rotation

Rotation is a transformation that turns every point of a figure around a fixed point in a plane or around a fixed axis in space by a specified angle. Distances and angles are preserved, and the transformation is orientation-preserving in the usual sense of a rotation.

In the plane, a rotation about a point P by angle θ maps a point X to X' so that PX = PX' and the directed angle between PX and PX' equals θ. It is represented in coordinates by the 2x2 matrix [cosθ −sinθ; sinθ cosθ].

In three dimensions, a rotation is about some axis by a chosen angle, or more generally an orientation-preserving isometry fixing a line. Rotations can be represented by 3x3 rotation matrices with determinant +1, by unit quaternions, or by axis-angle and Euler angles. The composition of rotations depends on order; in three dimensions rotations do not commute in general.

Group-theoretic perspective shows that the plane rotations about a fixed origin form the group SO(2), while three-dimensional rotations about the origin form SO(3). Core properties include distance and angle preservation, invertibility with the inverse rotation by the opposite angle, and the ability to be combined to produce new rotations.

Applications span computer graphics, robotics, physics, and astronomy, where rotations describe object orientation, angular momentum, and celestial mechanics. Common examples include the Earth’s daily rotation, the turning of clock hands by fixed angles, and the orientation of spacecraft.

See also: rotation matrix, axis-angle, quaternion, Euler angles, angular velocity, rotational symmetry.