What is rotation?

Rotation repositions an object along a circular path in the xy plane.To rotate, we need to specify a rotation angle θ and the rotation point(xr,yr) about which the object is rotated.Counter clockwise rotation about the pivot point is defined by positive values for the rotation angle and clockwise direction is defined by negative values.

Rotating a point P(x,y) about the coordinate origin

If r is the distance of P from origin and angle Ф is the angular position of P from the x axis and θ is the rotation angle, we can express the transformated coordinates P'(x',y') as
x'=r cos(Ф + θ)=r cos Ф cos θ -r sin Ф sin θ
y'=r sin(Ф + θ)=r cos Ф sin θ -r sin Ф cos θ
The point P(x,y)can be written in polar coordinates as
x=r cos Ф ,y=r sin Ф
Hence we obtain the transformation equations for rotating a point (x,y) through an angle θ about the origin as
x'=x cos θ - y sin θ
y'=x sin θ + y cos θ.
Rotations are rigid body transformations that move objects without deformation.

In matrix form,

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