Quaternion parameters

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... 

In evaluating the PF ineq. (37) note that the conjugation operationgkgtgkT1, or kik l, is to be regarded as a single operation. The quaternion parameters X, A for the product of... 

We now have all the necessary machinery for working out the matrix elements l in the MRs of the proper rotations R in any point group for any required value of The l are given in terms of the Cayley-Klein parameters a, b and their CCs by eq. (11.8.43). The parameters a, b may be evaluated from the quaternion parameters X, A for R, using... 

Table 12.4. Rotation parameters n or m, real (A, A), and complex (p, r) quaternion parameters, and the Cayley-Klein parameters a, b for the operators R D3.

Table 14.5. Rotation parameters

Exercise 16.8-3 Write down the quaternion parameters for R c S4 and hence verify the matrices in Table 16.21. 

The second method is to use the so called quaternion parameters (qp) (18). These are defined in terms of the Euler angles as... 

The quaternion parameters X, n, C, C can be defined in terms of Goldstein Euler angles (13) by... 

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