Euler angles

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The Euler angles are used to indicate the orientation of an object in space. See also Tait-Bryan angles.

Simple diagram
Two different ways to define the orientation, i.e. two different sets of angles.
The first way is the only one used in solid mechanics, the angles are noted (ψ,θ,φ). In materials sciences, Bunge, who is one of the reference in crystallite orientation, uses different names: (φ1,Φ,φ2). Additionally, he defines the orientation with another variant of Euler angles where the second rotation is performed around the new y-axis; these angles ares called (Ψ,Θ,Φ), although they are totally different from the angles named with the lower-case version of the same greek letters.
The determination of the orientation can be viewed as a generalization of the spherical coordinates: ρ is not of interest, but an additional angle ω must be defined. This definition can help understanding the concept, but is never used in practice; only the Euler angles are used.
Euler rotations of the Earth. Intrinsic (green), Precession (blue) and Nutation (red)
Three axes z-x-z-gimbal showing Euler angles. External frame and external axis 'x' are not shown. Axes 'Y' are perpendicular to each gimbal ring
Stereoscopic image of Euler angles, showing intermediate frames
Anaglyph image of Euler angles, showing intermediate frames
Intermediate frames
Euler rotation theorem