TheInfoList

Rotation in
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
is a concept originating in
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

. Any rotation is a
motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon in which an object changes its position (mathematics), position over time. Motion is mathematically described in terms of Displacem ...
of a certain
space Space is the boundless extent in which and events have relative and . In , physical space is often conceived in three s, although modern s usually consider it, with , to be part of a boundless known as . The concept of space is considere ...
that preserves at least one
point Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...
. It can describe, for example, the motion of a
rigid body In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...
around a fixed point. Rotation can have
sign A sign is an object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an object which does not exist at ...
(as in the ): 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 Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
, which have no fixed points, and (hyperplane) reflections, each of them having an entire -dimensional flat of fixed points in a -
dimensional File:Dimension levels.svg, thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum numb ...
space. Mathematically, a rotation is a
map A map is a symbol A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. A ...
. All rotations about a fixed point form a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
under
composition Composition or Compositions may refer to: Arts * Composition (dance), practice and teaching of choreography * Composition (music), an original piece of music and its creation *Composition (visual arts) The term composition means "putting togethe ...
called the rotation group (of a particular space). But in
mechanics Mechanics (Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximat ...

and, more generally, in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, this concept is frequently understood as a
coordinate transformation In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as Euclidean space. ...
(importantly, a transformation of an
orthonormal basisIn linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of u ...
), because for any motion of a body there is an inverse transformation which if applied to the
frame of reference In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...

results in the body being at the same coordinates. For example, in two dimensions rotating a body
clockwise Two-dimensional rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line ...

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 called
active and passive transformation In analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτ ...
s.

Related definitions and terminology

The ''rotation group'' is a
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
of rotations about a fixed point. This (common) fixed point is called the ''
center Center or centre may refer to: Mathematics *Center (geometry) In geometry, a centre (or center) (from Ancient Greek language, Greek ''κέντρον'') of an object is a point in some sense in the middle of the object. According to the speci ...
of rotation'' and is usually identified with the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
. The rotation group is a '' point stabilizer'' in a broader group of (orientation-preserving)
motions 300px, Motion involves a change in position In physics, motion is the phenomenon in which an object changes its position over time. Motion is mathematically described in terms of displacement, distance Distance is a numerical measurement of ...
. For a particular rotation: * The ''axis of rotation'' is a
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...

of its fixed points. They exist only in . * The ''
plane of rotationIn geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...
'' is a
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
that is invariant under the rotation. Unlike the axis, its points are not fixed themselves. The axis (where present) and the plane of a rotation are
orthogonal In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements ''u'' and ''v'' of a vector space with bilinear form ''B'' are orthogonal when . Depending on the bili ...
. A ''representation'' of rotations is a particular formalism, either algebraic or geometric, used to parametrize a rotation map. This meaning is somehow inverse to the meaning in the group theory. Rotations of (affine) spaces of points and of respective
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s are not always clearly distinguished. The former are sometimes referred to as ''affine rotations'' (although the term is misleading), whereas the latter are ''vector rotations''. See the article below for details.

Definitions and representations

In Euclidean geometry

A motion of a
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
is the same as its
isometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
: it leaves the distance between any two points unchanged after the transformation. But a (proper) rotation also has to preserve the orientation structure. The "
improper rotation In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
" term refers to isometries that reverse (flip) the orientation. In the language of
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
the distinction is expressed as ''direct'' vs ''indirect'' isometries in the
Euclidean group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, where the former comprise the
identity component In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. Any direct Euclidean motion can be represented as a composition of a rotation about the fixed point and a translation. There are no non- trivial rotations in one dimension. In
two dimensions 300px, Bi-dimensional Cartesian coordinate system Two-dimensional space (also known as bi-dimensional space) is a geometric setting in which two values (called parameter A parameter (from the Ancient Greek language, Ancient Greek wikt:παρ ...
, only a single
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

is needed to specify a rotation about the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
– the ''angle of rotation'' that specifies an element of the
circle group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
(also known as ). The rotation is acting to rotate an object
counterclockwise Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite sen ...
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
; see
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
for details. Composition of rotations
sums In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

their angles modulo 1
turn Turn may refer to: Arts and entertainment Dance and sports * Turn (dance and gymnastics), rotation of the body * Turn (swimming), reversing direction at the end of a pool * Turn (professional wrestling), a transition between face and heel * Turn, ...
, which implies that all two-dimensional rotations about ''the same'' point commute. Rotations about ''different'' points, in general, do not commute. Any two-dimensional direct motion is either a translation or a rotation; see
Euclidean plane isometry Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of Euclid, an ancient Greek mathematician. It is the name of: Geometry *Euclidean space, the two-dimensional plane and three-dimensional space of Euclidean geometry a ...
for details. Rotations in
three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the ...
differ from those in two dimensions in a number of important ways. Rotations in three dimensions are generally not
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, 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, in
general position In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commu ...
, is not a rotation but a screw operation. Rotations about the origin have three degrees of freedom (see
rotation formalisms in three dimensions In geometry, various formalisms exist to express a rotation (mathematics), rotation in three dimension (vector space), dimensions as a mathematical transformation (geometry), transformation. In physics, this concept is applied to classical mechanic ...
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: *
Euler angles The Euler angles are three angles introduced by Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics ( ...
(pictured at the left). Any rotation about the origin can be represented as the
composition Composition or Compositions may refer to: Arts * Composition (dance), practice and teaching of choreography * Composition (music), an original piece of music and its creation *Composition (visual arts) The term composition means "putting togethe ...
of three rotations defined as the motion obtained by changing one of the Euler angles while leaving the other two constant. They constitute a mixed axes of rotation system because angles are measured with respect to a mix of different
reference frames In physics, a frame of reference (or reference frame) consists of an abstract coordinate system and the set of physical reference points that uniquely fix (locate and orient) the coordinate system and standardize measurements within that frame . ...
, rather than a single frame that is purely external or purely intrinsic. Specifically, the first angle moves the
line of nodes An orbital node is either of the two points where an orbit In celestial mechanics, an orbit is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, o ...
around the external axis ''z'', the second rotates around the line of nodes and the third is an intrinsic rotation (a spin) around an axis fixed in the body that moves. Euler angles are typically denoted as , , , or , , ''ψ''. This presentation is convenient only for rotations about a fixed point. *
Axis–angle representation 150px, The angle and axis unit vector define a rotation, concisely represented by the rotation vector . In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical st ...
(pictured at the right) specifies an angle with the axis about which the rotation takes place. It can be easily visualised. There are two variants to represent it: ** as a pair consisting of the angle and a
unit vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
for the axis, or ** as a
Euclidean vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
obtained by multiplying the angle with this unit vector, called the ''rotation vector'' (although, strictly speaking, it is a
pseudovector In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function (mathematics), function of some vector (geometry), vectors or other geometric shapes, that resembles a vector, and behaves like a vector in ma ...

). * Matrices, versors (quaternions), and other
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

ic things: see the section ''Linear and Multilinear Algebra Formalism'' for details. A general rotation in four dimensions has only one fixed point, the centre of rotation, and no axis of rotation; see rotations 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 each
plane of rotationIn geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...
, through which points in the planes rotate. If these are and then all points not in the planes rotate through an angle between and . Rotations in four dimensions about a fixed point have six degrees of freedom. A four-dimensional direct motion in general position ''is'' a rotation about certain point (as in all even Euclidean dimensions), but screw operations exist also.

Linear and multilinear algebra formalism

When one considers motions of the Euclidean space that preserve the origin, the distinction between points and vectors, important in pure mathematics, can be erased because there is a canonical
one-to-one correspondence In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is paired with exactly on ...
between points and position vectors. The same is true for geometries other than
Euclidean Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of Euclid, an ancient Greek mathematician. It is the name of: Geometry *Euclidean space, the two-dimensional plane and three-dimensional space of Euclidean geometry a ...
, but whose space is an
affine space In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...
with a supplementary
structure A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. ...
; see an example below. Alternatively, the vector description of rotations can be understood as a parametrization of geometric rotations
up to Two mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
their composition with translations. In other words, one vector rotation presents many
equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry *Equivalence class (music) *''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equivalent ...
rotations about ''all'' points in the space. A motion that preserves the origin is the same as a
linear operator In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
on vectors that preserves the same geometric structure but expressed in terms of vectors. For
Euclidean vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s, this expression is their ''magnitude'' (
Euclidean norm Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ...
). In
components Component may refer to: In engineering, science, and technology Generic systems *System A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounde ...
, such operator is expressed with
orthogonal matrix In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is :Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose In linear algebra, t ...
that is multiplied to
column vector In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
s. As it 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of a rotation orthogonal matrix must be 1. The only other possibility for the determinant of an orthogonal matrix is , and this result means the transformation is a hyperplane reflection, a
point reflection In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...
(for odd ), or another kind of
improper rotation In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
. Matrices of all proper rotations form the
special orthogonal group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
.

Two dimensions

In two dimensions, to carry out a rotation using a matrix, the point to be rotated counterclockwise is written as a column vector, then multiplied by a
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation (mathematics), rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \th ...
calculated from the angle : :$\begin x\text{'} \\ y\text{'} \end = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end \begin x \\ y \end$. The coordinates of the point after rotation are , and the formulae for and are :$\begin x\text{'}&=x\cos\theta-y\sin\theta\\ y\text{'}&=x\sin\theta+y\cos\theta. \end$ The vectors $\begin x \\ y \end$ and $\begin x\text{'} \\ y\text{'} \end$ have the same magnitude and are separated by an angle as expected. Points on the plane can be also presented as
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s: the point in the plane is represented by the complex number :$z = x + iy$ This can be rotated through an angle by multiplying it by , then expanding the product using
Euler's formula Euler's formula, named after Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) incl ...

as follows: :$\begin e^ z &= \left(\cos \theta + i \sin \theta\right) \left(x + i y\right) \\ &= x \cos \theta + i y \cos \theta + i x \sin \theta - y \sin \theta \\ &= \left(x \cos \theta - y \sin \theta\right) + i \left( x \sin \theta + y \cos \theta\right) \\ &= x\text{'} + i y\text{'} , \end$ and equating real and imaginary parts gives the same result as a two-dimensional matrix: :$\begin x\text{'}&=x\cos\theta-y\sin\theta\\ y\text{'}&=x\sin\theta+y\cos\theta. \end$ Since complex numbers form a
commutative ring In ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical ana ...
, vector rotations in two dimensions are commutative, unlike in higher dimensions. They have only one
degree of freedom Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
, as such rotations are entirely determined by the angle of rotation.

Three dimensions

As in two dimensions, a matrix can be used to rotate a point to a point . The matrix used is a matrix, : $\mathbf = \begin a & b & c \\ d & e & f \\ g & h & i \end$ This is multiplied by a vector representing the point to give the result :$\mathbf \begin x \\ y \\ z \end = \begin a & b & c \\ d & e & f \\ g & h & i \end \begin x \\ y \\ z \end = \begin x\text{'} \\ y\text{'} \\ z\text{'} \end$ The set of all appropriate matrices together with the operation of
matrix multiplication In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

is the
rotation group SO(3) In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximatel ...
. The matrix is a member of the three-dimensional
special orthogonal group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, , that is it is an
orthogonal matrix In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is :Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose In linear algebra, t ...
with
determinant In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

1. That it is an orthogonal matrix means that its rows are a set of orthogonal
unit vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s (so they are an
orthonormal basisIn linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of u ...
) 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.

Quaternions

Unit quaternions, or ''versors'', 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. 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: :$\mathbf = \mathbf^,$ where is the versor, is its multiplicative inverse, inverse, and 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, :$\mathbf = e^,$ where 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 dimensions which describe proper rotations (determinant = +1), together with the operation of matrix multiplication, forms the
special orthogonal group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. 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 transformations are represented by matrices. They are not rotation matrices, but a transformation that represents a Euclidean rotation has a 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 $\mathrm\left(n\right)$ is known as the Spin group, $\mathrm\left(n\right)$. It can be conveniently described in terms of a Clifford algebra. Unit quaternions give the group $\mathrm\left(3\right) \cong \mathrm\left(2\right)$.

In non-Euclidean geometries

In spherical geometry, a direct motion of the n-sphere, -sphere (an example of the elliptic geometry) is the same as a rotation of -dimensional Euclidean space about the origin (). For odd , most of these motions do not have fixed points on the -sphere and, strictly speaking, are not rotations ''of the sphere''; such motions are sometimes referred to as ''William Kingdon Clifford, Clifford translations''. Rotations about a fixed point in elliptic and hyperbolic space, hyperbolic geometries are not different from Euclidean ones. Affine geometry and projective geometry have not a distinct notion of rotation.

In relativity

One application of this is special relativity, as it can be considered to operate in a four-dimensional space, spacetime, spanned by three space dimensions and one of time. In special relativity this space is linear and the four-dimensional rotations, called Lorentz transformations, have practical physical interpretations. The Minkowski space is not a metric space, and the term ''isometry'' is inapplicable to Lorentz transformation. If a rotation is only in the three space dimensions, i.e. in a plane that is entirely in space, then this rotation is the same as a spatial rotation in three dimensions. But a rotation in a plane spanned by a space dimension and a time dimension is a hyperbolic rotation, a transformation between two different Frame of reference, reference frames, which is sometimes called a "Lorentz boost". These transformations demonstrate the pseudo-Euclidean space, pseudo-Euclidean nature of the Minkowski space. They are sometimes described as ''squeeze mappings'' and frequently appear on Minkowski diagrams which visualize (1 + 1)-dimensional pseudo-Euclidean geometry on planar drawings. The study of relativity is concerned with the Lorentz group generated by the space rotations and hyperbolic rotations.Hestenes 1999, pp. 580–588. Whereas rotations, in physics and astronomy, correspond to rotations of celestial sphere as a sphere, 2-sphere in the Euclidean 3-space, Lorentz transformations from induce conformal map, conformal transformations of the celestial sphere. It is a broader class of the sphere transformations known as Möbius transformations.

Importance

Rotations define important classes of symmetry: rotational symmetry is an invariant (mathematics), invariance with respect to a ''particular rotation''. The circular symmetry is an invariance with respect to all rotation about the fixed axis. As was stated above, Euclidean rotations are applied to rigid body dynamics. Moreover, most of mathematical formalism in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

(such as the vector calculus) is rotation-invariant; see rotation for more physical aspects. Euclidean rotations and, more generally, Lorentz symmetry #In relativity, described above are thought to be symmetry (physics), symmetry laws of nature. In contrast, the Parity (physics), reflectional symmetry is not a precise symmetry law of nature.

Generalizations

The complex number, complex-valued matrices analogous to real orthogonal matrices are the unitary matrix, unitary matrices $\mathrm\left(n\right)$, which represent rotations in complex space. The set of all unitary matrices in a given dimension forms a unitary group $\mathrm\left(n\right)$ of degree ; and its subgroup representing proper rotations (those that preserve the orientation of space) is the special unitary group $\mathrm\left(n\right)$ of degree . These complex rotations are important in the context of spinors. The elements of $\mathrm\left(2\right)$ are used to parametrize ''three''-dimensional Euclidean rotations (see #Quaternions, above), as well as respective transformations of the spin (physics), spin (see representation theory of SU(2)).