Topology-Oriented Incremental Algorithm for the Robust Construction of the Voronoi Diagrams of Disks

Abstract

Voronoi diagrams are useful for spatial reasoning, and the robust and efficient construction of the ordinary Voronoi diagram of points is well known. However, its counterpart for circular disks in R 2 and spherical balls in R 3 remains a challenge. In this article, we propose a topology-oriented incremental algorithm which robustly and efficiently computes a Voronoi diagram by incrementing a new disk generator to an existing one. The key idea is to enforce the convexity of the Voronoi cell corresponding to the incrementing disk so that a simple variation of the algorithm for points proposed by Sugihara in 1992 can be applied. A benchmark using both random and degenerate disks shows that the proposed algorithm is superior to CGAL in both computational efficiency and algorithmic robustness.

Near optimal minimal convex hulls of disks

“…VD(D) has O(n) V-vertices, O(n) V-edges, and n V-cells and can be constructed by an optimal O(n log n) time for n disks using the plane sweep method [12,40] or the divideand-conquer method [24,36]. However, we prefer to use the topology-oriented incremental algorithm which guarantees robustness [25] (or the edge-flipping algorithm [22,23]) for its robust construction. Both algorithms take O(n 2 ) time in the worst case but O(n) time on average.…”

“…Both VD and VD can be constructed with a similar efficiency. Even if an optimal algorithm taking O(n log n) time is known, we prefer to use the topology-oriented incremental algorithm (with an average O(n) time and the worst case O(n 2 ) time) [25] with the winged-edge data structure. This is because of the guaranteed robustness with a sufficiently good efficiency -actually significantly faster than the optimal algorithm for large problem instances.…”

Support-free hollowing for 3D printing via Voronoi diagram of ellipses

“…Specifically, we perform an ellipse packing in the polygons on parallel section planes of a 3D model and protrude the ellipses of one plane to its neighboring planes in the shape volume to generate the hollowed results. To better pack the ellipses in a polygon, we construct the Voronoi diagram of ellipses to efficiently reason the free-space around the ellipses and other geometric features by taking advantage of the available algorithm for the efficient and robust construction of the Voronoi diagram of circular disks which approximate the ellipses [34]. The algorithm inherits the disk packing algorithm using the VD of disks [33].…”

“…TOI-D Algorithm. The proposed algorithm takes advantage of the Voronoi diagram of circular disks [42,43], particularly the recently reported topologyoriented incremental (TOI) algorithm for computing the Voronoi of circular disks, thus abbreviated as the TOI-D algorithm, which takes O(n 2 ) time in the worst case but O(n) time on average for n disks [34]. The idea is to approximate target geometric entities using circular disks in a sufficient resolution, construct the VD of the disks using the TOI-D algorithm, and merge some V-cells.…”

“…Time Complexity. The TOI-D algorithm for the construction of the VD of m disks takes O(m) time on average and O(m 2 ) time in the worst case[34]. With the polygon P of n P-vertices and n P-edges, n < m, the merge process takes O(m − n) time because each merge of two adjacent V-cells takes O(1) time.…”