Normal line ($\mathbf{x}=\mathbf{p}+t\mathbf{n}$)

At a point $\mathbf{p}$ of a smooth surface with nonzero normal vector $\mathbf{n}$, its normal line is $\{\mathbf{p}+t\mathbf{n}:t\in \mathbb{R}\}$. For a level set $F=c$ with $\nabla F(\mathbf{p})\ne 0$, one can use $\mathbf{n}=\nabla F(\mathbf{p})$.