two dimensional rotation group

Two Dimensional Rotation Group

For a real {% n \times n %} matrix {% R %}, {% R %} is defined to be a rotation matrix if

{% R^T R = 1 %}

and

{% det(R) = 1 %}

For a given infinitesimal rotation angle {% \theta %}, the rotation matrix is close to the identiy matrix. We can therefore write it as

{% R(\theta) \approx I + A %}

For this equation, we are following the Taylor Series Expansion keeping only terms that are first order in {% \theta %}.

{% R^T R \approx (I + A^T)(I+A) \approx (I + A^T + A) = I %}

which means that {% A^T = -A %}. There is only one matrix that satisifies this requirement

{% J = \begin{bmatrix} 0 & - 1 \\ 1 & 0 \\ \end{bmatrix} %}

A finite rotation can then be computed to be the limit of a bunch of small rotations.

{% R(\theta) = lim_{n \to \infty} R(\frac{\theta}{n})^n = lim_{n \to \infty} (1 + \frac{\theta J}{n})^n = e^{\theta J} %}