Rotation (mathematics)

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For a less mathematical treatment of rotations, see rotation.

In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. A rotation is different from a translation, which has no fixed points, and from a reflection, which "flips" the bodies it is transforming. A rotation and the above-mentioned transformations are isometries, they leave the distance between any two points unchanged after the transformation.

A rotation in two dimensions around a point O.
A rotation in two dimensions around a point O.

Contents

[edit] Two dimensions

A plane rotation around a point followed by another rotation around a different point results in a total motion which is either a rotation (as in this picture), or a translation.
A plane rotation around a point followed by another rotation around a different point results in a total motion which is either a rotation (as in this picture), or a translation.
A reflection against an axis followed by a reflection against a second axis not parallel to the first one results in a total motion that is a rotation around the point of intersection of the axes.
A reflection against an axis followed by a reflection against a second axis not parallel to the first one results in a total motion that is a rotation around the point of intersection of the axes.

It is important to understand the frame of reference when discussing rotations. From one point of view, you may be discussing rotating a vector, keeping the axes fixed. From another point of view, you may be rotating the coordinates, while keeping the vector fixed.

In the first point of view, a counterclockwise rotation of a coordinate or vector about the origin, where (x,y) is rotated θ and we want to know the coordinates after the rotation, (x',y'):

\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta \\ +\sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}.

or

\begin{align} x'&=x\cos\theta-y\sin\theta\\ y'&=+x\sin\theta+y\cos\theta \end{align}

In a counterclockwise rotation of the plane or axes about the origin, the coordinates in the new plane will be rotated clockwise in the new coordinates. In this case, if the coordinates in the old plane are (x,y) and the coordinates of the same vector in the new plane are (x',y'), then:

\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos \theta & +\sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}.

or

\begin{align} x'&=x\cos\theta+y\sin\theta\\ y'&=-x\sin\theta+y\cos\theta \end{align}

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

z = x + iy \,

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.

\begin{align} e^{i \theta} z &= (\cos \theta + i \sin \theta) (x + i y) \\ &= (x \cos \theta + i y \cos \theta + i x \sin \theta - y \sin \theta) \\ &= (x \cos \theta - y \sin \theta) + i (x \sin \theta + y \cos \theta) \\ &= x' + i y' . \end{align}

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

A rotation describes the motion of a rigid body around a point.
A rotation describes the motion of a rigid body around a point.
Main article: Rotation matrix

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, and quaternions avoid the problem of gimbal lock.

[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.

[edit] See also