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?

Shadow Curve http://cs.smith.edu/%7Eorourke/MathOverflow/ProjectionCurve.jpg

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!

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!