Asrotatie

Asrotatie, or axis rotation, is the motion of a rigid body in which all points describe circles about a fixed straight line called the axis. The axis remains fixed in space, and distances between points are preserved during the motion.

In three-dimensional space, a rotation by angle θ around a unit axis n leaves every point on the axis fixed while rotating the perpendicular component by θ. It can be represented by a rotation matrix R, given by Rodrigues' formula: R = I cos θ + (1 − cos θ) nn^T + [n]_× sin θ, where [n]_× is the cross‑product matrix.

Axis–angle representation: The rotation is specified by an axis n and an angle θ. Quaternions provide an efficient alternative to compose and apply such rotations computationally.

Rotations about a non-origin axis can be performed by translating the space so that the axis passes through the origin, performing the rotation, and translating back.

Applications of axis rotations appear in astronomy (rotation of planets about their axes, day-night cycles), geophysics, computer graphics (orienting cameras and objects), robotics, and mechanical engineering (shafts and gyroscopes).

See also: rotation matrix, axis-angle, quaternion, Euler angles.