User:Tohline/Appendix/Ramblings/ConcentricEllipsodalDaringAttack
Daring Attack
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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 the so-called T6 (concentric elliptic) coordinate system, here we take a somewhat daring attack on this problem, mixing our approach to identifying the expression for the third curvilinear coordinate. Broadly speaking, this entire study 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.
| Direction Cosine Components for T6 Coordinates | ||||||||||||||
| <math>~n</math> | <math>~\lambda_n</math> | <math>~h_n</math> | <math>~\frac{\partial \lambda_n}{\partial x}</math> | <math>~\frac{\partial \lambda_n}{\partial y}</math> | <math>~\frac{\partial \lambda_n}{\partial z}</math> | <math>~\gamma_{n1}</math> | <math>~\gamma_{n2}</math> | <math>~\gamma_{n3}</math> | ||||||
| <math>~1</math> | <math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} </math> | <math>~\lambda_1 \ell_{3D}</math> | <math>~\frac{x}{\lambda_1}</math> | <math>~\frac{q^2 y}{\lambda_1}</math> | <math>~\frac{p^2 z}{\lambda_1}</math> | <math>~(x) \ell_{3D}</math> | <math>~(q^2 y)\ell_{3D}</math> | <math>~(p^2z) \ell_{3D}</math> | ||||||
| <math>~2</math> | --- | --- | --- | --- | --- | <math>~\ell_q \ell_{3D} (xp^2z)</math> | <math>~\ell_q \ell_{3D} (q^2 y p^2z) </math> | <math>~- (x^2 + q^4y^2)\ell_q \ell_{3D}</math> | ||||||
| <math>~3</math> | <math>~\tan^{-1}\biggl( \frac{y^{1/q^2}}{x} \biggr)</math> | <math>~\frac{xq^2 y \ell_q}{\sin\lambda_3 \cos\lambda_3}</math> | <math>~-\frac{\sin\lambda_3 \cos\lambda_3}{x}</math> | <math>~+\frac{\sin\lambda_3 \cos\lambda_3}{q^2y}</math> | <math>~0</math> | <math>~-q^2 y \ell_q</math> | <math>~x\ell_q</math> | <math>~0</math> | ||||||
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New Approach
As before, let's adopt the first-coordinate expression,
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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> |
but for the third-coordinate expression we will abandon the trigonometric expression and instead simply use,
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<math>~\lambda_3</math> |
<math>~\equiv</math> |
<math>~\frac{y^{1/q^2}}{x} \, .</math> |
This modified third-coordinate expression means that the last row of the above table changes, as follows.
| Daring Attack | ||||||||
| <math>~n</math> | <math>~\lambda_n</math> | <math>~h_n</math> | <math>~\frac{\partial \lambda_n}{\partial x}</math> | <math>~\frac{\partial \lambda_n}{\partial y}</math> | <math>~\frac{\partial \lambda_n}{\partial z}</math> | <math>~\gamma_{n1}</math> | <math>~\gamma_{n2}</math> | <math>~\gamma_{n3}</math> |
| <math>~1</math> | <math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} </math> | <math>~\lambda_1 \ell_{3D}</math> | <math>~\frac{x}{\lambda_1}</math> | <math>~\frac{q^2 y}{\lambda_1}</math> | <math>~\frac{p^2 z}{\lambda_1}</math> | <math>~(x) \ell_{3D}</math> | <math>~(q^2 y)\ell_{3D}</math> | <math>~(p^2z) \ell_{3D}</math> |
| <math>~2</math> | --- | --- | --- | --- | --- | <math>~\ell_q \ell_{3D} (xp^2z)</math> | <math>~\ell_q \ell_{3D} (q^2 y p^2z) </math> | <math>~- (x^2 + q^4y^2)\ell_q \ell_{3D}</math> |
| <math>~3</math> | <math>~\frac{y^{1/q^2}}{x} </math> | <math>~\frac{xq^2 y \ell_q}{\lambda_3}</math> | <math>~-\frac{\lambda_3}{x}</math> | <math>~+\frac{\lambda_3}{q^2y}</math> | <math>~0</math> | <math>~-q^2 y \ell_q</math> | <math>~x\ell_q</math> | <math>~0</math> |
Notice that the direction cosine functions for the (as yet, unknown) second-coordinate function remain the same. This is because the direction-cosine functions associated with both <math>~\lambda_1</math> and <math>~\lambda_3</math> remain unchanged, so it must be true that the cross product of the first and third unit vectors leads to the same components for the second unit vector.
See Also
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© 2014 - 2021 by Joel E. Tohline |