Rotation (mathematics)

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Rotation (mathematics)

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
and linear algebra

Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
, a rotation is a transformation

Transformation (mathematics)

In mathematics, a transformation could be any function from a set X to itself. However, often the set X has some additional algebraic or geometric structure and the term "transformation" refers to a function from X to itself which preserves this structure....
in a plane or in space that describes the motion of a rigid body

Rigid body

In physics, a rigid body is an idealization of a solid Physical body of finite size in which deformation is neglected. In other words, the distance between any two given Point s of a rigid body remains constant in time regardless of external forces exerted on it....
around a fixed point. A rotation is different from a translation, which has no fixed points, and from a reflection

Reflection (mathematics)

In mathematics, a reflection is a function that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q....
, 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.

s important to understand the frame of reference when discussing rotations.

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Encyclopedia

In geometry

Geometry

Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
, a rotation is a transformation

Transformation (mathematics)

Rigid body

Reflection (mathematics)

Two dimensions

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 about the origin, where is rotated and we want to know the coordinates after the rotation, :

or

In a clockwise

Clockwise

A clockwise motion is one that proceeds 'like the clock's hands': from the top to the right, then down and then to the left, and back to the top....
rotation of the plane or axes about the origin, the coordinates in the new plane will be rotated counterclockwise in the new coordinates. In this case, if the coordinates in the old plane are and the coordinates of the same vector in the new plane are , then:

or

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 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 ? by pre-multiplying it with (see Euler's formula

Euler's formula

Euler's formula, named after Leonhard Euler, is a mathematics formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function....
, §2), viz.

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.

Three dimensions

In ordinary

Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
three-dimensional space, a coordinate rotation can be defined by three Euler angles

Euler angles

The Euler angles were developed by Leonhard Euler to describe the orientation of a rigid body in dimension Euclidean space. To give an object a specific orientation it may be subjected to a sequence of three rotations described by the 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

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
transformation called a rotation matrix

Rotation matrix

In matrix theory, a rotation matrix is a real number square matrix whose transpose is its invertible matrix and whose determinant is 1 The matrix is so-called because it geometrically corresponds to a linear map that sends vectors to a corresponding vector rotated about the origin by a fixed angle....
. Rotations about another point can be described by a 4×4 matrix acting on the homogeneous coordinates

Homogeneous coordinates

In mathematics, homogeneous coordinates, introduced by August Ferdinand M?bius in his 1827 work Der barycentrische Calc?l, allow affine transformations to be easily represented by a matrix....
.

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, and quaternions avoid the problem of gimbal lock

Gimbal lock

Gimbal lock is the loss of one degree of freedom that occurs when the axes of two of the three gimbals needed to apply or compensate for rotations in three dimensional space are driven to the same direction....
.

Generalizations

Orthogonal matrices

The set of all matrices M(v,?) described above together with the operation of matrix multiplication

Matrix multiplication

In mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix. This article gives an overview of the various ways to perform matrix multiplication....
is called rotation group
Rotation group

In classical mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of functional composition....
: SO(3).

More generally, coordinate rotations in any dimension are represented by orthogonal matrices

Orthogonal matrix

In matrix theory, a real number orthogonal matrix is a Matrix #Square matrices Q whose transpose is its inverse matrix:A special orthogonal matrix is an orthogonal matrix with determinant +1:...
. 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)

SO(4)

In mathematics, SO is the four-dimensional rotation group; that is, the group of rotations about a fixed point in four-dimensional Euclidean space....
.

Orthogonal matrices have real elements. The analogous complex-valued matrices are the unitary matrices

Unitary matrix

In mathematics, a unitary matrix is an n by n complex number matrix U satisfying the condition where is the identity matrix and is the conjugate transpose of U....
. The set of all unitary matrices in a given dimension n forms a unitary group

Unitary group

In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrix, with the group operation that of matrix multiplication....
of degree n, U(n); and the subgroup of U(n) representing proper rotations forms a special unitary group

Special unitary group

In mathematics, the special unitary group of degree n, denoted SU, is the group of n×n unitary matrix Matrix with determinant 1....
of degree n, SU(n). The elements of SU(2) are used in quantum mechanics

Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
to rotate spin

Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
.

Relativity

In special relativity

Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#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 transformation

In physics, the Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in constant motion with respect to the other....
, Lorentz group

Lorentz group

In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical field theory setting for all physics....
.