Editing Euler angles (section)

===Rotation matrix===
Any orientation can be achieved by composing three elemental rotations, starting from a known standard orientation. Equivalently, any [[rotation matrix]] ''R'' can be [[Matrix decomposition|decomposed]] as a product of three elemental rotation matrices. For instance:
<math display="block">R = X(\alpha) Y(\beta) Z(\gamma)</math>
is a rotation matrix that may be used to represent a composition of [[#Definition by extrinsic rotations|extrinsic rotations]] about axes ''z'', ''y'', ''x'', (in that order), or a composition of [[#Definition by intrinsic rotations|intrinsic rotations]] about axes ''x''-''y''′-''z''″ (in that order). However, both the definition of the elemental rotation matrices ''X'', ''Y'', ''Z'', and their multiplication order depend on the choices taken by the user about the definition of both rotation matrices and Euler angles (see, for instance, [[Rotation matrix#Ambiguities|Ambiguities in the definition of rotation matrices]]). Unfortunately, different sets of conventions are adopted by users in different contexts. The following table was built according to this set of conventions:

# Each matrix is meant to operate by pre-multiplying [[column vector]]s <math display="inline">\begin{bmatrix} x \\ y \\ z \end{bmatrix}</math> (see [[Rotation matrix#Ambiguities|Ambiguities in the definition of rotation matrices]])
# Each matrix is meant to represent an [[Active and passive transformation|active rotation]] (the composing and composed matrices are supposed to act on the coordinates of vectors defined in the initial fixed reference frame and give as a result the coordinates of a rotated vector defined in the same reference frame).
# Each matrix is meant to represent, primarily, a composition of [[intrinsic rotations]] (around the axes of the rotating reference frame) and, secondarily, the composition of three [[extrinsic rotations]] (which corresponds to the constructive evaluation of the R matrix by the multiplication of three truly elemental matrices, in reverse order).
# [[Right hand rule|Right handed]] reference frames are adopted, and the [[right hand rule]] is used to determine the sign of the angles ''α'', ''β'', ''γ''.
For the sake of simplicity, the following table of matrix products uses the following nomenclature:
# ''X'', ''Y'', ''Z'' are the matrices representing the elemental rotations about the axes ''x'', ''y'', ''z'' of the fixed frame (e.g., ''X''<sub>''α''</sub> represents a rotation about ''x'' by an angle ''α'').
# ''s'' and ''c'' represent sine and cosine (e.g., ''s''<sub>''α''</sub> represents the sine of ''α'').

{| class="wikitable" style="padding-left:1.5em;"
|-
! Proper Euler angles !! Tait–Bryan angles
|-
|<math>X_\alpha Z_\beta X_\gamma = \begin{bmatrix}
 c_\beta          & - c_\gamma s_\beta                            & s_\beta s_\gamma \\
 c_\alpha s_\beta & c_\alpha c_\beta c_\gamma - s_\alpha s_\gamma & - c_\gamma s_\alpha - c_\alpha c_\beta s_\gamma \\
 s_\alpha s_\beta & c_\alpha s_\gamma + c_\beta c_\gamma s_\alpha & c_\alpha c_\gamma - c_\beta s_\alpha s_\gamma
\end{bmatrix}</math>
|<math>X_\alpha Z_\beta Y_\gamma = \begin{bmatrix}
 c_\beta c_\gamma                              & - s_\beta        & c_\beta s_\gamma \\
 s_\alpha s_\gamma + c_\alpha c_\gamma s_\beta & c_\alpha c_\beta & c_\alpha s_\beta s_\gamma - c_\gamma s_\alpha \\
 c_\gamma s_\alpha s_\beta - c_\alpha s_\gamma & c_\beta s_\alpha & c_\alpha c_\gamma + s_\alpha s_\beta s_\gamma
\end{bmatrix}</math>
|-
|<math>X_\alpha Y_\beta X_\gamma = \begin{bmatrix}
 c_\beta            & s_\beta s_\gamma                              & c_\gamma s_\beta \\
 s_\alpha s_\beta   & c_\alpha c_\gamma - c_\beta s_\alpha s_\gamma & - c_\alpha s_\gamma - c_\beta c_\gamma s_\alpha \\
 - c_\alpha s_\beta & c_\gamma s_\alpha + c_\alpha c_\beta s_\gamma & c_\alpha c_\beta c_\gamma - s_\alpha s_\gamma
\end{bmatrix}</math>
|<math>X_\alpha Y_\beta Z_\gamma = \begin{bmatrix}
 c_\beta c_\gamma                              & - c_\beta s_\gamma                            & s_\beta \\
 c_\alpha s_\gamma + c_\gamma s_\alpha s_\beta & c_\alpha c_\gamma - s_\alpha s_\beta s_\gamma & - c_\beta s_\alpha \\
 s_\alpha s_\gamma - c_\alpha c_\gamma s_\beta & c_\gamma s_\alpha + c_\alpha s_\beta s_\gamma & c_\alpha c_\beta
\end{bmatrix}</math>
|-
|<math>Y_\alpha X_\beta Y_\gamma = \begin{bmatrix}
 c_\alpha c_\gamma - c_\beta s_\alpha s_\gamma   & s_\alpha s_\beta & c_\alpha s_\gamma + c_\beta c_\gamma s_\alpha \\
 s_\beta s_\gamma                                & c_\beta          & - c_\gamma s_\beta \\
 - c_\gamma s_\alpha - c_\alpha c_\beta s_\gamma & c_\alpha s_\beta & c_\alpha c_\beta c_\gamma - s_\alpha s_\gamma
\end{bmatrix}</math>
|<math>Y_\alpha X_\beta Z_\gamma = \begin{bmatrix}
 c_\alpha c_\gamma + s_\alpha s_\beta s_\gamma & c_\gamma s_\alpha s_\beta - c_\alpha s_\gamma &   c_\beta s_\alpha \\
 c_\beta s_\gamma                              & c_\beta c_\gamma                              & - s_\beta \\
 c_\alpha s_\beta s_\gamma - c_\gamma s_\alpha & c_\alpha c_\gamma s_\beta + s_\alpha s_\gamma & c_\alpha c_\beta
\end{bmatrix}</math>
|-
|<math>Y_\alpha Z_\beta Y_\gamma = \begin{bmatrix}
 c_\alpha c_\beta c_\gamma - s_\alpha s_\gamma    & - c_\alpha s_\beta & c_\gamma s_\alpha + c_\alpha c_\beta s_\gamma \\
 c_\gamma s_\beta                                 & c_\beta            & s_\beta s_\gamma \\
  - c_\alpha s_\gamma - c_\beta c_\gamma s_\alpha & s_\alpha s_\beta   & c_\alpha c_\gamma - c_\beta s_\alpha s_\gamma
\end{bmatrix}</math>
|<math>Y_\alpha Z_\beta X_\gamma = \begin{bmatrix}
 c_\alpha c_\beta   & s_\alpha s_\gamma - c_\alpha c_\gamma s_\beta & c_\gamma s_\alpha + c_\alpha s_\beta s_\gamma \\
 s_\beta            & c_\beta c_\gamma                              & - c_\beta s_\gamma \\
 - c_\beta s_\alpha & c_\alpha s_\gamma + c_\gamma s_\alpha s_\beta & c_\alpha c_\gamma - s_\alpha s_\beta s_\gamma
\end{bmatrix}</math>
|-
|<math>Z_\alpha Y_\beta Z_\gamma = \begin{bmatrix}
 c_\alpha c_\beta c_\gamma - s_\alpha s_\gamma &  - c_\gamma s_\alpha - c_\alpha c_\beta s_\gamma & c_\alpha s_\beta \\
 c_\alpha s_\gamma + c_\beta c_\gamma s_\alpha & c_\alpha c_\gamma - c_\beta s_\alpha s_\gamma    & s_\alpha s_\beta \\
 - c_\gamma s_\beta                            & s_\beta s_\gamma                                 & c_\beta
\end{bmatrix}</math>
|<math>Z_\alpha Y_\beta X_\gamma = \begin{bmatrix}
 c_\alpha c_\beta & c_\alpha s_\beta s_\gamma - c_\gamma s_\alpha & s_\alpha s_\gamma + c_\alpha c_\gamma s_\beta \\
 c_\beta s_\alpha & c_\alpha c_\gamma + s_\alpha s_\beta s_\gamma & c_\gamma s_\alpha s_\beta - c_\alpha s_\gamma \\
 - s_\beta        & c_\beta s_\gamma                              & c_\beta c_\gamma
\end{bmatrix}</math>
|-
|<math>Z_\alpha X_\beta Z_\gamma = \begin{bmatrix}
 c_\alpha c_\gamma - c_\beta s_\alpha s_\gamma & - c_\alpha s_\gamma - c_\beta c_\gamma s_\alpha &  s_\alpha s_\beta \\
 c_\gamma s_\alpha + c_\alpha c_\beta s_\gamma & c_\alpha c_\beta c_\gamma - s_\alpha s_\gamma   & - c_\alpha s_\beta \\
 s_\beta s_\gamma                              & c_\gamma s_\beta                                & c_\beta
\end{bmatrix}</math>
|<math>Z_\alpha X_\beta Y_\gamma = \begin{bmatrix}
 c_\alpha c_\gamma - s_\alpha s_\beta s_\gamma & - c_\beta s_\alpha & c_\alpha s_\gamma + c_\gamma s_\alpha s_\beta \\
 c_\gamma s_\alpha + c_\alpha s_\beta s_\gamma & c_\alpha c_\beta   & s_\alpha s_\gamma - c_\alpha c_\gamma s_\beta \\
 - c_\beta s_\gamma                            & s_\beta            & c_\beta c_\gamma
\end{bmatrix}</math>
|}

These tabular results are available in numerous textbooks.<ref>''E.g.'' Appendix I (p. 483) of: {{cite book | last1 = Roithmayr | first1 = Carlos M. | last2 = Hodges | first2 = Dewey H. | title = Dynamics: Theory and Application of Kane's Method | edition = 1st | publisher = Cambridge University Press | year = 2016 | isbn = 978-1107005693 }}</ref>  For each column the last row constitutes the most commonly used convention.

To change the formulas for [[Active and passive transformation|passive rotations]] (or find reverse active rotation), transpose the matrices (then each matrix transforms the initial coordinates of a vector remaining fixed to the coordinates of the same vector measured in the rotated reference system; same rotation axis, same angles, but now the coordinate system rotates, rather than the vector).

The following table contains formulas for angles ''α'', ''β'' and ''γ'' from elements of a rotation matrix <math>R</math>.<ref>{{Cite tech report | last = Henderson | first = D. M. | date = 1977-06-09 | title = Euler angles, quaternions, and transformation matrices for space shuttle analysis|url=https://ntrs.nasa.gov/citations/19770019231|institution = NASA | language = en | pages = 12–24}}</ref>
{| class="wikitable"
! colspan="2" |Proper Euler angles
! colspan="2" |Tait–Bryan angles
|-
|<math>X_\alpha Z_\beta X_\gamma</math>
|<math>\begin{align}
\alpha & = \arctan\left(\frac{R_{31}}{R_{21}}\right) \\
\beta & = \arccos\left(R_{11}\right) \\
\gamma & = \arctan\left(\frac{R_{13}}{-R_{12}}\right)
\end{align}</math>
|<math>X_\alpha Z_\beta Y_\gamma</math>
|<math>\begin{align}
\alpha & = \arctan\left(\frac{R_{32}}{R_{22}}\right) \\
\beta & = \arcsin\left(-R_{12}\right) \\
\gamma & = \arctan\left(\frac{R_{13}}{R_{11}}\right)
\end{align}</math>
|-
|<math>X_\alpha Y_\beta X_\gamma</math>
|<math>\begin{align}
\alpha & = \arctan\left(\frac{R_{21}}{-R_{31}}\right) \\
\beta & = \arccos\left(R_{11}\right) \\
\gamma & = \arctan\left(\frac{R_{12}}{R_{13}}\right)
\end{align}</math>
|<math>X_\alpha Y_\beta Z_\gamma</math>
|<math>\begin{align}
\alpha & = \arctan\left(\frac{-R_{23}}{R_{33}}\right) \\
\beta & = \arcsin\left(R_{13}\right) \\
\gamma & = \arctan\left(\frac{-R_{12}}{R_{11}}\right)
\end{align}</math>
|-
|<math>Y_\alpha X_\beta Y_\gamma</math>
|<math>\begin{align}
\alpha & = \arctan\left(\frac{R_{12}}{R_{32}}\right) \\
\beta & = \arccos\left(R_{22}\right) \\
\gamma & = \arctan\left(\frac{R_{21}}{-R_{23}}\right)
\end{align}</math>
|<math>Y_\alpha X_\beta Z_\gamma</math>
|<math>\begin{align}
\alpha & = \arctan\left(\frac{R_{13}}{R_{33}}\right) \\
\beta & = \arcsin\left(-R_{23}\right) \\
\gamma & = \arctan\left(\frac{R_{21}}{R_{22}}\right)
\end{align}</math>
|-
|<math>Y_\alpha Z_\beta Y_\gamma</math>
|<math>\begin{align}
\alpha & = \arctan\left(\frac{R_{32}}{-R_{12}}\right) \\
\beta & = \arccos\left(R_{22}\right) \\
\gamma & = \arctan\left(\frac{R_{23}}{R_{21}}\right)
\end{align}</math>
|<math>Y_\alpha Z_\beta X_\gamma</math>
|<math>\begin{align}
\alpha & = \arctan\left(\frac{-R_{31}}{R_{11}}\right) \\
\beta & = \arcsin\left(R_{21}\right) \\
\gamma & = \arctan\left(\frac{-R_{23}}{R_{22}}\right)
\end{align}</math>
|-
|<math>Z_\alpha Y_\beta Z_\gamma</math>
|<math>\begin{align}
\alpha & = \arctan\left(\frac{R_{23}}{R_{13}}\right) \\
\beta & = \arctan\left(\frac{\sqrt{1-R_{33}^2}}{R_{33}}\right) \\
\gamma & = \arctan\left(\frac{R_{32}}{-R_{31}}\right)
\end{align}</math>
|<math>Z_\alpha Y_\beta X_\gamma</math>
|<math>\begin{align}
\alpha & = \arctan\left(\frac{R_{21}}{R_{11}}\right) \\
\beta & = \arcsin\left(-R_{31}\right) \\
\gamma & = \arctan\left(\frac{R_{32}}{R_{33}}\right)
\end{align}</math>
|-
|<math>Z_\alpha X_\beta Z_\gamma</math>
|<math>\begin{align}
\alpha & = \arctan\left(\frac{R_{13}}{-R_{23}}\right) \\
\beta & = \arccos\left(R_{33}\right) \\
\gamma & = \arctan\left(\frac{R_{31}}{R_{32}}\right)
\end{align}</math>
|<math>Z_\alpha X_\beta Y_\gamma</math>
|<math>\begin{align}
\alpha & = \arctan\left(\frac{-R_{12}}{R_{22}}\right) \\
\beta & = \arcsin\left(R_{32}\right) \\
\gamma & = \arctan\left(\frac{-R_{31}}{R_{33}}\right)
\end{align}</math>
|}