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SO(3)

In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R³. By definition, a rotation is a linear transformation that preserves the length of vectors, and also preserves the orientation, or handedness, of space. (A transformation that preserves length but reverses orientation is sometimes called an improper rotation).

The composition of two rotations is a rotation, and every rotation has a unique inverse which is again a rotation. These properties give the set of all rotations the mathematical structure of a group with composition as the group operation. In so happens that this group has a natural manifold structure for which the group operations are smooth, so that the rotation group is actually a real Lie group. This group is often denoted SO(3) for reasons that are explained below.

Properties

Besides just preserving length, rotations also preserve the angles between vectors. This follows from the fact that the standard inner product between two vectors can be written purely in terms of length:

<math>\langle x, y \rangle = \frac{1}{2}\left(\|x+y\|^2 - \|x\|^2 - \|y\|^2\right)<math>

Hence, any transformation preserving length in R³ will preserve the inner product, and therefore angles, as well. It follows immediately that every rotation takes a orthonormal basis for R³ to another orthonormal basis.

It should be noted that rotations are often defined as linear transformations that preserve the inner product on R³. By the above argument, this is equivalent to requiring them to preserve length.

Another important property of the rotation group is that it 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.

Orthogonal matrices

Like any linear transformation, a rotation can always be represented by a matrix. Let R be a given rotation. With respect to the standard basis <math>(e_1, e_2, e_3)<math> of R³ the columns of R are given by <math>(Re_1, Re_2, Re_3)<math>. Since the standard basis is orthonormal, the columns of R form another orthonormal basis. This orthonormality condition can be expressed in the form

R^TR = 1

where R^T is denotes the transpose of R and 1 is the 3 × 3 identity matrix. Matrices for which this property holds are called orthogonal matrices. The group of all 3 × 3 orthogonal matrices is denoted O(3).

In addition to preserving length, 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 R, note that det R^T = det R implies (det R)² = 1 so that det R = ±1. The subgroup of orthogonal matrices with determinant +1 is called the special orthogonal group, denoted SO(3).

Thus every rotation can be represented uniquely by a orthogonal matrix with unit determinant. Moreover, since composition of rotations corresponds to matrix multiplication, the rotation group is isomorphic to the special orthogonal group SO(3).

Note that improper rotations correspond to orthogonal matrices with determinant −1. Improper rotations do not form a group since the product of two improper rotations is a proper rotation.

Axis of rotation

Every rotation in 3 dimensions fixes a unique 1-dimensional linear subspace of R³ which is called the axis of rotation. Each rotation will act like a normal 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. (Technically, one needs to specify an orientation for the axis and whether the rotation is taken to be clockwise or counterclockwise with respect to this orientation).

In terms of orthogonal matrices, the rotations about the standard coordinate axes through an angle φ are given by

<math>R_x(\phi) = \begin{pmatrix}1 & 0 & 0 \\ 0 & \cos\phi & -\sin\phi \\ 0 & \sin\phi & \cos\phi\end{pmatrix}<math>

<math>R_y(\phi) = \begin{pmatrix}\cos\phi & 0 & \sin\phi \\ 0 & 1 & 0 \\ -\sin\phi & 0 & \cos\phi\end{pmatrix}<math>

<math>R_z(\phi) = \begin{pmatrix}\cos\phi & -\sin\phi & 0 \\ \sin\phi & \cos\phi & 0 \\ 0 & 0 & 1\end{pmatrix}<math>

Given a unit vector n in R³ and an angle φ, let R(φ, n) represent a counterclockwise rotation about the axis through n (with orientation determined by n). Then

• R(0, n) is the identity transformation for any n
• R(φ, n) = R(−φ, −n)
• R(π + φ, n) = R(π − φ, −n)

Using these properties one can show that any rotation can be represented by a unique angle φ in the range 0 ≤ φ ≤ π and a unit vector n such that

• n is unique if 0 < φ < π
• n is arbitrary if φ = 0
• n is unique up to a sign if φ = π (that is, the rotations R(π, ± n) are identical)

Topology

To be completed.

Generalizations

The rotation group generalizes quite naturally to n-dimensional Euclidean space, Rⁿ. The group of all proper and improper rotations in n dimensions is called the orthogonal group, O(n), and the subgroup of proper rotations is called the special orthogonal group, SO(n).

In special relativity, one works in a 4-dimensional vector space, known as Minkowski space rather than 3-dimensional Euclidean space. Unlike Euclidean space, Minkowski space has an inner product with an indefinite signature. However, one can still define generalized rotations which preserve this inner product. Such generalized rotations are known as Lorentz transformations and the group of all such transformations is called the Lorentz group.

Related topics

• coordinate rotations
• Charts on SO(3)
• Euler angles
• quaternions and spatial rotations
• rigid body
• angular momentum