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A general rotation of a rigid body (e.g. a 3D reconstruction) can be described as a series of 3 rotations about the axes of the coordinate system (i.e. x, y and z axes). The angles of the 3 rotations are known as Euler angles (a subset are also known as Tait-Bryan angles).
Each rotation can be around one of three axes. Given that successive axes must be different, there are 3 x 2 x 2 = 12 possibilities. Although in practice only a few axis combinations are used, this can still be a source of confusion. In addition, a positive angle may be interpreted as clockwise or anti-clockwise around the axis (when viewed in a specified direction).
Rotations can be described in terms of external fixed axes, or internal axes of the rigid body which rotate with the body. It can be shown that a set of intrinsic rotations about axes x then y then z through angles α, β, γ is equivalent to a set of extrinsic rotations about axes z then y then x by angles γ, β, α.
A common convention is to rotate α about the z-axis, β about the new y-axis, and γ about the final z-axis. These are rotations with respect to an internal coordinate system. As above, this is equivalent to γ about external z, β about external y, and α about external z. In terms of rotation matrices, this would be written:
Definitions of nomenclature: