Traditional problems in computational geometry involve aspects that are both discrete and continuous. One such example is nearest-neighbor searching, where the input is discrete, but the result depends on distances, which vary continuously. In many real-world applications of geometric data structures, it is assumed that query results are continuous, free of jump discontinuities. This is at odds with many modern data structures in computational geometry, which employ approximations to achieve efficiency, but these approximations often suffer from discontinuities. In this paper, we present a general method for transforming an approximate but discontinuous data structure into one that produces a smooth approximation, while matching the asymptotic space efficiencies of the original. We achieve this by adapting an approach called the partition-of-unity method, which smoothly blends multiple local approximations into a single smooth global approximation. We illustrate the use of this technique in a specific application of approximating the distance to the boundary of a convex polytope in $\mathbb{R}^d$ from any point in its interior. We begin by developing a novel data structure that efficiently computes an absolute $\varepsilon$-approximation to this query in time $O(\log (1/\varepsilon))$ using $O(1/\varepsilon^{d/2})$ storage space. Then, we proceed to apply the proposed partition-of-unity blending to guarantee the smoothness of the approximate distance field, establishing optimal asymptotic bounds on the norms of its gradient and Hessian.

翻译：计算几何中的传统问题涉及既离散又连续的方面。其中一个例子是最近邻搜索，其输入是离散的，但结果依赖于距离，而距离是连续变化的。在许多几何数据结构的实际应用中，假设查询结果是连续的，不存在跳跃间断点。这与计算几何中许多现代数据结构相矛盾，这些数据结构通过近似实现效率，但这些近似往往存在不连续性。本文提出了一种通用方法，能将近似但不连续的数据结构转换为产生平滑近似的结构，同时保持与原结构相同的渐近空间效率。我们通过采用称为单位分解法（partition-of-unity method）的技术实现这一目标，该技术将多个局部近似平滑地融合为单个全局平滑近似。我们以在$\mathbb{R}^d$中凸多面体内部任意点处近似其边界距离的具体应用为例，展示了该技术的使用。首先，我们开发了一种新型数据结构，能在$O(\log (1/\varepsilon))$时间内，使用$O(1/\varepsilon^{d/2})$存储空间，高效计算该查询的绝对$\varepsilon$-近似值。然后，我们应用所提出的单位分解融合来保证近似距离场的平滑性，并建立了其梯度和海森矩阵范数的最优渐近界。