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Rotation inmathematics is a concept originating ingeometry. Any rotation is amotion of a certainspace that preserves at least onepoint. It can describe, for example, the motion of arigid body around a fixed point. Rotation can have asign (as in thesign of an angle): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude.A rotation is different from other types of motions:translations, which have no fixed points, and(hyperplane) reflections, each of them having an entire(n − 1)-dimensionalflat of fixed points in an-dimensional space.
Mathematically, a rotation is amap. All rotations about a fixed point form agroup undercomposition called therotation group (of a particular space). But inmechanics and, more generally, inphysics, this concept is frequently understood as acoordinate transformation (importantly, a transformation of anorthonormal basis), because for any motion of a body there is an inverse transformation which if applied to theframe of reference results in the body being at the same coordinates. For example, in two dimensions rotating a bodyclockwise about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed. These two types of rotation are calledactive and passive transformations.[1][2]
Therotation group is aLie group of rotations about afixed point. This (common) fixed point orcenter is called thecenter of rotation and is usually identified with theorigin. The rotation group is apoint stabilizer in a broader group of (orientation-preserving)motions.
For a particular rotation:
Arepresentation of rotations is a particular formalism, either algebraic or geometric, used to parametrize a rotation map. This meaning is somehow inverse tothe meaning in the group theory.
Rotations of(affine) spaces of points and of respectivevector spaces are not always clearly distinguished. The former are sometimes referred to asaffine rotations (although the term is misleading), whereas the latter arevector rotations. See the article below for details.
A motion of aEuclidean space is the same as itsisometry: it leavesthe distance between any two points unchanged after the transformation. But a (proper) rotation also has to preserve theorientation structure. The "improper rotation" term refers to isometries that reverse (flip) the orientation. In the language ofgroup theory the distinction is expressed asdirect vsindirect isometries in theEuclidean group, where the former comprise theidentity component. Any direct Euclidean motion can be represented as a composition of a rotation about the fixed point and a translation.
Inone-dimensional space, there are onlytrivial rotations. Intwo dimensions, only a singleangle is needed to specify a rotation about theorigin – theangle of rotation that specifies an element of thecircle group (also known asU(1)). The rotation is acting to rotate an objectcounterclockwise through an angleθ about theorigin; seebelow for details. Composition of rotationssums their anglesmodulo 1turn, which implies that all two-dimensional rotations aboutthe same pointcommute. Rotations aboutdifferent points, in general, do not commute. Any two-dimensional direct motion is either a translation or a rotation; seeEuclidean plane isometry for details.
Rotations inthree-dimensional space differ from those in two dimensions in a number of important ways. Rotations in three dimensions are generally notcommutative, so the order in which rotations are applied is important even about the same point. Also, unlike the two-dimensional case, a three-dimensional direct motion, ingeneral position, is not a rotation but ascrew operation. Rotations about the origin have three degrees of freedom (seerotation formalisms in three dimensions for details), the same as the number of dimensions.A three-dimensional rotation can be specified in a number of ways. The most usual methods are:
A general rotation infour dimensions has only one fixed point, the centre of rotation, and no axis of rotation; seerotations in 4-dimensional Euclidean space for details. Instead the rotation has two mutually orthogonal planes of rotation, each of which is fixed in the sense that points in each plane stay within the planes. The rotation has two angles of rotation, one for eachplane of rotation, through which points in the planes rotate. If these areω1 andω2 then all points not in the planes rotate through an angle betweenω1 andω2. Rotations in four dimensions about a fixed point have six degrees of freedom. A four-dimensional direct motion in general positionis a rotation about certain point (as in alleven Euclidean dimensions), but screw operations exist also.
When one considers motions of the Euclidean space that preservethe origin, thedistinction between points and vectors, important in pure mathematics, can be erased because there is a canonicalone-to-one correspondence between points andposition vectors. The same is true for geometries other thanEuclidean, but whose space is anaffine space with a supplementarystructure; seean example below. Alternatively, the vector description of rotations can be understood as a parametrization of geometric rotationsup to their composition with translations. In other words, one vector rotation presents manyequivalent rotations aboutall points in the space.
A motion that preserves the origin is the same as alinear operator on vectors that preserves the same geometric structure but expressed in terms of vectors. ForEuclidean vectors, this expression is theirmagnitude (Euclidean norm). Incomponents, such operator is expressed withn × northogonal matrix that is multiplied tocolumn vectors.
As itwas already stated, a (proper) rotation is different from an arbitrary fixed-point motion in its preservation of the orientation of the vector space. Thus, thedeterminant of a rotation orthogonal matrix must be 1. The only other possibility for the determinant of an orthogonal matrix is−1, and this result means the transformation is ahyperplane reflection, apoint reflection (foroddn), or another kind ofimproper rotation. Matrices of all proper rotations form thespecial orthogonal group.
In two dimensions, to carry out a rotation using a matrix, the point(x, y) to be rotated counterclockwise is written as a column vector, then multiplied by arotation matrix calculated from the angleθ:
The coordinates of the point after rotation arex′, y′, and the formulae forx′ andy′ are
The vectors and have the same magnitude and are separated by an angleθ as expected.
Points on theR2 plane can be also presented ascomplex numbers: the point(x, y) in the plane is represented by the complex number
This can be rotated through an angleθ by multiplying it byeiθ, then expanding the product usingEuler's formula as follows:
and equating real and imaginary parts gives the same result as a two-dimensional matrix:
Since complex numbers form acommutative ring, vector rotations in two dimensions are commutative, unlike in higher dimensions. They have only onedegree of freedom, as such rotations are entirely determined by the angle of rotation.[3]
As in two dimensions, a matrix can be used to rotate a point(x, y, z) to a point(x′, y′, z′). The matrix used is a3×3 matrix,
This is multiplied by a vector representing the point to give the result
The set of all appropriate matrices together with the operation ofmatrix multiplication is therotation group SO(3). The matrixA is a member of the three-dimensionalspecial orthogonal group,SO(3), that is it is anorthogonal matrix withdeterminant 1. That it is an orthogonal matrix means that its rows are a set of orthogonalunit vectors (so they are anorthonormal basis) as are its columns, making it simple to spot and check if a matrix is a valid rotation matrix.
Above-mentioned Euler angles and axis–angle representations can be easily converted to a rotation matrix.
Another possibility to represent a rotation of three-dimensional Euclidean vectors are quaternions described below.
Unitquaternions, orversors, are in some ways the least intuitive representation of three-dimensional rotations. They are not the three-dimensional instance of a general approach. They are more compact than matrices and easier to work with than all other methods, so are often preferred in real-world applications.[citation needed]
A versor (also called arotation quaternion) consists of four real numbers, constrained so thenorm of the quaternion is 1. This constraint limits the degrees of freedom of the quaternion to three, as required. Unlike matrices and complex numbers two multiplications are needed:
whereq is the versor,q−1 is itsinverse, andx is the vector treated as a quaternion with zeroscalar part. The quaternion can be related to the rotation vector form of the axis angle rotation by theexponential map over the quaternions,
wherev is the rotation vector treated as a quaternion.
A single multiplication by a versor,either left or right, is itself a rotation, but in four dimensions. Any four-dimensional rotation about the origin can be represented with two quaternion multiplications: one left and one right, by twodifferent unit quaternions.
More generally, coordinate rotations in any dimension are represented by orthogonal matrices. The set of all orthogonal matrices inn dimensions which describe proper rotations (determinant = +1), together with the operation of matrix multiplication, forms thespecial orthogonal groupSO(n).
Matrices are often used for doing transformations, especially when a large number of points are being transformed, as they are a direct representation of thelinear operator. Rotations represented in other ways are often converted to matrices before being used. They can be extended to represent rotations and transformations at the same time usinghomogeneous coordinates.Projective transformations are represented by4×4 matrices. They are not rotation matrices, but a transformation that represents a Euclidean rotation has a3×3 rotation matrix in the upper left corner.
The main disadvantage of matrices is that they are more expensive to calculate and do calculations with. Also in calculations wherenumerical instability is a concern matrices can be more prone to it, so calculations to restoreorthonormality, which are expensive to do for matrices, need to be done more often.
As was demonstrated above, there exist threemultilinear algebra rotation formalisms: one withU(1), or complex numbers, for two dimensions, and two others withversors, or quaternions, for three and four dimensions.
In general (even for vectors equipped with a non-Euclidean Minkowskiquadratic form) the rotation of a vector space can be expressed as abivector. This formalism is used ingeometric algebra and, more generally, in theClifford algebra representation of Lie groups.
In the case of a positive-definite Euclidean quadratic form, the doublecovering group of the isometry group is known as theSpin group,. It can be conveniently described in terms of a Clifford algebra. Unit quaternions give the group.
Inspherical geometry, a direct motion[clarification needed] of then-sphere (an example of theelliptic geometry) is the same as a rotation of(n + 1)-dimensional Euclidean space about the origin (SO(n + 1)). For oddn, most of these motions do not have fixed points on then-sphere and, strictly speaking, are not rotationsof the sphere; such motions are sometimes referred to asClifford translations.[citation needed] Rotations about a fixed point in elliptic andhyperbolic geometries are not different from Euclidean ones.[clarification needed]
Affine geometry andprojective geometry have not a distinct notion of rotation.
A generalization of a rotation applies inspecial relativity, where it can be considered to operate on a four-dimensional space,spacetime, spanned by three space dimensions and one of time. In special relativity, this space is calledMinkowski space, and the four-dimensional rotations, calledLorentz transformations, have a physical interpretation. These transformations preserve a quadratic form called thespacetime interval.
If a rotation of Minkowski space is in a space-like plane, then this rotation is the same as a spatial rotation in Euclidean space. By contrast, a rotation in a plane spanned by a space-like dimension and a time-like dimension is ahyperbolic rotation, and if this plane contains the time axis of the reference frame, is called a "Lorentz boost". These transformations demonstrate thepseudo-Euclidean nature of the Minkowski space. Hyperbolic rotations are sometimes described as "squeeze mappings" and frequently appear onMinkowski diagrams that visualize (1 + 1)-dimensional pseudo-Euclidean geometry on planar drawings. The study of relativity deals with theLorentz group generated by the space rotations and hyperbolic rotations.[4]
WhereasSO(3) rotations, in physics and astronomy, correspond to rotations ofcelestial sphere as a2-sphere in the Euclidean 3-space, Lorentz transformations fromSO(3;1)+ induceconformal transformations of the celestial sphere. It is a broader class of the sphere transformations known asMöbius transformations.
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Rotations define important classes ofsymmetry:rotational symmetry is aninvariance with respect to aparticular rotation. Thecircular symmetry is an invariance with respect to all rotation about the fixed axis.
As was stated above, Euclidean rotations are applied torigid body dynamics. Moreover, most of mathematical formalism inphysics (such as thevector calculus) is rotation-invariant; seerotation for more physical aspects. Euclidean rotations and, more generally, Lorentz symmetrydescribed above are thought to besymmetry laws of nature. In contrast, thereflectional symmetry is not a precise symmetry law of nature.
Thecomplex-valued matrices analogous to real orthogonal matrices are theunitary matrices, which represent rotations in complex space. The set of all unitary matrices in a given dimensionn forms aunitary group of degreen; and its subgroup representing proper rotations (those that preserve the orientation of space) is thespecial unitary group of degreen. These complex rotations are important in the context ofspinors. The elements of are used to parametrizethree-dimensional Euclidean rotations (seeabove), as well as respective transformations of thespin (seerepresentation theory of SU(2)).
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