Rotation (mathematics)
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- For a less mathematical treatment of rotations, see rotation.
In linear algebra and geometry, a rotation is a type of transformation from one system of coordinates to another system of coordinates such that distance between any two points remains invariant under the transformation. In other words, a rotation is a type of isometry – note however that there are isometries other than rotations, such as translations, reflections, and glide reflections.
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[edit] Two dimensions
In two dimensions, a counterclockwise rotation of the plane about the origin, where (x,y) is mapped to (x',y'), 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',y'):
In other words
Then the magnitude of the vector (x, y) is the same as the magnitude of vector (x′, y′).
[edit] 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
be such a complex number. Its real component is the abscissa and its imaginary component its ordinate.
Then z can be rotated counterclockwise by an angle θ by pre-multiplying it with eiθ (see Euler's formula, §2), viz.
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This can be seen to correspond to the rotation described in § 1.
Because multiplication of complex numbers is commutative, rotation in 2 dimensions is commutative, unlike in higher dimensions.
[edit] 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 3×3 matrix transformation called a rotation matrix. Rotations about another point can be described by a 4×4 matrix acting on the homogeneous coordinates.
[edit] Quaternions
Main article: Quaternions and spatial rotation
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.
[edit] Generalizations
[edit] 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.
[edit] Relativity
In special relativity a Lorentzian 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.
