PORTELA, Antonio Edilson Cardoso. Noções de geometria projetiva. 2017. 58 f. Dissertação (Mestrado Profissional em Matemática em Rede Nacional) - Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Jessyca Silva (jessyca@mat.ufc.br) on 2017-09-06T17:17:00Z No. of bitstreams: 1 2017_dis_aecportela.pdf: 1065928 bytes, checksum: 468c05aa35745f3fd2761f13aa26eff1 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Boa tarde, Estou devolvendo a Dissertação de ANTONIO EDILSON CARDOSO PORTELA, para que o mesmo realize algumas correções na formatação do trabalho. 1- SUMÁRIO ( A formatação do sumário está incorreta, primeiro, retire o último ponto final que aparece após a numeração dos capítulos e seções (Ex.: 3.1. Axioma....; deve ser corrigido para: 3.1 Axioma.....), o alinhamento dos títulos deve seguir o modelo abaixo 1 INTRODUÇÃO.....................00 2 O ESPAÇO...........................00 3 GEOMETRIA........................00 3.1 Axiomas...............................00 REFERÊNCIAS...................00 (OBS.: não altere a formatação do negrito, pois já estava correta) 2- TITULO DOS CAPÍTULOS E SEÇÕES ( retire o ponto final que aparece após o último dígito da numeração dos capítulos e seções, seguindo o modelo do sumário. Retire o recuo de parágrafo dos títulos das seções. Ex.: 3.1 Axioma.......) 3- REFERÊNCIAS ( substitua o termo REFERÊNCIAS BIBLIOGRÁFICAS apenas por REFERÊNCIAS, com fonte n 12, negrito e centralizado. Retire a numeração progressiva que aparece nos itens da referência. Atenciosamente, on 2017-09-06T17:56:50Z (GMT) / Submitted by Jessyca Silva (jessyca@mat.ufc.br) on 2017-09-11T14:48:40Z No. of bitstreams: 1 2017_dis_aecportela.pdf: 944228 bytes, checksum: 3ab4691817df04ba5d7818fd02e5095f (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-09-11T15:30:32Z (GMT) No. of bitstreams: 1 2017_dis_aecportela.pdf: 944228 bytes, checksum: 3ab4691817df04ba5d7818fd02e5095f (MD5) / Made available in DSpace on 2017-09-11T15:30:32Z (GMT). No. of bitstreams: 1 2017_dis_aecportela.pdf: 944228 bytes, checksum: 3ab4691817df04ba5d7818fd02e5095f (MD5) Previous issue date: 2017 / In this work, initially, some results of Linear Algebra are presented, in particular the study of the Vector Space R^n, which becomes, together with Analytical Geometry, the language used in the chapters that follow. We present a study from an axiomatic point of view, from the perspectives of Hilbert's axioms and we elaborate models of planes for the Euclidean, Elliptic and Projective Geometries. The validity of the Incidence and Order axioms for Euclidean Geometry is verified. In R^3, an approach is made to the study of the plane and the unitary sphere, highlighting the elliptical line obtained by the intersection of these sets, thus making an approach to the Elliptic Geometry. With the concepts and definitions studied in the Vector Space R^n, Three-dimensional Space and in the Euclidean and Elliptic Geometries we will approach the study of Projective Geometry, demonstrating propositions and verifying its axioms. / Neste trabalho, inicialmente, apresenta-se alguns resultados da Álgebra Linear, em especial o estudo do Espaço Vetorial R^n, que passa a ser, juntamente com a Geometria Analítica, a linguagem empregada nos capítulos que se seguem. Apresentamos um estudo de um ponto de vista axiomático, sob a ótica dos axiomas de Hilbert e elaboramos modelos de planos para as Geometrias Euclidiana, Elíptica e Projetiva. É verificada a validade dos axiomas de Incidência e Ordem para a Geometria Euclidiana. No R^3, é feita uma abordagem do estudo de plano e da esfera unitária, destacando a reta elíptica obtida pela interseção destes conjuntos, passando assim a fazer uma abordagem da Geometria Elíptica. Com os conceitos e definições estudadas no Espaço Vetorial R^n, Espaço tridimensional e nas Geometrias Euclidiana e Elíptica, abordaremos o estudo da Geometria Projetiva, demonstrando proposições e verificando os seus axiomas.

Submitted by Geandra Rodrigues (geandrar@gmail.com) on 2018-05-17T13:38:00Z No. of bitstreams: 1 marcelocunhafigueiredo.pdf: 942741 bytes, checksum: 463b01a7fb542d3aa4a0d2f5867445e7 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-05-22T13:54:17Z (GMT) No. of bitstreams: 1 marcelocunhafigueiredo.pdf: 942741 bytes, checksum: 463b01a7fb542d3aa4a0d2f5867445e7 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-05-22T13:55:00Z (GMT) No. of bitstreams: 1 marcelocunhafigueiredo.pdf: 942741 bytes, checksum: 463b01a7fb542d3aa4a0d2f5867445e7 (MD5) / Made available in DSpace on 2018-05-22T13:55:00Z (GMT). No. of bitstreams: 1 marcelocunhafigueiredo.pdf: 942741 bytes, checksum: 463b01a7fb542d3aa4a0d2f5867445e7 (MD5) Previous issue date: 2018-02-28 / As geometrias denominadas não-euclidianas fazem parte da história da Matemática e nossas leituras mostram que elas não têm tido muito espaço nas licenciaturas da matéria no Brasil. Contrapondo esse panorama, buscamos estruturar um material para atuais e futuros professores de matemática, vislumbrando um outro olhar para a Geometria. Nosso estudo tem como objetivo buscar nas literaturas sobre o tema formas de apresentação da Geometria Projetiva, para confecção de um produto educacional que mostre uma das possibilidades de axiomatização desta teoria. O curso proposto foi aplicado na prática junto a um grupo de licenciandos de uma universidade pública que já tinham estudado Geometria Euclidiana Plana, oportunizando uma pesquisa de campo. As atividades propostas no produto são de cunho investigativo, e buscam solucionar problemas de Geometria Euclidiana de forma alternativa, com auxílio de software de geometria dinâmica e após a apresentação de uma concepção geométrica projetiva. Nossa investigação teve como foco a questão: ‘Como um curso básico e introdutório de Geometria Projetiva pode contribuir para licenciandos repensarem a geometria estruturalmente, de modo ampliado em seus fundamentos?’ Da pesquisa empreendida resultaram dados, analisados com base na metodologia fenomenológica, através da qual se procede com as reduções a partir das manifestações genuínas dos sujeitos de pesquisa, obtendo-se categorizações em forma de convergências de significações. / The so named non-Euclidean geometries are a part of the History of Mathematics, but our readings indicate an absence of space for this subject in Brazilian Mathematics teaching degree courses. Opposing this scenario, we aimed to devise materials for both current and future Mathematics teachers under a different perspective for Geometry. Our goal was to search the literature for ways of presenting projective geometry in order to make up an educational product that shows a possibility of axiomatizing such theory. The proposed course was applied to a group of Mathematics teaching undergraduate students who had already gone through a Plane Euclidean Geometry class at a public University, which created the opportunity for a field study. The activities we propose in our product are of investigative nature, aiming to solve Euclidean Geometry problems in alternative ways with the help of a Dynamic Geometry software after the presentation of a Projective Geometry conception. Our investigation was focused on the question “In which ways can a basic and introductory class on Projective Geometry contribute for teaching degree students to structurally rethink Geometry by widening its foundations?”. The data we generated in our research was analyzed according to the phenomenological approach, in which reductions are made based from genuine manifestations of the study subjects and categorizations are obtained in the form of signification convergences.

Measuring 3D surface geometry with precision and accuracy is an important part of many engineering and scientific tasks. 3D Scanning techniques measure surface geometry by estimating the locations of sampled surface points. In recent years, Structured-Light 3D scanners have gained significant popularity owing to their ability to produce highly accurate scans in real-time at a low cost. In this thesis we describe an approach for Structured-Light 3D scanning using a digital camera and a digital projector. We utilise the projective geometric relationships between the projector and the camera to carry out both an implicit calibration of the system and to solve for 3D structure. Our approach to geometric calibration is flexible, reliable and amenable to robust estimation. In addition, we model and account for the radiometric non-linearities in the projector such as gamma distortion. Finally, we apply a post-processing step to efficiently smooth out high-frequency surface noise while retaining the structural details. Consequently, the proposed work reduces the computational load and set-up time of a Structured-Light 3D scanner; thereby speeding up the whole scanning process while retaining the ability to generate highly accurate results. We demonstrate the accuracy of our scanning results on real-world objects of varying degrees of surface complexity. Introduction The projective geometry for a pair of pin-hole viewing devices is completely defined by their intrinsic calibration and their relative motion or extrinsic calibration in the form of matrices. For a Euclidean reconstruction, the geometry elements represented by the calibration matrices must be parameterised and estimated in some form. The use of a projector as the ‘second viewing’ device has led to numerous approaches to model and estimate its intrinsic parameters and relative motion with respect to the camera's 3D co-ordinate system. Proposed thesis work assimilates the benefits of projective geometry constructs such as Homography and the invariance of the cross-ratios to simplify the system calibration and the 3D estimation processes by an implicit modeling of the projector's intrinsic parameters and its relative motion. Though linear modeling of the projective geometry between a camera-projector view-pair captures the most essential aspects of the underlying geometry, it does not accommodate system non-linearities due to radiometric distortions of a projector device. We propose an approach that uses parametric splines to model the systematic errors introduced by radiometric non-linearities and thus correct for them. For 3D surfaces reconstructed as point-clouds, noise manifests itself as some high-frequency variations for the resulting mesh. Various pre and/or post processing techniques are proposed in the literature to model and minimize the effects of noise. We use simple bilateral filtering of the depth-map for the reconstructed surface to smoothen the surface while retaining its structural details. Modeling Projective Relations In our approach for calibrating the projective-geometric structure of a projector-camera view-pair, the frame of reference for measurements is attached to the camera. The camera is calibrated using a commonly used method. To calibrate the scanner system, one common approach is to project sinusoidal patterns onto the reference planes to generate reference phase maps. By relating the phase-information between the projector and image pixels, a dense mapping is obtained. However, this is an over-parameterisation of the calibration information. Since the reference object is a plane, we can use the projective relationships induced by a plane to implicitly calibrate the projector geometry. For the estimation of the three-dimensional structure of the imaged object, we utilise the invariance of cross-ratios along with the calibration information of two reference planes. Our formulation is also extensible to utilise more than two reference plane to compute more than one estimate of the location of an unknown surface point. Such estimates are amenable to statistical analysis which allows us to derive both the shape of an object and associate reliability scores to each estimated point location. Radiometric Correction Structured-light based 3D scanners commonly employ phase-shifted sinusoidal patterns to solve for the correspondence problem. For scanners using projective geometry between a camera and a projector, the projector's radiometric non-linearities introduce systematic errors in establishing correspondences. Such errors manifest as visual artifacts which become pronounced when fewer phase-shifted sinusoidal patterns are used. While these artifacts can be avoided by using a large number of phase-shifts, doing so also increases the acquisition time. We propose to model and rectify such systematic errors using parametric representations. Consequently, while some existing methods retain the complete reference phase maps to account for such distortions, our approach describes the deviations using a few model parameters. The proposed approach can be used to reduce the number of phase-shifted sinusoidal patterns required for codification while suppressing systematic artifacts. Additionally, our method avoids the 1D search steps that are needed when a complete reference phase map is used, thus reducing the computational load for 3D estimation. The effectiveness of our method is demonstrated with reconstruction of some geometric surfaces and a cultural figurine. Filtering Noise For a structured-light 3D scanner, various sources of noise in the environment and the devices lead to inaccuracies in estimating the codewords (phase map) for an unknown surface, during reconstruction. We examine the effects of such noise factors on our proposed methods for geometric and radiometric estimation. We present a quantitative evaluation for our proposed method by scanning the objects of known geometric properties or measures and then computing the deviations from the expected results. In addition, we evaluate the errors introduced due to inaccuracies in system calibration by computing the variance statistics from multiple estimates for the reconstructed 3D points, where each estimate is computed using a different pair of reference planes. Finally, we discuss the efficacy of certain filtering techniques in reducing the high-frequency surface noise when applied to: (a) the images of the unknown surface at a pre-processing stage, or (b) the respective phase (or depth) map at a post-processing stage. Conclusion In this thesis, we motivate the need for a procedurally simple and computationally less demanding approach for projector calibration. We present a method that uses homographies induced by a pair of reference planes to calibrate a structured-light scanner. By using the projective invariance of the cross-ratio, we solved for the 3D geometry of a scanned surface. We demonstrate the fact that 3D geometric information can be derived using our approach with accuracy on the order of 0.1 mm. Proposed method reduces the image acquisition time for calibration and the computational needs for 3D estimation. We demonstrate an approach to effectively model radiometric distortions for the projector using cubic splines. Our approach is shown to give significant improvement over the use of complete reference phase maps and its performance is comparable to that of a sate-of-the-art method, both quantitatively as well as qualitatively. In contrast with that method, proposed method is computationally less expensive, procedurally simpler and exhibits consistent performance even at relatively higher levels of noise in phase estimation. Finally, we use a simple bilateral filtering on the depth-map for the region-of-interest. Bilateral filtering provides the best trade-off between surface smoothing and the preservation of its structural details. Our filtering approach avoids computationally expensive surface normal estimation algorithms completely while improving surface fidelity.

No description available.

The thesis proposes an unsupervised representation learning method to predict 3D human pose from a 2D skeleton via a VAEGAN (Variational Autoencoder Generative Adversarial Network) hybrid network. The method learns to lift poses from 2D to 3D using selfsupervision and adversarial learning techniques. The method does not use images, heatmaps, 3D pose annotations, paired/unpaired 2Dto3D skeletons, 3D priors, synthetic 2D skeletons, multiview or temporal information in any shape or form. The 2D skeleton input is taken by a VAE that encodes it in a latent space and then decodes that latent representation to a 3D pose. The 3D pose is then reprojected to 2D for a constrained, selfsupervised optimization using the input 2D pose. Parallelly, the 3D pose is also randomly rotated and reprojected to 2D to generate a ’novel’ 2D view for unconstrained adversarial optimization using a discriminator network. The combination of the optimizations of the original and the novel 2D views of the predicted 3D pose results in a ’realistic’ 3D pose generation. The thesis shows that the encoding and decoding process of the VAE addresses the major challenge of erroneous and incomplete skeletons from 2D detection networks as inputs and that the variance of the VAE can be altered to get various plausible 3D poses for a given 2D input. Additionally, the latent representation could be used for crossmodal training and many downstream applications. The results on Human3.6M datasets outperform previous unsupervised approaches with less model complexity while addressing more hurdles in scaling the task to the real world. / Uppsatsen föreslår en oövervakad metod för representationslärande för att förutsäga en 3Dpose från ett 2D skelett med hjälp av ett VAE GAN (Variationellt Autoenkodande Generativt Adversariellt Nätverk) hybrid neuralt nätverk. Metoden lär sig att utvidga poser från 2D till 3D genom att använda självövervakning och adversariella inlärningstekniker. Metoden använder sig vare sig av bilder, värmekartor, 3D poseannotationer, parade/oparade 2D till 3D skelett, a priori information i 3D, syntetiska 2Dskelett, flera vyer, eller tidsinformation. 2Dskelettindata tas från ett VAE som kodar det i en latent rymd och sedan avkodar den latenta representationen till en 3Dpose. 3D posen är sedan återprojicerad till 2D för att genomgå begränsad, självövervakad optimering med hjälp av den tvådimensionella posen. Parallellt roteras dessutom 3Dposen slumpmässigt och återprojiceras till 2D för att generera en ny 2D vy för obegränsad adversariell optimering med hjälp av ett diskriminatornätverk. Kombinationen av optimeringarna av den ursprungliga och den nya 2Dvyn av den förutsagda 3Dposen resulterar i en realistisk 3Dposegenerering. Resultaten i uppsatsen visar att kodningsoch avkodningsprocessen av VAE adresserar utmaningen med felaktiga och ofullständiga skelett från 2D detekteringsnätverk som indata och att variansen av VAE kan modifieras för att få flera troliga 3D poser för givna 2D indata. Dessutom kan den latenta representationen användas för crossmodal träning och flera nedströmsapplikationer. Resultaten på datamängder från Human3.6M är bättre än tidigare oövervakade metoder med mindre modellkomplexitet samtidigt som de adresserar flera hinder för att skala upp uppgiften till verkliga tillämpningar.

Text in Afrikaans / Real empowerment of school learners requires preparing them for the age of technology. This empowerment can be achieved by developing their higher-order thinking skills. This is clearly the intention of the proposed South African FET National Curriculum Statements Grades 10 to 12 (Schools). This research shows that one method of developing higher-order thinking skills is to adopt an integrated curriculum approach. The research is based on the assumption that an integrated curriculum approach will produce learners with a more integrated knowledge structure which will help them to solve problems requiring higher-order thinking skills. These assumptions are realistic because the empirical results of several comparative research studies show that an integrated curriculum helps to improve learners' ability to use higher-order thinking skills in solving nonroutine problems. The curriculum mentions four kinds of integration, namely integration across different subject areas, integration of mathematics with the real world, integration of algebraic and geometric concepts, and integration into and the use of dynamic geometry software in the learning and teaching of geometry. This research shows that from a psychological, pedagogical, mathematical and historical perspective, the theme conic sections can be used as an integrating factor in the new proposed FET mathematics curriculum. Conics are a powerful tool for making the new proposed curriculum more integrated. Conics can be used as an integrating factor in the FET band by means of mathematical exploration, visualisation, relating learners' experiences of various parts of mathematics to one another, relating mathematics to the rest of the learners' experiences and also applying conics to solve real-life problems. / Mathematical Sciences / D.Phil. (Wiskundeonderwys)

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Nesta dissertação é apresentada uma modelagem analítica para o processo evolucionário formulado pela Teoria da Evolução por Endossimbiose representado através de uma sucessão de estágios envolvendo diferentes interações ecológicas e metábolicas entre populações de bactérias considerando tanto a dinâmica populacional como os processos produtivos dessas populações. Para tal abordagem é feito uso do sistema de equações diferenciais conhecido como sistema de Volterra-Hamilton bem como de determinados conceitos geométricos envolvendo a Teoria KCC e a Geometria Projetiva. Os principais cálculos foram realizados pelo pacote de programação algébrica FINSLER, aplicado sobre o MAPLE. / This work presents an analytical approach for modeling the evolutionary process formulated by the Serial Endosymbiosis Theory represented by a succession of stages involving different metabolic and ecological interactions among populations of bacteria considering both the population dynamics and production processes of these populations. In such approach we make use of systems of differential equations known as Volterra-Hamilton systems as well as some geometric concepts involving the KCC Theory and the Projective Geometry. The main calculations were performed by the computer algebra software FINSLER based on MAPLE.

GaugThe first half of the thesis concerns Abelian vortices and Yang-Mills theory. It is proved that the 5 types of vortices recently proposed by Manton are actually symmetry reductions of (anti-)self-dual Yang-Mills equations with suitable gauge groups and symmetry groups acting as isometries in a 4-manifold. As a consequence, the twistor integrability results of such vortices can be derived. It is presented a natural definition of their kinetic energy and thus the metric of the moduli space was calculated by the Samols' localisation method. Then, a modified version of the Abelian–Higgs model is proposed in such a way that spontaneous symmetry breaking and the Bogomolny argument still hold. The Painlevé test, when applied to its soliton equations, reveals a complete list of its integrable cases. The corresponding solutions are given in terms of third Painlevé transcendents and can be interpreted as original vortices on surfaces with conical singularity. The last two chapters present the following results in projective differential geometry and Hamiltonians of hydrodynamic-type systems. It is shown that the projective structures defined by the Painlevé equations are not metrisable unless either the corresponding equations admit first integrals quadratic in first derivatives or they define projectively flat structures. The corresponding first integrals can be derived from Killing vectors associated to the metrics that solve the metrisability problem. Secondly, it is given a complete set of necessary and sufficient conditions for an arbitrary affine connection in 2D to admit, locally, 0, 1, 2 or 3 Killing forms. These conditions are tensorial and simpler than the ones in previous literature. By defining suitable affine connections, it is shown that the problem of existence of Killing forms is equivalent to the conditions of the existence of Hamiltonian structures for hydrodynamic-type systems of two components.

Text in Afrikaans / Real empowerment of school learners requires preparing them for the age of technology. This empowerment can be achieved by developing their higher-order thinking skills. This is clearly the intention of the proposed South African FET National Curriculum Statements Grades 10 to 12 (Schools). This research shows that one method of developing higher-order thinking skills is to adopt an integrated curriculum approach. The research is based on the assumption that an integrated curriculum approach will produce learners with a more integrated knowledge structure which will help them to solve problems requiring higher-order thinking skills. These assumptions are realistic because the empirical results of several comparative research studies show that an integrated curriculum helps to improve learners' ability to use higher-order thinking skills in solving nonroutine problems. The curriculum mentions four kinds of integration, namely integration across different subject areas, integration of mathematics with the real world, integration of algebraic and geometric concepts, and integration into and the use of dynamic geometry software in the learning and teaching of geometry. This research shows that from a psychological, pedagogical, mathematical and historical perspective, the theme conic sections can be used as an integrating factor in the new proposed FET mathematics curriculum. Conics are a powerful tool for making the new proposed curriculum more integrated. Conics can be used as an integrating factor in the FET band by means of mathematical exploration, visualisation, relating learners' experiences of various parts of mathematics to one another, relating mathematics to the rest of the learners' experiences and also applying conics to solve real-life problems. / Mathematical Sciences / D.Phil. (Wiskundeonderwys)

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Nesta dissertação é apresentada uma modelagem analítica para o processo evolucionário formulado pela Teoria da Evolução por Endossimbiose representado através de uma sucessão de estágios envolvendo diferentes interações ecológicas e metábolicas entre populações de bactérias considerando tanto a dinâmica populacional como os processos produtivos dessas populações. Para tal abordagem é feito uso do sistema de equações diferenciais conhecido como sistema de Volterra-Hamilton bem como de determinados conceitos geométricos envolvendo a Teoria KCC e a Geometria Projetiva. Os principais cálculos foram realizados pelo pacote de programação algébrica FINSLER, aplicado sobre o MAPLE. / This work presents an analytical approach for modeling the evolutionary process formulated by the Serial Endosymbiosis Theory represented by a succession of stages involving different metabolic and ecological interactions among populations of bacteria considering both the population dynamics and production processes of these populations. In such approach we make use of systems of differential equations known as Volterra-Hamilton systems as well as some geometric concepts involving the KCC Theory and the Projective Geometry. The main calculations were performed by the computer algebra software FINSLER based on MAPLE.