# User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates

(Difference between revisions)
 Revision as of 09:48, 28 October 2020 (view source)Tohline (Talk | contribs) (→Concentric Ellipsoidal (T6) Coordinates)← Older edit Revision as of 12:50, 28 October 2020 (view source)Tohline (Talk | contribs) (→Other Coordinate Pair in the Tangent Plane)Newer edit → Line 285: Line 285: + + +
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+ '''Tangent Plane''' +
The two-dimensional plane that is tangent to the $~\lambda_1$ = constant surface ''at this point'' is given by the expression, The two-dimensional plane that is tangent to the $~\lambda_1$ = constant surface ''at this point'' is given by the expression, Line 383: Line 389:
+ - Building on our experience developing [[User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates#T5_Coordinates|T5 Coordinates]], we will seek a second coordinate — and its accompanying unit vector — such that $~z = z_0$.  In this case, the relevant line in our tangent plane must obey the prescription, + + Building on our experience developing [[User:Tohline/Appendix/Ramblings/T3Integrals#Integrals_of_Motion_in_T3_Coordinates|T3 Coordinates]] and, more recently, [[User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates#T5_Coordinates|T5 Coordinates]], let's define the two "angles,"
- $~y$ + $~\Zeta$ + + $~\equiv$ + + $~\sinh^{-1}\biggl(\frac{qy}{x} \biggr)$ +       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, + + +
+ $~\frac{\partial \lambda_1}{\partial x}$ + + $~=$ + + $~$ +
+ $~\frac{\partial \lambda_1}{\partial y}$ + + $~=$ + + $~$ +
+ $~\frac{\partial \lambda_1}{\partial z}$ + + $~=$ + + $~$ +
+ $~\frac{\partial \lambda_2}{\partial x}$ + + $~=$ + + $~$ +
+ $~\frac{\partial \lambda_2}{\partial y}$ + + $~=$ + + $~$ +
+ $~\frac{\partial \lambda_3}{\partial x}$ + + $~=$ + + $~$ +
+ $~\frac{\partial \lambda_3}{\partial z}$ Line 395: Line 557: - $~x \biggl[- \frac{x_0}{q^2 y_0} \biggr] + \frac{1}{q^2 y_0}\biggl[ (\lambda_1^2)_0 - p^2 z_0^2 \biggr]$ + $~$
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

# 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,

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

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,

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

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,

 $~\frac{\partial \lambda_1}{\partial x}$ $~=$ $~$ $~\frac{\partial \lambda_1}{\partial y}$ $~=$ $~$ $~\frac{\partial \lambda_1}{\partial z}$ $~=$ $~$ $~\frac{\partial \lambda_2}{\partial x}$ $~=$ $~$ $~\frac{\partial \lambda_2}{\partial y}$ $~=$ $~$ $~\frac{\partial \lambda_3}{\partial x}$ $~=$ $~$ $~\frac{\partial \lambda_3}{\partial z}$ $~=$ $~$