where R ^ ^ ( Δ Δ θ θ , a ^ ^ ) is a rotation matrix dependent on the axis and angle. In group theoretic language, the rotation matrices are group elements, and the angles and axis Δ Δ θ θ a ^ ^ = Δ Δ θ θ ( a 1 , a 2 , a 3 ) are the parameters, of the three-dimensional special orthogonal group, SO(3). The rotation matrices about the standard Cartesian basis vector e ^ ^ x , e ^ ^ y , e ^ ^ z through angle Δ θ, and the corresponding generators of rotations J = (J x, J y, J z), are: