Talk:PlanetPhysics/Direction Cosine Matrix to Axis Angle of Rotation

Original TeX Content from PlanetPhysics Archive
%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: direction cosine matrix to axis angle of rotation %%% Primary Category Code: 45.40.-f %%% Filename: DirectionCosineMatrixToAxisAngleOfRotation.tex %%% Version: 2 %%% 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}

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\begin{document}

The angle of rotation can be found from the \htmladdnormallink{trace}{http://planetphysics.us/encyclopedia/Trace.html} of the direction cosine matrix to axis angle of rotation \htmladdnormallink{matrix}{http://planetphysics.us/encyclopedia/Matrix.html} $$ A_{11} + A_{22} + A_{33} = 3cos(\alpha) + (1 - cos(\alpha))(e_1^2 + e_2^2 + e_3^2)  $$

Noting that the axis of rotation is a \htmladdnormallink{unit vector}{http://planetphysics.us/encyclopedia/PureState.html} 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 \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} 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...

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