The candidates include:

- Euler angles (θ,φ,ψ), representing a product of rotations about the
*x*-,*y*- and*z*-axes; - a pair (
, θ) of a unit vector representing an axis, and an angle of rotation about it (see coordinate rotation);*n* - a quaternion
*q*of length 1 (cf. quaternions and spatial rotation); - the exponential of an anti-symmetric 3×3 matrix (see skew-symmetric matrix for details).

Looking more closely, the fourth representation gives parameters in **R**^{3}. The second gives parameters in *S*^{2}×*S*^{1}; if we replace the unit vector by the actual axis of rotation, so that ** n** and -

That makes four or five manifolds that are used to try to give charts on SO(3). The truth about it, so to speak, is that it is diffeomorphic to **R**P^{3}: the quaternion representation is precisely a two-to-one mapping from *S*^{3} to SO(3). This suggests that it has certain theoretical advantages; and also that conversions from other representations to it will encounter chart problems.

One area in which these considerations, in some form, become inevitable, is the kinematics of a rigid body. One can take as definition the idea of a curve in the Euclidean group *E(3)* of three-dimensional Euclidean space, starting at the identity (initial position). The translation subgroup *T* of *E(3)* is a normal subgroup, with quotient SO(3) if we look at the subgroup *E ^{+}(3)* of direct isometries only (which is reasonable in kinematics). Therefore any rigid body movement leads directly to SO(3), when we factor out the translational part.