AxisAngleOfRotationToDirectionCosineMatrix 2005-08-28 22:59:32 2005-08-28 22:59:32 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 For now, without proof you can get the direction cosine matrix (DCM) from the axis angle of rotation by \begin{equation} \left[ \begin{array}{ccc} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{array} \right] = \left[ \begin{array}{ccc} cos(\alpha) + e_1^2(1 - cos(\alpha) & e_1 e_2 (1 - cos(\alpha)) + e_3 sin(\alpha) & e_1 e_3(1 - cos(\alpha)) - e_2 sin(\alpha) \\ e_1 e_2(1 - cos(\alpha)) - e_3 sin(\alpha) & cos(\alpha) + e_2^2(1 - cos(\alpha)) & e_2 e_3(1-cos(\alpha)) + e_1 sin(\alpha) \\ e_1 e_3(1 - cos(\alpha)) + e_2 sin(\alpha) & e_2 e_3(1-cos(\alpha)) - e_1 sin(\alpha) & cos(\alpha) + e_3^2(1 - cos(\alpha)) \end{array} \right] \end{equation}