# User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates

(Difference between revisions)
 Revision as of 10:54, 27 October 2020 (view source)Tohline (Talk | contribs) (→Concentric Ellipsoidal (T6) Coordinates)← Older edit Revision as of 11:25, 27 October 2020 (view source)Tohline (Talk | contribs) (→Orthogonal Coordinates)Newer edit → Line 9: Line 9: ==Orthogonal Coordinates== ==Orthogonal Coordinates== + + ===Primary (''radial-like'') Coordinate=== We start by defining a "radial" coordinate whose values identify various concentric ellipsoidal shells, We start by defining a "radial" coordinate whose values identify various concentric ellipsoidal shells, Line 245: Line 247: --- ---

+ + + + + ===Other Coordinate Pair in the Tangent Plane=== + + Let's focus on a particular point on the $~\lambda_1$ = constant surface, $~(x_0, y_0, z_0)$, that necessarily satisfies the function, $~F(x_0, y_0, z_0) = 0$.  We have already derived the expression for the unit vector that is normal to the ellipsoidal surface at this point, namely, + + +
+ $~\hat{e}_1$ + + $~\equiv$ + + $~ + \hat\imath (x_0 \ell_{3D}) + \hat\jmath (q^2y_0 \ell_{3D}) + \hat\jmath (p^2 z_0 \ell_{3D}) \, , +$ +
+ + + + + + where, for this specific point on the surface, + + +
+ $~\ell_{3D}$ + + $~=$ + + $~\biggl[ x_0^2 + q^4y_0^2 + p^4 z_0^2 \biggr]^{- 1 / 2} \, .$ +
+ + + + + + + The two-dimensional plane that is tangent to the $~\lambda_1$ = constant surface ''at this point'' is given by the expression, + + +
+ $~0$ + + $~=$ + + $~ + (x - x_0) \frac{\partial \lambda_1}{\partial x} + (y - y_0)\frac{\partial \lambda_1}{\partial y} + (z - z_0)\frac{\partial \lambda_1}{\partial z} +$ +
+   + + $~=$ + + $~ + (x - x_0) \frac{\partial F}{\partial x} + (y - y_0)\frac{\partial F}{\partial y} + (z - z_0)\frac{\partial F}{\partial z} +$ +
+   + + $~=$ + + $~ + (x - x_0) \biggl( \frac{x}{\lambda_1}\biggr) + (y - y_0)\biggl( \frac{q^2 y }{ \lambda_1 } \biggr) + (z - z_0)\biggl( \frac{ p^2z }{ \lambda_1 } \biggr) \, . +$
+ + + + + + + + + + + + + + +

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

### Other Coordinate Pair in the Tangent Plane

Let's focus on a particular point on the $~\lambda_1$ = constant surface, $~(x_0, y_0, z_0)$, that necessarily satisfies the function, $~F(x_0, y_0, z_0) = 0$. We have already derived the expression for the unit vector that is normal to the ellipsoidal surface at this point, namely,

 $~\hat{e}_1$ $~\equiv$ $~ \hat\imath (x_0 \ell_{3D}) + \hat\jmath (q^2y_0 \ell_{3D}) + \hat\jmath (p^2 z_0 \ell_{3D}) \, ,$

where, for this specific point on the surface,

 $~\ell_{3D}$ $~=$ $~\biggl[ x_0^2 + q^4y_0^2 + p^4 z_0^2 \biggr]^{- 1 / 2} \, .$

The two-dimensional plane that is tangent to the $~\lambda_1$ = constant surface at this point is given by the expression,

 $~0$ $~=$ $~ (x - x_0) \frac{\partial \lambda_1}{\partial x} + (y - y_0)\frac{\partial \lambda_1}{\partial y} + (z - z_0)\frac{\partial \lambda_1}{\partial z}$ $~=$ $~ (x - x_0) \frac{\partial F}{\partial x} + (y - y_0)\frac{\partial F}{\partial y} + (z - z_0)\frac{\partial F}{\partial z}$ $~=$ $~ (x - x_0) \biggl( \frac{x}{\lambda_1}\biggr) + (y - y_0)\biggl( \frac{q^2 y }{ \lambda_1 } \biggr) + (z - z_0)\biggl( \frac{ p^2z }{ \lambda_1 } \biggr) \, .$