User:Tohline/Appendix/Ramblings/ConcentricEllipsodalT12Coordinates
Concentric Ellipsoidal (T12) 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 (T8) 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.
Note that, in a separate but closely related discussion, we made attempts to define this coordinate system, numbering the trials up through "T7." In this "T7" effort, we were able to define a set of three, mutually orthogonal unit vectors that should work to define a fully three-dimensional, concentric ellipsoidal coordinate system. But we were unable to figure out what coordinate function, <math>~\lambda_3(x, y, z)</math>, was associated with the third unit vector. In addition, we found the <math>~\lambda_2</math> coordinate to be rather strange in that it was not oriented in a manner that resembled the classic spherical coordinate system. Here we begin by redefining the <math>~\lambda_2</math> coordinate such that its associated <math>~\hat{e}_3</math> unit vector lies parallel to the x-y plane.
The 1st coordinate and its associated unit vector are as follows:
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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> |
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<math>~\hat{e}_1</math> |
<math>~=</math> |
<math>~ \ell_{3D} \biggl[ \hat\imath (x) + \hat\jmath (q^2y ) + \hat{k} (p^2 z) \biggr] \, , </math> |
where,
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<math>~\ell_{3D}</math> |
<math>~\equiv</math> |
<math>~ (x^2 + q^4y^2 + p^4 z^2)^{- 1 / 2} \, . </math> |
Generalized Prescription for 2nd Coordinate
Let's adopt the following generalized prescription for the 2nd coordinate:
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<math>~\lambda_2</math> |
<math>~\equiv</math> |
<math>~ x^a y^b z^c \, , </math> |
in which case,
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<math>~\hat{e}_2</math> |
<math>~=</math> |
<math>~ \frac{1}{\mathfrak{L}} \biggl[ \hat\imath \biggl(\frac{yz}{bc}\biggr) + \hat\jmath \biggl(\frac{xz}{ac}\biggr) + \hat{k} \biggl(\frac{xy}{ab}\biggr) \biggr] \, , </math> |
where,
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<math>~\mathfrak{L}^2</math> |
<math>~\equiv</math> |
<math>~ \frac{1}{a^2b^2c^2} \biggl[ a^2(yz)^2 + b^2(xz)^2 + c^2(xy)^2 \biggr] \, . </math> |
Now, to ensure that <math>~\hat{e}_2</math> is perpendicular to <math>~\hat{e}_1</math>, we need,
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<math>~\hat{e}_1 \cdot \hat{e}_2</math> |
<math>~=</math> |
<math>~0</math> |
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<math>~\Rightarrow~~~ 0</math> |
<math>~=</math> |
<math>~ \frac{\ell_{3D}}{\mathfrak{L}} \biggl[ \frac{xyz}{bc} + \frac{q^2xyz}{ac} + \frac{p^2xyz}{ab} \biggr] = \frac{\ell_{3D} (xyz)}{\mathfrak{L}(abc)} \biggl[ a + q^2b + p^2 c \biggr] </math> |
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<math>~\Rightarrow~~~ 0</math> |
<math>~=</math> |
<math>~ \biggl[ a + q^2b + p^2 c \biggr]\, . </math> |
Henceforth, we will refer to this algebraic relation as the "One-Two Perpendicular Constraint."
Necessary 3rd Coordinate
The unit vector associated with the 3rd coordinate is obtained from the cross product of the first two unit vectors. That is,
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<math>~\hat{e}_3</math> |
<math>~=</math> |
<math>~\hat{e}_1 \times \hat{e}_2</math> |
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<math>~=</math> |
<math>~ \hat\imath \biggl[ e_{1y} e_{2z} - e_{1z} e_{2y} \biggr] + \hat\jmath \biggl[ e_{1z}e_{2x} - e_{1x}e_{2z} \biggr] + \hat{k} \biggl[ e_{1x}e_{2y} - e_{1y}e_{2x} \biggr] </math> |
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<math>~=</math> |
<math>~\frac{\ell_{3D}}{\mathfrak{L}} \biggl\{ \hat\imath \biggl[ (q^2y) \biggl( \frac{xy}{ab} \biggr) - (p^2z) \biggl( \frac{xz}{ac} \biggr) \biggr] + \hat\jmath \biggl[ (p^2z) \biggl( \frac{yz}{bc} \biggr) - (x)\biggl( \frac{xy}{ab} \biggr) \biggr] + \hat{k} \biggl[ (x) \biggl( \frac{xz}{ac} \biggr) - (q^2y) \biggl( \frac{yz}{bc} \biggr) \biggr] \biggr\} </math> |
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<math>~=</math> |
<math>~\frac{\ell_{3D}}{\mathfrak{L}(abc)} \biggl\{ \hat\imath \biggl[ (cq^2y^2) - (b p^2z^2) \biggr]x + \hat\jmath \biggl[ (ap^2z^2) - (cx^2) \biggr]y + \hat{k} \biggl[ (bx^2) - (aq^2y^2) \biggr]z \biggr\} </math> |
Old Examples
T6 Coordinates
In the set that we have elsewhere referenced as T6 coordinates, we chose: a = - 1, b = q-2, c = 0.
T10 Coordinates
In the set that we have elsewhere referenced as T10 coordinates, we chose: a = 1, b = q-2, c = - 2p-2.
See Also
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© 2014 - 2021 by Joel E. Tohline |