Representing Isometries in General

Step 1: Translate the line to pass through the origin

[math]P\left(x,y\right)\longrightarrow P'\left(x,y-3\right)[/math]

Step 2: Rotate to align with the x-axis

To rotate, we use a rotation around the origin by [math]-\theta[/math] where [math]\theta=tan^{-1}\left(\sqrt{3}\right)=60^\circ[/math].

Step 3: Reflect over the x-axis

[math]P''\left(0.5x+\frac{\sqrt{3}}{2}\left(y-3\right),-\frac{\sqrt{3}}{2}x+0.5\left(y-3\right)\right)\longrightarrow P'''\left(0.5x+\frac{\sqrt{3}}{2}\left(y-3\right),\frac{\sqrt{3}}{2}x-0.5\left(y-3\right)\right)[/math]

Step 4: Rotate Back

Step 5: Translate Back

[math]P''''\left(\frac{1}{4}x+\frac{\sqrt{3}}{4}\left(y-3\right)-\frac{3}{4}x+\frac{\sqrt{3}}{4}\left(y-3\right),\frac{\sqrt{3}}{4}x+\frac{3}{4}\left(y-3\right)+\frac{\sqrt{3}}{4}x-\frac{1}{4}\left(y-3\right)\right)\longrightarrow P'''''\left(\frac{1}{4}x+\frac{\sqrt{3}}{4}\left(y-3\right)-\frac{3}{4}x+\frac{\sqrt{3}}{4}\left(y-3\right),\frac{\sqrt{3}}{4}x+\frac{3}{4}\left(y-3\right)+\frac{\sqrt{3}}{4}x-\frac{1}{4}\left(y-3\right)+3\right)[/math]