Line Transformations $f(x) = ax + b$ Rotate ( by $\gamma$ at $(x,y)$ ) $$ \alpha = \arctan(a) $$ $$ a' = \tan(\alpha \pm \gamma) $$ $$ b' = y - (x \cdot a') $$ Translate (by $\perp$ distance $d$) $$ a' = a $$ $$ b' = \pm d \cdot \sqrt{1 + a^2} + b $$ Translate (by $\vec{x}$ distance $d$) $$ z = \frac{-b}{a} \pm d $$ $$ a' = a $$ $$ b' = -(z \cdot a') $$ Translate (by $\vec{y}$ distance $d$) $$ a' = a $$ $$ b' = b \pm d $$