Rotation Group SO(3)
In mechanics and geometry, the 3D rotation group 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 linear transformation that preserves length of vectors (it is an isometry) and preserves orientation (i.e. handedness) of space. A length-preserving transformation which reverses orientation is an improper rotation, that is a reflection or more generally a rotoinversion.
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 manifold structure for which the group operations are smooth; so it is in fact a Lie group. The rotation group is often denoted SO(3) for reasons explained below.
Read more about Rotation Group SO(3): Length and Angle, Orthogonal and Rotation Matrices, Group Structure, Axis of Rotation, Topology, Lie Algebra, Representations of Rotations, Generalizations
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