Charts on SO(3)

In mathematics, the special orthogonal group in three dimensions, otherwise known as the group of coordinate rotations, is a naturally occurring example of a manifold. The various chartss on SO(3) set up rival coordinate systems: in this case there cannot be said to be a preferred set of parameters describing a rotation. There are three degrees of freedom, so that the dimension of SO(3) is three. In numerous applications one or other coordinate system is used, and the question arises how to convert from a given system to another.

The candidates include:

• Euler angles (θ,φ,ψ), representing a product of rotations about the x-, y- and z-axes;

• a pair (n, θ) of a unit vector representing an axis, and an angle of rotation about it (see coordinate rotation);

• 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).

There are problems in using these as more than local charts, to do with their multiple-valued nature, and singularities. That is, one must be careful above all to work only with diffeomorphisms in the definition of chart. This explains why, for example, the Euler angles appear to give a variable in the 3-torus, and the quaternions in a 3-sphere. The uniqueness of the representation by Euler angles breaks down at some points, while the quaternion representation is always a double cover, with q and -q giving the same rotation.

Looking more closely, the fourth representation gives parameters in R³. The second gives parameters in S²×S¹; if we replace the unit vector by the actual axis of rotation, so that n and -n give the same axis line, this becomes RP²×S¹, where RP² is the real projective plane.

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 RP³: the quaternion representation is precisely a two-to-one mapping from S³ 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.