3D rotation group


In mechanics and geometry, the 3D rotation group, often denoted SO, is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition, which combines two rotations by performing one after the other.
By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance, and orientation. 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.
Every non-trivial rotation is determined by its axis of rotation and its angle of rotation. Rotations are not commutative, making the 3D rotation group a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable, so it is in fact a Lie group. It is compact and has dimension 3.
Rotations are linear transformations of and can therefore be represented by matrices once a basis of has been chosen. Specifically, if we choose an orthonormal basis of, every rotation is described by an orthogonal 3 × 3 matrix with determinant 1. The group SO can therefore be identified with the group of these matrices under matrix multiplication. These matrices are known as "special orthogonal matrices", explaining the notation SO.
The group SO is used to describe the possible rotational symmetries of an object, as well as the possible orientations of an object in space. Its representations are important in physics, where they give rise to the elementary particles of integer spin.

Length and angle

Besides just preserving length, rotations also preserve the angles between vectors. This follows from the fact that the standard dot product between two vectors u and v can be written purely in terms of length :
It follows that every length-preserving linear transformation in preserves the dot product, and thus the angle between vectors. Rotations are often defined as linear transformations that preserve the inner product on, which is equivalent to requiring them to preserve length. See classical group for a treatment of this more general approach, where appears as a special case.

Orthogonal and rotation matrices

Every rotation maps an orthonormal basis of to another orthonormal basis. Like any linear transformation of finite-dimensional vector spaces, a rotation can always be represented by a matrix. Let be a given rotation. With respect to the standard basis of the columns of are given by. Since the standard basis is orthonormal, and since preserves angles and length, the columns of form another orthonormal basis. This orthonormality condition can be expressed in the form
where denotes the transpose of and is the identity matrix. Matrices for which this property holds are called orthogonal matrices. The group of all orthogonal matrices is denoted, and consists of all proper and improper rotations.
In addition to preserving length, proper rotations must also preserve orientation. A matrix will preserve or reverse orientation according to whether the determinant of the matrix is positive or negative. For an orthogonal matrix, note that implies, so that. The subgroup of orthogonal matrices with determinant is called the special orthogonal group, denoted.
Thus every rotation can be represented uniquely by an orthogonal matrix with unit determinant. Moreover, since composition of rotations corresponds to matrix multiplication, the rotation group is isomorphic to the special orthogonal group.
Improper rotations correspond to orthogonal matrices with determinant, and they do not form a group because the product of two improper rotations is a proper rotation.

Group structure

The rotation group is a group under function composition. It is a subgroup of the general linear group consisting of all invertible linear transformations of the real 3-space.
Furthermore, the rotation group is nonabelian. That is, the order in which rotations are composed makes a difference. For example, a quarter turn around the positive x-axis followed by a quarter turn around the positive y-axis is a different rotation than the one obtained by first rotating around y and then x.
The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem.

Complete classification of finite subgroups

The finite subgroups of are completely classified.
Every finite subgroup is isomorphic to either an element of one of two countably infinite families of planar isometries: the cyclic groups or the dihedral groups, or to one of three other groups: the tetrahedral group, the octahedral group, or the icosahedral group.

Axis of rotation

Every nontrivial proper rotation in 3 dimensions fixes a unique 1-dimensional linear subspace of which is called the axis of rotation. Each such rotation acts as an ordinary 2-dimensional rotation in the plane orthogonal to this axis. Since every 2-dimensional rotation can be represented by an angle φ, an arbitrary 3-dimensional rotation can be specified by an axis of rotation together with an angle of rotation about this axis..
For example, counterclockwise rotation about the positive z-axis by angle φ is given by
Given a unit vector n in and an angle φ, let R represent a counterclockwise rotation about the axis through n. Then
  • R is the identity transformation for any n
  • R = R
  • R = R.
Using these properties one can show that any rotation can be represented by a unique angle φ in the range 0 ≤ φ ≤ and a unit vector n such that
  • n is arbitrary if φ = 0
  • n is unique if 0 < φ <
  • n is unique up to a sign if φ = .
In the next section, this representation of rotations is used to identify SO topologically with three-dimensional real projective space.

Topology

The Lie group SO is diffeomorphic to the real projective space
Consider the solid ball in of radius . Given the above, for every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin. The identity rotation corresponds to the point at the center of the ball. Rotations through an angle between 0 and are on the same axis at the same distance. Rotation through angles between 0 and − correspond to the point on the same axis and distance from the origin but on the opposite side of the origin. The one remaining issue is that the two rotations through and through − are the same. So we identify antipodal points on the surface of the ball. After this identification, we arrive at a topological space homeomorphic to the rotation group.
Indeed, the ball with antipodal surface points identified is a smooth manifold, and this manifold is diffeomorphic to the rotation group. It is also diffeomorphic to the real 3-dimensional projective space so the latter can also serve as a topological model for the rotation group.
These identifications illustrate that SO is connected but not simply connected. As to the latter, in the ball with antipodal surface points identified, consider the path running from the "north pole" straight through the interior down to the south pole. This is a closed loop, since the north pole and the south pole are identified. This loop cannot be shrunk to a point, since no matter how it is deformed, the start and end point have to remain antipodal, or else the loop will "break open". In terms of rotations, this loop represents a continuous sequence of rotations about the z-axis starting at the identity, through the south pole, jumping to the north pole and ending again at the identity rotation.
Surprisingly, running through the path twice, i.e., running from the north pole down to the south pole, jumping back to the north pole, and then again running from the north pole down to the south pole, so that φ runs from 0 to 4, gives a closed loop which can be shrunk to a single point: first move the paths continuously to the ball's surface, still connecting north pole to south pole twice. The second path can then be mirrored over to the antipodal side without changing the path at all. Now we have an ordinary closed loop on the surface of the ball, connecting the north pole to itself along a great circle. This circle can be shrunk to the north pole without problems. The plate trick and similar tricks demonstrate this practically.
The same argument can be performed in general, and it shows that the fundamental group of SO is the cyclic group of order 2. In physics applications, the non-triviality of the fundamental group allows for the existence of objects known as spinors, and is an important tool in the development of the spin–statistics theorem.
The universal cover of SO is a Lie group called Spin. The group Spin is isomorphic to the special unitary group SU; it is also diffeomorphic to the unit 3-sphere S3 and can be understood as the group of versors. The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotations. The map from S3 onto SO that identifies antipodal points of S3 is a surjective homomorphism of Lie groups, with kernel. Topologically, this map is a two-to-one covering map.

Connection between SO(3) and SU(2)

In this section, we give two different constructions of a two-to-one and surjective homomorphism of SU onto SO.

Using quaternions of unit norm

The group is isomorphic to the quaternions of unit norm via a map given by
restricted to where,,, and,.
Let us now identify with the span of. One can then verify that if is in and is a unit quaternion, then
Furthermore, the map is a rotation of Moreover, is the same as. This means that there is a homomorphism from quaternions of unit norm to the 3D rotation group.
One can work this homomorphism out explicitly: the unit quaternion,, with
is mapped to the rotation matrix
This is a rotation around the vector by an angle, where and. The proper sign for is implied, once the signs of the axis components are fixed. The is apparent since both and map to the same.