# Euler Angles

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## 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).

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

## Euler Angles

(or Eulerian angles), the angles φ, θ, and ψ that define the position of a rectangular Cartesian coordinate system *OXYZ* with respect to another rectangular Cartesian coordinate system *Oxyz* of the same orientation (see Figure 1).

Let *OK* be an axis, called the line of nodes, that coincides with the line of intersection of the coordinate plane *Oxy* of the first system with the coordinate plane *OXY* of the second system and is directed so that the axes *Oz, OZ*, and *OK* form a trihedral of the same orientation. The Euler angle φ, or angle of spin, is the angle between the axes *Ox* and *OK*, which is measured in the plane *Oxy* from *Ox* in the direction of the shortest rotation from *Ox* to *Oy*. The Euler angle θ, or angle of nutation, is the angle between *Oz* and *OZ;* it does not exceed π. The Euler angle ψ, or angle of precession, is the angle between the axes *OK* and *OX*, which is measured in the plane *OXY* from *Ok* in the direction of the shortest rotation from *OX* to *OY*. When θ = 0 or π, the Euler angles are not defined.

The Euler angles were introduced by L. Euler in 1748 and are used extensively in the dynamics of solids—for example, in the theory of the gyroscope—and in celestial mechanics.