Table of Rotation Matrices for Fixed Axis (Euler Angle) Conventions
The following matrices assume fixed (world) axes and column vectors, with rotations acting on objects rather than on reference frames. A matrix like that for $x(1)z(2)y(3)$ is constructed as a product of three matrices, Rot($y$)Rot($z$)Rot($x$). To obtain a matrix for the same axis order but with referred frame (body) axes, use the matrix for $yzx$ with $y$ and $x$ swapped. In the matrices, $c_1$ represents $cos(\theta_1)$, $s_1$ represents $sin(\theta_1)$, and similarly for the other subscripts.
| xzx |
$\begin{bmatrix}
c_2 & - c_1 s_2 & s_1 s_2 \\
c_3 s_2 & c_1 c_2 c_3 - s_1 s_3 & - c_2 c_3 s_1 - c_1 s_3 \\
s_2 s_3 & c_3 s_1 + c_1 c_2 s_3 & c_1 c_3 - c_2 s_1 s_3
\end{bmatrix}$
|
xzy |
$\begin{bmatrix}
c_2 c_3 & s_1 s_3 - c_1 c_3 s_2 & c_3 s_1 s_2 + c_1 s_3 \\
s_2 & c_1 c_2 & - c_2 s_1 \\
-c_2 s_3 & c_3 s_1 + c_1 s_2 s_3 & c_1 c_3 - s_1 s_2 s_3
\end{bmatrix}$ |
| xyx |
$\begin{bmatrix}
c_2 & s_1 s_2 & c_1 s_2 \\
s_2 s_3 & c_1 c_3 - c_2 s_1 s_3 & - c_3 s_1 - c_1 c_2 s_3 \\
-c_3 s_2 & c_2 c_3 s_1 + c_1 s_3 & c_1 c_2 c_3 - s_1 s_3
\end{bmatrix}$ |
xyz |
$\begin{bmatrix}
c_2 c_3 & c_3 s_1 s_2 - c_1 s_3 & c_1 c_3 s_2 + s_1 s_3 \\
c_2 s_3 & c_1 c_3 + s_1 s_2 s_3 & c_1 s_2 s_3 - c_3 s_1 \\
-s_2 & c_2 s_1 & c_1 c_2
\end{bmatrix}$ |
| yxy |
$\begin{bmatrix}
c_1 c_3 - c_2 s_1 s_3 & s_2 s_3 & c_3 s_1 + c_1 c_2 s_3 \\
s_1 s_2 & c_2 & - c_1 s_2 \\
-c_2 c_3 s_1 - c_1 s_3 & c_3 s_2 & c_1 c_2 c_3 - s_1 s_3
\end{bmatrix}$ |
yxz |
$\begin{bmatrix}
c_1 c_3 - s_1 s_2 s_3 & - c_2 s_3 & c_3 s_1 + c_1 s_2 s_3 \\
c_3 s_1 s_2 + c_1 s_3 & c_2 c_3 & s_1 s_3 - c_1 c_3 s_2 \\
-c_2 s_1 & s_2 & c_1 c_2
\end{bmatrix}$ |
| yzy |
$\begin{bmatrix}
c_1 c_2 c_3 - s_1 s_3 & - c_3 s_2 & c_2 c_3 s_1 + c_1 s_3 \\
c_1 s_2 & c_2 & s_1 s_2 \\
-c_3 s_1 - c_1 c_2 s_3 & s_2 s_3 & c_1 c_3 - c_2 s_1 s_3\end{bmatrix}$ |
yzx |
$\begin{bmatrix}
c_1 c_2 & - s_2 & c_2 s_1 \\
c_1 c_3 s_2 + s_1 s_3 & c_2 c_3 & c_3 s_1 s_2 - c_1 s_3 \\
c_1 s_2 s_3 - c_3 s_1 & c_2 s_3 & c_1 c_3 + s_1 s_2 s_3\end{bmatrix}$ |
| zyz |
$\begin{bmatrix}
c_1 c_2 c_3 - s_1 s_3 & - c_2 c_3 s_1 - c_1 s_3 & c_3 s_2 \\
c_3 s_1 + c_1 c_2 s_3 & c_1 c_3 - c_2 s_1 s_3 & s_2 s_3 \\
-c_1 s_2 & s_1 s_2 & c_2
\end{bmatrix}$ |
zyx |
$\begin{bmatrix}
c_1 c_2 & - c_2 s_1 & s_2 \\
c_3 s_1 + c_1 s_2 s_3 & c_1 c_3 - s_1 s_2 s_3 & - c_2 s_3 \\
s_1 s_3 - c_1 c_3 s_2 & c_3 s_1 s_2 + c_1 s_3 & c_2 c_3
\end{bmatrix}$ |
| zxz |
$\begin{bmatrix}
c_1 c_3 - c_2 s_1 s_3 & - c_3 s_1 - c_1 c_2 s_3 & s_2 s_3 \\
c_2 c_3 s_1 + c_1 s_3 & c_1 c_2 c_3 - s_1 s_3 & - c_3 s_2 \\
s_1 s_2 & c_1 s_2 & c_2
\end{bmatrix}$ |
zxy |
$\begin{bmatrix}
c_1 c_3 + s_1 s_2 s_3 & c_1 s_2 s_3 - c_3 s_1 & c_2 s_3 \\
c_2 s_1 & c_1 c_2 & - s_2 \\
c_3 s_1 s_2 - c_1 s_3 & c_1 c_3 s_2 + s_1 s_3 & c_2 c_3
\end{bmatrix}$ |
