Any orientation can be achieved by composing three elemental rotations, i.e. rotations about the axes of a coordinate system. Euler angles can be defined by three of these rotations. They can also be defined by elemental geometry and the geometrical definition demonstrates that three rotations are always sufficient to reach any frame.
The three elemental rotations may be extrinsic (rotations about the axes xyz of the original coordinate system, which is assumed to remain motionless), or intrinsic (rotations about the axes of the rotating coordinate system XYZ, solidary with the moving body, which changes its orientation after each elemental rotation).
Euler angles are typically denoted as α, β, γ, or φ, θ, ψ. Different authors may use different sets of rotation axes to define Euler angles, or different names for the same angles. Therefore, any discussion employing Euler angles should always be preceded by their definition.
Without considering the possibility of using two different conventions for the definition of the rotation axes (intrinsic or extrinsic), there exist twelve possible sequences of rotation axes, divided in two groups:
Tait–Bryan angles are also called Cardan angles; nautical angles; heading, elevation, and bank; or yaw, pitch, and roll. Sometimes, both kinds of sequences are called "Euler angles". In that case, the sequences of the first group are called proper or classic Euler angles.
The axes of the original frame are denoted as x,y,z and the axes of the rotated frame as X,Y,Z. The geometrical definition (sometimes referred to as static) begins by defining the line of nodes as the intersection of the planes xy and XY (it can also be defined as the common perpendicular to the axes z and Z and then written as the vector product N = z Z). Using it, the three Euler angles can be defined as follows:
Euler angles between two reference frames are defined only if both frames have the same handedness.