Rotation (mathematics)
For example, in two dimensions rotating a body clockwise about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed.
A motion of a Euclidean space is the same as its isometry: it leaves the distance between any two points unchanged after the transformation.
The "improper rotation" term refers to isometries that reverse (flip) the orientation.
Any direct Euclidean motion can be represented as a composition of a rotation about the fixed point and a translation.
Any two-dimensional direct motion is either a translation or a rotation; see Euclidean plane isometry for details.
Rotations in three-dimensional space differ from those in two dimensions in a number of important ways.
Also, unlike the two-dimensional case, a three-dimensional direct motion, in general position, is not a rotation but a screw operation.
A four-dimensional direct motion in general position is a rotation about certain point (as in all even Euclidean dimensions), but screw operations exist also.
When one considers motions of the Euclidean space that preserve the origin, the distinction between points and vectors, important in pure mathematics, can be erased because there is a canonical one-to-one correspondence between points and position vectors.
In components, such operator is expressed with n × n orthogonal matrix that is multiplied to column vectors.
As it was already stated, a (proper) rotation is different from an arbitrary fixed-point motion in its preservation of the orientation of the vector space.
Matrices of all proper rotations form the special orthogonal group.
Points on the R2 plane can be also presented as complex numbers: the point (x, y) in the plane is represented by the complex number This can be rotated through an angle θ by multiplying it by eiθ, then expanding the product using Euler's formula as follows: and equating real and imaginary parts gives the same result as a two-dimensional matrix: Since complex numbers form a commutative ring, vector rotations in two dimensions are commutative, unlike in higher dimensions.
Above-mentioned Euler angles and axis–angle representations can be easily converted to a rotation matrix.
Another possibility to represent a rotation of three-dimensional Euclidean vectors are quaternions described below.
Unit quaternions, or versors, are in some ways the least intuitive representation of three-dimensional rotations.
They are more compact than matrices and easier to work with than all other methods, so are often preferred in real-world applications.
Unlike matrices and complex numbers two multiplications are needed: where q is the versor, q−1 is its inverse, and x is the vector treated as a quaternion with zero scalar part.
More generally, coordinate rotations in any dimension are represented by orthogonal matrices.
They can be extended to represent rotations and transformations at the same time using homogeneous coordinates.
[citation needed] Rotations about a fixed point in elliptic and hyperbolic geometries are not different from Euclidean ones.
These transformations preserve a quadratic form called the spacetime interval.
By contrast, a rotation in a plane spanned by a space-like dimension and a time-like dimension is a hyperbolic rotation, and if this plane contains the time axis of the reference frame, is called a "Lorentz boost".
Hyperbolic rotations are sometimes described as "squeeze mappings" and frequently appear on Minkowski diagrams that visualize (1 + 1)-dimensional pseudo-Euclidean geometry on planar drawings.
The circular symmetry is an invariance with respect to all rotation about the fixed axis.
As was stated above, Euclidean rotations are applied to rigid body dynamics.
of degree n; and its subgroup representing proper rotations (those that preserve the orientation of space) is the special unitary group
are used to parametrize three-dimensional Euclidean rotations (see above), as well as respective transformations of the spin (see representation theory of SU(2)).
