Let $S$ be a surface parametrized by $$r:\mathcal{U}\subseteq\mathbb{R}^{2}\rightarrow\mathbb{R}^{3}$$ Let $p\in S$ be any point and $T_{p}S$ be the tangent plain passing through $p$. Let $v\in T_{p}S$ such that $||v||=1$, and let $H_{v}$ the plane passing through $p$ and parallel to $v$ and $N(p)$, where $N(p)$ is the unitary vector orthogonal to $p$. Prove that, in a neighborhood $V$ of $p$, $H_{v}\cap S$ is a regular surface.
I though about a function $$f:\mathcal{U}\rightarrow\mathbb{R}$$ $$f(y):=⟨r(y)-p,v\wedge N(p)⟩$$ so I can study the critical point and I can caracterize the set of $w\in H_{v}\cap S$, but honestly I don't know to proceed with this way. Can soneone help me? Thanks before!