An interesting subcomplex of the Delaunay triangulation are $\alpha$-shapes, which give a more detailed representation of the shape of point sets than the convex hull. We extend an algorithm which computes all Delaunay simplices over all time windows to also compute the temporal $\alpha$-shape, which is a description of all $\alpha$-shapes over all time windows and all values of $\alpha$, in output-sensitive linear time. We present an interactive demo application based on a fast query data structure. Experimental results show that our algorithm is practical and can be used on real-world data sets.

翻译：Delaunay三角形的一个有趣的子复合体是 $\ alpha$- shape, 它比 convex 外壳更详细地描述点形的形状。 我们扩展了一个计算所有时间窗口中所有Delaunay implices的算法, 以同时计算时间 $\ alpha$- shape, 这是描述所有时间窗口中的所有 $\ alpha$- shape, 以及输出敏感线性时间中的所有值 $\ alpha$ 。 我们根据快速查询数据结构展示一个交互式演示应用程序。 实验结果显示, 我们的算法是实用的, 可以用于真实世界的数据集 。</s>