determining rotations and reflections in ${ℝ}^2$

Let $E:{ℝ}^2\to {ℝ}^2$ be a rotation about some point $(x_0,y_0)$, and let $\theta$ the angle of rotation for $E$. A formula for $E$ can be determined as follows.

First, translate the point $(x_0,y_0)$ to $(0,0)$. The map of this translation is $T(x,y)=(x-x_0,y-y_0)$.

Next, rotate by $\theta$ about the origin. The map of this rotation is $R(x,y)=(x\cos \theta -y\sin \theta ,x\sin \theta +y\cos \theta )$.

Finally, translate the point $(0,0)$ back to $(x_0,y_0)$. The map of this translation is $T^{-1}(x,y)=(x+x_0,y+y_0)$.

The fact that $E=T^{-1}\circ R\circ T$ can be used to obtain a formula for $E$:

Let $E:{ℝ}^2\to {ℝ}^2$ be a reflection about some line $y=mx+b$. Let $\theta =2\arctan m$. A formula for $E$ can be determined as follows.

First, translate the $y$ intercept $(0,b)$ to $(0,0)$. The map of this translation is $B(x,y)=(x,y-b)$.

Next, reflect about the line $y=mx$. The map of this reflection is $F(x,y)=(x\cos \theta +y\sin \theta ,x\sin \theta -y\cos \theta )$.

Finally, translate the point $(0,0)$ back to $(0,b)$. The map of this translation is $B^{-1}(x,y)=(x,y+b)$.

The fact that $E=B^{-1}\circ F\circ B$ can be used to obtain a formula for $E$:

Let $E:{ℝ}^2\to {ℝ}^2$ be a reflection about some line $x=c$. A formula for $E$ can be determined as follows.

First, translate the $x$ intercept $(c,0)$ to $(0,0)$. The map of this translation is $C(x,y)=(x-c,y)$.

Next, reflect about the line $x=0$. The map of this reflection is $M(x,y)=(-x,y)$.

Finally, translate the point $(0,0)$ back to $(c,0)$. The map of this translation is $C^{-1}(x,y)=(x+c,y)$.

The fact that $E=C^{-1}\circ M\circ C$ can be used to obtain a formula for $E$: