发表机构
Università della Svizzera italiana (USI)（瑞士意大利大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种坍塌过程，用于在三维空间中构造直线的最远Voronoi图，完整分类了四种非终止局部事件，并证明沿三割线的距离函数极值界限紧致，为最小切球提供直接方法。
AI 中文摘要
我们研究了一种坍塌过程，用于在给定图的无界特征球面图的情况下，构造三维空间中直线的最远Voronoi图。坍塌过程按照与最远直线的距离递减顺序扫过该图。它遵循收缩映射，即拓扑球面上的一个胞腔复形，编码了具有固定最远距离的点的轨迹。我们证明，在三维空间中，坍塌过程恰好有四种改变收缩映射结构的非终止局部事件类型：删除、交换、局部最小值和局部最大值事件；外加一种终止事件。这个列表是完整的。我们给出了这四种非终止事件的固有三维几何描述。首先，我们通过从一个顶点到其四条定义直线的四个切点的球面凸包，对两种与顶点相关的事件（删除和交换事件）进行分类。然后，我们分析了与三条直线的三割线上距离函数的局部极值相关的事件。我们证明，沿三割线的距离函数至多有4个局部最大值和8个局部最小值，并且这两个界限都是紧的。极值可以通过一个12次多项式找到。作为副产品，这提供了一种直接的方法来找到与三条给定直线相切的最小球体。在每个局部极值处，切球与三条直线的切点位于一个大圆上。坍塌过程和事件列表的完整性也适用于在严格凸距离函数下凸站点的最远Voronoi图，在类似的一般位置假设下。
英文摘要
We study a \emph{collapse process} to construct the farthest Voronoi diagram of lines in three dimensions, given a spherical map of the diagram's unbounded features. The collapse process sweeps through the diagram in order of decreasing distance from the farthest lines. It follows the \emph{shrinking map}, a cell complex on a topological sphere that encodes the locus of points with a fixed farthest distance. We show that, in three dimensions, the collapse process has exactly four non-terminal local event types that change the structure of the shrinking map: \emph{deletion, swap, local minimum}, and \emph{local maximum} events; plus one terminal event. This list is complete. We give intrinsic three-dimensional geometric descriptions of the four non-terminal events. First, we classify the two vertex-related events, deletion and swap events, by the spherical convex hull of the four tangent points from a vertex to its four defining lines. Then, we analyze the events related to the local extrema of the distance function along the trisector of three lines. We show that the distance function along a trisector has at most $4$ local maxima and $8$ local minima, and that both bounds are tight. The extrema can be found via a polynomial of degree $12$. As a byproduct, this gives a direct method for finding the smallest sphere tangent to three given lines. At each local extremum, the tangent sphere touches the three lines at points lying on a great circle. The collapse process and the completeness of the event list also apply, under similar general position assumptions, to the farthest Voronoi diagram of convex sites under strictly convex distance functions.
