Rotation

Consider the triangle ABC (Figure 1.2) and rotate it by 90 ° counterclockwise about the origin.

(1.24)

Fig.1.2. Rotation

If we use the matrix (3×2), consisting of x and y coordinates of triangle vertices, we can write:

(1.25)

That is the coordinates of the resulting triangle A*B*C*. Rotation by 180° about the origin is achieved by the following conversion:

(1.26)

and by 270⁰:

(1.27)

How to make a rotation around the origin through an arbitrary angle θ? To answer this question, we consider the position vector from the origin to the point P (Figure 1.3). Denote r - the length of the vector, and φ - the angle between the vector and the x axis. Position vector is rotated around the origin through an angle θ and gets to the point P*. Writing the position vectors for the P and P*, we obtain:

(1.28)

and

(1.29)

Fig.1.3. Rotation of a point P from position (x, y) to position (x*, y*) through an angle θ relative to the coordinate origin

Thus, the transformed point has the coordinates

(1.30)

Or in matrix form:

(1.31)

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