DirectionCosineMatrixToAxisAngleOfRotation 2005-08-28 23:31:22 2005-08-28 23:31:22 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 The angle of rotation can be found from the trace of the direction cosine matrix to axis angle of rotation matrix $$ A_{11} + A_{22} + A_{33} = 3cos(\alpha) + (1 - cos(\alpha))(e_1^2 + e_2^2 + e_3^2) $$ Noting that the axis of rotation is a unit vector and has a length of 1 means $$ e_1^2 + e_2^2 + e_3^2 = 1 $$ therefore $$ A_{11} + A_{22} + A_{33} = 1 + 2cos(\alpha) $$ rearranging gives \begin{equation} \alpha = cos^-1( \dfrac{1}{2} (A_{11} + A_{22} + A_{33} - 1)) \end{equation} Inverse cosine is a multivalued function and there are 2 possible solutions for $\alpha$. Normally, the convention is to choose the principle value such that $ 0 < \alpha < \pi $ As long as $\alpha$ is not zero, the unit vector is given by \begin{equation} \left[ \begin{array}{c} e_1 \\ e_2 \\ e_3 \end{array} \right] = \left[ \begin{array}{c} \dfrac{(A_{23} - A_{32})}{2 sin(\alpha)} \\ \dfrac{(A_{31} - A_{13})}{2 sin(\alpha)} \\ \dfrac{(A_{12} - A_{21})}{2 sin(\alpha)} \end{array} \right] \end{equation} Above equation should be proved at some time...