The four parameters , , , and describing a finite rotation about an arbitrary axis. The
Euler parameters are defined by

(1) | |||

(2) |

and are a Quaternion in scalar-vector representation

(3) |

Because Euler's Rotation Theorem states that an arbitrary rotation may be described by only three parameters, a
relationship must exist between these four quantities

(4) |

(5) |

(6) |

The Euler parameters may be given in terms of the Euler Angles by

(7) | |||

(8) | |||

(9) | |||

(10) |

(Goldstein 1980, p. 155).

Using the Euler parameters, the Rotation Formula becomes

(11) |

(12) |

(13) |

(14) | |||

(15) | |||

(16) | |||

(17) | |||

(18) | |||

(19) | |||

(20) | |||

(21) | |||

(22) |

**References**

Arfken, G. *Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press, pp. 198-200, 1985.

Goldstein, H. *Classical Mechanics, 2nd ed.* Reading, MA: Addison-Wesley, 1980.

Landau, L. D. and Lifschitz, E. M. *Mechanics, 3rd ed.* Oxford, England: Pergamon Press, 1976.

© 1996-9

1999-05-25