Charts On SO(3) - Parametrizations

Parametrizations

We can parameterize the space of rotations in several ways, but degenerations will always appear. For example if we use three angles (Euler angles), such parameterization is degenerate at some points on the hypersphere, leading to the problem of gimbal lock. We can avoid this by using four Euclidean coordinates w,x,y,z, with w2 + x2 + y2 + z2 = 1. The point (w,x,y,z) represents a rotation around the axis directed by the vector by an angle

This problem is similar to parameterize the bidimensional surface of a sphere with two coordinates, such as latitude and longitude. Latitude and longitude are ill-behaved (degenerate) at the north and south poles, though the poles are not intrinsically different from any other points on the sphere. At the poles (latitudes +90° and −90°), the longitude becomes meaningless. It can be shown that no two-parameter coordinate system can avoid such degeneracy.

The possible parametrizations candidates include:

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

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

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

• 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; the 3×3 skew-symmetric matrices are the Lie algebra SO₃, and this is the exponential map in Lie theory;

• Cayley rational parameters, based on the Cayley transform, usable in all characteristics;

• fractional linear transformations, acting on the Riemann sphere.