, the 3D rotation group, often denoted SO
(3), is the group
of all rotation
s about the origin
of three-dimensional Euclidean space
under the operation of composition
. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance
(so it is an isometry
), and orientation
(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
rotation; and the identity map
satisfies the definition of a rotation. Owing to the above properties (along composite rotations' associative property
), the set of all rotations is a group
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
for which the group operations are smoothly differentiable
; so it is in fact a Lie group
. It is compact
and has dimension 3.
Rotations are linear transformation
and can therefore be represented by matrices
once a basis
has been chosen. Specifically, if we choose an orthonormal basis
, 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
, results in the identity matrix
) with determinant
1. The group SO(3) can therefore be identified with the group of these matrices under 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
s are important in physics, where they give rise to the elementary particle
s of integer spin
Length and angle
Besides just preserving length, rotations also preserve the angle
s between vectors. This follows from the fact that the standard dot product
between two vectors u and v can be written purely in terms of length:
It follows that every length-preserving transformation in
preserves the dot product, and thus the angle between vectors. Rotations are often defined as linear transformations that preserve the inner product on
, which is equivalent to requiring them to preserve length. See classical group
for a treatment of this more general approach, where appears as a special case.
Orthogonal and rotation matrices
Every rotation maps an orthonormal basis
to another orthonormal basis. Like any linear transformation of finite-dimensional
vector spaces, a rotation can always be represented by a matrix
. Let be a given rotation. With respect to the standard basis
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
where denotes the transpose
of and is the identity matrix
. Matrices for which this property holds are called orthogonal matrices
. 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
of the matrix is positive or negative. For an orthogonal matrix , note that implies , so that . The subgroup
of orthogonal matrices with determinant is called the ''special orthogonal group
'', denoted .
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 .
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.
The rotation group is a group
under function composition
(or equivalently the product of linear transformations
). It is a subgroup
of the general linear group
consisting of all invertible
linear transformations of the real 3-space
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
which is called the ''axis of rotation'' (this is Euler's rotation theorem
). Each such rotation acts as an ordinary 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
with respect to this orientation).
For example, counterclockwise rotation about the positive ''z''-axis by angle ''φ'' is given by
Given a unit vector
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
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.
The Lie group SO(3) is diffeomorphic
to the real projective space
Consider the solid ball in
of radius (that is, all points of
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
s on the surface of the ball. After this identification, we arrive at a topological space homeomorphic
to the rotation group.
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.
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 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
and similar tricks demonstrate this practically.
The same argument can be performed in general, and it shows that the fundamental group
of SO(3) is a cyclic group
of order 2 (a fundamental group with two elements). In physics
applications, the non-triviality (more than one element) of the fundamental group allows for the existence of objects known as spinor
s, and is an important tool in the development of the spin–statistics theorem
The universal cover
of SO(3) is a Lie group
. The group Spin(3) is isomorphic to the special unitary group
SU(2); it is also diffeomorphic to the unit 3-sphere
and can be understood as the group of versor
s with absolute value
1). The connection between quaternions and rotations, commonly exploited in computer graphics
, is explained in quaternions and spatial rotation
s. The map from ''S''3
onto SO(3) that identifies antipodal points of ''S''3
is a surjective homomorphism
of Lie groups, with kernel
. Topologically, this map is a two-to-one covering map
. (See the plate trick
Connection between SO(3) and SU(2)
In this section, we give two different constructions of a two-to-one and surjective homomorphism
of SU(2) onto SO(3).
Using quaternions of unit norm
The group is isomorphic
to the quaternion
s of unit norm via a map given by
Let us now identify
with the span of
. One can then verify that if
is a unit quaternion, then
Furthermore, the map
is a rotation of
is the same as
. 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
is mapped to the rotation matrix
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
300px|Stereographic projection from the sphere of radius from the north pole onto the plane given by coordinatized by , here shown in cross section.
The general reference for this section is . The points on the sphere
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
Let the coordinates on be . The line passing through and can be parametrized as
Demanding that the of
equals , one finds
Hence the map
where, for later convenience, the plane is identified with the complex plane
For the inverse, write as
and demand to find and thus
If is a rotation, then it will take points on to points on by its standard action on the embedding space
By composing this action with one obtains a transformation of ,
Thus is a transformation of
associated to the transformation of
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
it represents). To identify this matrix, consider first a rotation about the through an angle ,
which, unsurprisingly, is a rotation in the complex plane. In an analogous way, if is a rotation about the through and angle , then
which, after a little algebra, becomes
These two rotations,
thus correspond to bilinear transform
s of , namely, they are examples of Möbius transformation
A general Möbius transformation is given by
generate all of and the composition rules of the Möbius transformations show that any composition of
translates to the corresponding composition of Möbius transformations. The Möbius transformations can be represented by matrices
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
These matrices are unitary and thus . In terms of Euler angles
[This is effected by first applying a rotation through about the to take the to the line , the intersection between the planes and , the latter being the rotated . Then rotate with through about to obtain the new from the old one, and finally rotate by through an angle about the ''new'' , where is the angle between and the new . In the equation, and 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 Here boldface means that the rotation is expressed in the ''original'' basis. Likewise,
one finds for a general rotation
For the converse, consider a general matrix
Make the substitutions
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 ,
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
just described is a smooth, and surjective group homomorphism
. It is hence an explicit description of the universal covering map
of from the universal covering group
Associated with every Lie group is its Lie algebra
, a linear space of the same dimension as the Lie group, closed under a bilinear alternating product called the Lie bracket
. The Lie algebra of is denoted by
and consists of all skew-symmetric
matrices. This may be seen by differentiating the orthogonality condition
[For an alternative derivation of , see Classical group.]
The Lie bracket of two elements of
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
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
denotes a counterclockwise rotation with angle φ about the axis specified by the unit vector
This can be used to show that the Lie algebra
(with commutator) is isomorphic to the Lie algebra
(with cross product
). Under this isomorphism, an Euler vector
corresponds to the linear map
In more detail, a most often suitable basis for
as a vector space is
The commutation relation
s of these basis elements are,
which agree with the relations of the three standard unit vectors
under the cross product.
As announced above, one can identify any matrix in this Lie algebra with an Euler vector
This identification is sometimes called the hat-map.
Under this identification, the
bracket corresponds in
to the cross product
The matrix identified with a vector
has the property that
where the left-hand side we have ordinary matrix multiplication. This implies
is in the null space
of the skew-symmetric matrix with which it is identified, because
A note on Lie algebras
In Lie algebra representation
s, 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,
of the algebra
That is, the Casimir invariant is given by
For unitary irreducible representations
, the eigenvalues of this invariant are real and discrete, and characterize each representation, which is finite dimensional, of dimensionality
. That is, the eigenvalues of this Casimir operator are
is integer or half-integer, and referred to as the spin
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 doublet
) representation. By taking Kronecker product
s 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 operator
s and ladder operator
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 bosonic representation
s, while half-integer values fermionic representation
s. The antihermitian
matrices used above are utilized as spin operator
s, after they are multiplied by , so they are now hermitian
(like the Pauli matrices). Thus, in this language,
Explicit expressions for these are,
is arbitrary and
For example, the resulting spin matrices for spin 1 (
Note, however, how these are in an equivalent, but different basis, the spherical basis
, than the above
in the Cartesian basis.
For spin (
For spin (
Isomorphism with 𝖘𝖚(2)
The Lie algebras
are isomorphic. One basis for
is given by
These are related to the Pauli matrices
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 constant
s remain the same, but the ''definition'' of them acquires a factor of . Likewise, commutation relations acquire a factor of . The commutation relations for the
where is the totally anti-symmetric symbol with . The isomorphism between
can be set up in several ways. For later convenience,
are identified by mapping
and extending by linearity.
The exponential map for , is, since is a matrix Lie group, defined using the standard matrix exponential
For any skew-symmetric matrix
, is always in . The proof uses the elementary properties of the matrix exponential
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
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
. This follows from the fact that every , since every rotation leaves an axis fixed (Euler's rotation theorem
), and is conjugate to a block diagonal matrix
of the form
such that , and that
together with the fact that is closed under the adjoint action
of , meaning that .
Thus, e.g., it is easy to check the popular identity
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 yields
. This is recognized as a matrix for a rotation around axis by the angle : cf. Rodrigues' rotation formula
Given , let
denote the antisymmetric part and let
Then, the logarithm of is given by
This is manifest by inspection of the mixed symmetry form of Rodrigues' formula,
where the first and last term on the right-hand side are symmetric.
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
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 and The series may still converge even if these conditions aren't fulfilled. A solution always exists since is onto in the cases under consideration.]
The infinite expansion in the BCH formula for reduces to a compact form,
for suitable trigonometric function coefficients .
The are given by
The inner product is the Hilbert–Schmidt inner product
and the norm is the associated norm. Under the hat-isomorphism,
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
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 2×2 derivation for SU(2)
The Pauli vector version
of the same BCH formula is the somewhat simpler group composition law of SU(2),
the spherical law of cosines
. (Note are angles, not the above.)
This is manifestly of the same format as above,
For uniform normalization of the generators in the Lie algebra involved, express the Pauli matrices in terms of -matrices, , so that
To verify then these are the same coefficients as above, compute the ratios of the coefficients,
Finally, given the identity .
For the general case, one might use Ref.
formulation of the composition of two rotations RB
also yields directly the rotation axis
and angle of the composite rotation RC
Let the quaternion associated with a spatial rotation R is constructed from its rotation axis
S and the rotation angle ''φ'' this axis. The associated quaternion is given by,
Then the composition of the rotation RR
is the rotation RC
with rotation axis and angle defined by the product of the quaternions
Expand this product to obtain
Divide both sides of this equation by the identity, which is the law of cosines on a sphere
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.
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
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
First, test the orthogonality condition, . The product is
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,
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,
Compare the products to ,
is second-order, we discard it: thus, to first order, multiplication of infinitesimal rotation matrices is ''commutative''. In fact,
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
algebra with versor
s and the map 3-sphere
→ SO(3) (see quaternions and spatial rotation
* in geometric algebra
as a rotor
* as a sequence of three rotations about three fixed axes; see Euler angle
See also Representations of SO(3)
The group of three-dimensional Euclidean rotations has an infinite-dimensional representation on the Hilbert space
are spherical harmonics
. Its elements are square integrable complex-valued functions
[The 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 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
[A 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,] [
where the integral is the unique invariant integral over normalized to , here expressed using the Euler angles parametrization. The inner product inside the integral is any inner product on .
The rotation group generalizes quite naturally to ''n''-dimensional Euclidean space, with its standard Euclidean structure. 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''), which is a Lie group 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 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 E+(3), the Euclidean group of direct isometries of Euclidean 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 chiral objects it is the same as the full symmetry group.
*Charts on SO(3)
*Representations of SO(3)
*Rodrigues' rotation formula
*Quaternions and spatial rotations
*Plane of rotation
*Three-dimensional rotation operator
* (translation of the original 1932 edition, ''Die Gruppentheoretische Methode in Der Quantenmechanik'').
Category:Rotation in three dimensions
Category:Euclidean solid geometry