Three Dimensional Rotation Group
Overview
For a real nxn matrix {% R %}, {% R %} is defined to be a rotation matrix if
{% RR^T = 1 %}
and
{% det(R) = 1 %}
Three Dimensional
The three dimensional rotation matrices are more complex, in particular because the rotation can
occur around any of three axes. (the case of a transformation around an axis that is intermediate
between the 3 axes can be achieved by composing these three rotations)
{%
R_x
\begin{bmatrix}
1 & 0 & 0 \\
0 & sin \theta & -sin \theta \\
0 & sin \theta & cos \theta \\
\end{bmatrix}
%}
{%
R_y
\begin{bmatrix}
cos \theta & 0 & sin \theta \\
0 & 1 & 0 \\
-sin \theta & 0 & cos \theta \\
\end{bmatrix}
%}
{%
R_z
\begin{bmatrix}
cos \theta & -sin \theta & 0 \\
sin \theta & cos \theta & 0 \\
0 & 0 & 1 \\
\end{bmatrix}
%}
A general three dimensional rotation matrix can be constructed by mutiplying the 3 basis rotations.
