Editing Gimbal lock (section)

===Loss of a degree of freedom with Euler angles===
A rotation in 3D space can be represented numerically with [[matrix (mathematics)|matrices]] in several ways. One of these representations is:
:<math>\begin{align}
R &= \begin{bmatrix}
1 & 0 & 0 \\
0 & \cos \alpha & -\sin \alpha \\
0 & \sin \alpha & \cos \alpha \end{bmatrix} \begin{bmatrix}
\cos \beta & 0 & \sin \beta \\
0 & 1 & 0 \\
-\sin \beta & 0 & \cos \beta \end{bmatrix} \begin{bmatrix}
\cos \gamma & -\sin \gamma & 0 \\
\sin \gamma & \cos \gamma & 0 \\
0 & 0 & 1 \end{bmatrix} \end{align}
</math>

An example worth examining happens when <math>\beta = \tfrac{\pi}{2}</math>. Knowing that <math>\cos \tfrac{\pi}{2} = 0</math> and <math>\sin \tfrac{\pi}{2} = 1</math>, the above expression becomes equal to:
:<math>\begin{align}
 R &= \begin{bmatrix}
1 & 0 & 0 \\
0 & \cos \alpha & -\sin \alpha \\
0 & \sin \alpha & \cos \alpha \end{bmatrix} \begin{bmatrix}
0 & 0 & 1 \\
0 & 1 & 0 \\
-1 & 0 & 0 \end{bmatrix} \begin{bmatrix}
\cos \gamma & -\sin \gamma & 0 \\
\sin \gamma & \cos \gamma & 0 \\
0 & 0 & 1 \end{bmatrix} \end{align}
</math>

Carrying out [[matrix multiplication]]:
:<math>\begin{align}
 R &= \begin{bmatrix}
0 & 0 & 1 \\
\sin \alpha & \cos \alpha & 0 \\
-\cos \alpha & \sin \alpha & 0 \end{bmatrix} \begin{bmatrix}
\cos \gamma & -\sin \gamma & 0 \\
\sin \gamma & \cos \gamma & 0 \\
0 & 0 & 1 \end{bmatrix} = \begin{bmatrix}
0 & 0 & 1 \\
\sin \alpha \cos \gamma + \cos \alpha \sin \gamma & -\sin \alpha \sin \gamma + \cos \alpha \cos \gamma & 0 \\
-\cos \alpha \cos \gamma + \sin \alpha \sin \gamma & \cos \alpha \sin \gamma + \sin \alpha \cos \gamma & 0 \end{bmatrix} \end{align}
</math>

And finally using the [[trigonometry formulas#Angle sum and difference identities|trigonometry formulas]]:
:<math>\begin{align}
 R &= \begin{bmatrix}
0 & 0 & 1 \\
\sin ( \alpha + \gamma ) & \cos (\alpha + \gamma) & 0 \\
-\cos ( \alpha + \gamma ) & \sin (\alpha + \gamma) & 0 \end{bmatrix} \end{align}
</math>

Changing the values of <math>\alpha</math> and <math>\gamma</math> in the above matrix has the same effects: the rotation angle <math>\alpha + \gamma</math> changes, but the rotation axis remains in the <math>Z</math> direction: the last column and the first row in the matrix won't change. The only solution for <math>\alpha</math> and <math>\gamma</math> to recover different roles is to change <math>\beta</math>.

It is possible to imagine an airplane rotated by the above-mentioned Euler angles using the '''X-Y-Z''' convention. In this case, the first angle - <math>\alpha</math> is the pitch. Yaw is then set to <math>\tfrac{\pi}{2}</math> and the final rotation - by <math>\gamma</math> - is again the airplane's pitch. Because of gimbal lock, it has lost one of the degrees of freedom - in this case the ability to roll.

It is also possible to choose another convention for representing a rotation with a matrix using Euler angles than the '''X-Y-Z''' convention above, and also choose other variation intervals for the angles, but in the end there is always at least one value for which a degree of freedom is lost.

The gimbal lock problem does not make Euler angles "invalid" (they always serve as a well-defined coordinate system), but it makes them unsuited for some practical applications.