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 $d$.

翻译：形状是理解点集的强大工具。α-形状作为凸包的推广提供了可调节的细节层次，正式定义了形状的概念。许多现实世界的点集具有固有的时间属性，因为自然过程常随时间发生，例如雷暴中的闪电或移动的动物群体。为探索这类每个点都关联时间戳的点集，交互式应用可利用α-形状，允许用户指定不同的时间窗口和α值。我们展示了如何以输出敏感的线性时间计算时间α-形状α_T——对所有时间窗口上所有α-形状的最小化描述，并给出了α_T的复杂度上界。利用α_T，我们可交互式可视化用户指定时间窗口内的α-形状，而无需持续计算所需的α-形状。实验结果表明，我们的方法性能至少比现有方法提升约52倍，且计算得到的描述在实践中具有合理规模。该算法基于现有算法，该算法以每个三角形O(1)的时间复杂度计算所有时间窗口上的Delaunay三角剖分。对于固定维度d，我们的方法可推广至更高维度并保持相同运行时间。