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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

Primary (radial-like) Coordinate

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,

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

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,

$~\ell_{3D}$

$~=$

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

Tangent Plane

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

$~0$	$~=$	$~ (x - x_0) \biggl[ \frac{\partial \lambda_1}{\partial x} \biggr]_0 + (y - y_0) \biggl[\frac{\partial \lambda_1}{\partial y} \biggr]_0 + (z - z_0) \biggl[\frac{\partial \lambda_1}{\partial z} \biggr]_0$
	$~=$	$~ (x - x_0) \biggl[ \frac{\partial F}{\partial x} \biggr]_0 + (y - y_0) \biggl[\frac{\partial F}{\partial y} \biggr]_0 + (z - z_0) \biggl[ \frac{\partial F}{\partial z} \biggr]_0$
	$~=$	$~ (x - x_0) \biggl( \frac{x}{\lambda_1}\biggr)_0 + (y - y_0)\biggl( \frac{q^2 y }{ \lambda_1 } \biggr)_0 + (z - z_0)\biggl( \frac{ p^2z }{ \lambda_1 } \biggr)_0$
$~\Rightarrow~~~ x \biggl( \frac{x}{\lambda_1}\biggr)_0 + y \biggl( \frac{q^2 y }{ \lambda_1 } \biggr)_0 + z \biggl( \frac{ p^2z }{ \lambda_1 } \biggr)_0$	$~=$	$~ x_0 \biggl( \frac{x}{\lambda_1}\biggr)_0 + y_0 \biggl( \frac{q^2 y }{ \lambda_1 } \biggr)_0 + z_0 \biggl( \frac{ p^2z }{ \lambda_1 } \biggr)_0$
$~\Rightarrow~~~ x x_0 + q^2 y y_0 + p^2 z z_0$	$~=$	$~ x_0^2 + q^2 y_0^2 + p^2 z_0^2$
$~\Rightarrow~~~ x x_0 + q^2 y y_0 + p^2 z z_0$	$~=$	$~ (\lambda_1^2)_0 \, .$

Fix the value of $~\lambda_1$ . This means that the relevant ellipsoidal surface is defined by the expression,

$~\lambda_1^2$

$~=$

$~x^2 + q^2y^2 + p^2z^2 \, .$

If $~z = 0$ , the semi-major axis of the relevant x-y ellipse is $~\lambda_1$ , and the square of the semi-minor axis is $~\lambda_1^2/q^2$ . At any other value, $~z = z_0 < c$ , the square of the semi-major axis of the relevant x-y ellipse is, $~(\lambda_1^2 - p^2z_0^2)$ and the square of the corresponding semi-minor axis is, $~(\lambda_1^2 - p^2z_0^2)/q^2$ . Now, for any chosen $~x_0^2 \le (\lambda_1^2 - p^2z_0^2)$ , the y-coordinate of the point on the $~\lambda_1$ surface is given by the expression,

$~y_0^2$

$~=$

$~\frac{1}{q^2}\biggl[ \lambda_1^2 - p^2 z_0 -x_0^2 \biggr] \, .$

The slope of the line that lies in the z = z₀ plane and that is tangent to the ellipsoidal surface at $~(x_0, y_0)$ is,

$~m \equiv \frac{dy}{dx}\biggr|_{z_0}$

$~=$

$~- \frac{x_0}{q^2y_0}$

Speculation1

Building on our experience developing T3 Coordinates and, more recently, T5 Coordinates, let's define the two "angles,"

$~\Zeta$

$~\equiv$

$~\sinh^{-1}\biggl(\frac{qy}{x} \biggr)$

and,

$~\Upsilon$

$~\equiv$

$~\sinh^{-1}\biggl(\frac{pz}{x} \biggr) \, ,$

in which case we can write,

$~\lambda_1^2$

$~=$

$~x^2(\cosh^2\Zeta + \sinh^2\Upsilon)\, .$

We speculate that the other two orthogonal coordinates may be defined by the expressions,

$~\lambda_2$	$~\equiv$	$~x \biggl[ \sinh\Zeta \biggr]^{1/(1-q^2)} = x \biggl[ \frac{qy}{x}\biggr]^{1/(1-q^2)} = x \biggl[ \frac{x}{qy}\biggr]^{1/(q^2-1)} = \biggl[ \frac{x^{q^2}}{qy}\biggr]^{1/(q^2-1)} \, ,$
$~\lambda_3$	$~\equiv$	$~x \biggl[ \sinh\Upsilon \biggr]^{1/(1-p^2)} = x \biggl[ \frac{pz}{x}\biggr]^{1/(1-p^2)} = x \biggl[ \frac{x}{pz}\biggr]^{1/(p^2-1)} = \biggl[ \frac{x^{p^2}}{pz}\biggr]^{1/(p^2-1)} \, .$

Some relevant partial derivatives are,

$~\frac{\partial \lambda_2}{\partial x}$	$~=$	$~\biggl[ \frac{1}{qy}\biggr]^{1/(q^2-1)} \biggl[ \frac{q^2}{q^2-1} \biggr]x^{1/(q^2-1)} = \biggl[ \frac{q^2}{q^2-1} \biggr]\biggl[ \frac{x}{qy}\biggr]^{1/(q^2-1)} = \biggl[ \frac{q^2}{q^2-1} \biggr]\frac{\lambda_2}{x} \, ;$
$~\frac{\partial \lambda_2}{\partial y}$	$~=$	$~\biggl[ \frac{x^{q^2}}{q}\biggr]^{1/(q^2-1)} \biggl[ \frac{1}{1-q^2} \biggr] y^{q^2/(1-q^2)} = - \biggl[ \frac{1}{q^2-1} \biggr] \frac{\lambda_2}{y} \, ;$
$~\frac{\partial \lambda_3}{\partial x}$	$~=$	$~ \biggl[ \frac{p^2}{p^2-1} \biggr]\frac{\lambda_3}{x} \, ;$
$~\frac{\partial \lambda_3}{\partial z}$	$~=$	$~ - \biggl[ \frac{1}{p^2-1} \biggr] \frac{\lambda_3}{z} \, .$

And the associated scale factors are,

$~h_2^2$	$~=$	$~ \biggl\{ \biggl[ \biggl( \frac{q^2}{q^2-1} \biggr)\frac{\lambda_2}{x} \biggr]^2 + \biggl[ - \biggl( \frac{1}{q^2-1} \biggr) \frac{\lambda_2}{y} \biggr]^2 \biggr\}^{-1}$
	$~=$	$~ \biggl\{ \biggl( \frac{q^2}{q^2-1} \biggr)^2 \frac{\lambda_2^2}{x^2} + \biggl( \frac{1}{q^2-1} \biggr)^2 \frac{\lambda_2^2}{y^2} \biggr\}^{-1}$
	$~=$	$~ \biggl\{x^2 + q^4 y^2 \biggr\}^{-1} \biggl[ \frac{(q^2 - 1)^2x^2 y^2}{\lambda_2^2} \biggr] \, ;$
$~h_3^2$	$~=$	$~ \biggl\{x^2 + p^4 z^2 \biggr\}^{-1} \biggl[ \frac{(p^2 - 1)^2x^2 z^2}{\lambda_3^2} \biggr] \, .$

We can now fill in the rest of our direction-cosines table, namely,

Direction Cosines for T6 Coordinates $~\gamma_{ni} = h_n \biggl( \frac{\partial \lambda_n}{\partial x_i}\biggr)$
$~n$	$~i = x, y, z$
$~1$	$~x\ell_{3D}$	$~q^2 y \ell_{3D}$	$~p^2 z \ell_{3D}$
$~2$	$~q^2 y \ell_q$	$~-x\ell_q$	$~0$
$~3$	$~p^2 z \ell_p$	$~0$	$~-x\ell_p$

Hence,

$~\hat{e}_2$	$~=$	$~ \hat\imath \gamma_{21} + \hat\jmath \gamma_{22} +\hat{k} \gamma_{23} = \hat\imath (q^2y\ell_q) - \hat\jmath (x\ell_q) \, ;$
$~\hat{e}_3$	$~=$	$~ \hat\imath \gamma_{31} + \hat\jmath \gamma_{32} +\hat{k} \gamma_{33} = \hat\imath (p^2z\ell_p) -\hat{k} (x\ell_p) \, .$

Check:

$~\hat{e}_2 \cdot \hat{e}_2$	$~=$	$~ (q^2y\ell_q)^2 + (x\ell_q)^2 = 1 \, ;$
$~\hat{e}_3 \cdot \hat{e}_3$	$~=$	$~ (p^2z\ell_p)^2 + (x\ell_p)^2 = 1 \, ;$
$~\hat{e}_2 \cdot \hat{e}_3$	$~=$	$~ (q^2y\ell_q)(p^2z\ell_p) \ne 0 \, .$

Speculation2

Try,

$~\lambda_2$

$~=$

$~ \frac{x}{y^{1/q^2} z^{1/p^2}} \, ,$

in which case,

$~\frac{\partial \lambda_2}{\partial x}$	$~=$	$~ \frac{\lambda_2}{x} \, ,$
$~\frac{\partial \lambda_2}{\partial y}$	$~=$	$~ \frac{x}{z^{1/p^2}} \biggl(-\frac{1}{q^2}\biggr) y^{-1/q^2 - 1} = -\frac{\lambda_2}{q^2 y} \, ,$
$~\frac{\partial \lambda_2}{\partial z}$	$~=$	$~ -\frac{\lambda_2}{p^2 z} \, .$

The associated scale factor is, then,

$~h_2^2$	$~=$	$~\biggl[ \biggl( \frac{\partial \lambda_2}{\partial x} \biggr)^2 + \biggl( \frac{\partial \lambda_2}{\partial y} \biggr)^2 + \biggl( \frac{\partial \lambda_2}{\partial z} \biggr)^2 \biggr]^{-1}$
	$~=$	$~\biggl[ \biggl( \frac{ \lambda_2}{x} \biggr)^2 + \biggl( -\frac{\lambda_2}{q^2y} \biggr)^2 + \biggl( - \frac{\lambda_2}{p^2z} \biggr)^2 \biggr]^{-1}$