Editing Rotation matrix (section)

=== Euler angles ===
Complexity of conversion escalates with [[Euler angles]] (used here in the broad sense). The first difficulty is to establish which of the twenty-four variations of Cartesian axis order we will use. Suppose the three angles are {{math|''θ''<sub>1</sub>}}, {{math|''θ''<sub>2</sub>}}, {{math|''θ''<sub>3</sub>}}; physics and chemistry may interpret these as
<!--
 Don't even THINK of "fixing" this; it is NOT a typo. "zyz" is correct.
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:<math> Q(\theta_1,\theta_2,\theta_3)=  Q_{\mathbf{z}}(\theta_1) Q_{\mathbf{y}}(\theta_2) Q_{\mathbf{z}}(\theta_3) , </math>
<!--                                               ^ --- That's right, this is MEANT to be a "z", not an "x". -->
while aircraft dynamics may use
:<math> Q(\theta_1,\theta_2,\theta_3)=  Q_{\mathbf{z}}(\theta_3) Q_{\mathbf{y}}(\theta_2) Q_{\mathbf{x}}(\theta_1) . </math>
One systematic approach begins with choosing the rightmost axis. Among all [[permutation]]s of {{math|(''x'',''y'',''z'')}}, only two place that axis first; one is an even permutation and the other odd. Choosing parity thus establishes the middle axis. That leaves two choices for the left-most axis, either duplicating the first or not. These three choices gives us {{nowrap|3 × 2 × 2 {{=}} 12}} variations; we double that to 24 by choosing static or rotating axes.

This is enough to construct a matrix from angles, but triples differing in many ways can give the same rotation matrix. For example, suppose we use the {{math|'''zyz'''}} convention above; then we have the following equivalent pairs:
:{| style="text-align:right"
| (90°,||45°,||−105°) || ≡ || (−270°,||−315°,||255°) || ''multiples of 360°''
|-
| (72°,||0°,||0°) || ≡ || (40°,||0°,||32°) || ''singular alignment''
|-
| (45°,||60°,||−30°) || ≡ || (−135°,||−60°,||150°) || ''bistable flip''
|}
Angles for any order can be found using a concise common routine ({{Harvnb|Herter|Lott|1993}}; {{Harvnb|Shoemake|1994}}).

The problem of singular alignment, the mathematical analog of physical [[gimbal lock]], occurs when the middle rotation aligns the axes of the first and last rotations. It afflicts every axis order at either even or odd multiples of 90°. These singularities are not characteristic of the rotation matrix as such, and only occur with the usage of Euler angles.

The singularities are avoided when considering and manipulating the rotation matrix as orthonormal row vectors (in 3D applications often named the right-vector, up-vector and out-vector) instead of as angles. The singularities are also avoided when working with quaternions.