This master thesis deals with conics in the real projective plane. The goal was to com- prehensibly introduce conics in the projective plane to high-school students and teachers. In order to fulfill this goal, the projective plane and homogenous coordinates were intro- duced, and harmonic set and priniple of duality were studied closely. The conics in the projective plane were approached from the perspective of history, and various definitions. Well-motivated introduction of a pole and a polar was emphasized.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This thesis will make a study of the conic, which can be defined as quadratic equations solutions with two variables, with the main objective recognition of same through a simplification of the quadratic form associated, whose procedure involves the diagonalization of symmetric matrices. Throughout this work, will address the prerequisites needed for the reader with little familiarity on the subject, can understand each stage of its development, as Euclidean spaces and matrix diagonalization. / Nesta dissertação faremos um estudo das cônicas, as quais podem ser definidas como soluções de equações do segundo grau com duas variáveis, tendo como objetivo principal o reconhecimento das mesmas por meio de uma simplificação da forma quadrática associada, cujo procedimento envolve a diagonalização de matrizes simétricas. Ao longo deste trabalho, serão abordados os pré-requisitos necessários para que o leitor, com pouca familiaridade no assunto, possa compreender cada etapa de seu desenvolvimento, como espaços euclidianos e diagonalização de matrizes.

Title: Utilization of the internet by teaching conics at high school Author: Bc. Věra Effenberger Department: Department of Mathematics Education Supervisor: RNDr. Jana Hromadová, Ph.D. Supervisor's e-mail address: Jana.Hromadova@mff.cuni.cz Abstract This diploma thesis is dealing with conics' problems. It is mainly destined for high school (or university) teachers of descriptive geometry and for students too. It can by used in aid of education of conics or by self-study, because it includes many of illustrative pictures and dynamic applets made in the program GeoGebra, which support the written theoretical text. In the work are enumerated definitions, properties and various constructions of individual conics. Further there is their origin as an intersection of a right circular cone (as the case may be of a right circular cylinder) with a plane, their osculating circle and conjugate diameters. Compilation of examples constitutes an addition of this work. The examples have various difficulty and also can serve as a control over got knowledge. Keywords: ellipse, hyperbola, parabola, foci, tangents, normals, construction of conics

Este trabalho é um estudo realizado em torno das principais curvas cônicas estudadas por alunos do ensino básico: parábola, elipse e hipérbole. A ideia central do trabalho é a autosuficiência, pois apresentamos todas as ferramentas matemáticas necessárias para o entedimento desses entes e suas aplicações, desde os axiomas iniciais da geometria plana até as definições formais das cônicas e demonstrações de suas propriedades. Espera-se que uma pessoa não especializada em matemática, ao ler o trabalho, entenda toda a matemática no entorno das aplicações dessas cônicas. / This work is a study carried out around the main conic curves studied by elementary school students: parabola, ellipse and hyperbola. The main idea of this work is to be self-contained, starting from the basic axioms from the geometry and after we present formal definitions, properties and applications of conics in the everyday life. It is expected that a person that is not a specialist in mathematics, are able to read and understand all the mathematics in the surroundings of the applications of these conics.

This is a thesis written for a master's degree at the Computer Vision Laboratory, University of Linköping. An abstract outer product is defined and used as a bridge to reach 2:nd and 4:th order tensors. Some applications of these in geometric analysis of range data are discussed and illustrated. In idealized setups, simple geometric objects, like spheres or polygons, are successfully detected. Finally, the generalization to n:th order tensors for storing and analysing geometric information is discussed.

<p>This is a thesis written for a master's degree at the Computer Vision Laboratory, University of Linköping. An abstract outer product is defined and used as a bridge to reach 2:nd and 4:th order tensors. Some applications of these in geometric analysis of range data are discussed and illustrated. In idealized setups, simple geometric objects, like spheres or polygons, are successfully detected. Finally, the generalization to n:th order tensors for storing and analysing geometric information is discussed.</p>

[pt] O estudo das Curvas Cônicas tem sido cada vez menos abordado no Ensino Médio e, nos poucos casos em que tal abordagem é apresentada, verifica-se uma prioridade indevida à memorização de equações. Por outro lado, embora a eficiência do Origami não seja divulgada com frequência no ensino de assuntos matemáticos de maior complexidade, existe uma geometria axiomática consistente por trás desta arte de dobrar papéis que a torna um instrumento de ensino capaz de explorar, com clareza, propriedades e definições de assuntos matemáticos. O presente trabalho pretende unir esses dois elementos, curvas cônicas e origami, com o intuito de desenvolver conceitos do primeiro a partir de construções do segundo. Desta forma, faz-se um relato histórico e conceitual sobre as Curvas Cônicas; descreve-se a importância do Origami e seu uso no ensino da Matemática; apresenta-se o estudo das sete possibilidades para uma única dobragem no Origami conhecidas como os axiomas de Huzita-Hatori com o objetivo de sugerir o uso das dobraduras no estudo da elipse, da parábola e da hipérbole no Ensino Médio das escolas do país. A fim de divulgar o Origami como um recurso eficiente e interessante no ensino das Cônicas e validar a pesquisa apresentada, uma oficina foi desenvolvida, aplicada, avaliada e aprimorada num pequeno grupo de estudantes de Licenciatura em Matemática e seus resultados estão aqui expostos. / [en] The study of Conic Curves has been each time less approached at High School and, in those few cases it is presented, it s possible to verify an improperly prioritized of equation memorizations. On the other hand, although the efficiency of the Origami is not often divulged at teaching mathematical subjects of greater complexity, there is a consistent axiomatic geometry behind this art of folding papers that makes it an a teaching tool able to explore, clearly, the properties and definitions of mathematical subjects. This study aims to join these two elements, conic curves and origami, in order to develop concepts from the first to building the second one. This way, it can make a historical and conceptual essay about the Conic Curves; describing the importance of the Origami and its use in Mathematics teaching; presenting the study of the seven possibilities for a single folding in Origami known as Huzita-Hatori s axioms in order to suggest the use of the folding in the study of ellipse, parable and hyperbole at High Schools all over the country. Divulging the Origami as an efficient and interesting resource in the teaching of the Conics and validate this research, a workshop was developed, applied, evaluated and improved in a small group of students of Degree in Mathematics and its results are exposed here.

Diploma thesis deals especially with practical examples on the topic of conics. It is divided into two parts. The first part includes chapters with examples on the topics discussed in the course Geometry I. In the second part affine properties of conics, which can be used in construction tasks, Pascal's theorem and Brianchon's theorem and their use are mentioned. The aim of this thesis is to ilustrate the conics on typical examples for those interested and then to show them the interesting properties, for which there is no space in the course Geometry I.

Submitted by Celso Magalhaes (celsomagalhaes@ufrrj.br) on 2017-06-08T12:58:44Z No. of bitstreams: 1 2016 - Alan Jorge Ciqueira Gon?alves.pdf: 2968550 bytes, checksum: c41aeac563a3e880f387fa38231d62f7 (MD5) / Made available in DSpace on 2017-06-08T12:58:44Z (GMT). No. of bitstreams: 1 2016 - Alan Jorge Ciqueira Gon?alves.pdf: 2968550 bytes, checksum: c41aeac563a3e880f387fa38231d62f7 (MD5) Previous issue date: 2015-08-31 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior - CAPES / This study aims to improve understanding of studies of conics. For this, we will use as theoretical foundation in the construction of the proposed activities, the geometric constructivist theory of Van Hiele. Moreover, in line with the new teaching, and learning tools, we use a dynamic geometry software, the GeoGebra / Este trabalho tem por objetivo melhorar a compreens?o do estudo de c?nicas. Para isto, usaremos como fundamenta??o te?rica na constru??o das atividades propostas a teoria construtivista geom?trica de Van Hiele. Al?m disso, em conson?ncia com as novas ferramentas de ensino e aprendizagem, utilizaremos um software de geometria din?mica, GeoGebra.

Este trabalho é um estudo realizado em torno das principais curvas cônicas estudadas por alunos do ensino básico: parábola, elipse e hipérbole. A ideia central do trabalho é a autosuficiência, pois apresentamos todas as ferramentas matemáticas necessárias para o entedimento desses entes e suas aplicações, desde os axiomas iniciais da geometria plana até as definições formais das cônicas e demonstrações de suas propriedades. Espera-se que uma pessoa não especializada em matemática, ao ler o trabalho, entenda toda a matemática no entorno das aplicações dessas cônicas. / This work is a study carried out around the main conic curves studied by elementary school students: parabola, ellipse and hyperbola. The main idea of this work is to be self-contained, starting from the basic axioms from the geometry and after we present formal definitions, properties and applications of conics in the everyday life. It is expected that a person that is not a specialist in mathematics, are able to read and understand all the mathematics in the surroundings of the applications of these conics.