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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. Two rotations with a common center commute as a matter of course. The product of rotations is not in general commutative. Successive rotations result in a rotation or a translation. However, all circles centered at the center of rotation are fixed. Except for the trivial case, rotations have no fixed lines.

Įxcept for the trivial rotation through a zero angle which is identical, rotations have a single fixed point - the center of rotation.

Rotation maps parallel lines onto parallel lines. Rotation is isometry: a rotation preserves distances. For example, if a polygon is traversed clockwise, its rotated image is likewise traversed clockwise. The following observations are noteworthy: In the applet, you rotate a pentagon whose shape is defined by draggable vertices.) (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. For any point P, its image P' = R O, α(P) lies at the same distance from O as P and, in addition (1)

The case α = 0 (mod 2 p) leads to a trivial transformation that moves no point.

Rotation is a geometric transformation R O, α defined by a point O called the center of rotation, or a rotocenter, and an angle α, known as the angle of rotation.