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

(1) | |||

(2) | |||

(3) | |||

(4) | |||

(5) |

and

(6) | |||

(7) |

where denotes the Complex Conjugate. In terms of the Euler Angles , , and , the Cayley-Klein parameters are given by

(8) | |||

(9) | |||

(10) | |||

(11) |

(Goldstein 1960, p. 155).

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

(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)

(13) |

(14) | |||

(15) |

of a linear space having complex axes. This matrix satisfies

(16) |

(17) |

(18) |

**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.

© 1996-9

1999-05-26