The structure of Euclidean, spherical, and hyperbolic geometries are compared, considering specifically postulates, curvature of the plane, and visual models. Implications for distance, the sum of the angles of triangles, and circumference to diameter ratios are investigated. / text

Convection in a Boussinesq fluid confined by a annular channel fast rotating about a vertical axis and uniformly heated from below, is one of our concerns in this thesis. An assumption that the channel has a sufficiently large radius in comparison with its gap-width is employed, so that the curvature effect can be neglected. The aspect ratio of the channel has great influence on the convective flow in it. Guided by the result of the linear stability analysis, we perform three-dimensional numerical simulations to investigate the convective flows under three different types of aspect ratios, which are namely the moderate or large aspect ratios, the very small aspect ratios and the moderately small aspect ratios. Also, we numerically study how convection in the channel is affected by inhomogeneous heat fluxes on sidewalls, which is a simple simulation of the thermal interaction between the Earth's core and mantle. Convection and dynamo action in a rapidly rotating, self-gravitating, Boussinesq fluid sphere is the other concern. We develop a finite element model for the dynamo problem in a whole sphere. This model is constructed by incorporating dynamo equations with globally implemented magnetic boundary conditions to a whole sphere convection model, which is also presented here. The coordinate singularity at the center usually encountered when applying the spectral method is no longer an obstacle and no nonphysical assumptions (i.e. hyper-diffusivities) are used in our model. A large effort has been made to efficiently parallelize the model. Consequently, it can take the full advantage of modern massively parallel computers. Based on this dynamo model, we investigate the dynamo process in a sphere and find that self-sustaining dynamos are more difficult to obtain in a sphere than in a spherical shell. They are activated at relatively high Rayleigh numbers. Moreover, the magnetic fields generated are not dipole-dominant, different from those generated in most dynamo simulations.

In projective geometry, the common self-polar triangle has often been used to discuss the location relationship of two planar conics. However, there are few researches on the properties of the common self-polar triangle, especially when the two planar conics are special conics. In this thesis, the properties of the common self-polar triangle of special conics are studied and their applications to computer vision are presented. Specifically, the applications focus on the two topics of the computer vision: camera calibration and homography estimation. This thesis first studies the common self-polar triangle of two sphere images and also that the common self-polar triangle of two concentric circles, and exploits its properties to solve the problem of camera calibration. For the sphere images, by recovering the constraints on the imaged absolute conic from the vertices of the common self-polar triangles, a novel method for estimating the intrinsic parameters of a camera from an image of three spheres has been developed. For the other case of concentric circles, it is shown in this thesis that the imaged circle center and the vanishing line of the support plane can be recovered simultaneously. Furthermore, many orthogonal vanishing points can be obtained from the common self-polar triangles. Consequently, two novel calibration methods have been developed. Based on our method, one of the state-of-the-art calibration methods has been well interpreted. This thesis then studies the common self-polar triangle of two separate ellipses, and applies it to planar homography estimation. For two images of the separate ellipses, by inducing four corresponding lines from the common self-polar triangle, a homography estimation method has been developed without ambiguity. Based on these results, a special case of planar rectification with two identical circles is also studied. It is shown that given one image of the two identical circles, the vanishing line of the support plane can be recovered from the common self-polar triangle and the imaged circle points can be obtained by intersecting the vanishing line with the image of the circle. Accordingly, a novel method for estimating the rectification homography has been developed and experimental results show the feasibility of our method.

Made available in DSpace on 2014-06-11T19:24:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-06-09Bitstream added on 2014-06-13T20:32:28Z : No. of bitstreams: 1 reis_jds_me_rcla.pdf: 2063932 bytes, checksum: b3a3de600d7749d917237284e24bd8f6 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Governo do Estado de São Paulo / Esta pesquisa tem como objetivo identificar materiais manipuláveis e descrever o seu uso em um processo de ensino e aprendizagem de Geometria Esférica. Para isso, foi desenvolvido um curso de extensão universitária sobre Geometria Esférica utilizando tais materiais e, desse modo, investigar esta utilização em um ambiente natural de sala de aula. Primeiramente, foram feitos estudos nos livros e dissertações que abordam as Geometrias Não-Euclidianas, bem como uma pesquisa sobre os recursos pedagógicos disponíveis que pudessem ser utilizados neste contexto, tais como softwares de geometria dinâmica, caleidoscópios, além de outros materiais manipuláveis. Após esta etapa, fizemos um estudo piloto para verificar a adequação e o encadeamento na aplicação das atividades. Em seguida, elaboramos e aplicamos o curso de extensão intitulado Geometria Esférica que foi direcionado a alunos do 3° ao 8° semestres da Graduação em Matemática da UNESP de Rio Claro. Os sujeitos de nossa pesquisa foram dez alunos deste programa de formação. Os dados coletados foram analisados qualitativamente, buscando compreender como estes materiais manipuláveis podem colaborar na aquisição de conceitos e propriedades básicas da Geometria Esférica. De acordo com os resultados, acreditamos que esta pesquisa pode auxiliar na busca por propostas alternativas para o ensino de Geometria, possibilitando uma melhor experiência de aprendizagem do futuro professor, enquanto aluno de graduação. / This research aims to identify handling materials and to describe their use in a teaching learning process of Spherical Geometry. For this, we developed a course on Spherical Geometry for students of higher education using those materials and, thus, investigate this use in a natural classroom environment. First, we studied books and dissertations about Non-Euclidean Geometries, as well as, we had done a search about available pedagogic sources that could be used in this context, such as softwares of dynamic geometry, kaleidoscope, besides others handling materials. After this stage, we made a pilot study to verify the adaptation and chaining in the application of the activities. Following, we elaborated and applied the course entitled Spherical Geometry that was addressed to the math students at the third to the eighth semesters of UNESP College, at Rio Claro city. The subjects of our research were ten students from this institution. The collected material were analyzed qualitatively, in order to understand how these handle materials can collaborate in the acquisition of concepts and basic proprieties of the Spherical Geometry. According to our results, we think that this research can assist in a search for alternatives purposes to the Geometry teaching, making possible a better experience of learning for the future teacher, while graduated student at a college.

Orientador: Claudemir Murari / Banca: Henrique Lazari / Banca: Ruy Madsen Barbosa / Resumo: Esta pesquisa tem como objetivo identificar materiais manipuláveis e descrever o seu uso em um processo de ensino e aprendizagem de Geometria Esférica. Para isso, foi desenvolvido um curso de extensão universitária sobre Geometria Esférica utilizando tais materiais e, desse modo, investigar esta utilização em um ambiente natural de sala de aula. Primeiramente, foram feitos estudos nos livros e dissertações que abordam as Geometrias Não-Euclidianas, bem como uma pesquisa sobre os recursos pedagógicos disponíveis que pudessem ser utilizados neste contexto, tais como softwares de geometria dinâmica, caleidoscópios, além de outros materiais manipuláveis. Após esta etapa, fizemos um estudo piloto para verificar a adequação e o encadeamento na aplicação das atividades. Em seguida, elaboramos e aplicamos o curso de extensão intitulado "Geometria Esférica" que foi direcionado a alunos do 3° ao 8° semestres da Graduação em Matemática da UNESP de Rio Claro. Os sujeitos de nossa pesquisa foram dez alunos deste programa de formação. Os dados coletados foram analisados qualitativamente, buscando compreender como estes materiais manipuláveis podem colaborar na aquisição de conceitos e propriedades básicas da Geometria Esférica. De acordo com os resultados, acreditamos que esta pesquisa pode auxiliar na busca por propostas alternativas para o ensino de Geometria, possibilitando uma melhor experiência de aprendizagem do futuro professor, enquanto aluno de graduação. / Abstract: This research aims to identify handling materials and to describe their use in a teaching learning process of Spherical Geometry. For this, we developed a course on Spherical Geometry for students of higher education using those materials and, thus, investigate this use in a natural classroom environment. First, we studied books and dissertations about Non-Euclidean Geometries, as well as, we had done a search about available pedagogic sources that could be used in this context, such as softwares of dynamic geometry, kaleidoscope, besides others handling materials. After this stage, we made a pilot study to verify the adaptation and chaining in the application of the activities. Following, we elaborated and applied the course entitled "Spherical Geometry" that was addressed to the math students at the third to the eighth semesters of UNESP College, at Rio Claro city. The subjects of our research were ten students from this institution. The collected material were analyzed qualitatively, in order to understand how these handle materials can collaborate in the acquisition of concepts and basic proprieties of the Spherical Geometry. According to our results, we think that this research can assist in a search for alternatives purposes to the Geometry teaching, making possible a better experience of learning for the future teacher, while graduated student at a college. / Mestre

Made available in DSpace on 2015-04-22T22:16:05Z (GMT). No. of bitstreams: 1 Celia Maria Nogueira Batista.pdf: 1926587 bytes, checksum: f61cde4e613a60695c9ce389a0eba8b1 (MD5) Previous issue date: 2007-12-26 / FAPEAM - Fundação de Amparo à Pesquisa do Estado do Amazonas / This thesis presents a new demonstration of the result FOR Plato not the fourth century BC in Ancient Greece, to que There hum Finite number of classes Poliedro Regular congruent, using a basic theory of spherical geometry, of the agreement with the work " The solid platonicos " make Dr. John Lucas Marques Barbosa University of Ceará, presented in Manaus. / Esta dissertação apresenta uma nova demostração do resultado obtido por Platão no século IV a.C, na Grécia antiga, de que existe um número finito de classes de Poliedro regulares congruentes, usando a teoria básica da Geometria Esférica, de acordo com o trabalho "Os sólidos Platônicos", do Dr. joão Lucas Marques Barbosa da Universidade do Céara, apresentado em Manaus.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O principal objetivo deste trabalho foi estudar, em triângulos construídos sobre uma superfície esféerica, versões para resultados conhecidos da geometria euclidiana plana e da trigonometria nos triângulos planos. Inicialmente apresentam-se os conceitos fundamentais da geometria esférica e alguns elementos de trigonometria triangular esférica. Para isso, iniciou-se com uma breve revisão de alguns desses resultados e também com algumas definições da geometria plana necessárias para a construção de resultados da geometria esférica. Feito isso, foram construídas, em um triângulo esférico, versões para a lei dos senos, a lei dos cossenos e outros resultados da trigonometria triangular plana. Também foi visto o Teorema de Girard, onde pode-se estudar a área de um triângulo construído sobre a superfície de uma esfera de raio R e a soma de seus ângulos internos, que ao contrário do que ocorre nos triângulos planos inscritos em um círculo de raio r, não é constante. Foi apresentado um contraexemplo, neste ambiente, em que o famoso teorema de Pitagoras não vale. Ao longo do texto são apresentados alguns exemplos com a utilização das relações trigonométricas estudadas, bem como alguns conceitos elementares de coordenadas geograficas e aplicações práaticas da trigonometria esférica na aviação e na geografia. Finalmente, observa-se que esse trabalho utiliza fortemente a matemática do Ensino Básico, facilitando assim a compreensão e o acesso de alunos e professores do Ensino Médio, bem como profissionais que fazem uso da matematica. / The main objective of this work was to study in triangles constructed on a spherical surface, versions of known results of the plane euclidean geometry and trigonometry in plans triangles. Initially it presents the fundamental concepts of spherical geometry and some elements of spherical triangular trigonometry. For this, begins with a brief review of some of these results and also with some denitions of plane geometry required for the construction of spherical geometry results. That done, are build, in a spherical triangle, versions for the law of sines, law of cosines and other results of the plane triangular trigonometry. Was also seen is the theorem of Girard, where can study the area of a triangle built on the surface of a sphere of radius R and the sum of its internal angles, which is not constant unlike what occurs in triangles plans built on disc of radius r. The Pythagorean theorem is not true in this environment and a counter-example will be presented. Throughout the text will be presented some examples with the use of trigonometric relations, as well as some elementary concepts of geographical coordinates and practical applications of spherical trigonometry in aviation and geography. Finally it is observed that this work strongly uses the mathematics of basic education, facilitating the understanding of the said theory, of students and teachers of basic education, as well as of the professionals who use math.

A numerical method for solving the buoyancy-driven magneto-convection equations in a rapidly rotating spherical shell is presented. The method is implemented through a FORTRAN 90 program, based on a FORTRAN 66 code written by Hollerbach [International Journal for Numerical Methods in Fluids, 32 (2000)] and partially translated in FORTRAN 90 by Riquier. The program uses the pseudo-spectral method and computes velocity as well as temperature fields in a rapidly rotating spherical shell. The code has been validated through comparisons with previous studies and parallelized using OpenMP. Comparisons with Hollerbach's method have been carried out and showed improvements in stability.

Made available in DSpace on 2016-04-27T16:57:07Z (GMT). No. of bitstreams: 1 Maria Lucia Torelli Doria de Andrade.pdf: 1813485 bytes, checksum: d412ceceb706049da015f5eeb9437db2 (MD5) Previous issue date: 2011-05-31 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The objective of the current work is to investigate ownership of Spherical Geometry elementary concepts by students in the second year of high school, from a teaching sequence. In addition, it aimed to solve the activities of this sequence, the subjects carry out research treatments and Semiotic Registers Representation conversions relevant to the mathematical Spherical Geometry objects to be studied. We seek to answer the following question: How a didactic sequence articulating different Registers Representation can evaluate high school students in learning Spherical Geometry concepts? Therefore, we applied to two high school students teaching sequence. We based it on Raymond Duval s Semiotic Registers Representation Theory and Guy Brousseau s Didactic Situations Theory as theoretical foundation of this research. The research approach was qualitative and, as methodology we adopted Engineering Curriculum assumptions. Watching the subjects research production in the course of the activities sequence, we found out these individuals performed conversions and registers treatments by semiotic representation according to Duval (2009), as well as they did the record conversion in natural language (activity statement) to record material (Styrofoam ball), which led them to solve the activity and understand Spherical Geometry straight line concept. In this sense, we could highlight semiotic representation material record, and infer that it contributed to the concepts ownership required by the research subjects / O presente trabalho tem como objetivo investigar a apropriação de conceitos elementares de Geometria Esférica por alunos do 2º ano do Ensino Médio, a partir de uma sequência de ensino. Além disso, objetivava que ao resolver as atividades desta sequência, os sujeitos de pesquisa realizassem os tratamentos e conversões dos Registros de Representação Semiótica pertinentes aos objetos matemáticos da Geometria Esférica a serem estudados. Buscamos responder a seguinte questão de pesquisada: Como uma sequência didática articulando diferentes registros de representação pode avaliar alunos do Ensino Médio na aprendizagem de conceitos de Geometria Esférica? Assim, aplicamos junto a dois alunos do Ensino Médio a sequência de ensino. Nos embasamos na Teoria dos Registros de Representação Semiótica de Raymond Duval e na Teoria das Situações Didáticas de Guy Brousseau como fundamentação teórica dessa pesquisa. A abordagem desta investigação foi qualitativa e, como metodologia adotamos pressupostos da Engenharia Didática. Observando as produções dos sujeitos de pesquisa no decorrer das atividades da sequência, constatamos que esses realizaram as conversões e os tratamentos dos registros de representação semiótica de acordo com Duval (2009), como quando fizeram a conversão do registro em língua natural (enunciado da atividade) para o registro material (bola de isopor), o que os levou a resolver a atividade e compreender o conceito de reta na Geometria Esférica. Nesse sentido, pudemos destacar o registro material de representação semiótica, e inferir que esse contribuiu para a apropriação dos conceitos requeridos por parte dos sujeitos de pesquisa

Esta dissertação traz ideias para a inserção de novos conteúdos na matemática escolar. Ela trata da exploração de geometrias não-euclidianas, através de dois ambientes de geometria dinâmica, o "Spherical Easel” e o "Disco de Poincaré". O primeiro é um software livre e o segundo foi desenvolvido utilizando-se o recurso de macro-construção do GeoGebra. Na concepção das atividades tratamos as idéias que correspondem ao mundo não-euclidiano fazendo comparações com aquelas que fazem parte da geometria euclidiana e para cada atividade há um comentário que explica a sua intenção de aprendizagem. É a partir de considerações teóricas sobre a natureza da geometria e sua evolução histórica, bem como sobre o processo de aprendizagem da geometria, que é feita a apresentação da proposta. / This dissertation brings ideas to the inclusion of new contents in school mathematics. They are related to the exploitation of non-Euclidean geometries through two dynamic geometry environments, the "Spherical Easel" and the "Poincaré Disk". The first one is a free software and the second one was developed using the GeoGebra macro-construction. In the design of the activities the approach of ideas that correspond to non-euclidian worlds was made through comparison with the euclidian world and for each activity there is a comment that explain its learning objective. The proposal is supported by theoretical considerations about the nature of geometry and its historical evolution, as well as about the geometry learning process.