According to Euler's Rotation Theorem, any Rotation may be described using three Angles. If
the Rotations are written in terms of Rotation Matrices *B*, *C*, and *D*,
then a general Rotation *A* can be written as

(1) |

(2) |

In the so-called ``-convention,'' illustrated above,

(3) | |||

(4) | |||

(5) |

so

To obtain the components of the Angular Velocity in the body axes, note that for a Matrix

(6) |

(7) | |||

(8) |

Now, corresponds to rotation about the axis, so look at the component of ,

(9) |

(10) |

(11) |

(12) |

The -convention Euler angles are given in terms of the Cayley-Klein Parameters by

(13) | |||

(14) | |||

(15) |

In the ``-convention,''

(16) |

(17) |

(18) |

(19) |

(20) |

(21) |

(22) | |||

(23) | |||

(24) |

and

In the ``'' (pitch-roll-yaw) convention, is pitch, is
roll, and is yaw.

(25) | |||

(26) |

and

A set of parameters sometimes used instead of angles are the Euler Parameters , , and ,
defined by

(27) | |||

(28) |

Using Euler Parameters (which are Quaternions), an arbitrary Rotation Matrix can be described by

(Goldstein 1960, p. 153).

If the coordinates of two pairs of points and are known, one rotated with respect to the other,
then the Euler rotation matrix can be obtained in a straightforward manner using Least Squares Fitting. Write the
points as arrays of vectors, so

(29) |

(30) |

(31) |

(32) |

(33) |

(34) |

(35) |

Using Nonlinear Least Squares Fitting then gives solutions which converge to .

**References**

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

Goldstein, H. ``The Euler Angles'' and ``Euler Angles in Alternate Conventions.'' §4-4 and Appendix B in
*Classical Mechanics, 2nd ed.* Reading, MA: Addison-Wesley, pp. 143-148 and 606-610, 1980.

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

© 1996-9

1999-05-25