User:Tohline/Appendix/Ramblings/ConcentricEllipsodalT8Coordinates
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Building on our [[User:Tohline/Appendix/Ramblings/DirectionCosinesgeneral introduction to ''Direction Cosines'']] in the context of orthogonal curvilinear coordinate systems, and on our previous development of [[User:Tohline/Appendix/Ramblings/T3IntegralsT3]] (concentric oblatespheroidal) and [[User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates#T5_CoordinatesT5]] (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_.232desire to construct a fully analytically prescribable model of a nonuniformdensity ellipsoidal configuration that is an analog to Riemann SType ellipsoids]].  Building on our [[User:Tohline/Appendix/Ramblings/DirectionCosinesgeneral introduction to ''Direction Cosines'']] in the context of orthogonal curvilinear coordinate systems, and on our previous development of [[User:Tohline/Appendix/Ramblings/T3IntegralsT3]] (concentric oblatespheroidal) and [[User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates#T5_CoordinatesT5]] (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_.232desire to construct a fully analytically prescribable model of a nonuniformdensity ellipsoidal configuration that is an analog to Riemann SType ellipsoids]].  
  Note that, in a [[User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates#Backgroundseparate but closely related discussion]], we made attempts to define this coordinate system, numbering the trials up through "T7."  +  Note that, in a [[User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates#Backgroundseparate 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 threedimensional, 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 xy plane. 
+  ==Realigning the Second Coordinate==  
+  
+  The first coordinate remains the same as before, namely,  
+  <table border="0" cellpadding="5" align="center">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~\lambda_1^2</math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~x^2 + q^2 y^2 + p^2 z^2 \, .</math>  
+  </td>  
+  </tr>  
+  </table>  
+  This may be rewritten as,  
+  <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>  
+  </tr>  
+  </table>  
+  where,  
+  <table border="0" cellpadding="5" align="center">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~a = \lambda_1 \, ,</math>  
+  </td>  
+  <td align="center"> </td>  
+  <td align="center">  
+  <math>~b = \frac{\lambda_1}{q} \, ,</math>  
+  </td>  
+  <td align="center"> </td>  
+  <td align="left">  
+  <math>~c = \frac{\lambda_1}{p} \, .</math>  
+  </td>  
+  </tr>  
+  </table>  
+  
+  By specifying the value of <math>~z = z_0 < c</math>, as well as the value of <math>~\lambda_1</math>, we are picking a plane that lies parallel to, but a distance <math>~z_0</math> above, the equatorial plane. The elliptical curve that defines the intersection of the <math>~\lambda_1</math>constant surface with this plane is defined by the expression,  
+  <table border="0" cellpadding="5" align="center">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~\lambda_1^2  p^2z_0^2</math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~x^2 + q^2 y^2 </math>  
+  </td>  
+  </tr>  
+  
+  <tr>  
+  <td align="right">  
+  <math>~\Rightarrow~~~1</math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~\biggl( \frac{x}{a_{2D}}\biggr)^2 + \biggl( \frac{y}{b_{2D}}\biggr)^2 \, ,</math>  
+  </td>  
+  </tr>  
+  </table>  
+  where,  
+  <table border="0" cellpadding="5" align="center">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~a_{2D} = \biggl(\lambda_1^2  p^2z_0^2 \biggr)^{1 / 2} \, ,</math>  
+  </td>  
+  <td align="center"> </td>  
+  <td align="left">  
+  <math>~b_{2D} = \frac{1}{q} \biggl(\lambda_1^2  p^2z_0^2 \biggr)^{1 / 2} \, .</math>  
+  </td>  
+  </tr>  
+  </table>  
=See Also=  =See Also= 
Revision as of 22:47, 18 January 2021
Contents 
Concentric Ellipsoidal (T8) Coordinates
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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 oblatespheroidal) and T5 (concentric elliptic) coordinate systems, here we explore the creation of a concentric ellipsoidal (T8) coordinate system. This is motivated by our desire to construct a fully analytically prescribable model of a nonuniformdensity ellipsoidal configuration that is an analog to Riemann SType ellipsoids.
Note that, in a 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 threedimensional, concentric ellipsoidal coordinate system. But we were unable to figure out what coordinate function, , was associated with the third unit vector. In addition, we found the 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 coordinate such that its associated unit vector lies parallel to the xy plane.
Realigning the Second Coordinate
The first coordinate remains the same as before, namely,



This may be rewritten as,



where,



By specifying the value of , as well as the value of , we are picking a plane that lies parallel to, but a distance above, the equatorial plane. The elliptical curve that defines the intersection of the constant surface with this plane is defined by the expression,






where,


See Also
© 2014  2020 by Joel E. Tohline 