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(Concentric Ellipsoidal (T6) Coordinates)
(Concentric Ellipsoidal (T6) Coordinates)
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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 (T6) 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 (T6) 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]].
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==Orthogonal Coordinates==
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 +
We start by defining a "radial" coordinate whose values identify various concentric ellipsoidal shells,
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<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\lambda_1</math>
 +
  </td>
 +
  <td align="center">
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<math>~\equiv</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} \, .</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
When <math>~\lambda_1 = a</math>, we obtain the standard definition of an ellipsoidal surface, it being understood that, <math>~q^2 = a^2/b^2</math> and <math>~p^2 = a^2/c^2</math>.  (We will assume that <math>~a > b > c</math>, that is, <math>~p^2 > q^2 > 1</math>.)
=See Also=
=See Also=

Revision as of 08:17, 26 October 2020

Contents

Concentric Ellipsoidal (T6) 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 (T6) 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.

Orthogonal Coordinates

We start by defining a "radial" coordinate whose values identify various concentric ellipsoidal shells,

~\lambda_1

~\equiv

~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} \, .

When ~\lambda_1 = a, we obtain the standard definition of an ellipsoidal surface, it being understood that, ~q^2 = a^2/b^2 and ~p^2 = a^2/c^2. (We will assume that ~a > b > c, that is, ~p^2 > q^2 > 1.)

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

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