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>