# User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates

(Difference between revisions)
 Revision as of 10:09, 27 October 2020 (view source)Tohline (Talk | contribs) (→Concentric Ellipsoidal (T6) Coordinates)← Older edit Revision as of 10:54, 27 October 2020 (view source)Tohline (Talk | contribs) (→Concentric Ellipsoidal (T6) Coordinates)Newer edit → Line 110: Line 110: - +
+   + + $~=$ + + $~ + \biggl[ \biggl( \frac{x}{\lambda_1} \biggr)^2 + + \biggl( \frac{q^2y}{\lambda_1} \biggr)^2 + + \biggl(\frac{p^2 z}{\lambda_1} \biggr)^2 \biggr]^{1 / 2} +$ +
+   + + $~=$ + + $~ + \frac{1}{\lambda_1 \ell_{3D}} +$ +
+ + + + + + + + + + + where, + - The properly normalized +
+ $~\ell_{3D}$ + + $~\equiv$ + + $~\biggl[ x^2 + q^4y^2 + p^4 z^2 \biggr]^{- 1 / 2} \, .$ +
+ + + + + - Next, we appreciate that the vector that is normal to theWhat is the expression for the unit vector normal to the surface at $~(x, y, z)$ when written in terms of Cartesian unit vectors? + It is therefore clear that the ''properly normalized'' normal unit vector that should be associated with any $~\lambda_1$ = constant ellipsoidal surface is, + - Well, to start with we know that $~\lambda_1^2$ is constant across the entire surface, so at any point on this specified surface we must find, +
+ $~\hat{e}_1$ + + $~\equiv$ + + $~ + \frac{ \bold\hat{n} }{ [ \bold{\hat{n}} \cdot \bold{\hat{n}} ]^{1 / 2} } + = + \hat\imath (x \ell_{3D}) + \hat\jmath (q^2y \ell_{3D}) + \hat\jmath (p^2 z \ell_{3D}) \, . +$ +
+ + + + + + From our [[User:Tohline/Appendix/Ramblings/DirectionCosines#Scale_Factors|accompanying discussion of direction cosines]], it is clear, as well, that the scale factor associated with the $~\lambda_1$ coordinate is,
- $~0$ + $~h_1^2$ Line 129: Line 187: - $~2x dx + 2q^2y dy + 2p^2z dz \, .$ + $~\lambda_1^2 \ell_{3D}^2 \, .$ +
+ + + We can also fill in the top line of our direction-cosines table, namely, + + + +
+ '''Direction Cosines for T6 Coordinates''' +
+ $~\gamma_{ni} = h_n \biggl( \frac{\partial \lambda_n}{\partial x_i}\biggr)$ +
$~n$$~i = x, y, z$ +
$~1$
+ $~x\ell_{3D}$
+
$~q^2 y \ell_{3D}$ + $~p^2 z \ell_{3D}$ +
$~2$ +
+ --- +
+
+
+ --- +
+
+
+ --- +
+
$~3$ +
+ --- +
+
+
+ --- +
+
+
+ --- +

+ + + + + + + + + + + + + + + + +

# Concentric Ellipsoidal (T6) 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 (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, $~p^2 > q^2 > 1$.)

A vector, $~\bold{\hat{n}}$, that is normal to the $~\lambda_1$ = constant surface is given by the gradient of the function,

 $~F(x, y, z)$ $~\equiv$ $~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} - \lambda_1 \, .$

In Cartesian coordinates, this means,

 $~\bold{\hat{n}}(x, y, z)$ $~=$ $~ \hat\imath \biggl( \frac{\partial F}{\partial x} \biggr) + \hat\jmath \biggl( \frac{\partial F}{\partial y} \biggr) + \hat{k} \biggl( \frac{\partial F}{\partial z} \biggr)$ $~=$ $~ \hat\imath \biggl[ x(x^2 + q^2 y^2 + p^2 z^2)^{- 1 / 2} \biggr] + \hat\jmath \biggl[ q^2y(x^2 + q^2 y^2 + p^2 z^2)^{- 1 / 2} \biggr] + \hat\jmath \biggl[ p^2 z(x^2 + q^2 y^2 + p^2 z^2)^{- 1 / 2} \biggr]$ $~=$ $~ \hat\imath \biggl( \frac{x}{\lambda_1} \biggr) + \hat\jmath \biggl( \frac{q^2y}{\lambda_1} \biggr) + \hat\jmath \biggl(\frac{p^2 z}{\lambda_1} \biggr) \, ,$

where it is understood that this expression is only to be evaluated at points, $~(x, y, z)$, that lie on the selected $~\lambda_1$ surface — that is, at points for which the function, $~F(x,y,z) = 0$. The length of this normal vector is given by the expression,

 $~[ \bold{\hat{n}} \cdot \bold{\hat{n}} ]^{1 / 2}$ $~=$ $~ \biggl[ \biggl( \frac{\partial F}{\partial x} \biggr)^2 + \biggl( \frac{\partial F}{\partial y} \biggr)^2 + \biggl( \frac{\partial F}{\partial z} \biggr)^2 \biggr]^{1 / 2}$ $~=$ $~ \biggl[ \biggl( \frac{x}{\lambda_1} \biggr)^2 + \biggl( \frac{q^2y}{\lambda_1} \biggr)^2 + \biggl(\frac{p^2 z}{\lambda_1} \biggr)^2 \biggr]^{1 / 2}$ $~=$ $~ \frac{1}{\lambda_1 \ell_{3D}}$

where,

 $~\ell_{3D}$ $~\equiv$ $~\biggl[ x^2 + q^4y^2 + p^4 z^2 \biggr]^{- 1 / 2} \, .$

It is therefore clear that the properly normalized normal unit vector that should be associated with any $~\lambda_1$ = constant ellipsoidal surface is,

 $~\hat{e}_1$ $~\equiv$ $~ \frac{ \bold\hat{n} }{ [ \bold{\hat{n}} \cdot \bold{\hat{n}} ]^{1 / 2} } = \hat\imath (x \ell_{3D}) + \hat\jmath (q^2y \ell_{3D}) + \hat\jmath (p^2 z \ell_{3D}) \, .$

From our accompanying discussion of direction cosines, it is clear, as well, that the scale factor associated with the $~\lambda_1$ coordinate is,

 $~h_1^2$ $~=$ $~\lambda_1^2 \ell_{3D}^2 \, .$

We can also fill in the top line of our direction-cosines table, namely,

 Direction Cosines for T6 Coordinates $~\gamma_{ni} = h_n \biggl( \frac{\partial \lambda_n}{\partial x_i}\biggr)$ $~n$ $~i = x, y, z$ $~1$ $~x\ell_{3D}$ $~q^2 y \ell_{3D}$ $~p^2 z \ell_{3D}$ $~2$ --- --- --- $~3$ --- --- ---