Editing Euler angles (section)

==Chained rotations equivalence==
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{{See also |chained rotations}}

Euler angles can be defined by elemental [[geometry]] or by composition of rotations (i.e. [[chained rotations]]). The geometrical definition demonstrates that three consecutive ''[[elemental rotation]]s'' (rotations about the axes of a [[coordinate system]]) are ''always'' sufficient to reach any target frame.

The three elemental rotations may be [[#Definition by extrinsic rotations|extrinsic]] (rotations about the axes ''xyz'' of the original coordinate system, which is assumed to remain motionless), or [[#Definition by intrinsic rotations|intrinsic]] (rotations about the axes of the rotating coordinate system ''XYZ'', solidary with the moving body, which changes its orientation with respect to the extrinsic frame after each elemental rotation).

In the sections below, an axis designation with a prime mark superscript (e.g., ''z''″) denotes the new axis after an elemental rotation.

Euler angles are typically denoted as [[Alpha|''α'']], [[Beta|''β'']], [[Gamma|''γ'']], or [[Psi (Greek)|''ψ'']], [[Theta|''θ'']], [[Phi|''φ'']]. 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:

* '''Proper Euler angles''' {{math|(''z''-''x''-''z'', ''x''-''y''-''x'', ''y''-''z''-''y'', ''z''-''y''-''z'', ''x''-''z''-''x'', ''y''-''x''-''y'')}}
* '''Tait–Bryan angles''' {{math|(''x''-''y''-''z'', ''y''-''z''-''x'', ''z''-''x''-''y'', ''x''-''z''-''y'', ''z''-''y''-''x'', ''y''-''x''-''z'')}}.

Tait–Bryan angles are also called '''Cardan angles'''; '''nautical angles'''; '''[[heading (navigation)|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.