Revision as of 20:03, 17 March 2021
Daring Attack
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> |
See Also
