Editing Euler angles (section)

===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.