Definition of a shadow in space, and how to derive a shadow for a given shape
$\newcommand{\Reals}{\mathbf{R}}$Imagine a light "at infinity" on the $z$-axis: Its rays travel along lines parallel to the $z$-axis. If $S \subset \Reals^{3}$, and if $(x_0, y_0)$ is a point of $\Reals^{2}$, then the ray of light $\{(x_0, y_0, t): t > 0\}$ touches $S$ if and only if there exists a $z > 0$ such that $(x_0, y_0, z) \in S$, if and only if $(x_0, y_0, 0)$ lies in the shadow of $S$.
