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
Famous quotes containing the words rotation and/or group:
“The lazy manage to keep up with the earths rotation just as well as the industrious.”
—Mason Cooley (b. 1927)
“The government of the United States at present is a foster-child of the special interests. It is not allowed to have a voice of its own. It is told at every move, Dont do that, You will interfere with our prosperity. And when we ask: where is our prosperity lodged? a certain group of gentlemen say, With us.”
—Woodrow Wilson (18561924)