User:Tohline/Appendix/Ramblings/ConcentricEllipsodalT8Coordinates
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 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, <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,
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<math>~\lambda_1^2</math> |
<math>~=</math> |
<math>~x^2 + q^2 y^2 + p^2 z^2 \, .</math> |
This may be rewritten as,
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<math>~1</math> |
<math>~=</math> |
<math>~\biggl( \frac{x}{a}\biggr)^2 + \biggl( \frac{y}{b}\biggr)^2 + \biggl(\frac{z}{c}\biggr)^2 \, ,</math> |
where,
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<math>~a = \lambda_1 \, ,</math> |
<math>~b = \frac{\lambda_1}{q} \, ,</math> |
<math>~c = \frac{\lambda_1}{p} \, .</math> |
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,
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<math>~\lambda_1^2 - p^2z_0^2</math> |
<math>~=</math> |
<math>~x^2 + q^2 y^2 </math> |
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<math>~\Rightarrow~~~1</math> |
<math>~=</math> |
<math>~\biggl( \frac{x}{a_{2D}}\biggr)^2 + \biggl( \frac{y}{b_{2D}}\biggr)^2 \, ,</math> |
where,
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<math>~a_{2D} = \biggl(\lambda_1^2 - p^2z_0^2 \biggr)^{1 / 2} \, ,</math> |
<math>~b_{2D} = \frac{1}{q} \biggl(\lambda_1^2 - p^2z_0^2 \biggr)^{1 / 2} \, .</math> |
At each point along this elliptic curve, the line that is tangent to the curve has a slope that can be determined by simply differentiating the equation that describes the curve, that is,
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<math>~0</math> |
<math>~=</math> |
<math>~\frac{2x dx}{a_{2D}^2} + \frac{2y dy}{b_{2D}^2}</math> |
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<math>~\Rightarrow~~~\frac{dy}{dx}</math> |
<math>~=</math> |
<math>~- \frac{2x}{a_{2D}^2} \cdot \frac{b_{2D}^2}{2y} = - \frac{x}{q^2y} \, .</math> |
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<math>~\Rightarrow~~~\Delta y</math> |
<math>~=</math> |
<math>~- \biggl( \frac{x}{q^2y} \biggr)\Delta x \, .</math> |
The unit vector that lies tangent to any point on this elliptical curve will be described by the expression,
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<math>~\hat{e}_2</math> |
<math>~=</math> |
<math>~ \hat\imath~ \biggl\{ \frac{\Delta x}{[ (\Delta x)^2 + (\Delta y)^2 ]^{1 / 2}} \biggr\} + \hat\jmath~ \biggl\{ \frac{\Delta y}{[ (\Delta x)^2 + (\Delta y)^2 ]^{1 / 2}} \biggr\} </math> |
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<math>~=</math> |
<math>~ \hat\imath~ \biggl\{ \frac{1}{[ 1 + x^2/(q^4y^2) ]^{1 / 2}} \biggr\} - \hat\jmath~ \biggl\{ \frac{x/(q^2y)}{[ 1 + x^2/(q^4y^2) ]^{1 / 2}} \biggr\} </math> |
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<math>~=</math> |
<math>~ \hat\imath~ \biggl\{ \frac{q^2y}{[ x^2 + q^4y^2 ]^{1 / 2}} \biggr\} - \hat\jmath~ \biggl\{ \frac{x}{[ x^2 + q^4y^2 ]^{1 / 2}} \biggr\} \, .</math> |
As we have discovered, the coordinate that gives rise to this unit vector is,
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<math>~\lambda_2</math> |
<math>~=</math> |
<math>~\frac{x}{y^{1/q^2}} \, .</math> |
See Also
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© 2014 - 2021 by Joel E. Tohline |