User:Tohline/Appendix/Ramblings/ConcentricEllipsodalT8Coordinates

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• [[User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates#Background|Trials up through T7 Coordinates]]
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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 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,

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

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,

 $~0$ $~=$ $~\frac{2x dx}{a_{2D}^2} + \frac{2y dy}{b_{2D}^2}$ $~\Rightarrow~~~\frac{dy}{dx}$ $~=$ $~- \frac{2x}{a_{2D}^2} \cdot \frac{b_{2D}^2}{2y} = - \frac{x}{q^2y} \, .$ $~\Rightarrow~~~\Delta y$ $~=$ $~- \biggl( \frac{x}{q^2y} \biggr)\Delta x \, .$

The unit vector that lies tangent to any point on this elliptical curve will be described by the expression,

 $~\hat{e}_2$ $~=$ $~ \hat\imath~ \biggl\{ \frac{\Delta x}{[ (\Delta x)^2 + (\Delta y)^2 ]^{1 / 2}} \biggr\} + \hat\jmath~ \biggl\{ \frac{\Delta y}{[ (\Delta x)^2 + (\Delta y)^2 ]^{1 / 2}} \biggr\}$ $~=$ $~ \hat\imath~ \biggl\{ \frac{1}{[ 1 + x^2/(q^4y^2) ]^{1 / 2}} \biggr\} - \hat\jmath~ \biggl\{ \frac{x/(q^2y)}{[ 1 + x^2/(q^4y^2) ]^{1 / 2}} \biggr\}$ $~=$ $~ \hat\imath~ \biggl\{ \frac{q^2y}{[ x^2 + q^4y^2 ]^{1 / 2}} \biggr\} - \hat\jmath~ \biggl\{ \frac{x}{[ x^2 + q^4y^2 ]^{1 / 2}} \biggr\} \, .$

As we have discovered, the coordinate that gives rise to this unit vector is,

 $~\lambda_2$ $~=$ $~\frac{x}{y^{1/q^2}} \, .$