Rotation group
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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. It 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.
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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
- RTR = 1
where RT 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 RT = 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
Consider the solid ball in R3 of radius π (that is, all points of R3 of distance π or less from the origin). Given the above, for every point in this ball there is a rotation, with axis through the point 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. Rotation through angles between 0 and -π 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 π and through -π are the same. So we identify (or "glue together") 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(3) 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 center 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 you deform the loop, 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 and ending at the identity rotation (i.e. a series of rotation through an angle φ where φ runs from 0 to 2π).
Surprisingly, if you run through the path twice (so that φ runs from 0 to 4π), i.e. from north pole down to south pole, jump back up to the north pole and run again down to the south pole, you get 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 half of the 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 same argument can be performed in general, and it shows that the fundamental group of SO(3) is cyclic of order 2. In physics applications, the non-triviality of the fundamental group allows for the existence of objects known as spinors.
The universal cover of SO(3) is a Lie group called Spin(3). The group Spin(3) is diffeomorphic to the unit 3-sphere S3 and can be understood as the group of unit quaternions (i.e. those with absolute value 1). The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotations. The resulting map
- S3 → SO(3)
is a surjective homomorphism of Lie groups, with kernel {±1}.
Representations of rotations
We have seen that there are a variety of ways to represent rotations:
- as orthogonal matrices with determinant 1,
- by axis and rotation angle
- via the unit quaternions (see quaternions and spatial rotations) and the map S3 → SO(3).
Another method is to specify an arbitrary rotation by a sequence of rotations about some fixed axes. See:
See charts on SO(3) for further discussion.
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.