Two dimensions

In two dimensions, a counterclockwise rotation of the plane about the origin, where $(x,y)$ is mapped to $(x\text{'},y\text{'})$, is given by the same formulas as a coordinate transformation with a clockwise rotation of the coordinate axes, resulting in a change of coordinates $(x,y)$ into $(x\text{'},y\text{'})$:

$\backslash begin\{bmatrix\}\; x\text{'}\; \backslash \backslash \; y\text{'}\; \backslash end\{bmatrix\}\; =$

In other words

$x\text{'}=x\backslash cos\backslash theta+y\backslash sin\backslash theta,\backslash ,$

$y\text{'}=-x\backslash sin\backslash theta+y\backslash cos\backslash theta.\backslash ,$

Then the magnitude of the vector (x, y) is the same as the magnitude of vector (x′, y′).

Complex plane

A complex number can be seen as a two-dimensional vector in the complex plane, with its tail at the origin and its head given by the complex number. Let

$z\; =\; x\; +\; iy\; \backslash ,$

Then z can be rotated counterclockwise by an angle θ by pre-multiplying it with $e^\{i\; \backslash theta\}$ (see Euler's formula, §2), viz.

This can be seen to correspond to the rotation described in § 1.

Three dimensions

In ordinary three dimensional space, a coordinate rotation can be defined by three Euler angles, or by a single angle of rotation and the direction of a vector about which to rotate.

Rotations about the origin are most easily calculated using a 3 by 3 matrix transformation called a rotation matrix. Rotations about another point can be described by a 4 by 4 matrix acting on the heterogeneous coordinates.

Quaternions

An alternative approach to rotation in three dimensions uses quaternions.

Quaternions provide another way of representing rotations and orientations in three dimensions. They are applied in computer graphics, control theory, signal processing and orbital mechanics. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions, which are also used to telemeter their current attitude. The rationale is that combining many quaternion transformations is more numerically stable than combining many matrix transformations.

Generalizations

Orthogonal matrices

The set of all matrices M(v,θ) described above together with the operation of matrix multiplication is called rotation group: SO(3).

More generally, coordinate rotations in any dimension are represented by orthogonal matrices. The set of all orthogonal matrices of the n-th dimension which describe proper rotations (determinant = +1), together with the operation of matrix multiplication, forms the special orthogonal group: SO(n). See also SO(4).

Orthogonal matrices have real elements. The analogous complex-valued matrices are the unitary matrices. The set of all unitary matrices in a given dimension n forms a unitary group of degree n, U(n); and the subgroup of U(n) representing proper rotations forms a special unitary group of degree n, SU(n). The elements of SU(2) are used in quantum mechanics to rotate spin.

Relativity

In special relativity a Lorenzian coordinate rotation which rotates the time axis is called a boost, and, instead of spatial distance, the interval between any two points remains invariant. Lorentzian coordinate rotations which do not rotate the time axis are three dimensional spatial rotations. See: Lorentz transformation, Lorentz group.

See also

• Spin (physics)
• Charts on SO(3)
• Euler angles
• Vortical
• Rotation group
• Coordinate rotations and reflections
• Rotation matrix

Euclidean symmetries | Rotational symmetry

Rotaatio