2D Computer Graphics - in Two Dimensions

In Two Dimensions

In two dimensions every rotation matrix has the following form:

R(\theta) = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix}.

This rotates column vectors by means of the following matrix multiplication:

\begin{bmatrix} x' \\ y' \\ \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix}\begin{bmatrix} x \\ y \\ \end{bmatrix}.

So the coordinates (x',y') of the point (x,y) after rotation are:

,
.

The direction of vector rotation is counterclockwise if θ is positive (e.g. 90°), and clockwise if θ is negative (e.g. -90°).

R(-\theta) = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end{bmatrix}\,.

Read more about this topic:  2D Computer Graphics

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