 Normal line to a plane curve

At a regular point of a differentiable plane curve, the normal line passes through that point perpendicular to its tangent line. If the tangent slope is a finite nonzero number $m$, the normal slope is $-1/m$, since the direction vectors $(1,m)$ and $(1,-1/m)$ have zero dot product. A horizontal tangent has a vertical normal, and a vertical tangent has a horizontal normal. For a regular level set $F(x,y)=c$, the nonzero gradient $\nabla F$ gives a normal direction because differentiating $F$ along the curve gives $\nabla F\cdot {\mathbf{r}}^{\prime }=0$.