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Rotation (mathematics)
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===Linear and multilinear algebra formalism===<!-- caution: an internal #-link --> {{main|Rotation matrix}} When one considers motions of the Euclidean space that preserve [[origin (mathematics)|the origin]], the [[point–vector distinction|distinction between points and vectors]], important in pure mathematics, can be erased because there is a canonical [[one-to-one correspondence]] between points and [[position (vector)|position vectors]]. The same is true for geometries other than [[Euclidean geometry|Euclidean]], but whose space is an [[affine space]] with a supplementary [[mathematical structure|structure]]; see [[#In relativity|an example below]]. Alternatively, the vector description of rotations can be understood as a parametrization of geometric rotations [[up to]] their composition with translations. In other words, one vector rotation presents many [[equivalence relation|equivalent]] rotations about ''all'' points in the space. A motion that preserves the origin is the same as a [[linear operator]] on vectors that preserves the same geometric structure but expressed in terms of vectors. For [[Euclidean vector]]s, this expression is their ''magnitude'' ([[Euclidean norm]]). In [[real coordinate space|components]], such operator is expressed with {{math|''n'' × ''n''}} [[orthogonal matrix]] that is multiplied to [[column vector]]s. As it [[#In Euclidean geometry|was already stated]], a (proper) rotation is different from an arbitrary fixed-point motion in its preservation of the orientation of the vector space. Thus, the [[determinant]] of a rotation orthogonal matrix must be 1. The only other possibility for the determinant of an orthogonal matrix is {{num|−1}}, and this result means the transformation is a [[reflection (mathematics)|hyperplane reflection]], a [[point reflection]] (for [[odd number|odd]] {{mvar|n}}), or another kind of [[improper rotation]]. Matrices of all proper rotations form the [[special orthogonal group]]. ====Two dimensions====<!-- caution: an internal #-link --> {{main|Rotations and reflections in two dimensions}} {{see also|Rotation of axes in two dimensions}} In two dimensions, to carry out a rotation using a matrix, the point {{math|(''x'', ''y'')}} to be rotated counterclockwise is written as a column vector, then multiplied by a [[rotation matrix]] calculated from the angle {{math|''θ''}}: :<math> \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}</math>. The coordinates of the point after rotation are {{math|''x′'', ''y′''}}, and the formulae for {{mvar|x′}} and {{mvar|y′}} are :<math>\begin{align} x'&=x\cos\theta-y\sin\theta\\ y'&=x\sin\theta+y\cos\theta. \end{align}</math> The vectors <math> \begin{bmatrix} x \\ y \end{bmatrix} </math> and <math> \begin{bmatrix} x' \\ y' \end{bmatrix} </math> have the same magnitude and are separated by an angle {{mvar|θ}} as expected. {{anchor|Complex numbers}} Points on the {{math|'''R'''<sup>2</sup>}} plane can be also presented as [[complex number]]s: the point {{math|(''x'', ''y'')}} in the plane is represented by the complex number :<math> z = x + iy </math> This can be rotated through an angle {{mvar|θ}} by multiplying it by {{math|''e''<sup>''iθ''</sup>}}, then expanding the product using [[Euler's formula]] as follows: :<math>\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}</math> and equating real and imaginary parts gives the same result as a two-dimensional matrix: :<math>\begin{align} x'&=x\cos\theta-y\sin\theta\\ y'&=x\sin\theta+y\cos\theta. \end{align}</math> Since complex numbers form a [[commutative ring]], vector rotations in two dimensions are commutative, unlike in higher dimensions. They have only one [[degrees of freedom (mechanics)|degree of freedom]], as such rotations are entirely determined by the angle of rotation.<ref>Lounesto 2001, p. 30.</ref> ====Three dimensions==== {{main|Rotation formalisms in three dimensions}} {{see also|Three-dimensional rotation operator}} {{further|3D rotation group}} As in two dimensions, a matrix can be used to rotate a point {{math|(''x'', ''y'', ''z'')}} to a point {{math|(''x′'', ''y′'', ''z′'')}}. The matrix used is a {{gaps|3|×|3}} matrix, : <math>\mathbf{A} = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}</math> This is multiplied by a vector representing the point to give the result :<math> \mathbf{A} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} x' \\ y' \\ z' \end{pmatrix} </math> The set of all appropriate matrices together with the operation of [[matrix multiplication]] is the [[rotation group SO(3)]]. The matrix {{math|'''A'''}} is a member of the three-dimensional [[special orthogonal group]], {{math|SO(3)}}, that is it is an [[orthogonal matrix]] with [[determinant]] 1. That it is an orthogonal matrix means that its rows are a set of orthogonal [[unit vector]]s (so they are an [[orthonormal basis]]) as are its columns, making it simple to spot and check if a matrix is a valid rotation matrix. [[#Euler angles|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. ====Quaternions==== {{Main|Quaternions and spatial rotation}} Unit [[quaternion]]s, or ''[[versor]]s'', 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|date=July 2010}} A versor (also called a ''rotation quaternion'') consists of four real numbers, constrained so the [[normed vector space|norm]] 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: :<math> \mathbf{x'} = \mathbf{qxq}^{-1},</math> where {{math|'''q'''}} is the versor, {{math|'''q'''<sup>−1</sup>}} is its [[multiplicative inverse|inverse]], and {{math|'''x'''}} is the vector treated as a quaternion with zero [[quaternion#Scalar and vector parts|scalar part]]. The quaternion can be related to the rotation vector form of the axis angle rotation by the [[exponential map (Lie theory)|exponential map]] over the quaternions, :<math> \mathbf{q} = e^{\mathbf{v}/2},</math> where {{math|'''v'''}} is the rotation vector treated as a quaternion. A single multiplication by a versor, [[left and right (algebra)|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 two ''different'' unit quaternions. ====Further notes==== More generally, coordinate rotations in any dimension are represented by orthogonal matrices. The set of all orthogonal matrices in {{mvar|n}} dimensions which describe proper rotations (determinant = +1), together with the operation of matrix multiplication, forms the [[special orthogonal group]] {{math|SO(''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 the [[Linear map|linear 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 using [[homogeneous coordinates]]. [[Projective transformation]]s are represented by {{gaps|4|×|4}} matrices. They are not rotation matrices, but a transformation that represents a Euclidean rotation has a {{gaps|3|×|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 where [[numerical stability|numerical instability]] is a concern matrices can be more prone to it, so calculations to restore [[orthonormality]], which are expensive to do for matrices, need to be done more often. ====More alternatives to the matrix formalism==== As was demonstrated above, there exist three [[multilinear algebra]] rotation formalisms: one with [[#Complex numbers|U(1), or complex numbers]], for two dimensions, and two others with [[#Quaternions|versors, or quaternions]], for three and four dimensions. In general (even for vectors equipped with a non-Euclidean Minkowski [[quadratic form]]) the rotation of a vector space can be expressed as a [[bivector]]. This formalism is used in [[geometric algebra]] and, more generally, in the [[Clifford algebra]] representation of Lie groups. In the case of a positive-definite Euclidean quadratic form, the double [[covering group]] of the isometry group <math>\mathrm{SO}(n)</math> is known as the [[Spin group]], <math>\mathrm{Spin}(n)</math>. It can be conveniently described in terms of a Clifford algebra. Unit quaternions give the group <math>\mathrm{Spin}(3) \cong \mathrm{SU}(2)</math>.
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