# Rotation Transform

Rotation is a geometric transformation RO, α defined by a point O called the center of rotation, or a rotocenter, and an angle α, known as the angle of rotation. The case α = 0 (mod 2p) leads to a trivial transformation that moves no point. For any point P, its image P' = RO, α(P) lies at the same distance from O as P and, in addition

 (1) ∠POP' = α.

(In the applet below, various rotations are controlled by a hollow blue point - the center of rotation, and a slider that determines the angle of rotation. In the applet, you rotate a pentagon whose shape is defined by draggable vertices.)

The following observations are noteworthy:

1. Rotation preserves the orientation. For example, if a polygon is traversed clockwise, its rotated image is likewise traversed clockwise.

2. Rotation is isometry: a rotation preserves distances.

3. Rotation preserves angles.

4. Rotation maps parallel lines onto parallel lines.

5. Except for the trivial rotation through a zero angle which is identical, rotations have a single fixed point - the center of rotation. Except for the trivial case, rotations have no fixed lines. However, all circles centered at the center of rotation are fixed.

6. Successive rotations result in a rotation or a translation.

7. The product of rotations is not in general commutative. Two rotations with a common center commute as a matter of course.

Two special rotations have acquired appellations of their own: a rotation through 180° is commonly referred to as a half-turn, a rotation through 90° is referred to as a quarter-turn. A half-turn is often referred to as a reflection in point.

Here is a short list of a problems that are solved with the help of the rotation transform: