Appendix/Ramblings/ConcentricEllipsoidalT8Coordinates - Revision history

Joel2: /* Concentric Ellipsoidal (T8) Coordinates */

2024-07-04T18:57:02Z

Concentric Ellipsoidal (T8) Coordinates

← Older revision		Revision as of 14:57, 4 July 2024
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	Building on our [[Appendix/Ramblings/DirectionCosines\|general introduction to ''Direction Cosines'']] in the context of orthogonal curvilinear coordinate systems, and on our previous development of [[Appendix/Ramblings/T3Integrals\|T3]] (concentric oblate-spheroidal) and [[Appendix/Ramblings/EllipticCylinderCoordinates#T5_Coordinates\|T5]] (concentric elliptic) coordinate systems, here we explore the creation of a concentric ellipsoidal (T8) coordinate system. This is motivated by our [[ThreeDimensionalConfigurations/Challenges#Trial_.232\|desire to construct a fully analytically prescribable model of a nonuniform-density ellipsoidal configuration that is an analog to Riemann S-Type ellipsoids]].		Building on our [[Appendix/Ramblings/DirectionCosines\|general introduction to ''Direction Cosines'']] in the context of orthogonal curvilinear coordinate systems, and on our previous development of [[Appendix/Ramblings/T3Integrals\|T3]] (concentric oblate-spheroidal) and [[Appendix/Ramblings/EllipticCylinderCoordinates#T5_Coordinates\|T5]] (concentric elliptic) coordinate systems, here we explore the creation of a concentric ellipsoidal (T8) coordinate system. This is motivated by our [[ThreeDimensionalConfigurations/Challenges#Trial_.232\|desire to construct a fully analytically prescribable model of a nonuniform-density ellipsoidal configuration that is an analog to Riemann S-Type ellipsoids]].

	Note that, in a [[Appendix/Ramblings/~~ConcentricEllipsodalCoordinates#Background~~\|separate but closely related discussion]], we made attempts to define this coordinate system, numbering the trials up through "T7." In this "T7" effort, we were able to define a set of three, mutually orthogonal unit vectors that should work to define a fully three-dimensional, concentric ellipsoidal coordinate system. But we were unable to figure out what coordinate function, <math>~\lambda_3(x, y, z)</math>, was associated with the third unit vector. In addition, we found the <math>~\lambda_2</math> coordinate to be rather strange in that it was not oriented in a manner that resembled the classic spherical coordinate system. Here we begin by redefining the <math>~\lambda_2</math> coordinate such that its associated <math>~\hat{e}_3</math> unit vector lies parallel to the x-y plane.		Note that, in a [[Appendix/Ramblings/ConcentricEllipsoidalCoordinates\|separate but closely related discussion]], we made attempts to define this coordinate system, numbering the trials up through "T7." In this "T7" effort, we were able to define a set of three, mutually orthogonal unit vectors that should work to define a fully three-dimensional, concentric ellipsoidal coordinate system. But we were unable to figure out what coordinate function, <math>~\lambda_3(x, y, z)</math>, was associated with the third unit vector. In addition, we found the <math>~\lambda_2</math> coordinate to be rather strange in that it was not oriented in a manner that resembled the classic spherical coordinate system. Here we begin by redefining the <math>~\lambda_2</math> coordinate such that its associated <math>~\hat{e}_3</math> unit vector lies parallel to the x-y plane.

	==Realigning the Second Coordinate==		==Realigning the Second Coordinate==

Joel2: /* Kappa (κ8) Coordinates */

2024-07-04T18:51:39Z

Kappa (κ8) Coordinates

← Older revision		Revision as of 14:51, 4 July 2024
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	</tr>		</tr>
	</table>		</table>
	and the partial derivatives of <math>~A</math> and <math>~B</math> are detailed in [[~~User:Tohline/~~Appendix/Ramblings/ConcentricEllipsodalCoordinates#ABderivatives\|an accompanying discussion]].		and the partial derivatives of <math>~A</math> and <math>~B</math> are detailed in [[Appendix/Ramblings/ConcentricEllipsodalCoordinates#ABderivatives\|an accompanying discussion]].
	</td>		</td>
	</tr>		</tr>
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	</table>		</table>

	The other boxes in the n = 2 row have been drawn from [[~~User:Tohline/~~Appendix/Ramblings/ConcentricEllipsodalT8Coordinates#Associated_h3_Scale_Factor\|our accompanying EUREKA! moment]] and the n = 3 row of the table that details "Direction Cosine Components for T8 Coordinates."		The other boxes in the n = 2 row have been drawn from [[Appendix/Ramblings/ConcentricEllipsodalT8Coordinates#Associated_h3_Scale_Factor\|our accompanying EUREKA! moment]] and the n = 3 row of the table that details "Direction Cosine Components for T8 Coordinates."

	==Attempt 1==		==Attempt 1==

Joel2: /* What is the Third Coordinate Function, λ3 */

2024-07-04T18:47:50Z

What is the Third Coordinate Function, λ3

← Older revision		Revision as of 14:47, 4 July 2024
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	</table>		</table>

	Then, after searching through the [[~~User:Tohline/~~Appendix/References#CRC\|CRC Mathematical Handbook's]] pages of familiar derivative expressions, we appreciate that		Then, after searching through the [[Appendix/References#CRC\|CRC Mathematical Handbook's]] pages of familiar derivative expressions, we appreciate that

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

Joel2: /* Associated h3 Scale Factor */

2024-07-04T18:28:49Z

Associated h3 Scale Factor

← Older revision		Revision as of 14:28, 4 July 2024
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	</tr>		</tr>
	</table>		</table>


	===What is the Third Coordinate Function, λ<sub>3</sub>===		===What is the Third Coordinate Function, λ<sub>3</sub>===

Joel2: /* Background */

2024-07-04T18:26:18Z

Background

← Older revision		Revision as of 14:26, 4 July 2024
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	==Background==		==Background==
	Building on our [[~~User:Tohline/~~Appendix/Ramblings/DirectionCosines\|general introduction to ''Direction Cosines'']] in the context of orthogonal curvilinear coordinate systems, and on our previous development of [[~~User:Tohline/~~Appendix/Ramblings/T3Integrals\|T3]] (concentric oblate-spheroidal) and [[~~User:Tohline/~~Appendix/Ramblings/EllipticCylinderCoordinates#T5_Coordinates\|T5]] (concentric elliptic) coordinate systems, here we explore the creation of a concentric ellipsoidal (T8) coordinate system. This is motivated by our [[~~User:Tohline/~~ThreeDimensionalConfigurations/Challenges#Trial_.232\|desire to construct a fully analytically prescribable model of a nonuniform-density ellipsoidal configuration that is an analog to Riemann S-Type ellipsoids]].		Building on our [[Appendix/Ramblings/DirectionCosines\|general introduction to ''Direction Cosines'']] in the context of orthogonal curvilinear coordinate systems, and on our previous development of [[Appendix/Ramblings/T3Integrals\|T3]] (concentric oblate-spheroidal) and [[Appendix/Ramblings/EllipticCylinderCoordinates#T5_Coordinates\|T5]] (concentric elliptic) coordinate systems, here we explore the creation of a concentric ellipsoidal (T8) coordinate system. This is motivated by our [[ThreeDimensionalConfigurations/Challenges#Trial_.232\|desire to construct a fully analytically prescribable model of a nonuniform-density ellipsoidal configuration that is an analog to Riemann S-Type ellipsoids]].

	Note that, in a [[~~User:Tohline/~~Appendix/Ramblings/ConcentricEllipsodalCoordinates#Background\|separate but closely related discussion]], we made attempts to define this coordinate system, numbering the trials up through "T7." In this "T7" effort, we were able to define a set of three, mutually orthogonal unit vectors that should work to define a fully three-dimensional, concentric ellipsoidal coordinate system. But we were unable to figure out what coordinate function, <math>~\lambda_3(x, y, z)</math>, was associated with the third unit vector. In addition, we found the <math>~\lambda_2</math> coordinate to be rather strange in that it was not oriented in a manner that resembled the classic spherical coordinate system. Here we begin by redefining the <math>~\lambda_2</math> coordinate such that its associated <math>~\hat{e}_3</math> unit vector lies parallel to the x-y plane.		Note that, in a [[Appendix/Ramblings/ConcentricEllipsodalCoordinates#Background\|separate but closely related discussion]], we made attempts to define this coordinate system, numbering the trials up through "T7." In this "T7" effort, we were able to define a set of three, mutually orthogonal unit vectors that should work to define a fully three-dimensional, concentric ellipsoidal coordinate system. But we were unable to figure out what coordinate function, <math>~\lambda_3(x, y, z)</math>, was associated with the third unit vector. In addition, we found the <math>~\lambda_2</math> coordinate to be rather strange in that it was not oriented in a manner that resembled the classic spherical coordinate system. Here we begin by redefining the <math>~\lambda_2</math> coordinate such that its associated <math>~\hat{e}_3</math> unit vector lies parallel to the x-y plane.

	==Realigning the Second Coordinate==		==Realigning the Second Coordinate==

Joel2: Created page with "FORCETOC =Concentric Ellipsoidal (T8) Coordinates= ==Background== Building on our general introduction to ''Direction Cosines'' in the context of orthogonal curvilinear coordinate systems, and on our previous development of T3 (concentric oblate-spheroidal) and User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates#T5_Coo..."

2024-01-13T00:49:34Z

Created page with "__FORCETOC__  =Concentric Ellipsoidal (T8) Coordinates= ==Background== Building on our general introduction to ''Direction Cosines'' in the context of orthogonal curvilinear coordinate systems, and on our previous development of T3 (concentric oblate-spheroidal) and User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates#T5_Coo..."

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