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Rotation group

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	Topics Rotation group Topic Home In mechanicsClassical mechanics Classical mechanics is used to describe the motion of macroscopic objects, from projectiles to parts of machinery, as well a... and geometryGeometry Geometry arose as the field of knowledge dealing with spatial relationships.... , the rotation group is the groupGroup (mathematics) In mathematics, a group is a set together with a binary operation satisfying certain axioms, detailed below.... of all rotationRotation Rotation is the movement of an object in a circular motion.... s about the origin of 3-dimensional Euclidean spaceEuclidean space Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "plane Euclidean geometry", wh... R³ under the operation of composition. By definition, a rotation about the origin is a linear transformationLinear transformation In mathematics, a linear transformation is a function between two vector spaces that preserves the operations of vector add... that preserves lengthLength Length is the long dimension of any object.... of vectorVector (spatial) In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a... s and preserves orientationOrientation (mathematics) In mathematics, an orientation on a real vector space is a choice of which ordered bases are "positively" oriented and which... (i.e. handedness) of space. A length-preserving transformation which reverses orientation is called an improper rotationImproper rotation In 3D geometry, an improper rotation, also called rotoreflection or rotary reflection is, depending on context, ... . Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity mapIdentity function In mathematics, an identity function, also called identity map or identity transformation, is a function which d... satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a groupGroup (mathematics) In mathematics, a group is a set together with a binary operation satisfying certain axioms, detailed below.... under composition. Moreover, the rotation group has a natural manifoldManifold A manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space, but in... structure for which the group operations are smoothSmooth function In mathematics, a smooth function is one that is infinitely differentiable, i.e., has derivatives of all finite orders:... ; so it is in fact a Lie groupLie group In mathematics, a Lie group is a continuous group, in the sense that the group elements have the topology of a manifold, an... . The rotation group is often denoted SO(3) for reasons explained below. Properties Besides just preserving length, rotations also preserve the angleAngle An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.... s between vectors. This follows from the fact that the standard dot productDot product In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over ... between two vectors u and v can be written purely in terms of length: Hence, any length-preserving transformation in R³ preserves the dot product, and thus the angle between vectors. It is a quick check that every rotation maps an orthonormal basisOrthonormal basis Summary In mathematics, an orthonormal basis of an inner product space V, or in particular of a Hilbert space H, is a set of... of R³ to another orthonormal basis. It should be noted that rotations are often defined as linear transformations that preserve the inner product on R³. By the above argument, this is equivalent to requiring them to preserve length. Another important property of the rotation group is that it is nonabelianNonabelian group In mathematics, a nonabelian group, also sometimes called a non-commutative group, is a group such that there are at ... . That is, the order in which rotations are composed makes a difference. For example, a quarter turn around the positive x-axis followed by a quarter turn around the positive y-axis is a different rotation than the one obtained by first rotating around y and then x. Axis of rotation Every rotation in 3 dimensions fixes a unique 1-dimensional linear subspace of R³Euclidean subspace In linear algebra, an Euclidean subspace is a set of vectors that is closed under addition and scalar multiplication.... which is called the axis of rotation (this is Euler's rotation theoremEuler's rotation theorem Euler's rotation theorem states that, in 3D space, for any two coordinate systems with a common origin, there is a single ei... ). Each rotation acts as a normal 2-dimensional rotation in the plane orthogonal to this axis. Since every 2-dimensional rotation can be represented by an angle φ, an arbitrary 3-dimensional rotation can be specified by an axis of rotation together with an angle of rotation about this axis. (Technically, one needs to specify an orientation for the axis and whether the rotation is taken to be clockwiseClockwise and counterclockwise 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,... or counterclockwise with respect to this orientation). For example, counterclockwise rotation about the positive z-axis by angle φ is given by Given a unit vectorUnit vector In mathematics, a unit vector in a normed vector space is a vector whose length is 1.... n in R³ and an angle φ, let R(φ, n) represent a counterclockwise rotation about the axis through n (with orientation determined by n). Then R(0, n) is the identity transformation for any n R(φ, n) = R(−φ, −n) R(π + φ, n) = R(π − φ, −n) Using these properties one can show that any rotation can be represented by a unique angle φ in the range 0 ≤ φ ≤ π and a unit vector n such that n is arbitrary if φ = 0 n is unique if 0 < φ < π n is unique up toUp to In mathematics, the phrase "up to xxxx" indicates that members of an equivalence class are to be regarded as a single entity... a sign if φ = π (that is, the rotations R(π, ±n) are identical) Topology Consider the solid ball in R³ of radius π (that is, all points of R³ of distance π or less from the origin). Given the above, for every point in this ball there is a rotation, with axis through the point and rotation angle equal to the distance of the point from the origin. The identity rotation corresponds to the point at the center of the ball. Rotation through angles between 0 and -π correspond to the point on the same axis and distance from the origin but on the opposite side of the origin. The one remaining issue is that the two rotations through π and through -π are the same. So we identifyQuotient space Overview In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluin... (or "glue together") antipodal pointAntipodal point In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite it s... s on the surface of the ball. After this identification, we arrive at a topological spaceTopological space Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity.... homeomorphic to the rotation group. Indeed, the ball with antipodal surface points identified is a smooth manifold, and this manifold is diffeomorphicDiffeomorphism In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds.... to the rotation group. It is also diffeomorphic to the real 3-dimensional projective spaceReal projective space In mathematics, real projective space, or RPn is the projective space of lines in Rn+1.... RP³, so the latter can also serve as a topological model for the rotation group. These identifications illustrate that SO(3) is connectedConnectedness Overview In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece".... but not simply connected. As to the latter, in the ball with antipodal surface points identified, consider the path running from the "north pole" straight through the center down to the south pole. This is a closed loop, since the north pole and the south pole are identified. This loop cannot be shrunk to a point, since no matter how you deform the loop, the start and end point have to remain antipodal, or else the loop will "break open". In terms of rotations, this loop represents a continuous sequence of rotations about the z-axis starting and ending at the identity rotation (i.e. a series of rotation through an angle φ where φ runs from 0 to 2π). Surprisingly, if you run through the path twice (so that φ runs from 0 to 4π), i.e. from north pole down to south pole, jump back up to the north pole and run again down to the south pole, you get a closed loop which can be shrunk to a single point: first move the paths continuously to the ball's surface, still connecting north pole to south pole twice. The second half of the path can then be mirrored over to the antipodal side without changing the path at all. Now we have an ordinary closed loop on the surface of the ball, connecting the north pole to itself along a great circle. This circle can be shrunk to the north pole without problems. The same argument can be performed in general, and it shows that the fundamental groupFundamental group Summary In mathematics, the fundamental group is one of the basic concepts of algebraic topology.... of SO(3) is cyclic groupCyclic group In group theory, a cyclic group or monogenous group is a group that can be generated by a single element, in the sense... of order 2. In physicsPhysics Physics , the most fundamental physical science, is concerned with the underlying principles of the natural world.... applications, the non-triviality of the fundamental group allows for the existence of objects known as spinorSpinor In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of mathematical ... s, and is an important tool in the development of the spin-statistics theoremSpin-statistics theorem The spin-statistics theorem in quantum mechanics relates the spin of a particle to the statistics obeyed by that particle.... . The universal cover of SO(3) is a Lie groupLie group In mathematics, a Lie group is a continuous group, in the sense that the group elements have the topology of a manifold, an... called Spin(3). The group Spin(3) is isomorphic to the special unitary groupSpecial unitary group Overview In mathematics, the special unitary group of degree n, denoted SU, is the group of n×n unitary matrices wi... SU(2); it is also diffeomorphic to the unit 3-sphere3-sphere Summary In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere.... S³ and can be understood as the group of unit quaternionsQuaternion In mathematics, quaternions are a non-commutative extension of complex numbers.... (i.e. those with absolute valueAbsolute value In mathematics, the absolute value of a real number is its numerical value without regard to its sign.... 1). The connection between quaternions and rotations, commonly exploited in computer graphicsComputer graphics Computer graphics is the field of visual computing, where one utilizes computers both to generate visual synthetically and... , is explained in quaternions and spatial rotationQuaternions and spatial rotation The algebra of quaternions is a useful mathematical tool for formulating the composition of arbitrary spatial rotations, and estab... s. The map from S³ onto SO(3) that identifies antipodal points of S³ is a surjective homomorphismHomomorphism Summary In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures .... of Lie groups, with kernelKernel (algebra) In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measure... . Topologically, this map is a two-to-one covering mapCovering map In mathematics, specifically topology, a covering map on a topological space X is a continuous surjective map p'' : ... . Lie algebra Since SO(3) is a Lie subgroupLie subgroup In mathematics, a subgroup H of a Lie group G is a Lie subgroup if it is also a submanifold of G.... of the general linear groupGeneral linear group In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with ... GL(3), its Lie algebraLie algebra In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups a... can identified with a Lie subalgebra of gl(3), the algebra of 3×3 matrices with the commutator given by The condition that a matrix A belong to SO(3) is that If is a one-parameter subgroup of SO(3), then differentiating () with respect to t gives and so the Lie algebra so(3) consists of all skew-symmetric 3×3 matrices. Representations of rotations We have seen that there are a variety of ways to represent rotations: as orthogonal matrices with determinant 1, by axis and rotation angle via the unit quaternionsQuaternion In mathematics, quaternions* are a non-commutative extension of complex numbers.... (see quaternions and spatial rotationQuaternions and spatial rotation The algebra of quaternions is a useful mathematical tool for formulating the composition of arbitrary spatial rotations, and estab... s) and the map S³ → SO(3). Another method is to specify an arbitrary rotation by a sequence of rotations about some fixed axes. See: Euler angles See charts on SO(3)Charts on SO(3) In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO, is a naturally occurri... for further discussion. Generalizations The rotation group generalizes quite naturally to n-dimensional Euclidean spaceEuclidean space Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "plane Euclidean geometry", wh... , Rⁿ. The group of all proper and improper rotations in n dimensions is called the orthogonal groupOrthogonal group In mathematics, the orthogonal group of degree n over a field F) is the group of n-by-n orthogonal matrices ... , O(n), and the subgroup of proper rotations is called the special orthogonal group, SO(n). In special relativitySpecial relativity The special theory of relativity was proposed in 1905 by Albert Einstein in his article "On the Electrodynamics of Moving Bo... , one works in a 4-dimensional vector space, known as Minkowski spaceMinkowski space In physics and mathematics, Minkowski space is the mathematical setting in which Einstein's theory of special relativity is ... rather than 3-dimensional Euclidean space. Unlike Euclidean space, Minkowski space has an inner product with an indefinite signatureMetric signature The signature of a metric tensor is the number of positive and negative eigenvalues of the metric.... . However, one can still define generalized rotations which preserve this inner product. Such generalized rotations are known as Lorentz transformationLorentz transformation A Lorentz transformation is a linear transformation that preserves the spacetime interval between any two events in Minkowsk... s and the group of all such transformations is called the Lorentz groupLorentz group In physics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for ... . The rotation group SO(3) can be described as a subgroup of E⁺(3), the Euclidean groupEuclidean group Overview In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Eucli... of direct isometries of R³. This larger group is the group of all motions of a rigid bodyRigid body In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected.... : each of these is a combination of a rotation about an arbitrary axis and a translation along the axis, or put differently, a combination of an element of SO(3) and an arbitrary translation. In general, the rotation group of an object is the symmetry groupSymmetry group The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operatio... within the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiralChirality (mathematics) In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to i... objects it is the same as the full symmetry group. See also