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. For other uses, see Geometry (disambiguation). ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In mathematics, a transformation in elementary terms is any of a variety of different functions from geometry, such as rotations, reflections and translations. ... In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... In erik, a reflection (also spelled reflexion) is a map that transforms an object into its mirror image. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ...

A rotation in two dimensions around a point O.

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 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 Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... In erik, a reflection (also spelled reflexion) is a map that transforms an object into its mirror image. ...

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'): A clockwise motion is one that proceeds like the clocks hands: from the top to the right, then down and then to the left, and back to the top. ...

or

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: A clockwise motion is one that proceeds like the clocks hands: from the top to the right, then down and then to the left, and back to the top. ... The Clockwise direction A clockwise motion is one that proceeds like the clocks hands: from the top to the right, then down and then to the left, and back to the top. ...

or

Then the magnitude of the vector (x, y) is the same as the magnitude of vector (x′, y′). A vector in physics and engineering typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a magnitude and a direction. The word vector is also now used for more general concepts (see also vector and generalizations below), but in this...

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. Abscissa means the x coordinate on an (x, y) graph; the input of a mathematical function against which the output is plotted. ... Ordinate means the y coordinate on an (x, y) graph; the plotted output of a mathematical function. ...

Then z can be rotated counterclockwise by an angle θ by pre-multiplying it with e^iθ (see Euler's formula, §2), viz. Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

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. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...

Three dimensions

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. A rotation matrix is a matrix which when multiplied by a vector has the effect of changing the direction of the vector but not its magnitude. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Euler angles are a means of representing the spatial orientation of an object. ...

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. In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... A rotation matrix is a matrix which when multiplied by a vector has the effect of changing the direction of the vector but not its magnitude. ... In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius, allow affine transformations to be easily represented by a matrix. ...

Quaternions

Main article: Quaternions and spatial rotation

An alternative approach to rotation in three dimensions uses quaternions. Quaternions provide a convenient mathematical notation for representing orientations and rotations of objects. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...

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. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...

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). This article gives an overview of the various ways to perform matrix multiplication. ... In mechanics and geometry, the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space, R3. ...

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). In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... SO(4) is the symbol used in mathematics for the group of rotations about a fixed point in four-dimensional Euclidean space (for short, the 4D rotation group). ...

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. In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse... In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices with complex entries, with the group operation that of matrix multiplication. ... In mathematics, the special unitary group of degree n, denoted SU(n), is the group of n×n unitary matrices with unit determinant. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ...

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. For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... 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. ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ...

See also