李泥兒

Graph Rotation

Suppose you want to rotate an object by an angle è counterclockwise. First, suppose you have a point

(x1, y1) that you want to rotate by an angle è to get to the point (x2, y2), as shown below.

The distance from the point to the origin is assumed to be r. We then have the following relations:

The point (x2, y2) is the same point rotated by an additional angle of è. Since this point also has a

distance r from the origin, its coordinates are given by

Substituting the components of x1 = r cos á and y1 = r sin á into the preceding equations gives

In matrix form, the equivalent rotation transformation that takes point (x1, y1) to (x2, y2) is given by

the following rotation matrix:

Proof: