Projective 3-space

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space $\R^3$ 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). 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.

Every non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation. 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 the 3D rotation group 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 transformations of $\R^3$ and can therefore be represented by matrices once a basis of $\R^3$ has been chosen. Specifically, if we choose an orthonormal basis 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, 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 representations are important in physics, where they give rise to the elementary particles of integer spin.

Length and angle

Besides just preserving length, rotations also preserve the angles 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 (see the law of cosines):
$\mathbf \cdot \mathbf = \frac \left(\, \mathbf + \mathbf\, ^2 - \, \mathbf\, ^2 - \, \mathbf\, ^2\right).$

It follows that every length-preserving linear 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 for a treatment of this more general approach, where appears as a special case.

Orthogonal and rotation matrices

Every rotation maps an orthonormal basis of $\R^3$ 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 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 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 .

Improper rotations 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 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 $\R^3$.

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.

Complete classification of finite subgroups

The finite subgroups of $\mathrm(3)$ are completely classified.

Every finite subgroup is isomorphic to either an element of one of two countably infinite families of planar isometries: the cyclic groups $C_n$ or the dihedral groups $D_$, or to one of three other groups: the tetrahedral group $\cong A_4$, the octahedral group $\cong S_4$, or the icosahedral group $\cong A_5$.

Axis of rotation

Every nontrivial proper rotation in 3 dimensions fixes a unique 1-dimensional linear subspace of $\R^3$ 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 or counterclockwise 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 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 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 to the real projective space $\mathbb^3(\R).$

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. Rotations through an angle between 0 and (not including either) are on the same axis at the same distance. 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 points 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 $\mathbb^3(\R),$ 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. 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 it is deformed, 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 the identity (center of the ball), through the south pole, jumping to the 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, running through the path twice, i.e., running from the north pole down to the south pole, jumping back to the north pole (using the fact that north and south poles are identified), and then again running from the north pole down to the south pole, so that ''φ'' runs from 0 to 4, gives 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 the 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 spinors, and is an important tool in the development of the spin–statistics theorem.

The universal cover of SO(3) is a Lie group called Spin(3). The group Spin(3) is isomorphic to the special unitary group SU(2); it is also diffeomorphic to the unit 3-sphere ''S''³ and can be understood as the group of versors (quaternions with absolute value 1). The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotations. 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 quaternions of unit norm via a map given by
$q = a\mathbf + b\mathbf + c\mathbf + d\mathbf = \alpha + \beta \mathbf \leftrightarrow \begin\alpha & \beta \\ -\overline\beta & \overline \alpha\end = U$
restricted to $a^2+ b^2 + c^2 + d^2 = , \alpha, ^2 +, \beta, ^2 = 1$ where $q \in \mathbb$, $a, b, c, d \in \R$, $U \in \operatorname(2)$, and $\alpha = a+bi \in\mathbb$, $\beta = c+di \in \mathbb$.

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
$\begin q &= w + x\mathbf + y\mathbf + z\mathbf , \\ 1 &= w^2 + x^2 + y^2 + z^2 , \end$
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.

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 $\Complex.$

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 $\Complex$ 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 $\Complex$ 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 an 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 transforms of , namely, they are examples of Möbius transformations.

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 anglesThis 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

:$\begin p:\operatorname(2) \to \operatorname(3)\\ \pm \Pi_u(g_) \mapsto g_ \end$

just described is a smooth, and surjective group homomorphism. It is hence an explicit description of the universal covering space of from the universal covering group .

Lie algebra

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 $\mathfrak(3)$ 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. 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 Euler vector $\boldsymbol\in\R^3$ corresponds to the linear map $\widetilde$ defined by $\widetilde(\boldsymbol) = \boldsymbol\times\boldsymbol.$

In more detail, most often a 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 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 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 is integer or half-integer, and referred to as the spin or angular momentum.

So, the 3 × 3 generators ''L'' displayed above act on the triplet (spin 1) representation, while the 2 × 2 generators below, ''t'', act on the doublet