Charts on SO(3)
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In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold. The various charts 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;
- Tait-Bryan 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);
- Euler-Rodrigues parameters, a 4-vector v of length 1, an older name for the following;
- a quaternion q of length 1 (cf. quaternions and spatial rotation, 3-sphere);
- a 3×3 skew-symmetric matrix, via exponentiation;
- Cayley rational parameters, based on the Cayley transform, usable in all characteristics;
- fractional linear transformations,
, acting on the Riemann sphere.
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 (cf. gimbal lock), while the quaternion representation is always a double cover, with q and −q giving the same rotation.
Looking more closely, the sixth representation gives parameters in R3. The third 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 real projective space 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.
