Question about definition of generator for dihedral groups?

Question

Dummit and Foote mention that a relation for the dihedral group is $rs = sr^{-1}$.

Now, I have interpreted the statement to mean $r$ is a rotation of $\frac{2\pi}{n}$ radians and $s$ is an involution. But take an involution of a hexagon at any point then, and go one $\frac{2\pi}{n}$ clockwise.

I have noticed that if we go $\frac{2\pi}{n}$ counterclockwise and then take the involution, we do not get to the same point. This only works for $n > 2$, though, and only if we perform the operation from the same vertex. $\textbf{Is there any specific proof why}$?

Answer 1

Just watch the central angles of the vertices, measured from (one direction of) the axis of $s$.

Say, a vertex $x$ has original angle $\alpha$, then if you first reflect it to $s$, its angle becomes $-\alpha$, then you rotate it, it becomes $-\alpha+\frac{2\pi}n$.

On the other hand, if you first rotate backwards, becomes $\alpha-\frac{2\pi}n$, then reflect it by $s$, yielding the same $-(\alpha-\frac{2\pi}n)$.