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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 [[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]].
-
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."  
+
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.
 +
==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">&nbsp; &nbsp;&nbsp; &nbsp;</td>
 +
  <td align="center">
 +
<math>~b = \frac{\lambda_1}{q} \, ,</math>
 +
  </td>
 +
<td align="center">&nbsp; &nbsp;&nbsp; &nbsp;</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">&nbsp; &nbsp;&nbsp; &nbsp;</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

Whitworth's (1981) Isothermal Free-Energy Surface
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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 oblate-spheroidal) 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 nonuniform-density ellipsoidal configuration that is an analog to Riemann S-Type 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 three-dimensional, concentric ellipsoidal coordinate system. But we were unable to figure out what coordinate function, ~\lambda_3(x, y, z), was associated with the third unit vector. In addition, we found the ~\lambda_2 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 ~\lambda_2 coordinate such that its associated ~\hat{e}_3 unit vector lies parallel to the x-y plane.

Realigning the Second Coordinate

The first coordinate remains the same as before, namely,

~\lambda_1^2

~=

~x^2 + q^2 y^2 + p^2 z^2 \, .

This may be rewritten as,

~1

~=

~\biggl( \frac{x}{a}\biggr)^2 + \biggl( \frac{y}{b}\biggr)^2 + \biggl(\frac{z}{c}\biggr)^2 \, ,

where,

~a = \lambda_1 \, ,

      

~b = \frac{\lambda_1}{q} \, ,

      

~c = \frac{\lambda_1}{p} \, .

By specifying the value of ~z = z_0 < c, as well as the value of ~\lambda_1, we are picking a plane that lies parallel to, but a distance ~z_0 above, the equatorial plane. The elliptical curve that defines the intersection of the ~\lambda_1-constant surface with this plane is defined by the expression,

~\lambda_1^2 - p^2z_0^2

~=

~x^2 + q^2 y^2

~\Rightarrow~~~1

~=

~\biggl( \frac{x}{a_{2D}}\biggr)^2 + \biggl( \frac{y}{b_{2D}}\biggr)^2 \, ,

where,

~a_{2D} = \biggl(\lambda_1^2 - p^2z_0^2 \biggr)^{1 / 2} \, ,

      

~b_{2D} = \frac{1}{q} \biggl(\lambda_1^2 - p^2z_0^2 \biggr)^{1 / 2} \, .

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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