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In mechanics and geometry, the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space, R3. By definition, a rotation about the origin is a linear transformation that preserves the length of vectors, and also preserves the orientation, or handedness, of space. A transformation that preserves length but reverses orientation is sometimes called an improper rotation. Classical mechanics is a branch of physics which studies the deterministic motion of objects. ...
Table of Geometry, from the 1728 Cyclopaedia. ...
A sphere rotating around its axis. ...
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. ...
In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
This article is about vectors that have a particular relation to the spatial coordinates. ...
In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ...
In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). ...
The composition of two rotations is a rotation, and every rotation has a unique inverse which is again a rotation. These properties give the set of all rotations the mathematical structure of a group. Moreover, the rotation group has a natural manifold structure for which the group operations are smooth, so that it is actually a Lie group. The rotation group is often denoted SO(3) for reasons that are explained below. In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...
This picture illustrates how the hours in a clock form a group. ...
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...
In mathematics, a smooth function is one that is infinitely (indefinitely) differentiable, i. ...
In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ...
Properties
Besides just preserving length, rotations also preserve the angles between vectors. This follows from the fact that the standard inner product between two vectors can be written purely in terms of length: An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ...
In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...
 Hence, any length-preserving transformation in R3 will preserve the inner product, and therefore angles as well. It is a quick check that every rotation maps an orthonormal basis of R3 to another orthonormal basis. In mathematics, an orthonormal basis of an inner product space V(i. ...
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 nonabelian. 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. Please refer to group theory for a general description of the topic. ...
Orthogonal matrices Like any linear transformation, a rotation can always be represented by a matrix. Let R be a given rotation. With respect to the standard basis (e1,e2,e3) of R3 the columns of R are given by (Re1,Re2,Re3). Since the standard basis is orthonormal, the columns of R form another orthonormal basis. This orthonormality condition can be expressed in the form In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...
In linear algebra, the standard basis for an -dimensional vector space is the basis obtained by taking the basis vectors where is the vector with a in the th coordinate and elsewhere. ...
In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized. ...
 where RT denotes the transpose of R and I is the 3 × 3 identity matrix. Matrices for which this property holds are called orthogonal matrices. The group of all 3 × 3 orthogonal matrices is denoted O(3). In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...
In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ...
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 addition to preserving length, rotations must also preserve orientation. A matrix will preserve or reverse orientation according to whether the determinant of the matrix is positive or negative. For an orthogonal matrix R, note that det RT = det R implies (det R)2 = 1 so that det R = ±1. The subgroup of orthogonal matrices with determinant +1 is called the special orthogonal group, denoted SO(3). In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...
Thus every rotation can be represented uniquely by an orthogonal matrix with unit determinant. Moreover, since composition of rotations corresponds to matrix multiplication, the rotation group is isomorphic to the special orthogonal group SO(3). This article gives an overview of the various ways to perform matrix multiplication. ...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...
Note that improper rotations correspond to orthogonal matrices with determinant −1. Improper rotations do not form a group since the product of two improper rotations is a proper rotation.
Axis of rotation Every rotation in 3 dimensions fixes a unique 1-dimensional linear subspace of R3 which is called the axis of rotation (this is Euler's rotation theorem). Each rotation will act like 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 clockwise or counterclockwise with respect to this orientation). The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ...
In mathematics, Eulers rotation theorem states that any rotation has an axis. ...
In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...
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. ...
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. ...
In terms of orthogonal matrices, the rotations about the standard coordinate axes through an angle φ are given by
   Given a unit vector 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 In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ...
- 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 unique if 0 < φ < π
- n is arbitrary if φ = 0
- n is unique up to a sign if φ = π (that is, the rotations R(π, ±n) are identical)
Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ...
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 identify (or "glue together") antipodal points on the surface of the ball. After this identification, we arrive at a topological space homeomorphic to the rotation group. In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ...
In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
This word should not be confused with homomorphism. ...
Indeed, the ball with antipodal surface points identified is a smooth manifold, and this manifold is diffeomorphic to the rotation group. It is also diffeomorphic to the real 3-dimensional projective space, so the latter can also serve as a topological model for the rotation group. In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...
In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...
In mathematics, real projective space, or RPn is the projective space of lines in Rn+1. ...
These identifications illustrate that SO(3) is connected 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). In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...
A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...
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 group of SO(3) is cyclic of order 2. In physics applications, the non-triviality of the fundamental group allows for the existence of objects known as spinors, and is an important tool in the development of the spin-statistics theorem. In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na...
Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space, and time. ...
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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 group called Spin(3). The group Spin(3) is isomorphic to the special unitary group SU(2); it is also diffeomorphic to the unit 3-sphere S3 and can be understood as the group of unit quaternions (i.e. those with absolute value 1). The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotations. The resulting map In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint...
In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ...
In mathematics the spinor group or spin group Spin(n) is a particular double cover of the special orthogonal group SO(n, R). ...
In mathematics, the special unitary group of degree n, denoted SU(n), is the group of n×n unitary matrices with unit determinant. ...
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ...
In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...
In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...
It has been suggested that CG artwork be merged into this article or section. ...
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- S3 → SO(3)
is a surjective homomorphism of Lie groups, with kernel {±1}. In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...
In abstract algebra, a homomorphism is a structure-preserving map. ...
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ...
Representations of rotations We have seen that there are a variety of ways to represent rotations: Another method is to specify an arbitrary rotation by a sequence of rotations about some fixed axes. See: In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...
To meet Wikipedias quality standards, this article or section may require cleanup. ...
See charts on SO(3) for further discussion. Euler angles are the classical way of representing rotations in 3-dimensional Euclidean space, named after Leonhard Euler. ...
In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold. ...
Generalizations The rotation group generalizes quite naturally to n-dimensional Euclidean space, Rn. The group of all proper and improper rotations in n dimensions is called the orthogonal group, O(n), and the subgroup of proper rotations is called the special orthogonal group, SO(n). 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. ...
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. ...
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. ...
In special relativity, one works in a 4-dimensional vector space, known as Minkowski space rather than 3-dimensional Euclidean space. Unlike Euclidean space, Minkowski space has an inner product with an indefinite signature. However, one can still define generalized rotations which preserve this inner product. Such generalized rotations are known as Lorentz transformations and the group of all such transformations is called the Lorentz group. The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest...
In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...
The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ...
A Lorentz transformation (LT) is a linear transformation that preserves the spacetime interval between any two events in Minkowski space, while leaving the origin fixed (=rotation of Minkowski space). ...
The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ...
The rotation group SO(3) can be described as a subgroup of E+(3), the Euclidean group of direct isometries of R3. This larger group is the group of all motions of a rigid body: 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 mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ...
In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. ...
In general, the rotation group of an object is the symmetry group within the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group. The symmetry group of an object (e. ...
In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone. ...
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