This paper presents a numerically robust algorithm to construct a Voronoi diagram of circles in the plane. The circles are allowed to have intersections among them, but one circle cannot fully contain another circle. The Voronoi diagram is a tessellation of the plane into Voronoi regions of given circles. Each circle has its Voronoi region which is defined by a set of points in the plane closer to the circle than any other circles. The distance from a point p to a circle ci of center pi and radius ri is ||p-pi||-ri, which is the closest Euclidean distance from p to the circle boundary. The proposed algorithm first constructs the point Voronoi diagram of centers of given circles, then it enlarges each point to the circle and expands its Voronoi region accordingly. This region-expansion process is done by local modifications and after completing this process for the whole circles the desired circle Voronoi diagram can be obtained. The proposed algorithm is numerically robust and we provide with a few examples to show its robustness. The algorithm runs in O(n2) time in the worst case and O(n) time on average where n is the number of the circles. The experiment shows that the region-expansion algorithm is robust and runs fast with strong linear time behavior.

Triangulation is one of the fundamental problems in computational geometry and computer graphics community, and it has huge application areas such as 3D printing, computer-aided engineering, surface reconstruction, surface visualization, and so on. The Delaunay refinement algorithm is a well-known method to generate quality triangular meshes when point cloud and/or constrained edges are given in two- or three-dimensional space. In this paper, we propose a simple but efficient algorithm to triangulate Voronoi surfaces of Voronoi diagram of spheres in 3-dimensional Euclidean space. The proposed algorithm is based on the Ruppert’s Delaunay refinement algorithm, and we modified the algorithm to be applied to the triangulation of Voronoi surfaces in two ways. First, a new method to deciding the location of a newly added vertex on the surface in 3-dimensional space is proposed. Second, a new efficient but effective way of estimating approximation error between Voronoi surface and triangulation. Because the proposed algorithm generates a triangular mesh for Voronoi surfaces with guaranteed quality, users can control the level of quality of the resulting triangulation that their application problems require. We have implemented and tested the proposed algorithm for random non-intersecting spheres, and the experimental result shows the proposed algorithm produces quality triangulations on Voronoi surfaces satisfying the quality criterion.

The Voronoi diagram of spheres and power diagram have been known as powerful tools to analyze spatial characteristics of weighted points, and these structures have variety range of applications including molecular spatial structure analysis, location based optimization, architectural design, etc. Due to the fact that both diagrams are based on different distance metrics, one has better usability than another depending on application problems. In this paper, we compare these diagrams in various situations from the user’s viewpoint, and show the Voronoi diagram of spheres is more effective in the problems based on the Euclidean distance metric such as nearest neighbor search, path bottleneck locating, and internal void finding.

Presented in this paper is an algorithm to compute the Voronoi diagram of a circle set from the Voronoi diagram of a point set. The circles are located in Euclidean plane, the radii of the circles are non-negative and not necessarily equal, and the circles are allowed to intersect each other. The idea of the algorithm is to use the topology of the point set Voronoi diagram as a seed so that the correct topology of the circle set Voronoi diagram can be obtained through a number of edge flipping operations. Then, the geometries of the Voronoi edges of the circle set Voronoi diagram are computed. The main advantages of the proposed algorithm are in its robustness, speed, and the simplicity in its concept as well as implementation.