rotare

Rotare is a term used in mathematics, computer science, and related fields to denote the operation of rotating a geometric object or coordinate system by a specified angle around a given axis. In 2D, rotare about the origin by angle theta is implemented by applying the rotation matrix R(theta) = [[cos theta, -sin theta], [sin theta, cos theta]] to position vectors. In 3D, rotations can be represented by matrices, quaternions, or axis-angle representations; they compose to form the rotation group SO(n), with preservation of length and orientation (for proper rotations).

Etymology: the word derives from the Latin rota, wheel, via English and other European languages where cognate terms refer to turning or revolving objects. In physics and engineering, rotare is sometimes used interchangeably with rotate, depending on language and domain.

Applications: rotare is central to computer graphics for orienting models, to robotics for changing end-effector pose, and to astronomy for coordinate transformations. Important practical notes include the non-commutativity of 3D rotations, the problem of gimbal lock when using Euler angles, and the use of quaternions for robust interpolation (SLERP).

Variants and related concepts: discrete rotations on pixel grids, interpolation between orientations, and rotation operations in higher dimensions. In mathematics, rotating coordinates corresponds to composing orthogonal transformations with determinant +1.

See also: rotation matrix, quaternion, axis-angle, Euler angles, SO(3).