Franz Aurenhammer

Voronoi diagrams—a survey of a fundamental geometric data structure

Computational geometry is concerned with the design and analysis of algorithms for geometrical problems. In addition, other more practically oriented, areas of computer science— such as computer graphics, computer-aided design, robotics, pattern recognition, and operations research—give rise to problems that inherently are geometrical. This is one reason computational geometry has attracted enormous research interest in the past decade and is a well-established area today. (For standard sources, we refer to the survey article by Lee and Preparata [19841 and to the textbooks by Preparata and Shames [1985] and Edelsbrunner [1987bl.) Readers familiar with the literature of computational geometry will have noticed, especially in the last few years, an increasing interest in a geometrical construct called the Voronoi diagram. This trend can also be observed in combinatorial geometry and in a considerable number of articles in natural science journals that address the Voronoi diagram under different names specific to the respective area. Given some number of points in the plane, their Voronoi diagram divides the plane according to the nearest-neighbor

Power diagrams: properties, algorithms and applications

The power pow

A Novel Type of Skeleton for Polygons

(x,s)

An optimal algorithm for constructing the weighted voronoi diagram in the plane

of a point x with respect to a sphere s in Euclidean d-space

Chapter 5 – Voronoi Diagrams*

E^d

Minkowski-type theorems and least-squares clustering

is given by

A criterion for the affine equivalence of cell complexes inRd and convex polyhedra inRd+1

d^2 (x,z) - r^2

Medial axis computation for planar free-form shapes

, where d denotes the Euclidean distance function, and z and r are the center and the radius of s. The power diagram of a finite set S of spheres in

Quickest Paths, Straight Skeletons, and the City Voronoi Diagram

E^d

Improved algorithms for disc and balls using power diagrams

is a cell complex that associates each