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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,

<math>~\lambda_1</math>

<math>~\equiv</math>

When <math>~\lambda_1 = a</math>, we obtain the standard definition of an ellipsoidal surface, it being understood that, <math>~q^2 = a^2/b^2</math> and <math>~p^2 = a^2/c^2</math>. (We will assume that <math>~a > b > c</math>, that is, <math>~p^2 > q^2 > 1</math>.)

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

<math>~\equiv</math>

<math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} - \lambda_1 \, .</math>

In Cartesian coordinates, this means,

<math>~\bold{\hat{n}}(x, y, z)</math>	<math>~=</math>	<math>~ \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) </math>
	<math>~=</math>	<math>~ \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] </math>
	<math>~=</math>	<math>~ \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) \, , </math>

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

<math>~[ \bold{\hat{n}} \cdot \bold{\hat{n}} ]^{1 / 2}</math>	<math>~=</math>	<math>~ \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} </math>
	<math>~=</math>	<math>~ \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} </math>
	<math>~=</math>	<math>~ \frac{1}{\lambda_1 \ell_{3D}} </math>

where,

<math>~\equiv</math>

<math>~\biggl[ x^2 + q^4y^2 + p^4 z^2 \biggr]^{- 1 / 2} \, .</math>

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

<math>~\equiv</math>

<math>~ \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}) \, . </math>

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

<math>~\lambda_1^2 \ell_{3D}^2 \, .</math>

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

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

Other Coordinate Pair in the Tangent Plane

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

<math>~\equiv</math>

<math>~ \hat\imath (x_0 \ell_{3D}) + \hat\jmath (q^2y_0 \ell_{3D}) + \hat\jmath (p^2 z_0 \ell_{3D}) \, , </math>

where, for this specific point on the surface,

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

Tangent Plane

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

<math>~0</math>	<math>~=</math>	<math>~ (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 </math>
	<math>~=</math>	<math>~ (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 </math>
	<math>~=</math>	<math>~ (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 </math>
<math>~\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 </math>	<math>~=</math>	<math>~ 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 </math>
<math>~\Rightarrow~~~ x x_0 + q^2 y y_0 + p^2 z z_0 </math>	<math>~=</math>	<math>~ x_0^2 + q^2 y_0^2 + p^2 z_0^2 </math>
<math>~\Rightarrow~~~ x x_0 + q^2 y y_0 + p^2 z z_0 </math>	<math>~=</math>	<math>~ (\lambda_1^2)_0 \, . </math>

Fix the value of <math>~\lambda_1</math>. This means that the relevant ellipsoidal surface is defined by the expression,

<math>~\lambda_1^2</math>

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

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

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

<math>~m \equiv \frac{dy}{dx}\biggr|_{z_0}</math>

Speculation1

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

<math>~\equiv</math>

<math>~\sinh^{-1}\biggl(\frac{qy}{x} \biggr)</math>

and,

<math>~\Upsilon</math>

<math>~\equiv</math>

<math>~\sinh^{-1}\biggl(\frac{pz}{x} \biggr) \, ,</math>

in which case we can write,

<math>~\lambda_1^2</math>

<math>~x^2(\cosh^2\Zeta + \sinh^2\Upsilon)\, .</math>

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

<math>~\lambda_2</math>	<math>~\equiv</math>	<math>~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)} \, ,</math>
<math>~\lambda_3</math>	<math>~\equiv</math>	<math>~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)} \, .</math>

Some relevant partial derivatives are,

<math>~\frac{\partial \lambda_2}{\partial x}</math>	<math>~=</math>	<math>~\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} \, ; </math>
<math>~\frac{\partial \lambda_2}{\partial y}</math>	<math>~=</math>	<math>~\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} \, ;</math>
<math>~\frac{\partial \lambda_3}{\partial x}</math>	<math>~=</math>	<math>~ \biggl[ \frac{p^2}{p^2-1} \biggr]\frac{\lambda_3}{x} \, ; </math>
<math>~\frac{\partial \lambda_3}{\partial z}</math>	<math>~=</math>	<math>~ - \biggl[ \frac{1}{p^2-1} \biggr] \frac{\lambda_3}{z} \, .</math>

And the associated scale factors are,

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

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

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

Hence,

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

Check:

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

User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates

Contents

Concentric Ellipsoidal (T6) Coordinates

Background

Orthogonal Coordinates

Primary (radial-like) Coordinate

Other Coordinate Pair in the Tangent Plane

Speculation1

Speculation2

See Also

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