[Math] What’s the angle of rotation of a product of two reflections

geometrygroup-theory

Let $F_1$ and $F_2$ be two arbitrary reflections about two lines in $\mathbb R^2$.

I've been trying to work out the angle of rotation of $R_1R_2$. To this end I drew pictures in which I reflect one point first along $R_1$ then along $R_2$. Then my plan was to calculate the angle of rotation between the point and its image but the problem I ran into was that I don't have the center of rotation.

Could someone help me and explain to me how to find the angle of $F_1
F_2$ (the product)?

Edit If possible using a geometric argument.

Best Answer

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The figure gives a simple geometric answer. Let $O$ the fixed point of the two reflections that is the fixed point of the rotation. Than:

$P'$ is the relfection of $P$ in the line $OM$ and $P''$ the reflection of $P'$ in $ON$ and we have $$ \angle POM=\angle P'OM $$ $$ \angle P'ON=\angle P''ON $$ and $\angle P'ON+\angle P'OM= \angle MON$, so $\angle POP''=2 \angle MON$

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