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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

$\displaystyle \alpha\alpha^*+\gamma\gamma^*$ $\textstyle =$ $\displaystyle 1$ (1)
$\displaystyle \alpha\alpha^*+\beta\beta^*$ $\textstyle =$ $\displaystyle 1$ (2)
$\displaystyle \beta\beta^*+\delta\delta^*$ $\textstyle =$ $\displaystyle 1$ (3)
$\displaystyle \alpha^*\beta+\gamma^*\delta$ $\textstyle =$ $\displaystyle 0$ (4)
$\displaystyle \alpha\delta-\beta\gamma$ $\textstyle =$ $\displaystyle 1$ (5)

$\displaystyle \beta$ $\textstyle =$ $\displaystyle -\gamma^*$ (6)
$\displaystyle \delta$ $\textstyle =$ $\displaystyle \alpha^*,$ (7)

where $z^*$ denotes the Complex Conjugate. In terms of the Euler Angles $\theta$, $\phi$, and $\psi$, the Cayley-Klein parameters are given by
$\displaystyle \alpha$ $\textstyle =$ $\displaystyle e^{i(\psi+\phi)/2}\cos({\textstyle{1\over 2}}\theta)$ (8)
$\displaystyle \beta$ $\textstyle =$ $\displaystyle ie^{i(\psi-\phi)/2}\sin({\textstyle{1\over 2}}\theta)$ (9)
$\displaystyle \gamma$ $\textstyle =$ $\displaystyle ie^{-i(\psi-\phi)/2}\sin({\textstyle{1\over 2}}\theta)$ (10)
$\displaystyle \delta$ $\textstyle =$ $\displaystyle e^{-(\psi+\phi)/2}\cos({\textstyle{1\over 2}}\theta)$ (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]
\end{displaymath} (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]
\end{displaymath} (13)

which characterizes the transformations
$\displaystyle u'$ $\textstyle =$ $\displaystyle \alpha u+\beta v$ (14)
$\displaystyle v'$ $\textstyle =$ $\displaystyle \gamma u+\delta v.$ (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}},
\end{displaymath} (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.
\end{displaymath} (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)
\end{displaymath} (18)

(Goldstein 1980, p. 156).

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


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

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© 1996-9 Eric W. Weisstein