Difference between revisions of "User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates"
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+\hat{k} \gamma_{33} | +\hat{k} \gamma_{33} | ||
= | = | ||
\hat\imath (p^2z\ | \hat\imath (p^2z\ell_p) | ||
-\hat{k} (x\ell_p) \, . | -\hat{k} (x\ell_p) \, . | ||
</math> | |||
</td> | |||
</tr> | |||
</table> | |||
Check: | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | |||
<td align="right"> | |||
<math>~\hat{e}_2 \cdot \hat{e}_2</math> | |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
(q^2y\ell_q)^2 | |||
+ (x\ell_q)^2 | |||
= | |||
1 \, ; | |||
</math> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td align="right"> | |||
<math>~\hat{e}_3 \cdot \hat{e}_3</math> | |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
(p^2z\ell_p)^2 | |||
+ (x\ell_p)^2 | |||
= | |||
1 \, ; | |||
</math> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td align="right"> | |||
<math>~\hat{e}_2 \cdot \hat{e}_3</math> | |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
(q^2y\ell_q)(p^2z\ell_p) \ne 0 \, . | |||
</math> | </math> | ||
</td> | </td> |
Revision as of 21:22, 28 October 2020
Concentric Ellipsoidal (T6) Coordinates
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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> |
<math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} \, .</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>~F(x, y, z)</math> |
<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>~\ell_{3D}</math> |
<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>~\hat{e}_1 </math> |
<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>~h_1^2</math> |
<math>~=</math> |
<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>~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>~\hat{e}_1 </math> |
<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>~\ell_{3D}</math> |
<math>~=</math> |
<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,
|
Building on our experience developing T3 Coordinates and, more recently, T5 Coordinates, let's define the two "angles,"
<math>~\Zeta</math> |
<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>~=</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>~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> |
See Also
© 2014 - 2021 by Joel E. Tohline |