In mechanics and geometry, the 3D rotation group, often denoted SO(3), 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 transformation that preserves the origin, Euclidean
distance, and orientation. Every non-trivial rotation is determined by
its axis of rotation and its angle of rotation. 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 structure as a manifold for
which the group operations are smooth; so it is in fact a Lie group. It is
compact and has dimension 3. Rotations are linear transformations of
R3 and can therefore be represented by matrices once a basis of R3 has been
chosen. Specifically, if we choose an orthonormal basis of R3, every rotation
is described by an orthogonal 3x3 matrix with determinant 1. The group SO(3) 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(3). The group SO(3) 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 any length-preserving
transformation in R3 preserves the dot product, and thus the angle between
vectors. Rotations are often defined as linear transformations that preserve the
inner product on R3, which is equivalent to requiring them to preserve length.
See classical group for a treatment of this more general approach, where SO(3)
appears as a special case. Orthogonal and rotation matrices
Every rotation maps an orthonormal basis of R3 to another orthonormal basis. Like
any linear transformation of finite-dimensional vector spaces, a
rotation can always be represented by a matrix. Let R be a given rotation. With
respect to the standard basis e1, e2, e3 of R3 the columns of R are given by.
Since the standard basis is orthonormal, and since R preserves angles and length,
the columns of R form another orthonormal basis. This orthonormality
condition can be expressed in the form where RT denotes the transpose of R and
I 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), 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 R, note that det RT = det R implies2 = 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 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 SO(3).
Improper rotations correspond to orthogonal matrices with determinant −1,
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 R3. 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.
Axis of rotation Every nontrivial proper rotation in 3
dimensions fixes a unique 1-dimensional linear subspace of R3 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 R3 and an angle φ, let R(φ, n) represent a
counterclockwise rotation about the axis through 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 arbitrary if φ = 0 n is unique if 0