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...

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...

Updating an abstract Voronoi diagram in linear time, after deletion of one site, has been an open problem for a long time. Similarly for various concrete Voronoi diagrams of generalized sites, other than points. In this paper we present a simple, expected linear-time algorithm to update an abstract Voronoi diagram after deletion. We introduce the concept of a Voronoi-like diagram, a relaxed ver...

The Voronoi diagram is a fundamental geometric structure that encodes proximity information. Given a set of geometric objects, called sites, their Voronoi diagram is a subdivision of the underlying space into maximal regions, such that all points within one region have the same nearest site. Problems in diverse application domains (such as VLSI CAD, robotics, facility location, etc.) demand var...

Despite its important applications in various disciplines in science and engineering, the Euclidean Voronoi diagram for spheres, also known as an additively weighted Voronoi diagram, in 3D space has not been studied as much as it deserves. In this paper, we present an algorithm to compute the Euclidean Voronoi diagram for 3D spheres with different radii. The presented algorithm follows Voronoi ...

We study the combinatorial complexity of Voronoi diagram of point sites on a general triangulated 2-manifold surface, based on the geodesic metric. Given a triangulated 2-manifold T of n faces and a set of m point sites S = {s1, s2, · · · , sm} ∈ T , we prove that the complexity of Voronoi diagram VT (S) of S on T is O(mn) if the genus of T is zero. For a genus-g manifold T in which the samples...

We present a systematic study of the expected complexity of the intersection of geometric objects. We first study the expected size of the intersection between a random Voronoi diagram and a generic geometric object that consists of a finite collection of line segments in the plane. Using this result, we explore the intersection complexity of a random Voronoi diagram with the following target o...

In this paper we study the Voronoi diagram of lines in R. The Voronoi diagram of three lines in general position was studied in [8]. In this paper we complete this work by presenting a complete characterization of the Voronoi diagram of three arbitrary lines in R. As in the general case, we prove that the arcs of trisectors are always monotonic in some direction and we show how to separate the ...

Voronoi diagram and its geometric dual, the Delaunay triangulation, both are practical geometric constructions which have been applied extensively in spatial analysis. Considering the low efficiency of the algorithm of indirectly building Voronoi diagram, this paper proposes an improved Voronoi diagram generation algorithm based on Delaunay triangulation of randomly distributed points in the Eu...

In this paper we address the L∞ Voronoi diagram of polygonal objects and present applications in VLSI layout and manufacturing. We show that in L∞ the Voronoi diagram of segments consists only of straight line segments and is thus much simpler to compute than its Euclidean counterpart. Moreover, it has a natural interpretation. In applications where Euclidean precision is not particularly impor...