Test 2005-08-16 21:12:01 2005-08-16 23:33:57 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A list of the Euler angle rotation matrics for different sequences 1-2-3 $ R_3(\psi)R_2(\theta)R_1(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\theta} & c_{\psi} s_{\theta} s_{\phi} + s_{\psi} c_{\phi} & -c_{\psi} s_{\theta} c_{\phi} + s_{\psi} s_{\phi} \\ -s_{\psi} c_{\theta} & -s_{\psi} s_{\theta} s_{\phi} + c_{\psi} c_{\phi} & s_{\psi} s_{\theta} c_{\phi} + c_{\psi} s_{\phi} \\ s_{\theta} & -c_{\theta} s_{\phi} & c_{\theta} c_{\phi} \end{array} \right] $ 1-3-2 $ R_2(\psi)R_3(\theta)R_1(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\theta} & c_{\psi} s_{\theta} c_{\phi} + s_{\psi} s_{\phi} & c_{\psi} s_{\theta} s_{\phi} - s_{\psi} c_{\phi} \\ -s_{\theta} & c_{\theta} c_{\phi} & c_{\theta} s_{\phi} \\ s_{\psi} c_{\theta} & s_{\psi} s_{\theta} c_{\phi} - c_{\psi} s_{\phi} & s_{\psi} s_{\theta} s_{\phi} + c_{\psi} c_{\phi} \end{array} \right] $ 1-2-1 $ R_1(\psi)R_2(\theta)R_1(\phi) = \left[ \begin{array}{ccc} c_{\theta} & -s_{\theta} s_{\phi} & s_{\theta} c_{\phi} \\ -s_{\psi} s_{\theta} & c_{\psi} c_{\phi} - s_{\psi} c_{\theta} s_{\phi} & c_{\psi} s_{\phi} + s_{\psi} c_{\theta} c_{\phi} \\ -s_{\theta} c_{\psi} & -s_{\psi} c_{\phi} - c_{\psi} c_{\theta} s_{\phi} & -s_{\psi} s_{\phi} + c_{\psi} c_{\theta} c_{\phi} \end{array} \right] $ 1-3-1 $ R_1(\psi)R_3(\theta)R_1(\phi) = \left[ \begin{array}{ccc} c_{\theta} & s_{\theta} c_{\phi} & s_{\theta} s_{\phi} \\ -c_{\psi} s_{\theta} & c_{\psi} c_{\theta} c_{\phi} - s_{\psi} s_{\phi} & c_{\psi} c_{\theta} s_{\phi} + s_{\psi} c_{\phi} \\ s_{\psi} s_{\theta} & -s_{\psi} c_{\theta} c_{\phi} - c_{\psi} s_{\phi} & - s_{\psi} c_{\theta} s_{\phi} + c_{\psi} c_{\phi} \end{array} \right] $ 2-1-3 $ R_3(\psi)R_1(\theta)R_2(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\phi} + s_{\psi} s_{\theta} s_{\phi} & s_{\psi} c_{\theta} & -c_{\psi} s_{\phi} + s_{\psi} s_{\theta} c_{\phi} \\ -s_{\psi} c_{\phi} + c_{\psi} s_{\theta} s_{\phi} & c_{\psi} c_{\theta} & s_{\psi} s_{\phi} + c_{\psi} s_{\theta} c_{\phi} \\ c_{\theta} s_{\phi} & -s_{\theta} & c_{\theta} c_{\phi} \end{array} \right] $ 2-3-1 $ R_1(\psi)R_3(\theta)R_2(\phi) = \left[ \begin{array}{ccc} c_{\theta} c_{\phi} & s_{\theta} & -c_{\theta} s_{\phi} \\ -c_{\psi} s_{\theta} c_{\phi} + s_{\psi} s_{\phi} & c_{\psi} c_{\theta} & c_{\psi} s_{\theta} s_{\phi} + s_{\psi} s_{\theta} s_{\phi} \\ s_{\psi} s_{\theta} c_{\phi} + c_{\psi} s_{\phi} & -s_{\psi} c_{\theta} & -s_{\psi} s_{\theta} s_{\phi} +c_{\psi} c_{\phi} \end{array} \right] $ 2-1-2 $ R_2(\psi)R_1(\theta)R_2(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\phi} - s_{\psi} c_{\theta} s_{\phi} & s_{\psi} s_{\theta} & -c_{\psi} s_{\phi} - s_{\psi} c_{\theta} c_{\phi} \\ s_{\theta} s_{\phi} & c_{\theta} & s_{\theta} c_{\phi} \\ s_{\psi} c_{\phi} + c_{\psi} c_{\theta} s_{\phi} & -c_{\psi} s_{\theta} & -s_{\psi} s_{\phi} + c_{\psi} c_{\theta} c_{\phi} \end{array} \right] $ 2-3-2 $ R_2(\psi)R_3(\theta)R_2(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\theta} c_{\phi} - s_{\psi} s_{\phi} & c_{\psi} s_{\theta} & c_{\psi} c_{\theta} s_{\phi} + s_{\psi} c_{\phi} \\ -s_{\theta} c_{\phi} & c_{\theta} & -s_{\theta} s_{\phi} \\ -s_{\psi} c_{\theta} c_{\phi} & -s_{\psi} s_{\theta} & - s_{\psi} c_{\theta} s_{\phi} + c_{\psi} c_{\phi} \end{array} \right] $ 3-1-2 $ 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] $ 3-2-1 $ R_1(\psi)R_2(\theta)R_3(\phi) = \left[ \begin{array}{ccc} c_{\theta} c_{\phi} & c_{\theta} s_{\phi} & s_{\theta} \\ - c_{\psi} s_{\phi} - s_{\psi} s_{\theta} c_{\phi} & c_{\psi} c_{\phi} - s_{\psi} s_{\theta} s_{\phi} & s_{\psi} c_{\theta} \\ s_{\psi} c_{\phi} - c_{\psi} s_{\theta} s_{\phi} & -s_{\psi} c_{\phi} - c_{\psi} s_{\theta} s_{\phi} & c_{\psi} c_{\theta} \end{array} \right] $ 3-1-3 $ R_3(\psi)R_1(\theta)R_3(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\phi} - s_{\psi} s_{\phi} c_{\theta} & c_{\psi} s_{\phi} + s_{\psi} c_{\theta} c_{\phi} & s_{\psi} s_{\theta} \\ -s_{\psi} c_{\phi} - c_{\psi} s_{\phi} c_{\theta} & -s_{\psi} s_{\phi} + c_{\psi} c_{\theta} c_{\phi} & c_{\psi} s_{\theta} \\ s_{\theta} s_{\phi} & -s_{\theta} c_{\phi} & c_{\theta} \end{array} \right] $ 3-2-3 $ R_3(\psi)R_2(\theta)R_3(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\theta} c_{\phi} - s_{\psi} s_{\phi} & c_{\psi} c_{\theta} s_{\phi} + s_{\psi} c_{\phi} & c_{\psi} \\ - s_{\psi} c_{\theta} c_{\phi} - c_{\psi} s_{\phi} & -s_{\psi} c_{\theta} s_{\phi} + c_{\psi} c_{\phi} & -s_{\psi} s_{\theta} \\ -s_{\theta} c_{\phi} & -s_{\theta} s_{\phi} & c_{\theta} \end{array} \right] $ For more info on Euler Sequences, notation and convention see the generic entry on Euler Angle Sequences. \\ $ R_{323}(\phi, \theta, \psi) = R_3(\psi) R_2(\theta) R_3(\phi) $ \\ The rotation matrices are \begin{equation} R_3(\psi) = \left[ \begin{array}{ccc} c_{\psi} & s_{\psi} & 0 \\ -s_{\psi} & c_{\psi} & 0 \\ 0 & 0 & 1 \end{array} \right] \end{equation} \begin{equation} R_2(\theta) = \left[ \begin{array}{ccc} c_{\theta} & 0 & -s_{\theta} \\ 0 & 1 & 0 \\ s_{\theta} & 0 & 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 matrix multiplication from right to left \\ $ R_2(\theta)R_3(\phi) = \left[ \begin{array}{ccc} c_{\theta} & 0 & -s_{\theta} \\ 0 & 1 & 0 \\ s_{\theta} & 0 & 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_{\theta} c_{\phi} & c_{\theta} s_{\phi} & s_{\theta} \\ -s_{\phi} & c_{\phi} & 0 \\ -s_{\theta} c_{\phi} & -s_{\theta} s_{\phi} & c_{\theta} \end{array} \right] $ \\ Finaly leaving us with the Euler 323 sequence \\ $ R_3(\psi)R_2(\theta)R_3(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\theta} c_{\phi} - s_{\psi} s_{\phi} & c_{\psi} c_{\theta} s_{\phi} + s_{\psi} c_{\phi} & c_{\psi} \\ - s_{\psi} c_{\theta} c_{\phi} - c_{\psi} s_{\phi} & -s_{\psi} c_{\theta} s_{\phi} + c_{\psi} c_{\phi} & -s_{\psi} s_{\theta} \\ -s_{\theta} c_{\phi} & -s_{\theta} s_{\phi} & c_{\theta} \end{array} \right] $