Editing Euler angles (section)

==Applications==

===Vehicles and moving frames===
{{Main|rigid body}}
{{See also|axes conventions}}

Their main advantage over other orientation descriptions is that they are directly measurable from a gimbal mounted in a vehicle. As gyroscopes keep their rotation axis constant, angles measured in a gyro frame are equivalent to angles measured in the lab frame. Therefore, gyros are used to know the actual orientation of moving spacecraft, and Euler angles are directly measurable. Intrinsic rotation angle cannot be read from a single gimbal, so there has to be more than one gimbal in a spacecraft. Normally there are at least three for redundancy. There is also a relation to the well-known [[gimbal lock]] problem of [[mechanical engineering]].<ref>The relation between the Euler angles and the Cardan suspension is explained in chap. 11.7 of the following textbook: U. Krey, A. Owen, ''Basic Theoretical Physics – A Concise Overview'', New York, London, Berlin, Heidelberg, Springer (2007)&nbsp;.</ref>

When studying rigid bodies in general, one calls the ''xyz'' system ''space coordinates'', and the ''XYZ'' system ''body coordinates''. The space coordinates are treated as unmoving, while the body coordinates are considered embedded in the moving body. Calculations involving [[acceleration]], [[angular acceleration]], [[angular velocity]], [[angular momentum]], and [[kinetic energy]] are often easiest in body coordinates, because then the moment of inertia tensor does not change in time.  If one also diagonalizes the rigid body's moment of inertia tensor (with nine components, six of which are independent), then one has a set of coordinates (called the principal axes) in which the moment of inertia tensor has only three components.

The angular velocity of a rigid body takes a [[Angular velocity#Components from Euler angles|simple form]] using Euler angles in the moving frame. Also the [[Euler's equations (rigid body dynamics)|Euler's rigid body equations]] are simpler because the inertia tensor is constant in that frame.

===Crystallographic texture===
[[File:MAUD-MTEX-TiAl-hasylab-2003-Liss.png|thumb|Pole figures displaying crystallographic texture of gamma-TiAl in an alpha2-gamma alloy, as measured by high energy X-rays.<ref name=Liss>{{ cite journal |vauthors=Liss KD, Bartels A, Schreyer A, Clemens H |title=High energy X-rays: A tool for advanced bulk investigations in materials science and physics |journal=Textures Microstruct. |year=2003 |volume=35 |issue=3/4 |pages=219–52 |doi=10.1080/07303300310001634952 |doi-access=free }}</ref>]]
In materials science, crystallographic [[texture (crystalline)|texture]] (or preferred orientation) can be described using Euler angles. In texture analysis, the Euler angles provide a mathematical depiction of the orientation of individual crystallites within a polycrystalline material, allowing for the quantitative description of the macroscopic material.<ref>{{Citation
  |last1 = Kocks
  |first1 = U.F.
  |last2 = Tomé
  |first2 = C.N.
  |last3 = Wenk
  |first3 = H.-R.
  |title = Texture and Anisotropy: Preferred Orientations in Polycrystals and their effect on Materials Properties
  |publisher = [[Cambridge University Press|Cambridge]]
  |year = 2000
  |isbn = 978-0-521-79420-6}}</ref>  
The most common definition of the angles is due to Bunge and corresponds to the ''ZXZ'' convention.  It is important to note, however, that the application generally involves axis transformations of tensor quantities, i.e. passive rotations.  Thus the matrix that corresponds to the Bunge Euler angles is the transpose of that shown in the table above.<ref>{{Citation
  |last = Bunge
  |first = H.
  |title = Texture Analysis in Materials Science: Mathematical Methods
  |publisher = [[Cuvillier Verlag]]
  |year = 1993
  |asin = B0014XV9HU}}</ref>

===Others===
[[File:Automation of foundry with robot.jpg|thumb|right|Industrial robot operating in a foundry]]
Euler angles, normally in the Tait–Bryan convention, are also used in [[Industrial robot|robotics]] for speaking about the degrees of freedom of a [[Robotic arm|wrist]]. They are also used in [[electronic stability control]] in a similar way.

Gun fire control systems require corrections to gun-order angles (bearing and elevation) to compensate for deck tilt (pitch and roll). In traditional systems, a stabilizing gyroscope with a vertical spin axis corrects for deck tilt, and stabilizes the optical sights and radar antenna. However, gun barrels point in a direction different from the line of sight to the target, to anticipate target movement and fall of the projectile due to gravity, among other factors. Gun mounts roll and pitch with the deck plane, but also require stabilization. Gun orders include angles computed from the vertical gyro data, and those computations involve Euler angles.

Euler angles are also used extensively in the quantum mechanics of angular momentum. In quantum mechanics, explicit descriptions of the representations of SO(3) are very important for calculations, and almost all the work has been done using Euler angles. In the early history of quantum mechanics, when physicists and chemists had a sharply negative reaction towards abstract group theoretic methods (called the ''Gruppenpest''), reliance on Euler angles was also essential for basic theoretical work.

Many mobile computing devices contain [[accelerometer]]s which can determine these devices' Euler angles with respect to the earth's gravitational attraction.  These are used in applications such as games, [[bubble level]] simulations, and [[kaleidoscope]]s.{{citation needed|reason=accelerometers can measure orientation, yes; is there any evidence they use Euler angles for this?|date=May 2011}}