Editing Euler angles (section)

==Classic Euler angles==
The Euler angles are three angles introduced by Swiss mathematician [[Leonhard Euler]] (1707–1783) to describe the [[Orientation (geometry)|orientation]] of a [[rigid body]] with respect to a fixed [[coordinate system]].<ref name="Euler"/>
{{multiple image
  | image1    = gimbaleuler.svg
  | width1    = 180
  | image2    = gimbaleuler2.svg
  | width2    = 160
  | footer    = '''Left:''' A [[gimbal]] set, showing a ''z''-''x''-''z'' rotation sequence. External frame shown in the base. Internal axes in red color. '''Right:''' A simple diagram showing similar Euler angles.
}}

===Geometrical definition===
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]] (N) 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'' &times; ''Z''). Using it, the three '''Euler angles''' can be defined as follows:

* <math>\alpha</math> (or <math>\varphi</math>) is the signed angle between the ''x'' axis and the ''N'' axis (''x''-convention – it could also be defined between ''y'' and ''N'', called ''y''-convention).
* <math>\beta</math> (or <math>\theta</math>) is the angle between the ''z'' axis and the ''Z'' axis.
* <math>\gamma</math> (or <math>\psi</math>) is the signed angle between the ''N'' axis and the ''X'' axis (''x''-convention).

Euler angles between two reference frames are defined only if both frames have the same [[Orientation (mathematics)|handedness]].

===Definition by intrinsic rotations===
{{anchor|Intrinsic rotations}}

Intrinsic rotations are elemental rotations that occur about the axes of a coordinate system ''XYZ'' attached to a moving body. Therefore, they change their orientation after each elemental rotation. The ''XYZ'' system rotates, while ''xyz'' is fixed. Starting with ''XYZ'' overlapping ''xyz'', a composition of three intrinsic rotations can be used to reach any target orientation for ''XYZ''.

Euler angles can be defined by intrinsic rotations. The rotated frame ''XYZ'' may be imagined to be initially aligned with ''xyz'', before undergoing the three elemental rotations represented by Euler angles. Its successive orientations may be denoted as follows:

* ''x''-''y''-''z'' or ''x''<sub>0</sub>-''y''<sub>0</sub>-''z''<sub>0</sub> (initial)
* ''x''′-''y''′-''z''′ or ''x''<sub>1</sub>-''y''<sub>1</sub>-''z''<sub>1</sub> (after first rotation)
* ''x''″-''y''″-''z''″ or ''x''<sub>2</sub>-''y''<sub>2</sub>-''z''<sub>2</sub> (after second rotation)
* ''X''-''Y''-''Z'' or ''x''<sub>3</sub>-''y''<sub>3</sub>-''z''<sub>3</sub> (final)

For the above-listed sequence of rotations, the [[line of nodes]] ''N'' can be simply defined as the orientation of ''X'' after the first elemental rotation. Hence, ''N'' can be simply denoted ''x''′. Moreover, since the third elemental rotation occurs about ''Z'', it does not change the orientation of ''Z''. Hence ''Z'' coincides with ''z''″. This allows us to simplify the definition of the Euler angles as follows:

* ''α'' (or ''φ'') represents a rotation around the ''z'' axis,
* ''β'' (or ''θ'') represents a rotation around the ''x''′ axis, 
* ''γ'' (or ''ψ'') represents a rotation around the ''z''″ axis.

===Definition by extrinsic rotations===
{{anchor|Extrinsic  rotations}}

Extrinsic rotations are elemental rotations that occur about the axes of the fixed coordinate system ''xyz''. The ''XYZ'' system rotates, while ''xyz'' is fixed. Starting with ''XYZ'' overlapping ''xyz'', a composition of three extrinsic rotations can be used to reach any target orientation for ''XYZ''. The Euler or Tait–Bryan angles (''α'', ''β'', ''γ'') are the amplitudes of these elemental rotations. For instance, the target orientation can be reached as follows (note the reversed order of Euler angle application):

* The ''XYZ'' system rotates about the ''z'' axis by ''γ''. The ''X'' axis is now at angle ''γ'' with respect to the ''x'' axis.
* The ''XYZ'' system rotates again, but this time about the ''x'' axis by ''β''. The ''Z'' axis is now at angle ''β'' with respect to the ''z'' axis.
* The ''XYZ'' system rotates a third time, about the ''z'' axis again, by angle ''α''.

In sum, the three elemental rotations occur about ''z'', ''x'' and ''z''. This sequence is often denoted ''z''-''x''-''z'' (or 3-1-3). Sets of rotation axes associated with both proper Euler angles and Tait–Bryan angles are commonly named using this notation (see above for the six possibilities for each).

If each step of the rotation acts on the rotating coordinate system XYZ, the rotation is intrinsic (''Z-X'-Z<nowiki>''</nowiki>''). ''Intrinsic'' rotation can also be denoted 3-1-3.

===Signs, ranges and conventions===
Angles are commonly defined according to the [[right-hand rule]]. Namely, they have positive values when they represent a rotation that appears clockwise when looking in the positive direction of the axis, and negative values when the rotation appears counter-clockwise. The opposite convention (left hand rule) is less frequently adopted.

About the ranges (using [[Interval (mathematics)#Notations for intervals|interval notation]]):

* for ''α'' and ''γ'', the range is defined [[Modular arithmetic|modulo]] 2{{pi}} [[radian]]s. For instance, a valid range could be {{closed-closed|&minus;{{pi}}, {{pi}}}}.
* for ''β'', the range covers {{pi}} radians (but can not be said to be modulo&nbsp;{{pi}}). For example, it could be {{closed-closed|0, {{pi}}}} or {{closed-closed|&minus;{{pi}}/2, {{pi}}/2}}.

The angles ''α'', ''β'' and ''γ'' are uniquely determined except for the singular case that the ''xy'' and the ''XY'' planes are identical, i.e. when the ''z'' axis and the ''Z'' axis have the same or opposite directions. Indeed, if the ''z'' axis and the ''Z'' axis are the same, ''β''&nbsp;=&nbsp;0 and only (''α''&nbsp;+&nbsp;''γ'') is uniquely defined (not the individual values), and, similarly, if the ''z'' axis and the ''Z'' axis are opposite, ''β''&nbsp;=&nbsp;{{pi}} and only (''α''&nbsp;&minus;&nbsp;''γ'') is uniquely defined (not the individual values). These ambiguities are known as [[gimbal lock]] in applications.

There are six possibilities of choosing the rotation axes for proper Euler angles. In all of them, the first and third rotation axes are the same. The six possible sequences are:

# ''z''<sub>1</sub>-''x''′-''z''<sub>2</sub>″ (intrinsic rotations) or ''z''<sub>2</sub>-''x''-''z''<sub>1</sub> (extrinsic rotations)
# ''x''<sub>1</sub>-''y''′-''x''<sub>2</sub>″ (intrinsic rotations) or ''x''<sub>2</sub>-''y''-''x''<sub>1</sub> (extrinsic rotations)
# ''y''<sub>1</sub>-''z''′-''y''<sub>2</sub>″ (intrinsic rotations) or ''y''<sub>2</sub>-''z''-''y''<sub>1</sub> (extrinsic rotations)
# ''z''<sub>1</sub>-''y''′-''z''<sub>2</sub>″ (intrinsic rotations) or ''z''<sub>2</sub>-''y''-''z''<sub>1</sub> (extrinsic rotations)
# ''x''<sub>1</sub>-''z''′-''x''<sub>2</sub>″ (intrinsic rotations) or ''x''<sub>2</sub>-''z''-''x''<sub>1</sub> (extrinsic rotations)
# ''y''<sub>1</sub>-''x''′-''y''<sub>2</sub>″ (intrinsic rotations) or ''y''<sub>2</sub>-''x''-''y''<sub>1</sub> (extrinsic rotations)

===Precession, nutation and intrinsic rotation===
[[Image:Praezession.svg|thumb|170px|right|Euler basic motions of the Earth. Intrinsic rotation (green), Precession (blue) and Nutation (red)]]
[[Precession]], [[nutation]], and [[rotation|intrinsic rotation]] are defined as the movements obtained by changing one of the Euler angles while leaving the other two constant. These motions are not all expressed in terms of the external frame, or all in terms of the co-moving rotated body frame, but in a mixture. They constitute a '''mixed axes of rotation''' system{{dash}}precession moves the line of nodes around the external axis ''z'', nutation rotates around the line of nodes ''N'', and intrinsic rotation is around ''Z'', an axis fixed in the body that moves.

Note: If an object undergoes a certain change of orientation this can be described as a combination of precession, nutation, and internal rotation, but how much of each depends on what XYZ coordinate system one has chosen for the object.

As an example, consider a [[Spinning top|top]]. If we define the Z axis to be the symmetry axis of the top, then the top spinning around its own axis of symmetry corresponds to intrinsic rotation. It also rotates around its pivotal axis, with its center of mass orbiting the pivotal axis; this rotation is a precession. Finally, the top may wobble up and down (if it is not what is called a [[symmetric top]]); the change of inclination angle is nutation. The same example can be seen with the movements of the earth.

Though all three movements can be represented by rotation matrices, only precession can be expressed in general as a matrix in the basis of the space without dependencies on the other angles.

These movements also behave as a gimbal set. Given a set of frames, able to move each with respect to the former according to just one angle, like a gimbal, there will exist an external fixed frame, one final frame and two frames in the middle, which are called "intermediate frames". The two in the middle work as two gimbal rings that allow the last frame to reach any orientation in space.