A Collapse Process for Farthest Voronoi Diagrams of Lines in Three Dimensions

2026-09-24

AI Summary

This study addresses the inherent difficulty of constructing the farthest Voronoi diagram of three-dimensional lines, where unbounded features yield a topologically complex spherical mapping. To overcome this, the work proposes a collapse algorithm based on distance-decreasing sweep-shrink mappings that dynamically reconstructs the diagram by identifying four non-terminal event types: deletion, swap, and two categories of extrema. This approach integrates computational geometric topology analysis, spherical convex hull computation, and polynomial root-finding techniques to handle high-dimensional geometric constraints. The authors establish the completeness of the event classification and derive a tight upper bound on the number of extrema along trisector curves equidistant from three lines. Furthermore, they present a direct method for computing minimum enclosing balls and generalize the collapse mechanism to farthest Voronoi diagrams of convex sites under strictly convex distance functions.

Abstract

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.

Problem

• Farthest Voronoi Diagram
• Three Dimensions
• Collapse Process
• Shrinking Map
• Lines

Innovation

• Farthest Voronoi Diagram
• Collapse Process
• Shrinking Map
• Event Classification
• Tangent Sphere