Two Dimensional Rotation Group
Overview
For a real {% n \times n %} matrix {% R %}, {% R %} is defined to be a rotation matrix if
{% R^TR = 1 %}
and
{% det(R) = 1 %}
Rotation Matrix
A rotation in the plane through the angle {% \theta %} is given by the following.
{% R =
\begin{bmatrix}
cos \theta & -sin \theta \\
sin \theta & cos \theta \\
\end{bmatrix}
%}
The plane rotation is a
lie group
of a single parameter, {% \theta %}. The derivative of the matrix with respect to {% \theta %}
is
{% J =
\begin{bmatrix}
0 & - 1 \\
1 & 0 \\
\end{bmatrix}
%}
This is the generator of the group.
The two dimensional rotation can then be shown to be equivalent to the
exponential
of the generator.
(see
rotation exponential)
Topics
