The Voronoi Diagram of Rotating Rays with Applications to Floodlight Illumination

Voronoi and Delaunay (Delone) cells of the root and weight lattices of the Coxeter-Weyl groups W(An) and W(Dn) are constructed. The face-centred cubic (f.c.c.) and body-centred cubic (b.c.c.) lattices are obtained in this context. Basic definitions are introduced such as parallelotope, fundamental simplex, contact polytope, root polytope, Voronoi cell, Delone cell, n-simplex, n-octahedron (cross polytope), n-cube and n-hemicube and their volumes are calculated. The Voronoi cell of the root lattice is constructed as the dual of the root polytope which turns out to be the union of Delone cells. It is shown that the Delone cells centred at the origin of the root lattice An are the polytopes of the fundamental weights ω1, ω2,…, ωn and the Delone cells of the root lattice Dn are the polytopes obtained from the weights ω1, ωn-1 and ωn. A simple mechanism explains the tessellation of the root lattice by Delone cells. It is proved that the (n-1)-facet of the Voronoi cell of the root lattice An is an (n-1)-dimensional rhombohedron and similarly the (n-1)-facet of the Voronoi cell of the root lattice Dn is a dipyramid with a base of an (n-2)-cube. The volume of the Voronoi cell is calculated via its (n-1)-facet which in turn can be obtained from the fundamental simplex. Tessellations of the root lattice with the Voronoi and Delone cells are explained by giving examples from lower dimensions. Similar considerations are also worked out for the weight lattices An* and Dn*. It is pointed out that the projection of the higher-dimensional root and weight lattices on the Coxeter plane leads to the h-fold aperiodic tiling, where h is the Coxeter number of the Coxeter-Weyl group. Tiles of the Coxeter plane can be obtained by projection of the two-dimensional faces of the Voronoi or Delone cells. Examples are given such as the Penrose-like fivefold symmetric tessellation by the A4 root lattice and the eightfold symmetric tessellation by the D5 root lattice.

An ordinary geographical information system has a collection of nearest neighbourhood operations, such as generating a buffer zone and searching for the nearest facility from a given location, and this collection serves as a useful tool box for spatial analysis. Computationally, these operations are undertaken through the ordinary Voronoi diagram. This paper extends this tool box by generalizing the ordinary Voronoi diagram. The tool box consists of 35 nearest neighbourhood operations based upon twelve generalized Voronoi diagrams: the order-fe Voronoi diagram, the ordered order-fc Voronoi diagram, the farthest-point Voronoi diagram, the kth-nearest-point Voronoi diagram, the weighted Voronoi diagram, the line Voronoi diagram, the area Voronoi diagram, the Manhattan Voronoi diagram, the spherical Voronoi diagram, the Voronoi diagram in a river, the polyhedral Voronoi diagram, and the network Voronoi diagram. Each operation is illustrated with examples and the literature of computational methods.

Chapter 18 - Voronoi Diagrams

Path planning for mobile robots is a complex problem. However, it becomes more challenging when it comes to planning paths in dynamic environments. This is because the robot needs to reach an agreement between the need of having efficient and optimal paths and the need to deal with unexpected obstacles. The proposed algorithm for this work is based on two concepts, the Voronoi Diagram used for the environment representation and the Deformation Retracts which are integrated into the system to enable the path planner to deal with the effect of the moving obstacle by deforming the Voronoi Diagram. The fusion of the aforementioned two concepts, Voronoi Diagrams and Deformation Retracts, which are from two related mathematical disciplines (Computational geometry and Algebraic topology), has not yet been considered in robotics applications. The proposed system first extracts the collision-free space by computing a Generalised Voronoi Diagram (GVD) and generates a pre-planned robot path, then the deformation retract is applied on the free space of the Voronoi Diagram created after an interference due to a moving obstacle. The map is deformed, and the initial path is updated to an alternative path if it exists. One important feature of this algorithm is that it is complete because it generates a solution (path) and the dimension of the map has been reduced to one which represents the retracted free space in the environment. This makes the new system applicable to robot navigation in complex environments, and in other research areas such as computer games, virtual reality, and computational geometry to mention but a few. Simulation results of some environments demonstrate the effectiveness of the new algorithm. The findings of this work have shown that Voronoi Diagram and Deformation Retracts Path planning for mobile robots is a complex problem. However, it becomes more challenging when it comes to planning paths in dynamic environments. This is because the robot needs to reach an agreement between the need of having efficient and optimal paths and the need to deal with unexpected obstacles. The proposed algorithm for this work is based on two concepts, the Voronoi Diagram used for the environment representation and the Deformation Retracts which are integrated into the system to enable the path planner to deal with the effect of the moving obstacle by deforming the Voronoi Diagram. The fusion of the aforementioned two concepts, Voronoi Diagrams and Deformation Retracts, which are from two related mathematical disciplines (Computational geometry and Algebraic topology), has not yet been considered in robotics applications. The proposed system first extracts the collision-free space by computing a Generalised Voronoi Diagram (GVD) and generates a pre-planned robot path, then the deformation retract is applied on the free space of the Voronoi Diagram created after an interference due to a moving obstacle. The map is deformed, and the initial path is updated to an alternative path if it exists. One important feature of this algorithm is that it is complete because it generates a solution (path) and the dimension of the map has been reduced to one which represents the retracted free space in the environment. This makes the new system applicable to robot navigation in complex environments, and in other research areas such as computer games, virtual reality, and computational geometry to mention but a few. Simulation results of some environments demonstrate the effectiveness of the new algorithm. The findings of this work have shown that Voronoi Diagram and Deformation Retracts techniques are a good combination for solving path planning problem using Deformable Voronoi Diagram for mobile robot in a dynamic environment.

A random tessellation is said to be stationary if its distribution is invariant under translations in ℝ. Assuming stationarity it is possible to define what is meant by a typical cell and a typical k-facet of the tessellation. The objective in this chapter is to formalize these concepts and study their relationships for stationary Voronoi and Delaunay tessellations. In fact all results presented for stationary Voronoi tessellations hold as well for arbitrary normal stationary tessellations with convex cells. ‘Normality’ in this context means that every k-facet lies in the boundaries of exactly d−k+1 cells, k = 0,…, d−1. Many real-life non-artificial tessellations for d = 1, 2, 3 possess this property. Indeed Voronoi tessellations in general quadratic position are normal, cf. Proposition 2.1.1. Though Delaunay tessellations are not, many results for Voronoi tessellations carry over because of the duality.

The Voronoi tessellation is used to study the geometrical arrangement of disordered packings of equal spheres. The statistics of the characteristics of the cells are compared to those of 3d natural foams. In the case of binary mixtures or polydisperse assemblies of spheres, the Voronoi tessellation is replaced by the radical tessellation. Important differences exist.

A tessellation is a partition of a space into subregions. Usually these subregions are required to be convex and polygonal, and tessellations of the plane () are more frequently discussed. In addition to the classical Voronoi and Delaunay tessellations, other tessellations are reviewed here, including Johnson–Mehl tessellations, hyperplane tessellations, dead leaves tessellations, Boolean models, and generalized Voronoi tessellations such as weighted Voronoi tessellations and order‐k Voronoi tessellations.

Approximation of Generalized Voronoi Diagrams by Ordinary Voronoi Diagrams

A spatial database is a database that is optimized for storing and querying data that represents objects as points, lines or polygons. A spatial query is a mechanism for retrieving objects stored in the database and consists of a specific question with certain parameters in a map. In general, a spatial query is intended to retrieve the objects either as a set of points of interests or a region for the answer. To get the answer, the query can be processed in two different ways: point-to-point calculation or region-based calculation. In point-to-point calculation, the query can be solved by choosing appropriate objects on the map that can be used to answer the query. In region-based calculation, the query can be solved by constructing the region that contains the correct objects that will answer the query. Region-based calculation has one major advantage over point-to-point calculation: this method does not need to check each object one-by-one; hence this method will not suffer from performance degradation where there is a high number of objects. A region-based calculation method that is commonly used to solve spatial queries is the Voronoi diagram. This method divides the map into smaller spaces based on the nearest distance to an object. A Voronoi diagram can mimic human visual intuition, where humans can easily identify whether an object is located inside or outside a closed shape. Even though a Voronoi diagram has been applied widely for various spatial query types, this diagram has some problems, which are: (1) Most queries use a Voronoi diagram only to prune the map to reduce the objects verification time, (2) The region to answer a spatial query cannot be retrieved directly from a complete Voronoi diagram even though the region for a spatial query is part of a Voronoi diagram. Each type of query needs a specific method in order to generate the region. Therefore this thesis will present a new variation of the Voronoi diagram named highest order Voronoi diagram (HSVD) that can be used directly to identify the region for various types of spatial queries. To show the flexibility of this structure, we applied it to commonly known nearest neighbours and reverse nearest neighbours with their queries variations. We also applied this structure to answer polychromatic queries and extend this method for hierarchical queries. Our analysis shows that the HSVD structure is very flexible and can adapt to various types of spatial queries without having to rebuild the current structure to answer variations in the queries.

Locational optimization problems solved through Voronoi diagrams

Approximation of Generalized Voronoi Diagrams by Ordinary Voronoi Diagrams

Visualizing two-dimensional data projections with group-wise coloring and confidence ellipses is a standard approach in biomedical data analysis. However, this method can obscure subtle group overlaps or atypical cases. Voronoi tessellation, which is widely used in crystallography to analyze local structure, offers a parameter-free geometric alternative that can improve the evaluation of group structure in raw or projected data. We implemented Voronoi tessellation as a plot type for two-dimensional biomedical data and compared it with confidence ellipses on three artificial datasets and three biomedical datasets. For datasets with well-separated classes, both visualization techniques effectively delineated groups. Voronoi tessellation more clearly highlighted cases with points overlapping the opposite group, revealed internal group heterogeneity, and enabled quantification of structural discordance via a Voronoi island count as a visualization-intrinsic metric with no equivalent in confidence ellipse approaches. In datasets with moderate or absent group separation, Voronoi tessellation more effectively exposed the lack of meaningful structure, whereas confidence ellipses more clearly indicated distant outliers. Voronoi tessellation also facilitated the identification of clustering failures. Thus, Voronoi tessellation enhances the detection of deviations from expected group patterns and provides geometric insights that complement statistical summaries from confidence ellipses. Therefore, integrating Voronoi tessellation into standard data analysis workflows is a valuable addition for visualizing biomedical data and supports hypothesis validation and exploratory analyses in both raw data visualization and dimensionality reduction or clustering. An R library "VoronoiBiomedPlot" is available at the Comprehensive R Archive Network (CRAN) at https://cran.r-project.org/package=VoronoiBiomedPlot.

We study the Voronoi diagrams of a finite set of Cauchy distributions and their dual complexes from the viewpoint of information geometry by considering the Fisher-Rao distance, the Kullback-Leibler divergence, the chi square divergence, and a flat divergence derived from Tsallis entropy related to the conformal flattening of the Fisher-Rao geometry. We prove that the Voronoi diagrams of the Fisher-Rao distance, the chi square divergence, and the Kullback-Leibler divergences all coincide with a hyperbolic Voronoi diagram on the corresponding Cauchy location-scale parameters, and that the dual Cauchy hyperbolic Delaunay complexes are Fisher orthogonal to the Cauchy hyperbolic Voronoi diagrams. The dual Voronoi diagrams with respect to the dual flat divergences amount to dual Bregman Voronoi diagrams, and their dual complexes are regular triangulations. The primal Bregman Voronoi diagram is the Euclidean Voronoi diagram and the dual Bregman Voronoi diagram coincides with the Cauchy hyperbolic Voronoi diagram. In addition, we prove that the square root of the Kullback-Leibler divergence between Cauchy distributions yields a metric distance which is Hilbertian for the Cauchy scale families.

Voronoi diagrams are fundamental data structures that have been extensively studied in Computational Geometry. A Voronoi diagram can be defined as the minimization diagram of a finite set of continuous functions. Usually, each of those functions is interpreted as the distance function to an object. The as- sociated Voronoi diagram subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define many variants of Voronoi diagrams depending on the class of objects, the distance functions and the embedding space. Affine di- agrams, i.e. diagrams whose cells are convex polytopes, are well understood. Their properties can be deduced from the properties of polytopes and they can be constructed efficiently. The situation is very different for Voronoi dia- grams with curved regions. Curved Voronoi diagrams arise in various contexts where the objects are not punctual or the distance is not the Euclidean dis- tance. We survey the main results on curved Voronoi diagrams. We describe in some detail two general mechanisms to obtain effective algorithms for some classes of curved Voronoi diagrams. The first one consists in linearizing the diagram and applies, in particular, to diagrams whose bisectors are algebraic hypersurfaces. The second one is a randomized incremental paradigm that can construct affine and several planar non-affine diagrams. We finally introduce the concept of Medial Axis which generalizes the concept of Voronoi diagram to infinite sets. Interestingly, it is possible to efficiently construct a certified approximation of the medial axis of a bounded set from the Voronoi diagram of a sample of points on the boundary of the set.

In this paper the problem of obtaining the Voronoi diagram which approximates a given tessellation of the plane is formulated as the optimization problem, where the objective function is the discrepancy of the Voronoi diagram and the given tessellation. The objective function is generally non-convex and nondifferentiable, so we adopt the primitive descent algorithm and its variants as a solution algorithm. Of course, we have to be content with the locally minimum solutions. However the results of the computational examples suggest that satisfactory good solutions can be obtained by our algorithm. This problem includes the problem to restore the generators from a given Voronoi diagram (Le., the inverse problem of constructing a Voronoi diagram from the given points) when the given diagram is itself a Voronoi diagram. We can get the approximate position of the generators from a given Voronoi diagram in practical timl:; it take~ db out 10 s to restore the generators from a Voronoi diagram generated from thirty-two points on a computer of speed about 17 MIPS. Two other practical examples are presented where our algorithm is efficient, one being a problem in ecology and the other being one in urban planning. We can get the Voronoi diagrams which approximate the given tessellations l which have 32 regions and are defmed by 172 points in the former example, 11 regions and 192 points in the latter example) within 10s in these two examples on the same computer.