Editing Euler angles (section)

===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)