Stabbing Circles for Sets of Segments in the Plane

Stabbing a set S of n segments in the plane by a line is a well-known problem. In this paper we consider the variation where the stabbing object is a circle instead of a line. We show that the problem is tightly connected to cluster Voronoi diagrams, in particular, the Hausdorff and the farthest-color Voronoi diagram. Based on these diagrams, we provide a method to compute all the combinatorially different stabbing circles for S, and the stabbing circles with maximum and minimum radius. We give conditions under which our method is fast. These conditions are satisfied if the segments in S are parallel, resulting in a \(O(n \log ^2{n})\) time algorithm. We also observe that the stabbing circle problem for S can be solved in optimal \(O(n^2)\) time and space by reducing the problem to computing the stabbing planes for a set of segments in 3D.

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