Given a set of sites in a simple polygon, a geodesic Voronoi diagram partitions the polygon into regions based on distances to sites under the geodesic metric. We present algorithms for computing the geodesic nearest-point, higher-order and farthest-point Voronoi diagrams of m point sites in a simple n-gon, which improve the best known ones form ≤ n/polylogn. Moreover, the algorithms for the ne...

Given a set S of n static points and a free point p in the Euclidean plane, we study a new variation of the minimum enclosing circle problem, in which a dynamic weight that equals to the reciprocal of the distance from the free point p to the undetermined circle center is included. In this work, we prove the optimal solution of the new problem is unique and lies on the boundary of the farthest-...

We propose an implicit representation for the farthest Voronoi Diagram of a set P of n points in the plane lying outside a set R of m disjoint axes-parallel rectangular obstacles. The distances are measured according to the L1 shortest path (geodesic) metric. In particular, we design a data structure of size O(N) in O(N log N) time that supports O(N logN)-time farthest point queries (where N = ...

We present an algorithm that computes the diameter of a set of n points in the cylinder in optimal time O(n log n); this algorithm uses as a fundamental tool the farthest point Voronoi diagram.

To support the need for interactive spatial analysis, it is often necessary to rethink the data structures and algorithms underpinning applications. This paper describes the development of an interactive environment in which a number of different Voronoi models of space can be manipulated together in real time, to (1) study their behaviour, (2) select appropriate models for specific analysis ta...

Given a set S of n static points and a mobile point p in R, we study the variations of the smallest circle that encloses S ∪ {p} when p moves along a straight line `. In this work, a complete characterization of the locus of the center of the minimum enclosing circle (MEC) of S ∪{p}, for p ∈ `, is presented. The locus is a continuous and piecewise differentiable linear function, and each of its...

Let be a set of points in convex position in . The farthest-point Voronoi diagram of partitions into convex cells. We consider the intersection of the diagram with the boundary of the convex hull of . We give an algorithm that computes an implicit representation of in expected time. More precisely, we compute the combinatorial structure of , the coordinates of its vertices, and the equation of ...

This paper describes a series of dynamic update methods that can be applied to a family of Voronoi diagram types, so that changes can be updated incrementally, without the usual recourse to complete reconstruction of their underlying data structure. More efficient incremental update methods are described for the ordinary Voronoi diagram, the farthest-point Voronoi diagram, the order-k Voronoi d...

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 ...