# User:Tohline/Appendix/Ramblings/ConcentricEllipsodalT8Coordinates

(Difference between revisions)
 Revision as of 22:11, 18 January 2021 (view source)Tohline (Talk | contribs) (Created page with ' =Concentric Ellipsoidal (T8) Coordinates= {{LSU_HBook_header}} ==Back…')← Older edit Revision as of 22:47, 18 January 2021 (view source)Tohline (Talk | contribs) (→Background)Newer edit → Line 8: Line 8: 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, $~\lambda_3(x, y, z)$, 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, + + +
+ $~a_{2D} = \biggl(\lambda_1^2 - p^2z_0^2 \biggr)^{1 / 2} \, ,$ +        + $~b_{2D} = \frac{1}{q} \biggl(\lambda_1^2 - p^2z_0^2 \biggr)^{1 / 2} \, .$ +

# 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,

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