Talk:PlanetPhysics/Euler 312 Sequence

Original TeX Content from PlanetPhysics Archive
%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Euler 312 Sequence %%% Primary Category Code: 45.40.-f %%% Filename: Euler312Sequence.tex %%% Version: 4 %%% Owner: bloftin %%% Author(s): bloftin %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

\setlength{\topmargin}{0.00in} \setlength{\headsep}{0.00in} \setlength{\headheight}{0.00in} \setlength{\evensidemargin}{0.00in} \setlength{\oddsidemargin}{0.00in} \setlength{\textwidth}{6.5in} \setlength{\textheight}{9.00in} \setlength{\voffset}{0.00in} \setlength{\hoffset}{0.00in} \setlength{\marginparwidth}{0.00in} \setlength{\marginparsep}{0.00in} \setlength{\parindent}{0.00in} \setlength{\parskip}{0.15in}

\usepackage{html}

\usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb}

\begin{document}

For more info on Euler Sequences, notation and convention see the generic entry on \htmladdnormallink{Euler angle sequences}{http://planetphysics.us/encyclopedia/EulerAngleSequence.html}. \\

$ R_{312}(\phi, \theta, \psi) = R_2(\psi) R_1(\theta) R_3(\phi) $ \\

The rotation \htmladdnormallink{matrices}{http://planetphysics.us/encyclopedia/Matrix.html} are

\begin{equation} R_2(\psi) = \left[ \begin{array}{ccc} c_{\psi} & 0 & -s_{\psi} \\ 0 & 1 & 0 \\ s_{\psi} & 0 & c_{\psi} \end{array} \right] \end{equation}

\begin{equation} R_1(\theta) = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & c_{\theta} & s_{\theta} \\ 0 & -s_{\theta} & c_{\theta} \end{array} \right] \end{equation}

\begin{equation} R_3(\phi) = \left[ \begin{array}{ccc} c_{\phi} & s_{\phi} & 0 \\ -s_{\phi} & c_{\phi} & 0 \\ 0 & 0 & 1 \end{array} \right] \end{equation}

Carrying out the multiplication from right to left \\

$ R_1(\theta)R_3(\phi) = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & c_{\theta} & s_{\theta} \\ 0 & -s_{\theta} & c_{\theta} \end{array} \right] \left[ \begin{array}{ccc} c_{\phi} & s_{\phi} & 0 \\ -s_{\phi} & c_{\phi} & 0 \\ 0 & 0 & 1 \end{array} \right] = \left[  \begin{array}{ccc} c_{\phi} & s_{\phi} & 0 \\ -s_{\phi} c_{\theta} & c_{\theta} c_{\phi} & s_{\theta} \\ s_{\theta} s_{\phi} & -s_{\theta} c_{\phi} & c_{\theta} \end{array} \right] $ \\

Finaly leaving us with the Euler 312 sequence \\

$ R_2(\psi)R_1(\theta)R_3(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\phi} - s_{\psi} s_{\theta} s_{\phi} & c_{\psi} s_{\phi} + s_{\psi} s_{\theta} c_{\phi} & -s_{\psi} c_{\theta} \\ -s_{\phi} c_{\theta} & c_{\theta} c_{\phi} & s_{\theta} \\ s_{\psi} c_{\phi} + c_{\psi} s_{\theta} s_{\phi} & s_{\psi} s_{\phi} - c_{\psi} s_{\theta} c_{\phi} & c_{\psi} c_{\theta} \end{array} \right] $

\end{document}