Charts On SO(3) - The Space of Rotations

The Space of Rotations

In geometry the 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 called an improper rotation. Every improper rotation of three-dimensional Euclidean space is rotation followed by a reflection in a plane through the origin.

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, 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.

The space of rotations is isomorphic with the set of rotation operators and the set of "orthonormal matrices with determinant +1". It is also isomorphic with the set of quaternions with their internal product, and also equivalent to the set of rotation vectors, with a difficult internal composition operation given by the product of their equivalent matrices.

Rotation vectors notation arise from the Euler's rotation theorem which states that any rotation in three dimensions can be described by a rotation by some angle about some axis. Considering this, we can then specify the axis of one of these rotations by two angles, and we can use the radius of the vector to specify the angle of rotation. These vectors represent a ball in 3D with an unusual topology.

This 3D solid sphere is equivalent to the surface of a 4D sphere, which is also a 3D variety. For doing this equivalence, we will have to define how will we represent a rotation with this 4D-embedded surface.