rotation group SO(3)

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R³ under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation (i.e. handedness of space). Every non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties (along with the associative property, which rotations obey), the set of all rotations is a group under composition. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smooth; so it is in fact a Lie group. It is compact and has dimension 3.