In the age where quick turn around time and high speed manufacturing methods are becoming more important, quality assurance is a consistent bottleneck in production. With the development of cheap and fast computer hardware, it has become viable to use machine vision for the collection of data points from a machined part. The generation of these large sample points have necessitated a need for a comprehensive algorithm that will be able to provide accurate results while being computationally efficient. Current established methods are least-squares (LSQ) and non-linear programming (NLP). The LSQ method is often deemed too inaccurate and is prone to providing bad results, while the NLP method is computationally taxing. A novel method of using support vector regression (SVR) to solve the NP-hard problem of cylindricity of machined parts is proposed. This method was evaluated against LSQ and NLP in both accuracy and CPU processing time. An open-source, user-modifiable programming package was developed to test the model. Analysis of test results show the novel SVR algorithm to be a viable alternative in exploring different methods of cylindricity in real-world manufacturing.

The sewing operation is an integral component of both Geometric Function Theory and Conformal Field Theory and in this thesis we explore the interplay between the two fields. We will first generalize Huang's Geometric Sewing Equation to the quasi-symmetric case. That is, given specific maps g(z) and f^{-1}(z), we show the existence of the sewing maps F_{1}(z) and F_{2}(z). Second, we display an algebraic procedure using convergent matrix operations showing that the coefficients of the Conformal Welding Theorem maps F(z) and G(z) are dependent on the coefficients of the map phi(z). We do this for both the analytic and quasi-symmetric cases, and it is done using a special block/vector decomposition of a matrix representation called the power matrix. Lastly, we provide a partial result: given specific maps g(z) and f^{-1}(z) with analytic extensions, as well as a particular analytic map phi(z), it is possible to provide a method of determining the coefficients of the complementary maps.

The sewing operation is an integral component of both Geometric Function Theory and Conformal Field Theory and in this thesis we explore the interplay between the two fields. We will first generalize Huang's Geometric Sewing Equation to the quasi-symmetric case. That is, given specific maps g(z) and f^{-1}(z), we show the existence of the sewing maps F_{1}(z) and F_{2}(z). Second, we display an algebraic procedure using convergent matrix operations showing that the coefficients of the Conformal Welding Theorem maps F(z) and G(z) are dependent on the coefficients of the map phi(z). We do this for both the analytic and quasi-symmetric cases, and it is done using a special block/vector decomposition of a matrix representation called the power matrix. Lastly, we provide a partial result: given specific maps g(z) and f^{-1}(z) with analytic extensions, as well as a particular analytic map phi(z), it is possible to provide a method of determining the coefficients of the complementary maps.

Orientador: Simão Nicolau Stelmastchuk / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-17T21:15:56Z (GMT). No. of bitstreams: 1 Pereira_CarlosAndreBogea_M.pdf: 2160729 bytes, checksum: f46cc601d486b802f179e9fd8befb099 (MD5) Previous issue date: 2011 / Resumo: Ao nosso entender, a Probabilidade Geométrica quantifica a probabilidade de ocorrer alguns fenômenos associados a entes geométricos. O primeiro estudo, talvez o mais famoso, a ser realizado neste sentido é o problema das agulhas de Buffon. A idéia deste estudo é simples. Traçadas duas retas paralelas a uma distância d, qual é a probabilidade de uma agulha de tamanho l tocar uma das retas? Neste trabalho nos dedicamos, inicialmente, a estudar este problema e sua resolução. Um segundo tópico do nosso trabalho foi baseado no seguinte problema: Suponha que uma antena transmissora de algum sinal, por exemplo, de celular, emite seus sinais uniformemente a uma distância a, em um plano. Se estou num ponto P do plano, qual a probabilidade de entrar na zona de emissão de sinal da antena se me deslocar até um raio b? Para a resolução deste problema nós utilizamos probabilidade contínua, coordenadas polares e integração de várias variáveis. Como aplicações deste estudo temos os casos das distribuições de probabilidade uniforme e normal. Um terceiro problema tratado foi o seguinte: no espaço tridimensional temos uma fonte de emissão T, por exemplo, algum gerador de campo magnético, a qual distribui uniformemente sua energia até um raio a. Suponha, dada uma partícula num ponto P do espaço. Se tal partícula se deslocar aleatoriamente um raio igual a b qual é a probabilidade dela entrar na zona de influência da fonte de emissão T? Neste problema usamos coordenadas esféricas, integral de superfície e distribuição de probabilidade continua para o seu estudo. Também, aplicamos aos casos de distribuição de probabilidade uniforme e normal / Abstract: In our view, Geometric Probability quantifies the probability of occurs some phenomena associated with geometric entities. The first study, perhaps the most famous, to be performed of this type is the problem of Buffon's needle. The idea of this study is simple. Two parallel lines drawn at a distance d, which is the probability that a needle of length l achieve one of the straights? In this work we decided initially to study this problem and its resolution. A second topic of our study was based on the following problem: Suppose an antenna transmitting a signal, eg mobile, send their signals uniformly until a distance a in a plane. If I'm at a point P of the plane, which is the probability to enter the zone of emission signal from the antenna if I move up to a radius b? To solve this problem we use continuous probability, polar coordinates and integration of several variables. As applications of this study we have the cases of probability distributions, uniform and normal. A third problem approached was the following: in a three-dimensional space we have an emission source T, for example, a magnetic field generator, which evenly distributes its energy up to a radius a. Given a particle at a point P in space. If this particle moves randomly a radius equal to b what is the probability of it entering the zone of influence of the emission source T? In this problem we use spherical coordinates, surface integral and continuous probability distribution for its study. Also apply to cases of uniform and normal probability distribution / Mestrado / Matematica / Mestre em Matemática

Neste estudo é proposto um método para resolução de problemas de tangências, especificamente para o décimo caso do problema de Apolônio. Este método é baseado na transformação geométrica inversão e no uso do conceito de feixes de circunferências. Além de permitir a resolução de todas as configurações do problema, ele é aplicável também à resolução de outros problemas. Através do trabalho indicamos a importância do tema Desenho Geométrico no ensino de Desenho que há muito tempo enfatiza apenas o desenvolvimento da visualização espacial. Destacamos ao longo do texto como o ensino de Desenho Geométrico pode ser utilizado eficazmente para fomentar o raciocínio lógico-dedutivo dos estudantes através da prática de demonstrações. / A new method to solve the tenth case of Appolonius problem is presented in this study. This method is based on the geometric transformation called inversion and the concept of coaxal circumferences. Besides allowing the resolution of all configurations of the problem, it can also be used to solve other problems. We indicate the importance of the subject about geometric constructions in teaching Drawing, which, since a long time ago has given attention only to the development of the spatial visualization ability. We detach along the text how the teaching of geometric construction can be used efficiently to foment the deductive logical reasoning of the students through the practice of demonstrations.

Este trabalho visa resgatar a importância do ensino do desenho geométrico em especial dos Métodos dos Lugares Geométricos, aplicado à resolução de problemas de construção geométrica plana. A abordagem apresentada é tradicional, com o uso da régua e do compasso. Nesse sentido, o trabalho é composto da apresentação(conceito e construção), de vários Lugares Geométricos que podem ser considerados fundamentais para a resolução de problemas elementares de Desenho Geométrico, e a apresentação de construções das cônicas como algo mais elaborado destes lugares geométricos considerados fundamentais. Para a fixação dos conceitos, cada Lugar Geométrico (L.G.) contará com alguns exemplos de aplicação e, ao final dos capítulos, serão apresentados alguns exercícios propostos (para o leitor que se interessar em praticar os conceitos e as construções abordadas). Finalizando será feito um breve comentário das origens do desenho geométrico, bem como seu ensino no Brasil, evidenciando a resolução de problemas como método eficaz para o ensino da geometria. / This study reviews the importance of education in special geometric design of the \"Methods of Geometric Places\", applied to the resolution of flat geometric construction problems. The presented approach is traditional, using ruler and compass. In this sense, the work consists of the presentation (concept and construction), several Geometric places that may be considered fundamental to solving elementary geometric design problems, and the presentation of conical constructions as something more elaborate of these loci considered fundamental. For fixing the concepts, each geometric place will feature some application examples and at the end of chapters, some proposed exercises will be presented ( to the reader who is interested iun practing the concepts and addressed buildings). Finalizing will be a brief review of the origins of geometric design and its teaching in Brazil, emphasizing problem solving as an effective method for teaching geometry.

Diploma thesis offers a new approach to the measuring of geometry errors in the machine tool workspace with the aid of laser tracker. There is a method of data acquisition and also the Matlab programs for data processing suggested in the thesis. This method can determine the accuracy and repeatibility of positioning and angular displacement of the numerical controlled axes of the measured machine tool and it compiles its mathematical model. All the gathered knowledge is used to evaluation of geometric accuracy of the virtual machined workpiece.

This dissertation explores geometric measure theory; the first part explores a question posed by Paul Erdös -- Is there a number c > 0 such that if E is a Lebesgue measurable subset of the plane with λ²(E) (planar measure)> c, then E contains the vertices of a triangle with area equal to one? -- other related geometric questions that arise from the topic. In the second part, "we parametrize the theorems from general topology characterizing the continuous images and the homeomorphic images of the Cantor set, C" (abstract, para. 5).

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B-spline surfaces have been widely used in aircraft design to represent different types of components in a uniform format. Unlike the visual trimming of B-spline surfaces, which hides unwanted portions in rendering, the geometric trimming approach provides a mathematically clean representation. This dissertation focuses on the geometric trimming of fuselage and wing components represented by B-spline surfaces. To trim two intersecting surfaces requires finding their intersections effectively. Most of the existing algorithms focus on providing intersections suitable for rendering. In this dissertation, an intersection algorithm suitable for geometric trimming of B-spline surfaces is presented. The number of intersection points depends on the number of isoparametric curves selected, and thus is controllable and independent of the error bound of intersection points. Trimming curves are classified and a new scheme for trimming by a closed trimming curve is provided to improve the accuracy. The surface trimmed by a closed trimming curve is subdivided into four patches and the trimming curve is converted into two open trimming curves. Two surface patches are created by knot insertion, which match the original surface exactly. The other two surface patches are trimmed by the converted open trimming curves. Factors affecting the trimming process are discussed and metrics are provided to measure trimming errors. Exact trimming is precluded due to the high degree of intersections. The process may lead to significant deviation from the corresponding portion on the original surface. Optimizations are employed to minimize approximation errors and obtain higher accuracy. The hybrid Parallel Tempering and Simulated Annealing optimization method, which is an effective algorithm to overcome the slow convergence waiting dilemma and initial value sensitivity, is applied for the minimization of B-spline surface representation errors. The results confirm that trimming errors are successfully reduced. / Ph. D.