We present novel exploration algorithms and control law that enable the construction of Voronoi diagram in an initially unknown area using one vehicle equipped with range scanners. Our control law and exploration algorithms are provably complete, i.e., convergence is guaranteed. The control laws use range measurements to make the vehicle track Voronoi edges between two obstacles. The exploratio...

Computational geometry classically assumes real-number arithmetic which does not exist in actual computers. A solution consists in using integer coordinates for data and exact arithmetic for computations. This approach implies that if the results of an algorithm are the input of another, these results must be rounded to match this hypothesis of integer coordinates. In this paper, we treat the c...

The main problem of educational centers in a mega city like Tehran, capital of Iran, is that no enforced service areas exist to guide school selection or allow students to make the most convenient commutes to the nearest schools. Without the defined school service areas, parents seeking better and more reputable schools often have no choice but to send children to schools outside the local area...

Consider a set of n points in d-dimensional Euclidean space, d 2, each of which is continuously moving along a given individual trajectory. At each instant in time, the points deene a Voronoi diagram. As the points move, the Voronoi diagram changes continuously, but at certain critical instants in time, topological events occur that cause a change in the Voronoi diagram. In this paper, we prese...

Ron Wein et al. [4] introduced the Visibility-Voronoi diagram for clearance c, denoted by V V , which is a hybrid between the visibility graph and the Voronoi diagram of polygons in the plane. It evolves from the visibility graph to the Voronoi diagram as the parameter c grows from 0 to ∞. This diagram can be used for planning natural-looking paths for a robot translating amidst polygonal obsta...

We revisit the L∞ Hausdorff Voronoi diagram of clusters of points, equivalently, the L∞ Hausdorff Voronoi diagram of rectangles, and present a plane sweep algorithm for its construction that generalizes and improves upon previous results. We show that the structural complexity of the L∞ Hausdorff Voronoi diagram is Θ(n+m), where n is the number of given clusters and m is the number of essential...

In this paper we consider fast nearest-neighbor search techniques based on the projections of Voronoi regions. The Voronoi diagram of a given set of points provides an implicit geometric interpretation of nearest-neighbor search and serves as an important basis for several proximity search algorithms in computational geometry and in developing structure-based fast vector quantization techniques...

Restricted Voronoi diagrams are a fundamental geometric structure used in many applications such as surface reconstruction from point sets or optimal transport. Given a set of sites V = {vk}k=1 ⊂ R d and a mesh X with vertices in Rd connected by triangles, the restricted Voronoi diagram partitions X by computing for each site the portion of X for which the site is the nearest. The restricted Vo...

We give lower bounds for the combinatorial complexity of the Voronoi diagram of polygonal curves under the discrete Fréchet distance. We show that the Voronoi diagram of n curves in R with k vertices each, has complexity Ω(n) for dimension d = 1, 2 and Ω(nd(k−1)+2) for d > 2.