rotates

Rotates is the third-person singular present tense form of the verb rotate, describing the action of turning around an axis or changing orientation. In physics and engineering, rotational motion is distinguished from translational motion by the change in orientation of an object rather than its position alone. The term is used across disciplines to refer to objects or coordinate frames undergoing such turning motions.

In two dimensions, a rotation about the origin by angle theta maps a point (x, y) to (x cos theta − y sin theta, x sin theta + y cos theta). This operation preserves distance to the origin and preserves orientation, without reflection. In three dimensions, rotations occur about an axis by a given angle and can be more complex to describe, often using matrices, quaternions, or Euler angles.

Mathematically, rotations are represented by orthogonal matrices with determinant 1 (special orthogonal group). Composition of rotations is generally non-commutative, so the order of operations matters. Common representations include rotation matrices, Euler angles (three angles describing successive rotations about coordinate axes), and quaternions (a four-parameter system that avoids some issues of Euler angles, such as gimbal lock).

Applications of rotations are widespread: computer graphics and animation use rotations to orient objects; robotics relies on rotational kinematics for arm and joint motion; aerospace models spacecraft attitude; astronomy considers the rotation of celestial bodies. In abstract terms, the set of all possible rotations in a plane forms the group SO(2), while in three dimensions it forms SO(3). Rotates, as a verb form, thus captures a fundamental geometric transformation encountered across science and engineering.