User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates
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,
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<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,
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<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,
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<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> |
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<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> |
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<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,
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<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> |
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<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> |
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<math>~=</math> |
<math>~ \frac{1}{\lambda_1 \ell_{3D}} </math> |
where,
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<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,
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<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,
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<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,
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Direction Cosines for T6 Coordinates
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| <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> |
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| <math>~3</math> |
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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,
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<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,
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<math>~\ell_{3D}</math> |
<math>~=</math> |
<math>~\biggl[ x_0^2 + q^4y_0^2 + p^4 z_0^2 \biggr]^{- 1 / 2} \, .</math> |
The two-dimensional plane that is tangent to the <math>~\lambda_1</math> = constant surface at this point is given by the expression,
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<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> |
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<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> |
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<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> |
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<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> |
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<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> |
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<math>~\Rightarrow~~~ x x_0 + q^2 y y_0 + p^2 z z_0 </math> |
<math>~=</math> |
<math>~ (\lambda_1^2)_0 \, . </math> |
Building on our experience developing T5 Coordinates, we will seek a second coordinate — and its accompanying unit vector — such that <math>~z = z_0</math>. In this case, the relevant line in our tangent plane must obey the prescription,
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<math>~y</math> |
<math>~=</math> |
<math>~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]</math> |
See Also
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© 2014 - 2021 by Joel E. Tohline |