Measuring spatial interaction behavior in team sports using superimposed Voronoi diagrams

In team sports, the spatial distribution of players on the field is determined by the interaction behavior established at both player and team levels. The distribution patterns observed during a game emerge from specific technical and tactical methods adopted by the teams, and from individual, environmental and task constraints that influence players' behaviour. By understanding how specific patterns of spatial interaction are formed, one can characterize the behavior of the respective teams and players. Thus, in the present work we suggest a novel spatial method for describing teams' spatial interaction behaviour, which results from superimposing the Voronoi diagrams of two competing teams. We considered theoretical patterns of spatial distribution in a well-defined scenario (5 vs 4+ GK played in a field of 20x20m) in order to generate reference values of the variables derived from the superimposed Voronoi diagrams (SVD). These variables were tested in a formal application to empirical data collected from 19 Futsal trials with identical playing settings. Results suggest that it is possible to identify a number of characteristics that can be used to describe players' spatial behavior at different levels, namely the defensive methods adopted by the players.

Spatial dynamics of team sports exposed by Voronoi diagrams

Team sports represent complex systems: players interact continuously during a game, and exhibit intricate patterns of interaction, which can be identified and investigated at both individual and collective levels. We used Voronoi diagrams to identify and investigate the spatial dynamics of players' behavior in Futsal. Using this tool, we examined 19 plays of a sub-phase of a Futsal game played in a reduced area (20 m(2)) from which we extracted the trajectories of all players. Results obtained from a comparative analysis of player's Voronoi area (dominant region) and nearest teammate distance revealed different patterns of interaction between attackers and defenders, both at the level of individual players and teams. We found that, compared to defenders, larger dominant regions were associated with attackers. Furthermore, these regions were more variable in size among players from the same team but, at the player level, the attackers' dominant regions were more regular than those associated with each of the defenders. These findings support a formal description of the dynamic spatial interaction of the players, at least during the particular sub-phase of Futsal investigated. The adopted approach may be extended to other team behaviors where the actions taken at any instant in time by each of the involved agents are associated with the space they occupy at that particular time.

Generalized Higher-Order Voronoi Diagrams on Polyhedral Surfaces

We present an algorithm for computing exact shortest paths, and consequently distances, from a generalized source (point, segment, polygonal chain or polygonal region) on a possibly non-convex polyhedral surface in which polygonal chain or polygon obstacles are allowed. We also present algorithms for computing discrete Voronoi diagrams of a set of generalized sites (points, segments, polygonal chains or polygons) on a polyhedral surface with obstacles. To obtain the discrete Voronoi diagrams our algorithms, exploiting hardware graphics capabilities, compute shortest path distances defined by the sites

Zone design of specific sizes using adaptive additively weighted voronoi diagrams

Territory or zone design processes entail partitioning a geographic space, organized as a set of areal units, into different regions or zones according to a specific set of criteria that are dependent on the application context. In most cases, the aim is to create zones of approximately equal sizes (zones with equal numbers of inhabitants, same average sales, etc.). However, some of the new applications that have emerged, particularly in the context of sustainable development policies, are aimed at defining zones of a predetermined, though not necessarily similar, size. In addition, the zones should be built around a given set of seeds. This type of partitioning has not been sufficiently researched; therefore, there are no known approaches for automated zone delimitation. This study proposes a new method based on a discrete version of the adaptive additively weighted Voronoi diagram that makes it possible to partition a two-dimensional space into zones of specific sizes, taking both the position and the weight of each seed into account. The method consists of repeatedly solving a traditional additively weighted Voronoi diagram, so that each seed?s weight is updated at every iteration. The zones are geographically connected using a metric based on the shortest path. Tests conducted on the extensive farming system of three municipalities in Castile-La Mancha (Spain) have established that the proposed heuristic procedure is valid for solving this type of partitioning problem. Nevertheless, these tests confirmed that the given seed position determines the spatial configuration the method must solve and this may have a great impact on the resulting partition.

Study of cerebral gene expression densities using Voronoi analysis

As the available public cerebral gene expression image data increasingly grows, the demand for automated methods to analyze such large amount of data also increases. An important study that can be carried out on these data is related to the spatial relationship between gene expressions. Similar spatial density distribution of expression between genes may indicate they are functionally correlated, thus the identification of these similarities is useful in suggesting directions of investigation to discover gene interactions and their correlated functions. In this paper, we describe the use of a high-throughput methodology based on Voronoi diagrams to automatically analyze and search for possible local spatial density relationships between gene expression images. We tested this method using mouse brain section images from the Allen Mouse Brain Atlas public database. This methodology provided measurements able to characterize the similarity of the density distribution between gene expressions and allowed the visualization of the results through networks and Principal Component Analysis (PCA). These visualizations are useful to analyze the similarity level between gene expression patterns, as well as to compare connection patterns between region networks. Some genes were found to have the same type of function and to be near each other in the PCA visualizations. These results suggest cerebral density correlations between gene expressions that could be further explored. (C) 2011 Elsevier B.V. All rights reserved.

Voronoi cells via linear inequality systems

The theory and methods of linear algebra are a useful alternative to those of convex geometry in the framework of Voronoi cells and diagrams, which constitute basic tools of computational geometry. As shown by Voigt and Weis in 2010, the Voronoi cells of a given set of sites T, which provide a tesselation of the space called Voronoi diagram when T is finite, are solution sets of linear inequality systems indexed by T. This paper exploits systematically this fact in order to obtain geometrical information on Voronoi cells from sets associated with T (convex and conical hulls, tangent cones and the characteristic cones of their linear representations). The particular cases of T being a curve, a closed convex set and a discrete set are analyzed in detail. We also include conclusions on Voronoi diagrams of arbitrary sets.

Computing Distance Functions from Generalized Sources on Weighted Polyhedral Surfaces

We present algorithms for computing approximate distance functions and shortest paths from a generalized source (point, segment, polygonal chain or polygonal region) on a weighted non-convex polyhedral surface in which obstacles (represented by polygonal chains or polygons) are allowed. We also describe an algorithm for discretizing, by using graphics hardware capabilities, distance functions. Finally, we present algorithms for computing discrete k-order Voronoi diagrams

Visibility and proximity on triangulated surfaces

En aquesta tesi es solucionen problemes de visibilitat i proximitat sobre superfícies triangulades considerant elements generalitzats. Com a elements generalitzats considerem: punts, segments, poligonals i polígons. Les estrategies que proposem utilitzen algoritmes de geometria computacional i hardware gràfic. Comencem tractant els problemes de visibilitat sobre models de terrenys triangulats considerant un conjunt d'elements de visió generalitzats. Es presenten dos mètodes per obtenir, de forma aproximada, mapes de multi-visibilitat. Un mapa de multi-visibilitat és la subdivisió del domini del terreny que codifica la visibilitat d'acord amb diferents criteris. El primer mètode, de difícil implementació, utilitza informació de visibilitat exacte per reconstruir de forma aproximada el mapa de multi-visibilitat. El segon, que va acompanyat de resultats d'implementació, obté informació de visibilitat aproximada per calcular i visualitzar mapes de multi-visibilitat discrets mitjançant hardware gràfic. Com a aplicacions es resolen problemes de multi-visibilitat entre regions i es responen preguntes sobre la multi-visibilitat d'un punt o d'una regió. A continuació tractem els problemes de proximitat sobre superfícies polièdriques triangulades considerant seus generalitzades. Es presenten dos mètodes, amb resultats d'implementació, per calcular distàncies des de seus generalitzades sobre superfícies polièdriques on hi poden haver obstacles generalitzats. El primer mètode calcula, de forma exacte, les distàncies definides pels camins més curts des de les seus als punts del poliedre. El segon mètode calcula, de forma aproximada, distàncies considerant els camins més curts sobre superfícies polièdriques amb pesos. Com a aplicacions, es calculen diagrames de Voronoi d'ordre k, i es resolen, de forma aproximada, alguns problemes de localització de serveis. També es proporciona un estudi teòric sobre la complexitat dels diagrames de Voronoi d'ordre k d'un conjunt de seus generalitzades en un poliedre sense pesos.

Wet tundra polygon centers and distance between polygon centroids on Samoylov Island, Lena Delta, Siberia, Russia, with links to shape files

Subgrid processes occur in various ecosystems and landscapes but, because of their small scale, they are not represented or poorly parameterized in climate models. These local heterogeneities are often important or even fundamental for energy and carbon balances. This is especially true for northern peatlands and in particular for the polygonal tundra, where methane emissions are strongly influenced by spatial soil heterogeneities. We present a stochastic model for the surface topography of polygonal tundra using Poisson-Voronoi diagrams and we compare the results with available recent field studies. We analyze seasonal dynamics of water table variations and the landscape response under different scenarios of precipitation income. We upscale methane fluxes by using a simple idealized model for methane emission. Hydraulic interconnectivities and large-scale drainage may also be investigated through percolation properties and thresholds in the Voronoi graph. The model captures the main statistical characteristics of the landscape topography, such as polygon area and surface properties as well as the water balance. This approach enables us to statistically relate large-scale properties of the system to the main small-scale processes within the single polygons.

Representação da área de responsabilidade de jogadores de futebol através do Diagrama de Voronoi

Universidade Estadual de Campinas . Faculdade de Educação Física

CCT diagrams of tricalcium silicate decomposition

The mechanical properties of Portland cement are closely related to the chemical composition of the clinker and particularly to the concentration of tricalcium silicate, C3S. In the industrial production process, the clinker must be rapidly quenched, to avoid its decomposition into dicalcium silicate and lime and also to avoid the transformation from higher temperature phases to lower temperature phases. This study investigated the kinetics of thermal decomposition of the C3S. Samples of laboratory-made C3S were thermally treated under specific conditions to determine the continuous cooling transformation (CCT) diagram of the material. The CCT diagram of the C3S showed decomposition rates with values that were much higher than the values traditionally accepted in the literature.

Phase diagrams of a CTAB/organic solvent/buffer system applied to extraction of enzymes by reverse micelles

A partial pseudo-ternary phase diagram has been studied for the cethyltrimethylammonium bromide/isooctane:hexanol:butanol/potassium phosphate buffer system, where the two-phase diagram consisting of the reverse micelle phase (L-2) in equilibrium with the solvent is indicated. Based on these diagrams two-phase systems of reverse micelles were prepared with different compositions of the compounds and used for extraction and recovery of two enzymes, and the percentage of enzyme recovery yield monitored. The enzymes glucose-6-phosphate dehydrogenase (G6PD) and xylose redutase (XR) obtained from Candida guilliermondii yeast were used in the extraction procedures. The recovery yield data indicate that micelles having different composition give selective extraction of enzymes. The method can thus be used to optimize enzyme extraction processes. (c) 2007 Elsevier B.V. All rights reserved.

Operating space diagrams: a tool for designers of wastewater treatment plants

Recent years have seen the introduction of new and varied designs of activated sludge plants. With increasing needs for higher efficiencies and lower costs, the possibility of a plant that operates more effectively has created the need for tools that can be used to evaluate and compare designs at the design stage. One such tool is the operating space diagram. It is the aim of this paper to present this tool and demonstrate its application and relevance to design using a simple case study. In the case study, use of the operating space diagram suggested changes in design that would improve the flexibility of the process. It also was useful for designing suitable control strategies.