Statistics of cross sections of Voronoi tessellations

In this paper we investigate relationships between the volumes of cells of three-dimensional Voronoi tessellations and the lengths and areas of sections obtained by intersecting the tessellation with a randomly oriented plane. Here, in order to obtain analytical results, Voronoi cells are approximated to spheres. First, the probability density function for the lengths of the radii of the sections is derived and it is shown that it is related to the Meijer $G$-function; its properties are discussed and comparisons are made with the numerical results. Next the probability density function for the areas of cross sections is computed and compared with the results of numerical simulations.Comment: 10 pages and 6 figure

A Method for the Optimized Placement of Bus Stops Based on Voronoi Diagrams

In this paper a new method for placing bus stops is presented. The method is suitable for permanently installed new bus stops and temporarily chosen collection points for call busses as well. Moreover, our implementation of the Voronoi algorithm chooses new locations for bus stops in such a way that more bus stops are set in densely populated areas and less in less populated areas. To achieve this goal, a corresponding weighting is applied to each possible placement point, based on the number of inhabitants around this point and the points of interest, such as medical centers and department stores around this point. Using the area of Roding, a small town in Bavaria, for a case study, we show that our method is especially suitable for for rural areas, where there are few multi-family houses or apartment blocks and the area is not densely populated

A Method for the Optimized Placement of Bus Stops Based on Voronoi Diagrams

In this paper a new method for placing bus stops is presented. The method is suitable for permanently installed new bus stops and temporarily chosen collection points for call busses as well. Moreover, our implementation of the Voronoi algorithm chooses new locations for bus stops in such a way that more bus stops are set in densely populated areas and less in less populated areas. To achieve this goal, a corresponding weighting is applied to each possible placement point, based on the number of inhabitants around this point and the points of interest, such as medical centers and department stores around this point. Using the area of Roding, a small town in Bavaria, for a case study, we show that our method is especially suitable for for rural areas, where there are few multi-family houses or apartment blocks and the area is not densely populated

Do Voters Vote Ideologically?, Third Version

In this paper we address the following question: To what extent is the hypothesis that voters vote “ideologically” (i.e., they always vote for the candidate who is ideologically “closest” to them) testable or falsifiable? We show that using data only on how individuals vote in a single election, the hypothesis that voters vote ideologically is irrefutable, regardless of the number of candidates competing in the election. On the other hand, using data on how the same individuals vote in multiple elections, the hypothesis that voters vote ideologically is potentially falsifiable, and we provide general conditions under which the hypothesis can be tested.voting, spatial models, falsifiability, testing

Constructing L∞ Voronoi diagrams in 2D and 3D

Voronoi diagrams and their computation are well known in the Euclidean L2 space. They are easy to sample and render in generalized Lp spaces but nontrivial to construct geometrically. Especially the limit of this norm with p → ∞ lends itself to many quad- and hex-meshing related applications as the level-set in this space is a hypercube. Many application scenarios circumvent the actual computation of L∞ diagrams altogether as known concepts for these diagrams are limited to 2D, uniformly weighted and axis-aligned sites. Our novel algorithm allows for the construction of generalized L∞ Voronoi diagrams. Although parts of the developed concept theoretically extend to higher dimensions it is herein presented and evaluated for the 2D and 3D case. It further supports individually oriented sites and allows for generating weighted diagrams with anisotropic weight vectors for individual sites. The algorithm is designed around individual sites, and initializes their cells with a simple meshed representation of a site's level-set. Hyperplanes between adjacent cells cut the initialization geometry into convex polyhedra. Non-cell geometry is filtered out based on the L∞ Voronoi criterion, leaving only the non-convex cell geometry. Eventually we conclude with discussions on the algorithms complexity, numerical precision and analyze the applicability of our generalized L∞ diagrams for the construction of Centroidal Voronoi Tessellations (CVT) using Lloyd's algorithm