This paper studies local properties of Voronoi diagrams of sets of disjoint compact convex sites in R. It is established that bisectors are C surfaces and trisectors are C curves, and that as a point moves along a trisector its clearance sphere develops monotonically (Lemma 2.4). This monotonicity property is useful in establishing the existence of Voronoi vertices bounding edges in certain sit...

This paper studies the Voronoi diagrams on 2-manifold meshes based on geodesic metric (a.k.a. geodesic Voronoi diagrams or GVDs), which have polyline generators. We show that our general setting leads to situations more complicated than conventional 2D Euclidean Voronoi diagrams as well as point-source based GVDs, since a typical bisector contains line segments, hyperbolic segments and paraboli...

A Voronoi diagram is an interdisciplinary concept that has been applied to many fields. In geographic information systems (GIS), existing capabilities for generating Voronoi diagrams normally focus on ordinary (not weighted) point (not linear or area) features. For better integration of Voronoi diagram models and GIS, a raster-based approach is developed, and implemented seamlessly as an ArcGIS...

The special case where wi = 1, i = 1, . . . , n, was proposed by J. Sylvester in 1857 and amounts to finding the smallest circle that contains all the given points. The general case was introduced by Francis [3]. The most efficient algorithm known to date for the unweighted case is an O(n logn) algorithm by Shamos and Hoey [lo].' This algorithm utilizes the data structure so-called "Farthest po...

In this rst installment of a two-part paper, the underlying theory for an algorithm that computes the Voronoi diagram and medial axis of a planar domain bounded by free-form (polynomial or rational) curve segments is presented. An incremental approach to computing the Voronoi diagram is used, wherein a single boundary segment is added to an existing boundary-segment set at each step. The introd...

To efficiently store and analyse spatial data at a global scale, the digital expression of the Earth’s data must be global, continuous and conjugate, i.e., a spherical dynamic data model is needed. The Voronoi data structure is the only published attempt and only solution (which is currently available) for dynamic GIS. The complexity of the Voronoi algorithms for line and area data sets in a ve...

The Voronoi diagram is a widely used partition of space which may be implemented in a variety of unconventional computing prototypes, sharing in common the uniform propagation of information via fronts emanating from data point sources. The giant single-celled amoeboid organism Physarum polycephalum constructs minimising transport networks but can also approximate the Voronoi diagram using two ...

We present a novel algorithm to compute a homotopy preserving bounded-error approximate Voronoi diagram of a 3D polyhedron. Our approach uses spatial subdivision to generate an adaptive volumetric grid and computes an approximate Voronoi diagram within each grid cell. Moreover, we ensure each grid cell satisfies a homotopy preserving criterion by computing an arrangement of 2D conics within a p...