Talk:PlanetPhysics/Euler 123 Sequence
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[edit source]%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Euler 123 sequence %%% Primary Category Code: 45.40.-f %%% Filename: Euler123Sequence.tex %%% Version: 3 %%% 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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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_{123}(\phi, \theta, \psi) = R_3(\psi) R_2(\theta) R_1(\phi) $ \\
The rotation \htmladdnormallink{matrices}{http://planetphysics.us/encyclopedia/Matrix.html} 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_1(\phi) =
\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & c_{\phi} & s_{\phi} \\
0 & -s_{\phi} & c_{\phi} \end{array} \right]
\end{equation}
Carrying out the \htmladdnormallink{matrix multiplication}{http://planetphysics.us/encyclopedia/Matrix.html} from right to left \\
$
R_2(\theta)R_1(\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}
1 & 0 & 0 \\
0 & c_{\phi} & s_{\phi} \\
0 & -s_{\phi} & c_{\phi} \end{array} \right] = \left[ \begin{array}{ccc}
c_{\theta} & s_{\theta}s_{\phi} & -s_{\theta} c_{\phi} \\
0 & c_{\phi} & s_{\phi} \\
s_{\theta} & -c_{\theta} s_{\phi} & c_{\theta} c_{\phi} \end{array} \right] $ \\
Finaly leaving us with the Euler 123 sequence \\
$ 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] $
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