Euler angles
Three angular parameters that specify the orientation of a body with respect to reference axes. They are used for describing rotating systems such as gyroscopes, tops, molecules, and nonspherical nuclei. They are not symmetrical in the three angles but are simpler to use than other rotational parameters.
Unfortunately, different definitions of Euler's angles are used, and therefore it is confusing to compare equations in different references. The definition given here is the majority convention according to H. Margenau and G. Murphy.
Let OXYZ be a right-handed cartesian (right-angled) set of fixed coordinate axes and Oxyz a set attached to the rotating body (see illustration).
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The orientation of Oxyz can be produced by three successive rotations about the fixed axes starting with Oxyz parallel to OXYZ. Rotate through (1) the angle ψ counterclockwise about OZ, (2) the angle Θ counterclockwise about OX, and (3) the angle &phgr; counterclockwise about OZ again. The line of intersection OK of the xy and XY planes is called the line of nodes.
Denote a rotation about OZ, for example, by Z (angle). Then the complete rotation is, symbolically, given by the equation below where the rightmost operation is done
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