1. Rotation Around the X-Axis
Rotating a point with homogeneous coordinates \((x, y, z, w)\) around the X-axis by an angle \(\alpha\) can be achieved using
\[ \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & \cos\alpha & -\sin\alpha & 0 \\
0 & \sin\alpha & \cos\alpha & 0 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}
\cdot
\begin{bmatrix}
x \\
y \\
z \\
w \\
\end{bmatrix}\]
2. Rotation Around the Y-Axis
Rotating a point with homogeneous coordinates \((x, y, z, w)\) around the Y-axis by an angle \(\alpha\) can be computed as follows:
\[ \begin{bmatrix}
\cos\alpha & 0 & \sin\alpha & 0 \\
0 & 1 & 0 & 0 \\
-\sin\alpha & 0 & \cos\alpha & 0 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}
\cdot
\begin{bmatrix}
x \\
y \\
z \\
w \\
\end{bmatrix}\]
3. Rotation Around the Z-Axis
Rotating a point with homogeneous coordinates \((x, y, z, w)\) around the Z-axis by an angle \(\alpha\):
\[ \begin{bmatrix}
\cos\alpha & -\sin\alpha & 0 & 0 \\
\sin\alpha & \cos\alpha & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}
\cdot
\begin{bmatrix}
x \\
y \\
z \\
w \\
\end{bmatrix}\]