  Shape is a powerful tool to understand point sets. A formal notion of shape
is given by $\alpha$-shapes, which generalize the convex hull and provide
adjustable level of detail. Many real-world point sets have an inherent
temporal property as natural processes often happen over time, like lightning
strikes during thunderstorms or moving animal swarms. To explore such point
sets, where each point is associated with one timestamp, interactive
applications may utilize $\alpha$-shapes and allow the user to specify
different time windows and $\alpha$-values. We show how to compute the temporal
$\alpha$-shape $\alpha_T$, a minimal description of all $\alpha$-shapes over
all time windows, in output-sensitive linear time. We also give complexity
bounds on $|\alpha_T|$. We use $\alpha_T$ to interactively visualize
$\alpha$-shapes of user-specified time windows without having to constantly
compute requested $\alpha$-shapes. Experimental results suggest that our
approach outperforms an existing approach by a factor of at least $\sim$52 and
that the description we compute has reasonable size in practice. The basis for
our algorithm is an existing algorithm which computes all Delaunay triangles
over all time windows using $\mathcal{O}(1)$ time per triangle. Our approach
generalizes to higher dimensions with the same runtime for fixed $d$.
