Editing Euler angles (section)

==Tait–Bryan angles==
{{See also|Aircraft principal axes}}
[[Image:taitbrianzyx.svg|thumb|left|200px|Tait–Bryan angles. ''z''-''y''′-''x''″ sequence (intrinsic rotations; ''N'' coincides with ''y'''). The angle rotation sequence is ''ψ'', ''θ'', ''φ''. Note that in this case {{nowrap|''ψ'' > 90°}} and ''θ'' is a negative angle.]]

The second type of formalism is called '''Tait–Bryan angles''', after Scottish mathematical physicist [[Peter Guthrie Tait]] (1831–1901) and English applied mathematician [[George H. Bryan]] (1864–1928). It is the convention normally used for aerospace applications, so that zero degrees elevation represents the horizontal attitude. Tait–Bryan angles represent the orientation of the aircraft with respect to the world frame. When dealing with other vehicles, different [[axes conventions]] are possible.

===Definitions===
[[Image:taitbrianangles.svg|thumb|right|325px|Tait–Bryan angles. ''z''-''x''′-''y''″ sequence (intrinsic rotations; ''N'' coincides with ''x''′)]]
The definitions and notations used for Tait–Bryan angles are similar to those described above for proper Euler angles ([[#Geometrical definition|geometrical definition]], [[#Definition by intrinsic rotations|intrinsic rotation definition]], [[#Definition by extrinsic rotations|extrinsic rotation definition]]). The only difference is that Tait–Bryan angles represent rotations about three distinct axes (e.g. ''x''-''y''-''z'', or ''x''-''y''′-''z''″), while proper Euler angles use the same axis for both the first and third elemental rotations (e.g., ''z''-''x''-''z'', or ''z''-''x''′-''z''″).

This implies a different definition for the [[line of nodes]] in the geometrical construction. In the proper Euler angles case it was defined as the intersection between two homologous Cartesian planes (parallel when Euler angles are zero; e.g. ''xy'' and ''XY''). In the Tait–Bryan angles case, it is defined as the intersection of two non-homologous planes (perpendicular when Euler angles are zero; e.g. ''xy'' and ''YZ'').

===Conventions===
[[File:Plane with ENU embedded axes.svg|thumb|right|250px|Heading, elevation and bank angles (''Z''-''Y''′-''X''″) for an aircraft using onboard ENU axes both onboard and for the ground tracking station. The fixed reference frame ''x''-''y''-''z'' represents such a tracking station. Onboard axes ''Y'' and ''Z'' are not shown. ''X'' shown in green color. According to RHS rules the shown ''y''-axis is negative.]]

The three elemental rotations may occur either about the axes of the original coordinate system, which remains motionless ([[#Extrinsic rotations|extrinsic rotations]]), or about the axes of the rotating coordinate system, which changes its orientation after each elemental rotation ([[#Intrinsic rotations|intrinsic rotations]]).

There are six possibilities of choosing the rotation axes for Tait–Bryan angles. The six possible sequences are:

* ''x''-''y''′-''z''″ (intrinsic rotations) or ''z''-''y''-''x'' (extrinsic rotations)
* ''y''-''z''′-''x''″ (intrinsic rotations) or ''x''-''z''-''y'' (extrinsic rotations)
* ''z''-''x''′-''y''″ (intrinsic rotations) or ''y''-''x''-''z'' (extrinsic rotations)
* ''x''-''z''′-''y''″ (intrinsic rotations) or ''y''-''z''-''x'' (extrinsic rotations)
* ''z''-''y''′-''x''″ (intrinsic rotations) or ''x''-''y''-''z'' (extrinsic rotations): the intrinsic rotations are known as: yaw, pitch and roll
* ''y''-''x''′-''z''″ (intrinsic rotations) or ''z''-''x''-''y'' (extrinsic rotations)

===Signs and ranges===
[[Image:Yaw Axis Corrected.svg|thumb|right|250px|The [[Aircraft principal axes|principal axes]] of an aircraft according to the air norm [[Deutsches Institut für Normung|DIN]] 9300. Notice that fixed and mobile frames must be coincident with angles zero. Therefore, this norm would also force a compatible [[Axes conventions|axes convention]] in the reference system]]
Tait–Bryan convention is widely used in engineering with different purposes. There are several [[axes conventions]] in practice for choosing the mobile and fixed axes, and these conventions determine the signs of the angles. Therefore, signs must be studied in each case carefully.

The range for the angles ''ψ'' and ''φ'' covers 2{{pi}} radians. For ''θ'' the range covers {{pi}} radians.

===Alternative names===
These angles are normally taken as one in the external reference frame ([[Heading (navigation)|heading]], [[Bearing (navigation)|bearing]]), one in the intrinsic moving frame ([[Banked turn|bank]]) and one in a middle frame, representing an [[Elevation (ballistics)|elevation]] or inclination with respect to the horizontal plane, which is equivalent to the line of nodes for this purpose.

===As chained rotations===
[[File:Roll_pitch_yaw_mnemonic.svg|thumb|Mnemonics to remember angle names]]
For an aircraft, they can be obtained with three rotations around its [[aircraft principal axes|principal axes]] if done in the proper order and starting from a frame coincident with the reference frame. 

* A [[Yaw (rotation)|yaw]] will obtain the bearing, 
* a [[Pitching moment|pitch]] will yield the elevation, and
* a roll gives the bank angle. 

Therefore, in aerospace they are sometimes called '''yaw, pitch, and roll'''. Notice that this will not work if the rotations are applied in any other order or if the airplane axes start in any position non-equivalent to the reference frame.

Tait–Bryan angles, following ''z''-''y''′-''x''″ (intrinsic rotations) convention, are also known as '''nautical angles''', because they can be used to describe the orientation of a ship or aircraft, or '''Cardan angles''', after the Italian mathematician and physicist [[Gerolamo Cardano]], who first described in detail the [[Cardan suspension]] and the [[Cardan joint]].