Rotate

Rotate refers to a rigid motion that turns every point of a figure around a fixed point in a plane or around a fixed axis in space. The center or axis remains fixed while other points describe circular paths. Rotations preserve distances and angles, and they preserve orientation in the plane.

In two dimensions, a rotate by angle θ around the origin sends a point (x, y) to (x cos θ − y sin θ, x sin θ + y cos θ). This action can be represented by the rotation matrix [cos θ, −sin θ; sin θ, cos θ]. Rotations can be composed by adding their angles (mod 2π).

In three dimensions, a rotate by angle θ occurs about a specified axis. General 3D rotation can be described by axis-angle form, a rotation matrix, or a quaternion. Unlike 2D rotations, 3D rotations are not commutative; the order of successive rotations matters. Rodrigues’ formula provides a direct way to compute a rotation matrix from an axis and angle, and quaternions offer a compact, numerically stable representation for combining rotations.

Algebraically, the set of all rotations about a fixed origin forms the special orthogonal group SO(n); in 2D this group is isomorphic to the circle group S^1. Rotations are inverse to themselves and can be reversed by rotating by the opposite angle.

Applications of rotation include computer graphics and animation, robotics and kinematics, aerospace navigation, crystallography, and image processing. Common notations involve rotation matrices, Euler angles, and quaternions, each with its own advantages and potential drawbacks, such as gimbal lock in Euler-angle representations.