Using Mather's theorem on stable mappings/using multijet transversality, we know that the set of immersion with normal crossing $\mathcal M\subseteq C^\infty(S^1,\mathbb R^2)$ is dense and open in the Whitney $C^\infty$-topology. Now, given an embedding (a knot) $f:S^1\to\mathbb R^3$, I want to show that there is a projection $\pi:\mathbb R^3\to \mathbb R^2$ such that $\pi\circ f$ is a knot diagram with normal crossing, i.e. that $\pi\circ f\in\mathcal M$.
My thought: The set of projections $\mathbb R^3\to \mathbb R^2$ is parametrized by $S^2$, i.e. given any vector $p\in S^2$, we can denote $\pi_p$ as the projection onto $p^\perp$. Then, we can get a map $S^2\to C^\infty(S^1,\mathbb R^2)$ via
$$p\mapsto\pi_p\circ f$$
Knowing that $\mathcal M$ is open dense in $C^\infty(S^1,\mathbb R^2)$, how to show that the preimage of $\mathcal M$ of the above map is non-empty?