32:28Z

Tipped Orbit Planes

← Older revision		Revision as of 20:32, 28 April 2021
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	</table>		</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,		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,
	~~<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>~~

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<math>~		<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\}		-~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\}		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>		</math>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	y \cdot \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{a^2}{c^2} \~~cdot~~ \~~sin~~^2\theta		-~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\}
	z \cdot \zeta_2 \biggl[ \frac{c^2 }{a^2 + c^2} \biggr] \frac{a^2}{c^2} \~~cdot~~ \~~cos~~^2\theta
	\~~, .~~
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</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,		Therefore,

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	<td align="left">		<td align="left">
	<math>~		<math>~
	y \cdot \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{a^2}{c^2} \~~cdot~~ \~~sin~~^2\theta		-~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\}
	z \cdot \zeta_2 \biggl[ \frac{c^2 }{a^2 + c^2} \biggr] \frac{a^2}{c^2} \~~cdot~~ \~~cos~~^2\theta
	~+~		~+~
	\biggl\{		\biggl\{
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	<td align="left">		<td align="left">
	<math>~		<math>~
	y \cdot \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{a^2}{c^2} \~~cdot~~ \~~sin~~^2\theta		-~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
	y\cdot \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr]
	\frac{~~a^2}~~{~~b^2c^2~~} ~~(c^2\cos^2\theta + b^2\sin^2\theta)~~
	+
	~~z \cdot \zeta_2 \biggl[ \frac{c^2~~ }{~~a^2 + c^2}~~ \~~biggr] \frac~~{~~a^2~~}~~{c^2~~} \~~cdot \cos~~^2~~\theta~~
	~+~		~+~
	z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr]		z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \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)~~		-~z \cdot \zeta_2 \biggl[ \frac{c^2 }{a^2 + c^2} \biggr] \frac{a^2}{c^2} \biggl\{ \tan^2\theta \biggr\}
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	y \cdot \zeta_3 \biggl[ \frac{a^2 }{a^2 + b^2} \biggr] \frac{~~1}{c~~^2} ~~\biggl\~~{ ~~b^2 \sin^2\theta~~		-~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\}
	(c^2\~~cos^2\theta + b^2~~\~~sin~~^2\theta~~) \biggr\}~~
	+
	~~z \cdot \zeta_2 \biggl[~~ \frac{a^2 }{a~~^2 + c~~^2} \~~biggr]~~ \frac{1}{~~b^2~~} \~~biggl\{ b~~^2\~~cos^2~~\~~theta~~
	~+~		~+~
	(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\{ \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2
			-~ \frac{a^2}{c^2} \cdot \tan^2\theta \biggr\}
	</math>		</math>
	</td>		</td>

Tohline: /* Demonstration */

2021-04-28T17:50:48Z

Demonstration

← Older revision		Revision as of 17:50, 28 April 2021
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	<math>~		<math>~
	\frac{\zeta_3}{\cos\theta}\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggl\{		\frac{\zeta_3}{\cos\theta}\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggl\{
	-[z\cos\theta - y\sin\theta] \~~sin\theta \cos~~\theta \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 \frac{a^2}{c^2}		+[z\cos\theta - y\sin\theta] \tan\theta \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 \frac{a^2}{c^2}
	\biggr\}		\biggr\}
	</math>		</math>
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	<td align="left">		<td align="left">
	<math>~		<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\~~sin~~^2\theta \biggr\}		\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 ~~\cos^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>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>~		<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\{ \~~sin~~^2\theta \biggr\}		-~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\{ \~~cos~~^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>		</math>
	</td>		</td>

Tohline: /* Demonstration */

2021-04-28T17:42:45Z

Demonstration

← Older revision		Revision as of 17:42, 28 April 2021
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	<td align="left">		<td align="left">
	<math>~		<math>~
	-[z\cos\theta - y\sin\theta] \~~sin\theta \cos~~\theta \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 \frac{a^2}{c^2}		+[z\cos\theta - y\sin\theta] \tan\theta \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 \frac{a^2}{c^2}
	\, .</math>		\, .</math>
	</td>		</td>

Tohline: /* Tipped Orbit Planes */

2021-04-28T17:39:50Z

Tipped Orbit Planes

← Older revision		Revision as of 17:39, 28 April 2021
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~- \frac{z' b^2 \~~sin\theta\cos~~\theta}{c^2 \cos^2\theta + b^2\sin^2\theta}		<math>~+ \frac{z' b^2 \tan\theta}{c^2 \cos^2\theta + b^2\sin^2\theta}
	=		=
	-z' \~~sin\theta \cos~~\theta \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 \frac{a^2}{c^2}		+z' \tan\theta \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 \frac{a^2}{c^2}
	=		=
	z' \cos\theta (1.40038)		\biggl( \frac{z'}{ \cos\theta }\biggr)(-1.40038)
	\, .</math>		\, .</math>
	</td>		</td>
	</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>.

	====Demonstration====		====Demonstration====

Tohline: /* Tipped Orbit Planes */

2021-04-27T23:14:13Z

Tipped Orbit Planes

← Older revision		Revision as of 23:14, 27 April 2021
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	<math>~		<math>~
	y \cdot \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{a^2}{c^2} \cdot \sin^2\theta		y \cdot \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{a^2}{c^2} \cdot \sin^2\theta
			~-~
			y\cdot \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr]
			\frac{a^2}{b^2c^2} (c^2\cos^2\theta + b^2\sin^2\theta)
	+		+
	z \cdot \zeta_2 \biggl[ \frac{c^2 }{a^2 + c^2} \biggr] \frac{a^2}{c^2} \cdot \cos^2\theta		z \cdot \zeta_2 \biggl[ \frac{c^2 }{a^2 + c^2} \biggr] \frac{a^2}{c^2} \cdot \cos^2\theta
	~+~		~+~
	~~\biggl\{~~
	z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr]		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]~~		\frac{a^2}{b^2c^2} (c^2\cos^2\theta + b^2\sin^2\theta)
	~~\biggr\}~~ \frac{a^2}{b^2c^2} (c^2\cos^2\theta + b^2\sin^2\theta)
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	y \cdot \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{a^2}{c^2} \~~cdot~~ \sin^2\theta		y \cdot \zeta_3 \biggl[ \frac{a^2 }{a^2 + b^2} \biggr] \frac{1}{c^2} \biggl\{ b^2 \sin^2\theta
			~-~
			(c^2\cos^2\theta + b^2\sin^2\theta) \biggr\}
	+		+
	z \cdot \zeta_2 \biggl[ \frac{c^2 }{a^2 + c^2} \biggr] \frac{a^2}{c^2~~} \cdot~~ \cos^2\theta		z \cdot \zeta_2 \biggl[ \frac{a^2 }{a^2 + c^2} \biggr] \frac{1}{b^2} \biggl\{ b^2\cos^2\theta
	~+~		~+~
	~~z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr]~~		(c^2\cos^2\theta + b^2\sin^2\theta) \biggr\}
	~~\frac{a^2}{b^2c^2}~~ (c^2\cos^2\theta + b^2\sin^2\theta)
	~~~-~~~
	~~y\cdot \zeta_3\biggl[ \frac{b^2}{a^2 + b^2}~~ \biggr]
	\~~frac{a^2~~}~~{b^2c^2} (c^2\cos^2\theta + b^2\sin^2\theta)~~
	</math>		</math>
	</td>		</td>

Tohline: /* Tipped Orbit Planes */

2021-04-27T23:01:29Z

Tipped Orbit Planes

← Older revision		Revision as of 23:01, 27 April 2021
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	-~y\cdot \zeta_3\biggl[ \frac{a^2}{a^2 + b^2} \biggr] \frac{b^2}{a^2}		-~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}		+~ 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] \sin\theta \cos\theta \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 \frac{a^2}{c^2}
	\biggr\}		\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\sin^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 \cos^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{b^2}{a^2 + b^2} \biggr] \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 \frac{a^2}{c^2} \biggl\{ \sin^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\{ \cos^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} \cdot \sin^2\theta
			+
			z \cdot \zeta_2 \biggl[ \frac{c^2 }{a^2 + c^2} \biggr] \frac{a^2}{c^2} \cdot \cos^2\theta
			\, .
			</math>
			</td>
			</tr>
			</table>
			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} \cdot \sin^2\theta
			+
			z \cdot \zeta_2 \biggl[ \frac{c^2 }{a^2 + c^2} \biggr] \frac{a^2}{c^2} \cdot \cos^2\theta
			~+~
			\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} \cdot \sin^2\theta
			+
			z \cdot \zeta_2 \biggl[ \frac{c^2 }{a^2 + c^2} \biggr] \frac{a^2}{c^2} \cdot \cos^2\theta
			~+~
			\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\} \frac{a^2}{b^2c^2} (c^2\cos^2\theta + b^2\sin^2\theta)
			</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} \cdot \sin^2\theta
			+
			z \cdot \zeta_2 \biggl[ \frac{c^2 }{a^2 + c^2} \biggr] \frac{a^2}{c^2} \cdot \cos^2\theta
			~+~
			z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr]
			\frac{a^2}{b^2c^2} (c^2\cos^2\theta + b^2\sin^2\theta)
			~-~
			y\cdot \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr]
			\frac{a^2}{b^2c^2} (c^2\cos^2\theta + b^2\sin^2\theta)
	</math>		</math>
	</td>		</td>

← Older revision		Revision as of 21:51, 11 May 2021
Line 999:		Line 999:
	* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]		* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]
	* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]		* [[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:		* Related discussions of models viewed from a rotating reference frame:
	** [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]		** [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]

← Older revision		Revision as of 16:23, 30 April 2021
Line 998:		Line 998:
	* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]		* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]
	* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]		* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]
			* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]
	* 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]]		** [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]