Given a smooth function $f: \mathbb R^3 \rightarrow \mathbb R$, a surface $S = \{f=0\}$ casts a (orthogonally projected) shadow to a plane with unit normal $\mathbb n$ that, provided $S$ is convex, has a boundary given by the subset $\{\langle \nabla f, \mathbb n \rangle = 0\}$, projected to the plane. So for an implicitly defined convex surface, we can obtain an implicitly defined curve for its shadow boundary on any plane.
I'm hoping to do the same thing, but for a parametric surface and obtain a corresponding parametric curve for the boundary.
Specifically I'm looking at quadric surfaces and to describe their shadows with (what I expect to be) conic sections, but a more general treatment would be neat.
So to set this up, we have a parameterization $s: \mathbb R^2 \rightarrow \mathbb R^3$ of $S$. Then I guess I'm looking for a curve $\alpha : \mathbb R \rightarrow \mathbb R^2$ such that $\langle \nabla f, \mathbb n \rangle \circ \alpha = 0$. Then I could simply project $s \circ \alpha$ to the plane. But in a parametric representation $s$ of $S$, I don't necessarily have enough information to form such an $f$? Then how to proceed? Is there a simple method I'm overlooking?
