Editing 3D rotation group (section)

==Axis of rotation==
{{main|Axis–angle representation}}
Every nontrivial proper rotation in 3 dimensions fixes a unique 1-dimensional [[linear subspace]] of <math>\R^3</math> 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 and counterclockwise|clockwise]] or [[counterclockwise]] with respect to this orientation).

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

:<math>R_z(\phi) = \begin{bmatrix}\cos\phi & -\sin\phi & 0 \\ \sin\phi & \cos\phi & 0 \\ 0 & 0 & 1\end{bmatrix}.</math>

Given a [[unit vector]] '''n''' in <math>\R^3</math> 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''({{pi}} + ''φ'', '''n''') = ''R''({{pi}} − ''φ'', −'''n''').

Using these properties one can show that any rotation can be represented by a unique angle ''φ'' in the range 0 ≤ φ ≤ {{pi}} and a unit vector '''n''' such that
* '''n''' is arbitrary if ''φ'' = 0
* '''n''' is unique if 0 < ''φ'' < {{pi}}
* '''n''' is unique up to a [[sign (mathematics)|sign]] if ''φ'' = {{pi}} (that is, the rotations ''R''({{pi}}, ±'''n''') are identical).
In the next section, this representation of rotations is used to identify SO(3) topologically with three-dimensional real projective space.