Cayley-Klein Parameters

The parameters $\alpha$, $\beta$, $\gamma$, and $\delta$ which, like the three Euler Angles, provide a way to uniquely characterize the orientation of a solid body. These parameters satisfy the identities

$$\alpha\alpha^*+\gamma\gamma^* = 1$$ (1)

$$\alpha\alpha^*+\beta\beta^* = 1$$ (2)

$$\beta\beta^*+\delta\delta^* = 1$$ (3)

$$\alpha^*\beta+\gamma^*\delta = 0$$ (4)

$$\alpha\delta-\beta\gamma = 1$$ (5)

and

$$\beta = -\gamma^*$$ (6)

$$\delta = \alpha^*,$$ (7)

where $z^*$ denotes the Complex Conjugate. In terms of the Euler Angles $\theta$, $\phi$, and $\psi$, the Cayley-Klein parameters are given by

$$\alpha = e^{i(\psi+\phi)/2}\cos({\textstyle{1\over 2}}\theta)$$ (8)

$$\beta = ie^{i(\psi-\phi)/2}\sin({\textstyle{1\over 2}}\theta)$$ (9)

$$\gamma = ie^{-i(\psi-\phi)/2}\sin({\textstyle{1\over 2}}\theta)$$ (10)

$$\delta = e^{-(\psi+\phi)/2}\cos({\textstyle{1\over 2}}\theta)$$ (11)

(Goldstein 1960, p. 155).

The transformation matrix is given in terms of the Cayley-Klein parameters by

$${\hbox{\sf A}}=\left[{\matrix{{\textstyle{1\over 2}}(\alpha^... ...lpha\gamma+\beta\delta) & \alpha\delta+\beta\gamma\cr}}\right]$$ (12)

(Goldstein 1960, p. 153).

The Cayley-Klein parameters may be viewed as parameters of a matrix (denoted Q for its close relationship with Quaternions)

$${\hbox{\sf Q}}=\left[{\matrix{\alpha & \beta\cr \gamma & \delta\cr}}\right]$$ (13)

which characterizes the transformations

$$u' = \alpha u+\beta v$$ (14)

$$v' = \gamma u+\delta v.$$ (15)

of a linear space having complex axes. This matrix satisfies

$${\hbox{\sf Q}}^\dagger{\hbox{\sf Q}}={\hbox{\sf Q}}{\hbox{\sf Q}}^\dagger={\hbox{\sf I}},$$ (16)

where I is the Identity Matrix and ${\hbox{\sf A}}^\dagger$ the Matrix Transpose, as well as

$$\vert{\hbox{\sf Q}}\vert^*\vert{\hbox{\sf Q}}\vert=1.$$ (17)

In terms of the Euler Parameters $e_i$ and the Pauli Matrices $\sigma_i$, the Q-matrix can be written as

$${\hbox{\sf Q}}=e_0{\hbox{\sf I}}+i(e_1\sigma_1+e_2\sigma_2+e_3\sigma_3)$$ (18)

(Goldstein 1980, p. 156).

See also Euler Angles, Euler Parameters, Pauli Matrices, Quaternion

References

Goldstein, H. ``The Cayley-Klein Parameters and Related Quantities.'' §4-5 in Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, pp. 148-158, 1980.