The farthest line segment Voronoi diagram shows properties different from both the closest-segment Voronoi diagram and the farthest-point Voronoi diagram. Surprisingly, this structure did not receive attention in the computational geometry literature. We analyze its combinatorial and topological properties and outline an O(n log n) time construction algorithm that is easy to implement. No restr...

A digital edge is a digitization of a straight segment joining two points of integer coordinates. Such a digital set may be analytically defined by the rational slope of the straight segment. We show in this paper that the convex hull, the Euclidean farthest-point Voronoi diagram as well as the dual farthest-point Delaunay triangulation of a digital edge can be fully described by the continued ...

w0-Nearest Points and w0-Farthest Point in Normed Linear Spaces

We reflect upon an analogue of the cut locus, a notion classically studied in Differential Geometry, for finite graphs. The cut locus C(x) of a vertex x shall be the graph induced by the set of all vertices y with the property that no shortest path between x and z, z 6= y, contains y. The cut locus coincides with the graph induced by the vertices realizing the local maxima of the distance funct...

We introduce the Fast Marching farthest point sampling (FastFPS) approach for the progressive sampling of planar domains and curved manifolds in triangulated, point cloud or implicit form. By using Fast Marching methods2, 3, 6 for the incremental computation of distance maps across the sampling domain, we obtain a farthest point sampling technique superior to earlier point sampling principles i...

In this paper we discuss farthest-point problems in which a set or sequence S of n points in the plane is given in advance and can be preprocessed to answer various queries efficiently. First, we give a data structure that can be used to compute the point farthest from a query line segment in O(log2 n) time. Our data structure needs O(n logn) space and preprocessing time. To the best of our kno...