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.