Rotation group - Citizendia

In mechanics and geometry, the rotation group is the group of all rotations about the origin of 3-dimensional Euclidean space R³ under the operation of composition. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation In Mathematics, a composite function represents the application of one function to the results of another

By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation (i. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Length is the long Dimension of any object The length of a thing is the distance between its ends its linear extent as measured from end to end See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which e. handedness) of space. A length-preserving transformation which reverses orientation is called an improper rotation. In 3D Geometry, an improper rotation, also called rotoreflection or rotary reflection is depending on context a Linear transformation or

Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that Owing to the above properties, the set of all rotations is a group under composition. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Moreover, the rotation group has a natural manifold structure for which the group operations are smooth; so it is in fact a Lie group. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group The rotation group is often denoted SO(3) for reasons explained below.

1 Properties
2 Orthogonal and rotation matrices
3 Axis of rotation
4 Topology
5 Lie algebra
6 Representations of rotations
7 Generalizations
8 See also

Properties

Besides just preserving length, rotations also preserve the angles between vectors. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called This follows from the fact that the standard dot product between two vectors u and v can be written purely in terms of length:

$\mathbf{u}\cdot\mathbf{v} = \tfrac{1}{2}\left(\|\mathbf{u}+\mathbf{v}\|^2 - \|\mathbf{u}\|^2 - \|\mathbf{v}\|^2\right).$

Hence, any length-preserving transformation in R³ preserves the dot product, and thus the angle between vectors. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R 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. In Mathematics, a nonabelian group, also sometimes called a non-commutative group, is a group ( G, *) such that there are at least two elements 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.

Orthogonal and rotation matrices

Main articles: Orthogonal matrix and Rotation matrix

Like any linear transformation, a rotation can always be represented by a matrix. In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T In Matrix theory, a rotation matrix is a real Square matrix whose Transpose is its inverse and whose Determinant is +1 In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Let R be a given rotation. With respect to the standard basis $(e 1, e 2, e 3)$ of R³ the columns of R are given by $(R e 1, R e 2, R e 3)$ . In Mathematics, the standard basis (also called natural basis or canonical basis) of the n- dimensional Euclidean space Since the standard basis is orthonormal, the columns of R form another orthonormal basis. In Linear algebra, two vectors in an Inner product space are orthonormal if they are orthogonal and both of unit length This orthonormality condition can be expressed in the form

$R^TR = I\,$

where R^T denotes the transpose of R and I is the 3 × 3 identity matrix. This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main Matrices for which this property holds are called orthogonal matrices. In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T The group of all 3 × 3 orthogonal matrices is denoted O(3).

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. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n For an orthogonal matrix R, note that det R^T = 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 Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of

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). In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

Improper rotations correspond to orthogonal matrices with determinant −1, and they do not form a group because 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). In Linear algebra, an Euclidean subspace (or subspace of R n) is a set of vectors that is closed under addition In Kinematics, Euler's rotation theorem states that in Three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed Each rotation acts as a normal 2-dimensional rotation in the plane orthogonal to this axis. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i 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). 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 and back to the top 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 and back to the top

For example, counterclockwise rotation about the positive z-axis by angle φ is given by

$R_z(\phi) = \begin{bmatrix}\cos\phi & -\sin\phi & 0 \\ \sin\phi & \cos\phi & 0 \\ 0 & 0 & 1\end{bmatrix}$

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). In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length 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 to 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). 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 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. In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying 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 After this identification, we arrive at a topological space homeomorphic to the rotation group. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Topological equivalence redirects here see also Topological equivalence (dynamical systems.

Indeed, the ball with antipodal surface points identified is a smooth manifold, and this manifold is diffeomorphic to the rotation group. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable It is also diffeomorphic to the real 3-dimensional projective space RP³, so the latter can also serve as a topological model for the rotation group. In Mathematics, real projective space, or RP n is the Projective space of lines in R n +1

These identifications illustrate that SO(3) is connected but not simply connected. In Mathematics, connectedness is used to refer to various properties meaning in some sense "all one piece" In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be 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 group of SO(3) is cyclic group of order 2. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an 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. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In Mathematics and Physics, in particular in the theory of the Orthogonal groups spinors are elements of a complex vector space introduced to expand the 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). In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics the spin group Spin( n) is the double cover of the Special orthogonal group SO( n) such that there exists a Short 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. Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n In Mathematics, a 3-sphere is a higher-dimensional analogue of a Sphere. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician e. those with absolute value 1). In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotations. Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data Unit quaternions provide a convenient mathematical notation for representing Orientations and Rotations of objects in three dimensions The map from S³ onto SO(3) that identifies antipodal points of S³ is a surjective homomorphism of Lie groups, with kernel {±1}. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector 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 Topologically, this map is a two-to-one covering map. In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism

Lie algebra

See also: Infinitesimal rotation

Since SO(3) is a Lie subgroup of the general linear group GL(3), its Lie algebra can identified with a Lie subalgebra of gl(3), the algebra of 3×3 matrices with the commutator given by

[A, B] = A B - B A . In Linear algebra, a skew-symmetric (or antisymmetric) matrix is a Square matrix A whose Transpose is also its negative In Mathematics, a Subgroup H of a Lie group G is a Lie subgroup if the inclusion map from H to G is smooth In Mathematics, the general linear group of degree n is the set of n \times n invertible matrices, together with the operation In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie

The condition that a matrix A belong to SO(3) is that

(*)

A A T = I .

If $t\mapsto A(t)$ is a one-parameter subgroup of SO(3), then differentiating (*) with respect to t gives

A'(0) + A'(0) T = 0

and so the Lie algebra so(3) consists of all skew-symmetric 3×3 matrices.

Representations of rotations

Main article: Rotation representation (mathematics)

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 quaternions (see quaternions and spatial rotations) and the map S³ → SO(3). In Geometry a rotation representation expresses the orientation of an object (or coordinate frame relative to a coordinate Reference frame. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician Unit quaternions provide a convenient mathematical notation for representing Orientations and Rotations of objects in three dimensions

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) for further discussion. The Euler angles were developed by Leonhard Euler to describe the orientation of a Rigid body (a body in which the relative position of all its points is constant 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). In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n

In special relativity, one works in a 4-dimensional vector space, known as Minkowski space rather than 3-dimensional Euclidean space. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity Unlike Euclidean space, Minkowski space has an inner product with an indefinite signature. The signature of a Metric tensor (or more generally a nondegenerate Symmetric bilinear form, thought of as Quadratic form) is the number of positive 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. In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to In Physics (and mathematics the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting

The rotation group SO(3) can be described as a subgroup of E⁺(3), the Euclidean group of direct isometries of R³. In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional 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 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. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is For chiral objects it is the same as the full symmetry group. 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

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