In this thesis, we will present examples of Voronoi diagrams that are not tessellations. Moreover, we will find sufficient conditions on subspaces of E2, S2 and the Poincaré disk and the sets of sites that guarantee that the Voronoi diagrams are pre-triangulations. We will also study g-spaces, which are metric spaces with ‘extendable’ geodesics joining any 2 points and give properties for a set of sites in a g-space that again guarantees that the Voronoi diagram is a pre-triangulation.

Following thesis regards itself with Voronoi diagram creation in 3D using a graphics card. It focuses on and compares certain algorithms that construct the diagram when given set of points in space. For this purpose there have been two algorithms implemented.  First one creates Delaunay tetrahedralization using parallel splitting and flipping of present tetrahedra. Then it transforms it into a Voronoi diagram. The second algorithms utilizes planes to cut a mesh until required shapes are created.  Testing shows the advantages and disadvantages of these algorithms and their relative performance. Main takeaway from this work for these algorithms is the relative sensitivity of the second method to the use of inappropriate shape in relation to given set of points. For the other algorithm its slower start and relative unsuitability for use with smaller sets of points is apparent, but it is greatly optimized for big sets.

The purpose of this research is to design a proof of concept system that is capable of locating a hidden radio transmitter and to investigate methods of multi-agent formation control with a specific interest in the effectiveness of these methods on the overall objective of locating this transmitter. A system is proposed and developed in which autonomous agents work together to locate this transmitter and their responsiveness is analyzed while using formations based both on a behavioral system and a system derived from centroidal Voronoi tessellations. Many software adaptations to the existing MASnet program are required, as well as some hardware adaptations, including development of a robust simulation platform that may be used in conjunction with the MASnet system, and exploration of a distributed formation system. While this work does not accomplish the overall goal of the MASnet platform -- to be able to locate and control a diffusion process -- it does further understanding of the way autonomous agents interact with their environment and develop tools that aid future research in the program, as well as introduce exciting new areas to which the platform can be applied.

This thesis introduces a design combining an emerging area in robotics with a well established mathematical research topic: swarm intelligence and Voronoi tessellations, respectively. The main objective for this research is to design an economical and robust swarm system to achieve distributed control. This research combines swarm intelligence with Voronoi tessellations to localize a source and create formations. Extensive software coding must be implemented for this design, such as the development of a discrete centroidal Voronoi tessellation (CVT) algorithm. The ultimate purpose of this research is to advance the existing Mobile Actuator and Sensor Network (MASnet) platform to eventually develop a cooperative robot team that can sense, predict, and nally neutralize a diusion process. Previous work on the MASnet platform has served as a foundation for this research. While growing closer to the MASnet goal, results also provide stimulating discoveries for mathematical and swarm research areas.

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Detta arbete undersöker hur skalbarheten i ett peer-to-peer-nätverk (P2P) byggt med voronoi-based overlay network (VON) som topologi, påverkas av entitetshantering och felhantering, när det används i nätverkslösnigen till ett realtidsspel med många spelare. Under arbetet skapades en testplattform som används under ett experiment för att utvärdera huruvida aspekter såsom ansvarsuppdelning och nodkraschhantering påverkar antalet meddelanden som behöver skickas och därigenom skalbarheten i nätverket. Experimentet undersöker flera fall, med olika mycket betoning på entitetshantering och felhantering och resultaten visar att nätverket behåller sin skalbarhet och att totala antalet meddelanden som skickas håller sig mestadels opåverkad mellan fallen, trots hanteringen. / <p>Det finns övrigt digitalt material (t.ex. film-, bild- eller ljudfiler) eller modeller/artefakter tillhörande examensarbetet som ska skickas till arkivet.</p><p>There are other digital material (eg film, image or audio files) or models/artifacts that belongs to the thesis and need to be archived.</p>

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This research is motivated by an issue frequently encountered in water quality monitoring and ecological assessment. One concern for researchers and watershed resource managers is how the biological community in a watershed is affected by human activities. The conventional single model approach based on regression and logistic regression usually fails to adequately model the relationship between biological responses and environmental stressors since the study samples are collected over a large spatial region and the response-stressor relationships are usually weak in this situation. In this dissertation, we propose two alternative modeling approaches to partition the whole region of study into disjoint subregions and model the response-stressor relationships within subregions simultaneously. In our examples, these modeling approaches found stronger relationships within subregions and should help the resource managers improve impairment assessment and decision making. The first approach is an adjusted Bayesian classification and regression tree (ABCART). It is based on the Bayesian classification and regression tree approach (BCART) and is modified to accommodate spatial partitions in ecological studies. The second approach is a Voronoi diagram based partition approach. This approach uses the Voronoi diagram technique to randomly partition the whole region into subregions with predetermined minimum sample size. The optimal partition/cluster is selected by Monte Carlo simulation. We propose several model selection criteria for optimal partitioning and modeling according to the nature of the study and extend it to multivariate analysis to find the underlying structure of response-stressor relationships. We also propose a multivariate hotspot detection approach (MHDM) to find the region where the response-stressor relationship is the strongest according to an R-square-like criterion. Several sets of ecological data are studied in this dissertation to illustrate the implementation of the above partition modeling approaches. The findings from these studies are consistent with other studies. / Ph. D.

Representações computacionais de formas podem ser criadas em ferramentas CAD ou geradas a partir de um objeto físico já existente. Esta última abordagem oferece como vantagens rapidez e fidelidade ao objeto original, que são os aspectos fundamentais em muitas aplicações, como Simulações Numéricas de Equações Diferenciais Parciais e Imagens Médicas. A reconstrução (ou geração de malhas superficiais) a partir de pontos amostrados de uma superfície de um objeto é um problema clássico de representação de formas. Nesta dissertação apresentamos um vasto levantamento bibliográfico deste tipo de reconstrução, classificando e descrevendo os principais trabalhos presentes na literatura. A partir do levantamento bibliográfico, selecionamos um conjunto de algoritmos sobre os quais foram realizadas comparações teóricas e empíricas cujos resultados são apresentados. Para finalizar, apresentamos aplicações de nossas implementações em Simulação Numérica de Equações Diferenciais Parciais e processamento de Imagens / Computational representations of shapes can be developed using CAD applications or created from data acquired from a real physical object. This latter is advantageous with respect to time and fidelity to the original object which are essential to several applications, such as Numerical Simulation of Partial Differential Equations and Medical Imaging. A classical shape representation problem is that of reconstruction (or superficial mesh generation) from points sampled over the surface of an object. In this Master\'s thesis we describe a broad survey of these reconstruction methods. We focus in the classification and characterization of the main algorithms proposed in the literature. From this survey, we selected some algorithms and conducted some theoretical and practical comparisons. We conclude this work describing applications of the algorithms implemented in Numerical Simulations of Differential Partial Equations and Image Processing