Euler-Rodrigues parameters

A general rotation R( o n) in ft3 requires the specification of three independent parameters which can be chosen in various ways. The natural and familiar way is to specify the angle of rotation and the direction of the unit vector n. (The normalization condition on n means that there are only three independent parameters.) A second parameterization R(a b) introduced above involves the Cayley-Klein parameters a, b. A third common parameterization is in terms of the three Euler angles a, (3, and 7 (see Section 11.7). Yet another parameterization using the quaternion or Euler-Rodrigues parameters will be introduced in Chapter 12. 

From the pole conventions in Section 11.3 it follows that R(— n) = K( (j> n), and this restricts the rotation angle to the range 0 < [Pg.234]

Parameters that satisfy eq. (12) are referred to as standard quaternion (or Euler-Rodrigues) parameters. Note that A = sin(j(/>)n belongs to the positive hemisphere h for positive rotations and to —h for negative rotations. For binary rotations, n, and so X = 0, A = 1, and A belongs to h because there is no rotation R( n n). Therefore for any point group, h must be defined so as to contain the poles of all positive rotations, including binary rotations. Due to the range of , standard quaternion parameters must satisfy either... 

---