Rotations in three dimensions

Lie groups are continuous groups. An example of a continuous group, that I manage to visualize, are rotations in three-dimensional space.

Fig. 1: An arbitrary rotation is undone by three rotations about the z-, y- and x-axes. 

Rotations form a group since:

• The result of two rotations \(a\) and \(b\), conducted one after the other, written formally as \(a \circ b\), is again a rotation ("closure").
• A series of rotations can be grouped at will, we'll always arrive at the same result. I.e. \( (a \circ b) \circ c = a \circ (b \circ c)\) ("associativity").
• Rotating an object by zero degrees is regarded a rotation as well. This operation, denoted by \(e\), commutes with every other rotation \( e \circ a = a \circ e\) ("identity element").
• Every rotation through an angle \(\alpha\) can be undone by a rotation through \(-\alpha\) ("inverse element").

Every rotation in three-dimensional space can be decomposed into a rotation about the x-axis, a rotation about the y-axis and a rotation about the z-axis (Fig. 1).

Fig.2: Three-dimensional rotations are not commutative.

Rotations in three-dimensional space are non-commutative. I.e. in general the order of the rotations does matter for the final position (Fig. 2).

Algebraically rotations are expressed in terms of 3x3 matrices.
\[
R_x(\alpha) =
\begin{pmatrix}
1 & 0 & 0 \\
0 & \cos\alpha & \sin\alpha \\
0 & -\sin\alpha & \cos\alpha
\end{pmatrix} \\
R_y(\alpha) =
\begin{pmatrix}
\cos\alpha  & 0 & \sin\alpha \\
0 & 1 & 0 \\
-\sin\alpha & 0 & \cos\alpha
\end{pmatrix} \\
R_z(\alpha) =
\begin{pmatrix}
\cos\alpha  & \sin\alpha & 0 \\
-\sin\alpha & \cos\alpha & 0 \\
0 & 0 & 1
\end{pmatrix}
\]
Here, the matrix \(R_x(\alpha)\) describes the rotation about the x-axis by an angle \(\alpha\). E.g., the point (x,y,z) rotated by \(\alpha\) about the x-axis, by \(\beta\) about the y-axis and by \(\gamma\) about the z-axis is given by
\[
R_z(\gamma) \cdot
R_y(\beta) \cdot
R_x(\alpha) \cdot
\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}
\]