Let $S$ be the surface of a compact, convex, smooth ($C^\infty$) body in $\mathbb{R}^3$,
with strictly positive Gaussian curvature at every point of $S$.
Fix a direction $z$ in a Cartesian coordinate system,
and consider all the lines parallel to $z$ and tangent to $S$,
which form a topological cylinder enclosing $S$,
touching $S$ on the *shadow boundary* resulting from a light source at $z=+\infty$
(yellow in the figure below).
Parametrize these lines from $s=0$ to $s=1$ around the cylinder,
and let $h(s)$ be the height of the point of tangency
to $S$ above the $xy$-plane, orthogonal to $z$.
My question is:

Can $h(s)$ have an arbitrarily large number of local maxima and minima?

I am interested to learn if this shadow-boundary curve is "well-behaved" in some sense, for smooth convex bodies. Thanks for pointers/suggestions/counterexmaples!