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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 approximately 10.7 million ...
and
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 ...

geometry
, the 3D rotation group, often denoted SO(3), is the
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 ...
of all
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 extending from the center an ...

rotation
s about the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * , a Wolverine comic book mini-series published by Marvel Comics in 2002 * , a 1999 ''Buffy the Vampire Slayer'' comic book series * , a major ''Judge Dred ...
of
three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Greek wikt:παρά#Ancient Greek, παρά, ''par ...
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
\R^3 under the operation of
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 ...
. By definition, a rotation about the origin is a transformation that preserves the origin,
Euclidean distance 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). ...
(so it is an
isometry In mathematics, an isometry (or congruence (geometry), congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be Bijection, bijective. "We shall find it convenient to use the wor ...
), and
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building design ...
(i.e. ''handedness'' of space). Every non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
rotation; and the
identity map
identity map
satisfies the definition of a rotation. Owing to the above properties (along composite rotations'
associative property 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). ...
), the set of all rotations is 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. Rotations are not commutative (for example, rotating ''R'' 90° in the x-y plane followed by ''S'' 90° in the y-z plane is not the same as ''S'' followed by ''R''), making it a nonabelian group. Moreover, the rotation group has a natural structure as a
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

manifold
for which the group operations are
smoothly differentiable
smoothly differentiable
; so it is in fact a
Lie group In mathematics, a Lie group (pronounced "Lee") is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operati ...
. It is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British N ...
and has dimension 3. Rotations are
linear transformation 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 ...
s of \R^3 and can therefore be represented by
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
once a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other ba ...
of \R^3 has been chosen. Specifically, if we choose 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 ...
of \R^3, every rotation is described by an orthogonal 3 × 3 matrix (i.e. a 3 × 3 matrix with real entries which, when multiplied by its
transpose 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 an ...

transpose
, results in the
identity matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

identity matrix
) with
determinant 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 ...

determinant
1. The group SO(3) can therefore be identified with the group of these matrices under
matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the nu ...

matrix multiplication
. These matrices are known as "special orthogonal matrices", explaining the notation SO(3). The group SO(3) is used to describe the possible rotational symmetries of an object, as well as the possible orientations of an object in space. Its
representation Representation may refer to: Law and politics *Representation (politics) Political representation is the activity of making citizens "present" in public policy making processes when political actors act in the best interest of citizens. This defin ...
s are important in physics, where they give rise to the
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include the fundamental fermions (quarks, leptons, antiquarks, and a ...
s of integer
spin Spin or spinning may refer to: Businesses * or South Pacific Island Network * , an American scooter-sharing system * , a chain of table tennis lounges Computing * , 's tool for formal verification of distributed software systems * , a Mach-like ...
.


Length and angle

Besides just preserving length, rotations also preserve the
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 ...

angle
s between vectors. This follows from the fact that the standard
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
between two vectors u and v can be written purely in terms of length: :\mathbf\cdot\mathbf = \frac\left(\, \mathbf+\mathbf\, ^2 - \, \mathbf\, ^2 - \, \mathbf\, ^2\right). It follows that every length-preserving transformation in \R^3 preserves the dot product, and thus the angle between vectors. Rotations are often defined as linear transformations that preserve the inner product on \R^3, which is equivalent to requiring them to preserve length. See
classical 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 gen ...
for a treatment of this more general approach, where appears as a special case.


Orthogonal and rotation matrices

Every rotation maps 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 ...
of \R^3 to another orthonormal basis. Like any linear transformation of
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e. the number of vectors) of a Basis (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after ...
vector spaces, a rotation can always be represented by a
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
. Let be a given rotation. With respect to the
standard basis 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 ...
of \R^3 the columns of are given by . Since the standard basis is orthonormal, and since preserves angles and length, the columns of form another orthonormal basis. This orthonormality condition can be expressed in the form :R^\mathsfR = RR^\mathsf = I, where denotes the
transpose 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 an ...

transpose
of and is the
identity matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

identity matrix
. Matrices for which this property holds are called
orthogonal matrices 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 ...
. The group of all orthogonal matrices is denoted , and consists of all proper and improper rotations. In addition to preserving length, proper rotations must also preserve orientation. A matrix will preserve or reverse orientation according to whether the
determinant 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 ...

determinant
of the matrix is positive or negative. For an orthogonal matrix , note that implies , so that . The
subgroup In 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 ...
of orthogonal matrices with determinant is called the ''special
orthogonal 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 gen ...
'', denoted . Thus every rotation can be represented uniquely by an orthogonal matrix with unit determinant. Moreover, since composition of rotations corresponds to
matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the nu ...

matrix multiplication
, the rotation group is
isomorphic 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 ...

isomorphic
to the special orthogonal group .
Improper rotation 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 ...
s correspond to orthogonal matrices with determinant , and they do not form a group because the product of two improper rotations is a proper rotation.


Group structure

The rotation group is 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
function composition 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 ...
(or equivalently the product of linear transformations). It is a
subgroup In 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 ...
of the
general linear 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 ge ...
consisting of all invertible linear transformations of the real 3-space \R^3. Furthermore, the rotation group 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''. The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem.


Axis of rotation

Every nontrivial proper rotation in 3 dimensions fixes a unique 1-dimensional
linear subspace 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 \R^3 which is called the ''axis of rotation'' (this is
Euler's rotation theorem In geometry, 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, is equivalent to a single rotation about some axis that runs through the fixed poi ...
). Each such rotation acts as an ordinary 2-dimensional rotation in the plane
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 ...
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 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 ...
about this axis. (Technically, one needs to specify an orientation for the axis and whether the rotation is taken to be
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 ...
or
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 ...
with respect to this orientation). For example, counterclockwise rotation about the positive ''z''-axis by angle ''φ'' is given by :R_z(\phi) = \begin\cos\phi & -\sin\phi & 0 \\ \sin\phi & \cos\phi & 0 \\ 0 & 0 & 1\end. Given 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 ...
n in \R^3 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''( + ''φ'', n) = ''R''( − ''φ'', −n). Using these properties one can show that any rotation can be represented by a unique angle ''φ'' in the range 0 ≤ φ ≤ and a unit vector n such that * n is arbitrary if ''φ'' = 0 * n is unique if 0 < ''φ'' < * n is unique up to a
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 ...
if ''φ'' = (that is, the rotations ''R''(, ±n) are identical). In the next section, this representation of rotations is used to identify SO(3) topologically with three-dimensional real projective space.


Topology

The Lie group SO(3) is
diffeomorphic 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). ...
to the
real projective spaceIn mathematics, real projective space, or RP''n'' or \mathbb_n(\mathbb), is the topological space of lines passing through the origin 0 in R''n''+1. It is a compact space, compact, smooth manifold of dimension ''n'', and is a special case Gr(1, R''n' ...
\mathbb^3(\R). Consider the solid ball in \R^3 of radius (that is, all points of \R^3 of distance or less from the origin). Given the above, for every point in this ball there is a rotation, with axis through the point and the origin, 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 − 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 and through − are the same. So we identify (or "glue together")
antipodal point 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 on the surface of the ball. After this identification, we arrive at a
topological space 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 gener ...
homeomorphic In the mathematical Mathematics (from 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 populat ...
to the rotation group. Indeed, the ball with antipodal surface points identified is a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
, and this manifold is
diffeomorphic 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). ...
to the rotation group. It is also diffeomorphic to the real 3-dimensional projective space \mathbb^3(\R), so the latter can also serve as a topological model for the rotation group. These identifications illustrate that SO(3) is connected but not
simply connected In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
. As to the latter, in the ball with antipodal surface points identified, consider the path running from the "north pole" straight through the interior 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 (by example) at identity (center of ball), through south pole, jump to north pole and ending again at the identity rotation (i.e. a series of rotation through an angle ''φ'' where ''φ'' runs from 0 to 2). Surprisingly, if you run through the path twice, i.e., run from north pole down to south pole, jump back to the north pole (using the fact that north and south poles are identified), and then again run from north pole down to south pole, so that ''φ'' runs from 0 to 4, 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 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
plate trick 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 ...
and similar tricks demonstrate this practically. The same argument can be performed in general, and it shows that the
fundamental group In the 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 ...

fundamental group
of SO(3) is a
cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

cyclic group
of order 2 (a fundamental group with two elements). 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 ...

physics
applications, the non-triviality (more than one element) of the fundamental group allows for the existence of objects known as
spinor In geometry and physics, spinors are elements of a complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most re ...
s, and is an important tool in the development of the
spin–statistics theorem In quantum mechanics, the spin–statistics theorem relates the spin (physics), intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ''ħ'', all ...
. The
universal cover 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). ...
of SO(3) is a
Lie group In mathematics, a Lie group (pronounced "Lee") is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operati ...
called Spin(3). The group Spin(3) is isomorphic to the
special unitary group In mathematics, the special unitary group of degree , denoted , is the Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures ...
SU(2); it is also diffeomorphic to the unit
3-sphere 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''3 and can be understood as the group of
versor 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 (
quaternion 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 ...

quaternion
s with
absolute value 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 ...

absolute value
1). The connection between quaternions and rotations, commonly exploited in
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great dea ...

computer graphics
, is explained in
quaternions and spatial rotation Unit Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or bea ...
s. The map from ''S''3 onto SO(3) that identifies antipodal points of ''S''3 is a
surjective 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 ...
homomorphism 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 analysis, analysis. I ...
of Lie groups, with
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
. Topologically, this map is a two-to-one
covering map 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). ...

covering map
. (See the
plate trick 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 ...
.)


Connection between SO(3) and SU(2)

In this section, we give two different constructions of a two-to-one and
surjective 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 ...
homomorphism 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 analysis, analysis. I ...
of SU(2) onto SO(3).


Using quaternions of unit norm

The group is
isomorphic 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 ...
to the
quaternion 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 ...

quaternion
s of unit norm via a map given by :q = a\mathbf + b\mathbf + c\mathbf + d\mathbf = \alpha + j\beta \leftrightarrow \begin\alpha & -\overline \beta \\ \beta & \overline \alpha\end = U restricted to a^2+b^2+c^2+d^2=, \alpha, ^2 +, \beta, ^2 = 1 where\quad q \in \mathbb,\quad a,b,c,d \in \R,\quad U \in \operatorname(2) and \alpha=a+ib \in\mathbb, \beta = c+id \in \mathbb. Let us now identify \R^3 with the span of \mathbf,\mathbf,\mathbf. One can then verify that if v is in \R^3 and q is a unit quaternion, then :qvq^\in \R^3. Furthermore, the map v\mapsto qvq^ is a rotation of \R^3. Moreover, (-q)v(-q)^ is the same as qvq^. This means that there is a homomorphism from quaternions of unit norm to the 3D rotation group . One can work this homomorphism out explicitly: the unit quaternion, , with :\begin q &= w + \mathbfx + \mathbfy + \mathbfz , \\ 1 &= w^2 + x^2 + y^2 + z^2 , \end is mapped to the rotation matrix : Q = \begin 1 - 2 y^2 - 2 z^2 & 2 x y - 2 z w & 2 x z + 2 y w \\ 2 x y + 2 z w & 1 - 2 x^2 - 2 z^2 & 2 y z - 2 x w \\ 2 x z - 2 y w & 2 y z + 2 x w & 1 - 2 x^2 - 2 y^2 \end. This is a rotation around the vector by an angle , where and . The proper sign for is implied, once the signs of the axis components are fixed. The is apparent since both and map to the same .


Using Möbius transformations

The general reference for this section is . The points on the sphere :\mathbf = \left \ can, barring the north pole , be put into one-to-one bijection with points on the plane defined by , see figure. The map is called
stereographic projection 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 o ...

stereographic projection
. Let the coordinates on be . The line passing through and can be parametrized as :L(t) = N + t(N - P) = \left(0,0,\frac\right) + t \left ( \left(0,0,\frac\right) - (x, y, z) \right ), \quad t\in \R. Demanding that the of L(t_0) equals , one finds :t_0 = \frac1. We have L(t_0)=(\xi,\eta,-1/2). Hence the map :\begin S:\mathbf \to M \\ P = (x,y,z) \longmapsto P'= (\xi, \eta) = \left(\frac, \frac\right) \equiv \zeta = \xi + i\eta \end where, for later convenience, the plane is identified with the complex plane \C. For the inverse, write as :L = N + s(P'-N) = \left(0,0,\frac\right) + s\left( \left(\xi, \eta, -\frac\right) - \left(0,0,\frac\right)\right), and demand to find and thus :\begin S^:M \to \mathbf \\ P'= (\xi, \eta) \longmapsto P = (x,y,z) = \left(\frac, \frac, \frac\right) \end If is a rotation, then it will take points on to points on by its standard action on the embedding space \R^3. By composing this action with one obtains a transformation of , :\zeta=P' \longmapsto P \longmapsto \Pi_s(g)P = gP \longmapsto S(gP) \equiv \Pi_u(g)\zeta = \zeta'. Thus is a transformation of \C associated to the transformation of \R^3. It turns out that represented in this way by can be expressed as a matrix (where the notation is recycled to use the same name for the matrix as for the transformation of \C it represents). To identify this matrix, consider first a rotation about the through an angle , :\begin x' &= x\cos \phi - y \sin \phi,\\ y' &= x\sin \phi + y \cos \phi,\\ z' &= z. \end Hence :\zeta' = \frac = \frac = e^\zeta = \frac, which, unsurprisingly, is a rotation in the complex plane. In an analogous way, if is a rotation about the through and angle , then :w' = e^w, \quad w = \frac, which, after a little algebra, becomes :\zeta' = \frac. These two rotations, g_, g_, thus correspond to
bilinear transform The bilinear transform (also known as Tustin's method) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa. The bilinear transform is a special c ...
s of , namely, they are examples of
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex number, complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers sa ...
s. A general Möbius transformation is given by :\zeta' = \frac, \quad \alpha\delta - \beta\gamma \ne 0. The rotations, g_, g_ generate all of and the composition rules of the Möbius transformations show that any composition of g_, g_ translates to the corresponding composition of Möbius transformations. The Möbius transformations can be represented by matrices :\begin\alpha & \beta\\ \gamma & \delta\end, \qquad \alpha\delta - \beta\gamma = 1, since a common factor of cancels. For the same reason, the matrix is ''not'' uniquely defined since multiplication by has no effect on either the determinant or the Möbius transformation. The composition law of Möbius transformations follow that of the corresponding matrices. The conclusion is that each Möbius transformation corresponds to two matrices . Using this correspondence one may write :\begin \Pi_u(g_\phi) &= \Pi_u\left begin \cos \phi & -\sin \phi & 0\\ \sin \phi & \cos \phi & 0\\ 0 & 0 & 1 \end\right= \pm \begin e^ & 0\\ 0 & e^ \end,\\ \Pi_u(g_\theta) &= \Pi_u\left begin 1 & 0 & 0\\ 0 & \cos \theta & -\sin \theta\\ 0 & \sin \theta & \cos \theta \end\right= \pm \begin \cos\frac & i\sin\frac\\ i\sin\frac & \cos\frac \end. \end These matrices are unitary and thus . In terms of
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 ( ...
This is effected by first applying a rotation g_ through about the to take the to the line , the intersection between the planes and , the latter being the rotated . Then rotate with g_ through about to obtain the new from the old one, and finally rotate by g_ through an angle about the ''new'' , where is the angle between and the new . In the equation, g_ and g_ are expressed in a temporary ''rotated basis'' at each step, which is seen from their simple form. To transform these back to the original basis, observe that \mathbf_ = g_g_g_^. Here boldface means that the rotation is expressed in the ''original'' basis. Likewise, :\mathbf_ = g_g_g_^ g_ g_ \left g_g_g_^ g_ \right . Thus :\mathbf_\mathbf_\mathbf_ = g_g_g_^ g_g_ \left _ g_ g_^ g_ \right * g_g_g_^* g_ = g_g_g_. one finds for a general rotation one has For the converse, consider a general matrix :\pm\Pi_u(g_) = \pm\begin \alpha & \beta\\ -\overline & \overline \end \in \operatorname(2). Make the substitutions :\begin \cos\frac &= , \alpha, , & \sin\frac &= , \beta, , & (0 \le \theta \le \pi),\\ \frac &= \arg \alpha, & \frac &= \arg \beta. & \end With the substitutions, assumes the form of the right hand side ( RHS) of , which corresponds under to a matrix on the form of the RHS of with the same . In terms of the complex parameters , :g_ = \begin \frac\left( \alpha^2 - \beta^2 + \overline - \overline\right) & \frac\left(-\alpha^2 - \beta^2 + \overline + \overline\right) & -\alpha\beta - \overline\overline\\ \frac\left(\alpha^2 - \beta^2 - \overline + \overline\right) & \frac\left(\alpha^2 + \beta^2 + \overline + \overline\right) & -i\left(+\alpha\beta - \overline\overline\right)\\ \alpha\overline + \overline\beta & i\left(-\alpha\overline + \overline\beta\right) & \alpha\overline - \beta\overline \end. To verify this, substitute for the elements of the matrix on the RHS of . After some manipulation, the matrix assumes the form of the RHS of . It is clear from the explicit form in terms of Euler angles that the map : \begin p:\operatorname(2) \to \operatorname(3)\\ \Pi_u(\pm g_) \mapsto g_\end just described is a smooth, and surjective
group homomorphism 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 ...

group homomorphism
. It is hence an explicit description of the universal covering map of from the
universal covering group In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map is a continuous (topology), continuous group homomorphism. The map ''p'' is called the c ...
.


Lie algebra

Associated with every Lie group is its
Lie algebra 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 ...
, a linear space of the same dimension as the Lie group, closed under a bilinear alternating product called the
Lie bracket In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightar ...
. The Lie algebra of is denoted by \mathfrak(3) and consists of all skew-symmetric matrices. This may be seen by differentiating the orthogonality condition, .For an alternative derivation of \mathfrak(3), see
Classical 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 gen ...
.
The Lie bracket of two elements of \mathfrak(3) is, as for the Lie algebra of every matrix group, given by the matrix commutator, , which is again a skew-symmetric matrix. The Lie algebra bracket captures the essence of the Lie group product in a sense made precise by the Baker–Campbell–Hausdorff formula. The elements of \mathfrak(3) are the "infinitesimal generators" of rotations, i.e. they are the elements of the tangent space of the manifold SO(3) at the identity element. If R(\phi, \boldsymbol) denotes a counterclockwise rotation with angle φ about the axis specified by the unit vector \boldsymbol, then :\forall \boldsymbol \in \R^3: \qquad \left. \frac \_ R(\phi,\boldsymbol) \boldsymbol = \boldsymbol \times \boldsymbol. This can be used to show that the Lie algebra \mathfrak(3) (with commutator) is isomorphic to the Lie algebra \R^3 (with cross product). Under this isomorphism, an Axis–angle representation#Rotation vector, Euler vector \boldsymbol\in\R^3 corresponds to the linear map \widetilde defined by \widetilde(\boldsymbol) = \boldsymbol\times\boldsymbol. In more detail, a most often suitable basis for \mathfrak(3) as a vector space is : \boldsymbol_x = \begin0&0&0\\0&0&-1\\0&1&0\end, \quad \boldsymbol_y = \begin0&0&1\\0&0&0\\-1&0&0\end, \quad \boldsymbol_z = \begin0&-1&0\\1&0&0\\0&0&0\end. The commutation relations of these basis elements are, : [\boldsymbol_x, \boldsymbol_y] = \boldsymbol_z, \quad [\boldsymbol_z, \boldsymbol_x] = \boldsymbol_y, \quad [\boldsymbol_y, \boldsymbol_z] = \boldsymbol_x which agree with the relations of the three standard basis, standard unit vectors of \R^3 under the cross product. As announced above, one can identify any matrix in this Lie algebra with an Euler vector \boldsymbol = (x,y,z) \in\R^3, :\widehat =\boldsymbol\cdot \boldsymbol = x \boldsymbol_x + y \boldsymbol_y + z \boldsymbol_z =\begin0&-z&y\\z&0&-x\\-y&x&0\end \in \mathfrak(3). This identification is sometimes called the hat-map. Under this identification, the \mathfrak(3) bracket corresponds in \R^3 to the cross product, :\left [\widehat,\widehat \right ] = \widehat. The matrix identified with a vector \boldsymbol has the property that :\widehat\boldsymbol = \boldsymbol \times \boldsymbol, where the left-hand side we have ordinary matrix multiplication. This implies \boldsymbol is in the null space of the skew-symmetric matrix with which it is identified, because \boldsymbol \times \boldsymbol = \boldsymbol.


A note on Lie algebras

In Lie algebra representations, the group SO(3) is compact and simple of rank 1, and so it has a single independent Casimir element, a quadratic invariant function of the three generators which commutes with all of them. The Killing form for the rotation group is just the Kronecker delta, and so this Casimir invariant is simply the sum of the squares of the generators, \boldsymbol_x, \boldsymbol_y, \boldsymbol_z, of the algebra : [\boldsymbol_x, \boldsymbol_y] = \boldsymbol_z, \quad [\boldsymbol_z, \boldsymbol_x] = \boldsymbol_y, \quad [\boldsymbol_y, \boldsymbol_z] = \boldsymbol_x. That is, the Casimir invariant is given by :\boldsymbol^2\equiv \boldsymbol\cdot \boldsymbol =\boldsymbol_x^2+\boldsymbol_y^2+\boldsymbol_z^2 \propto \boldsymbol. For unitary irreducible Lie algebra representation, representations , the eigenvalues of this invariant are real and discrete, and characterize each representation, which is finite dimensional, of dimensionality 2j+1. That is, the eigenvalues of this Casimir operator are :\boldsymbol^2=- j(j+1) \boldsymbol_ where j is integer or half-integer, and referred to as the
spin Spin or spinning may refer to: Businesses * or South Pacific Island Network * , an American scooter-sharing system * , a chain of table tennis lounges Computing * , 's tool for formal verification of distributed software systems * , a Mach-like ...
or angular momentum. So, above, the 3 × 3 generators, ''L'', displayed act on the triplet (spin 1) representation, while the 2 × 2 ones, ''t'', act on the Spinor, doublet (spin-½) representation. By taking Kronecker products of with itself repeatedly, one may construct all higher irreducible representations . That is, the resulting generators for higher spin systems in three spatial dimensions, for arbitrarily large , can be calculated using these spin operators and ladder operators. For every unitary irreducible representations there is an equivalent one, . All infinite-dimensional irreducible representations must be non-unitary, since the group is compact. In quantum mechanics, the Casimir invariant is the "angular-momentum-squared" operator; integer values of spin characterize boson, bosonic representations, while half-integer values fermion, fermionic representations. The Skew-Hermitian matrix, antihermitian matrices used above are utilized as spin operators, after they are multiplied by , so they are now Hermitian matrix, hermitian (like the Pauli matrices). Thus, in this language, : [\boldsymbol_x, \boldsymbol_y] = i\boldsymbol_z, \quad [\boldsymbol_z, \boldsymbol_x] = i\boldsymbol_y, \quad [\boldsymbol_y, \boldsymbol_z] = i\boldsymbol_x. and hence :\boldsymbol^2= j(j+1) \boldsymbol_. Explicit expressions for these are, :\begin \left (\boldsymbol_z^\right )_ &= (j+1-a)\delta_\\ \left (\boldsymbol_x^\right )_ &=\frac \left (\delta_+\delta_ \right ) \sqrt\\ \left (\boldsymbol_y^\right )_ &=\frac \left (\delta_-\delta_ \right ) \sqrt\\ \end where j is arbitrary and 1 \le a, b \le 2j+1. For example, the resulting spin matrices for spin 1 (j = 1) are: :\begin \boldsymbol_x &= \frac \begin 0 &1 &0\\ 1 &0 &1\\ 0 &1 &0 \end \\ \boldsymbol_y &= \frac \begin 0 &-i &0\\ i &0 &-i\\ 0 &i &0 \end \\ \boldsymbol_z &= \begin 1 &0 &0\\ 0 &0 &0\\ 0 &0 &-1 \end \end Note, however, how these are in an equivalent, but different basis, the Spherical basis#Change of basis matrix, spherical basis, than the above i\boldsymbol in the Cartesian basis.Specifically, \boldsymbol \boldsymbol_\boldsymbol^\dagger=i\boldsymbol_\alpha for :\boldsymbol= \frac \begin -1 & 0 & 1 \\ -i & 0 &- i \\ 0 & \sqrt & 0\end. For spin (j=\tfrac): :\begin \boldsymbol_x &= \frac \begin 0 &\sqrt &0 &0\\ \sqrt &0 &2 &0\\ 0 &2 &0 &\sqrt\\ 0 &0 &\sqrt &0 \end \\ \boldsymbol_y &= \frac \begin 0 &-i\sqrt &0 &0\\ i\sqrt &0 &-2i &0\\ 0 &2i &0 &-i\sqrt\\ 0 &0 &i\sqrt &0 \end \\ \boldsymbol_z &=\frac \begin 3 &0 &0 &0\\ 0 &1 &0 &0\\ 0 &0 &-1 &0\\ 0 &0 &0 &-3 \end. \end For spin (j = \tfrac): :\begin \boldsymbol_x &= \frac \begin 0 &\sqrt &0 &0 &0 &0 \\ \sqrt &0 &2\sqrt &0 &0 &0 \\ 0 &2\sqrt &0 &3 &0 &0 \\ 0 &0 &3 &0 &2\sqrt &0 \\ 0 &0 &0 &2\sqrt &0 &\sqrt \\ 0 &0 &0 &0 &\sqrt &0 \end \\ \boldsymbol_y &= \frac \begin 0 &-i\sqrt &0 &0 &0 &0 \\ i\sqrt &0 &-2i\sqrt &0 &0 &0 \\ 0 &2i\sqrt &0 &-3i &0 &0 \\ 0 &0 &3i &0 &-2i\sqrt &0 \\ 0 &0 &0 &2i\sqrt &0 &-i\sqrt \\ 0 &0 &0 &0 &i\sqrt &0 \end \\ \boldsymbol_z &= \frac \begin 5 &0 &0 &0 &0 &0 \\ 0 &3 &0 &0 &0 &0 \\ 0 &0 &1 &0 &0 &0 \\ 0 &0 &0 &-1 &0 &0 \\ 0 &0 &0 &0 &-3 &0 \\ 0 &0 &0 &0 &0 &-5 \end. \end


Isomorphism with 𝖘𝖚(2)

The Lie algebras \mathfrak(3) and \mathfrak(2) are isomorphic. One basis for \mathfrak(2) is given by :\boldsymbol_1 = \frac\begin0 & -i\\ -i & 0\end, \quad \boldsymbol_2 = \frac\begin0 & -1\\ 1 & 0\end, \quad \boldsymbol_3 = \frac\begin-i & 0\\ 0 & i\end. These are related to the Pauli matrix, Pauli matrices by :\boldsymbol_i \longleftrightarrow \frac \sigma_i. The Pauli matrices abide by the physicists' convention for Lie algebras. In that convention, Lie algebra elements are multiplied by , the exponential map (below) is defined with an extra factor of in the exponent and the structure constants remain the same, but the ''definition'' of them acquires a factor of . Likewise, commutation relations acquire a factor of . The commutation relations for the \boldsymbol_i are :[\boldsymbol_i, \boldsymbol_j] = \varepsilon_\boldsymbol_k, where is the totally anti-symmetric symbol with . The isomorphism between \mathfrak(3) and \mathfrak(2) can be set up in several ways. For later convenience, \mathfrak(3) and \mathfrak(2) are identified by mapping :\boldsymbol_x \longleftrightarrow \boldsymbol_1, \quad \boldsymbol_y \longleftrightarrow \boldsymbol_2, \quad \boldsymbol_z \longleftrightarrow \boldsymbol_3, and extending by linearity.


Exponential map

The exponential map for , is, since is a matrix Lie group, defined using the standard matrix exponential series, :\begin \exp : \mathfrak(3) \to \operatorname(3) \\ A \mapsto e^A = \sum_^\infty \frac A^k = I + A + \tfrac A^2 + \cdots.\end For any skew-symmetric matrix , is always in . The proof uses the elementary properties of the matrix exponential :\left(e^A\right)^\textsf e^A = e^ e^A = e^ = e^ = e^ = e^A \left(e^A\right)^\textsf = e^0 = I. since the matrices and commute, this can be easily proven with the skew-symmetric matrix condition. This is not enough to show that is the corresponding Lie algebra for , and shall be proven separately. The level of difficulty of proof depends on how a matrix group Lie algebra is defined. defines the Lie algebra as the set of matrices :\left\, in which case it is trivial. uses for a definition derivatives of smooth curve segments in through the identity taken at the identity, in which case it is harder. For a fixed , is a one-parameter subgroup along a geodesic in . That this gives a one-parameter subgroup follows directly from properties of the exponential map. The exponential map provides a diffeomorphism between a neighborhood of the origin in the and a neighborhood of the identity in the . For a proof, see Closed subgroup theorem. The exponential map is
surjective 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 ...
. This follows from the fact that every , since every rotation leaves an axis fixed (
Euler's rotation theorem In geometry, 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, is equivalent to a single rotation about some axis that runs through the fixed poi ...
), and is conjugate to a block diagonal matrix of the form :D = \begin\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0 & 0 & 1\end = e^, such that , and that :Be^B^ = e^, together with the fact that is closed under the adjoint representation, adjoint action of , meaning that . Thus, e.g., it is easy to check the popular identity :e^ e^ e^ = e^. As shown above, every element is associated with a vector , where is a unit magnitude vector. Since is in the null space of , if one now rotates to a new basis, through some other orthogonal matrix , with as the axis, the final column and row of the rotation matrix in the new basis will be zero. Thus, we know in advance from the formula for the exponential that must leave fixed. It is mathematically impossible to supply a straightforward formula for such a basis as a function of , because its existence would violate the hairy ball theorem; but direct exponentiation is possible, and Axis–angle representation#Exponential map from 𝖘𝖔(3) to SO(3), yields :\begin \exp(\tilde) &= \exp(\theta(\boldsymbol)) = \exp\left(\theta \begin 0 & -z & y \\ z & 0 & -x \\ -y & x & 0 \end\right)\\[4pt] &= \boldsymbol + 2cs(\boldsymbol) + 2s^2 (\boldsymbol)^2 \\[4pt] &= \begin 2 \left(x^2 - 1\right) s^2 + 1 & 2 x y s^2 - 2 z c s & 2 x z s^2 + 2 y c s \\ 2 x y s^2 + 2 z c s & 2 \left(y^2 - 1\right) s^2 + 1 & 2 y z s^2 - 2 x c s \\ 2 x z s^2 - 2 y c s & 2 y z s^2 + 2 x c s & 2 \left(z^2 - 1\right) s^2 + 1 \end, \end where c = \cos\frac and s = \sin\frac. This is recognized as a matrix for a rotation around axis by the angle : cf. Rodrigues' rotation formula.


Logarithm map

Given , let A = \tfrac \left(R - R^\mathrm\right) denote the antisymmetric part and let \, A\, = \sqrt. Then, the logarithm of is given by :\log R = \fracA. This is manifest by inspection of the mixed symmetry form of Rodrigues' formula, :e^X = I + \fracX + 2\fracX^2, \quad \theta = \, X\, , where the first and last term on the right-hand side are symmetric.


Baker–Campbell–Hausdorff formula

Suppose and in the Lie algebra are given. Their exponentials, and , are rotation matrices, which can be multiplied. Since the exponential map is a surjection, for some in the Lie algebra, , and one may tentatively write : Z = C(X, Y), for some expression in and . When and commute, then , mimicking the behavior of complex exponentiation. The general case is given by the more elaborate BCH formula, a series expansion of nested Lie brackets. For matrices, the Lie bracket is the same operation as the commutator, which monitors lack of commutativity in multiplication. This general expansion unfolds as follows,For a full proof, see Derivative of the exponential map. Issues of convergence of this series to the correct element of the Lie algebra are here swept under the carpet. Convergence is guaranteed when \, X\, + \, Y\, < \log 2 and \, Z\, < \log 2. The series may still converge even if these conditions aren't fulfilled. A solution always exists since is onto in the cases under consideration. :Z = C(X, Y) = X + Y + \frac [X, Y] + \tfrac [X, [X, Y - \frac [Y, [X, Y + \cdots. The infinite expansion in the BCH formula for reduces to a compact form, :Z = \alpha X + \beta Y + \gamma[X, Y], for suitable trigonometric function coefficients . The are given by :\alpha = \phi \cot\left(\frac\right) \gamma, \qquad \beta = \theta \cot\left(\frac\right)\gamma, \qquad \gamma = \frac\frac, where :\begin c &= \frac\sin\theta\sin\phi - 2\sin^2\frac\sin^2\frac\cos(\angle(u, v)),\quad a = c \cot\left(\frac\right), \quad b = c \cot\left(\frac\right), \\ d &= \sqrt, \end for :\theta = \frac\, X\, ,\quad \phi = \frac\, Y\, ,\quad \angle(u, v) = \cos^\frac. The inner product is the Hilbert–Schmidt inner product and the norm is the associated norm. Under the hat-isomorphism, :\langle u, v\rangle = \frac\operatornameX^\mathrmY, which explains the factors for and . This drops out in the expression for the angle. It is worthwhile to write this composite rotation generator as :\alpha X + \beta Y + \gamma[X, Y]\underset X + Y + \frac [X, Y] + \frac [X, [X, Y - \frac [Y, [X, Y + \cdots, to emphasize that this is a ''Lie algebra identity''. The above identity holds for all faithful representations of . The kernel (algebra), kernel of a Lie algebra homomorphism is an ideal (Lie algebra), ideal, but , being simple (abstract algebra), simple, has no nontrivial ideals and all nontrivial representations are hence faithful. It holds in particular in the doublet or spinor representation. The same explicit formula thus follows in a simpler way through Pauli matrices, cf. the Pauli matrices#Exponential of a Pauli vector, 2×2 derivation for SU(2). The Pauli matrices#Exponential of a Pauli vector, Pauli vector version of the same BCH formula is the somewhat simpler group composition law of SU(2), : e^e^ = \exp\left( \frac \sin a' \sin b' \left(\left(i\cot b'\hat + i \cot a' \hat\right)\cdot\vec + \frac \left[i \hat \cdot \vec, i \hat \cdot \vec\right]\right) \right), where :\cos c' = \cos a' \cos b' - \hat \cdot\hat \sin a' \sin b', the spherical law of cosines. (Note are angles, not the above.) This is manifestly of the same format as above, :Z = \alpha' X + \beta' Y + \gamma' [X, Y], with :X = i a'\hat \cdot \mathbf, \quad Y = ib'\hat \cdot \mathbf \in \mathfrak(2), so that :\begin \alpha' &= \frac\frac\cos b' \\ \beta' &= \frac\frac\cos a' \\ \gamma' &= \frac\frac\frac\frac. \end For uniform normalization of the generators in the Lie algebra involved, express the Pauli matrices in terms of -matrices, , so that :a' \mapsto -\frac, \quad b' \mapsto - \frac. To verify then these are the same coefficients as above, compute the ratios of the coefficients, :\begin \frac &= \theta\cot\frac &= \frac\\ \frac &= \phi\cot\frac &= \frac. \end Finally, given the identity . For the general case, one might use Ref. The
quaternion 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 ...

quaternion
formulation of the composition of two rotations RB and RA also yields directly the Axis of rotation, rotation axis and angle of the composite rotation RC = RBRA. Let the quaternion associated with a spatial rotation R is constructed from its Axis of rotation, rotation axis S and the rotation angle ''φ'' this axis. The associated quaternion is given by, :S = \cos\frac + \sin\frac \mathbf. Then the composition of the rotation RR with RA is the rotation RC = RBRA with rotation axis and angle defined by the product of the quaternions :A = \cos\frac + \sin\frac\mathbf\quad\text\quad B = \cos\frac + \sin\frac\mathbf, that is : C = \cos\frac + \sin\frac\mathbf = \left(\cos\frac + \sin\frac\mathbf\right)\left(\cos\frac + \sin\frac\mathbf\right). Expand this product to obtain : \cos\frac + \sin\frac \mathbf = \left( \cos\frac\cos\frac - \sin\frac\sin\frac \mathbf\cdot \mathbf \right) + \left( \sin\frac\cos\frac \mathbf + \sin\frac\cos\frac \mathbf + \sin\frac\sin\frac \mathbf \times \mathbf \right). Divide both sides of this equation by the identity, which is the spherical law of cosines, law of cosines on a sphere, :\cos\frac = \cos\frac\cos\frac - \sin\frac\sin\frac \mathbf\cdot \mathbf, and compute :\tan\frac \mathbf = \frac. This is Rodrigues' formula for the axis of a composite rotation defined in terms of the axes of the two rotations. He derived this formula in 1840 (see page 408). The three rotation axes A, B, and C form a spherical triangle and the dihedral angles between the planes formed by the sides of this triangle are defined by the rotation angles.


Infinitesimal rotations

The matrices in the Lie algebra are not themselves rotations; the skew-symmetric matrices are derivatives. An actual "differential rotation", or ''infinitesimal rotation matrix'' has the form : I + A \, d\theta, where is vanishingly small and . These matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals . To understand what this means, consider : dA_ = \begin 1 & 0 & 0 \\ 0 & 1 & -d\theta \\ 0 & d\theta & 1 \end. First, test the orthogonality condition, . The product is : dA_\mathbf^\textsf \, dA_\mathbf = \begin 1 & 0 & 0 \\ 0 & 1 + d\theta^2 & 0 \\ 0 & 0 & 1 + d\theta^2 \end, differing from an identity matrix by second order infinitesimals, discarded here. So, to first order, an infinitesimal rotation matrix is an orthogonal matrix. Next, examine the square of the matrix, : dA_^2 = \begin 1 & 0 & 0 \\ 0 & 1 - d\theta^2 & -2d\theta \\ 0 & 2\,d\theta & 1 - d\theta^2 \end. Again discarding second order effects, note that the angle simply doubles. This hints at the most essential difference in behavior, which we can exhibit with the assistance of a second infinitesimal rotation, :dA_\mathbf = \begin 1 & 0 & d\phi \\ 0 & 1 & 0 \\ -d\phi & 0 & 1 \end. Compare the products to , :\begin dA_\,dA_ &= \begin 1 & 0 & d\phi \\ d\theta\,d\phi & 1 & -d\theta \\ -d\phi & d\theta & 1 \end \\ dA_\,dA_ &= \begin 1 & d\theta\,d\phi & d\phi \\ 0 & 1 & -d\theta \\ -d\phi & d\theta & 1 \end. \\ \end Since d\theta \, d\phi is second-order, we discard it: thus, to first order, multiplication of infinitesimal rotation matrices is ''commutative''. In fact, : dA_\,dA_ = dA_\,dA_, again to first order. In other words, the order in which infinitesimal rotations are applied is irrelevant. This useful fact makes, for example, derivation of rigid body rotation relatively simple. But one must always be careful to distinguish (the first order treatment of) these infinitesimal rotation matrices from both finite rotation matrices and from Lie algebra elements. When contrasting the behavior of finite rotation matrices in the BCH formula above with that of infinitesimal rotation matrices, where all the commutator terms will be second order infinitesimals one finds a bona fide vector space. Technically, this dismissal of any second order terms amounts to Group contraction.


Realizations 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 * in
quaternion 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 ...

quaternion
algebra with
versor 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 and the map
3-sphere 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''3 → SO(3) (see
quaternions and spatial rotation Unit Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or bea ...
s) * in geometric algebra as a Rotor (mathematics), rotor * as a sequence of three rotations about three fixed axes; see Euler angles.


Spherical harmonics

See also Representation of a Lie group#An example: The rotation group SO.283.29, Representations of SO(3) The group of three-dimensional Euclidean rotations has an infinite-dimensional representation on the Hilbert space :L^2\left(\mathbf^2\right) = \operatorname \left\, where Y^\ell_m are spherical harmonics. Its elements are square integrable complex-valued functionsThe elements of are actually equivalence classes of functions. two functions are declared equivalent if they differ merely on a set of measure zero. The integral is the Lebesgue integral in order to obtain a ''complete'' inner product space. on the sphere. The inner product on this space is given by If is an arbitrary square integrable function defined on the unit sphere , then it can be expressed as where the expansion coefficients are given by The Lorentz group action restricts to that of and is expressed as This action is unitary, meaning that The can be obtained from the of above using Clebsch–Gordan coefficients, Clebsch–Gordan decomposition, but they are more easily directly expressed as an exponential of an odd-dimensional -representation (the 3-dimensional one is exactly ). In this case the space decomposes neatly into an infinite direct sum of irreducible odd finite-dimensional representations according to Section 4.3.5. This is characteristic of infinite-dimensional unitary representations of . If is an infinite-dimensional unitary representation on a separable space, separableA Hilbert space is separable if and only if it has a countable basis. All separable Hilbert spaces are isomorphic. Hilbert space, then it decomposes as a direct sum of finite-dimensional unitary representations. Such a representation is thus never irreducible. All irreducible finite-dimensional representations can be made unitary by an appropriate choice of inner product, :\langle f, g\rangle_U \equiv \int_ \langle\Pi(R)f, \Pi(R)g\rangle \, dg = \frac \int_0^ \int_0^\pi \int_0^ \langle \Pi(R)f, \Pi(R)g\rangle \sin \theta \, d\phi \, d\theta \, d\psi, \quad f,g \in V, where the integral is the unique invariant integral over normalized to , here expressed using the
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 ( ...
parametrization. The inner product inside the integral is any inner product on .


Generalizations

The rotation group generalizes quite naturally to ''n''-dimensional
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
, \R^n with its standard Euclidean structure. The group of all proper and improper rotations in ''n'' dimensions is called the
orthogonal 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 gen ...
O(''n''), and the subgroup of proper rotations is called the special orthogonal group SO(''n''), which is a
Lie group In mathematics, a Lie group (pronounced "Lee") is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operati ...
of dimension . 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 metric signature, 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 rotation group SO(3) can be described as a subgroup of SE(3), E+(3), the Euclidean group of Euclidean group#Direct and indirect isometries, direct isometries of Euclidean \R^3. 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 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 chirality (mathematics), chiral objects it is the same as the full symmetry group.


See also

*Orthogonal group *Angular momentum *Coordinate rotations *Charts on SO(3) *Representation of a Lie group#An example: The rotation group SO(3), Representations of SO(3) *
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 ( ...
*Rodrigues' rotation formula *Infinitesimal rotation *Pin group *Quaternions and spatial rotations *Rigid body *Spherical harmonics *Plane of rotation *
Lie group In mathematics, a Lie group (pronounced "Lee") is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operati ...
*Pauli matrix *Plate trick *Three-dimensional rotation operator


Footnotes


References


Bibliography

* * *

* * * * * * (translation of the original 1932 edition, ''Die Gruppentheoretische Methode in Der Quantenmechanik''). *. {{DEFAULTSORT:Rotation Group Lie groups Rotational symmetry Rotation in three dimensions Euclidean solid geometry