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Mode of access: World Wide Web. / Thesis (Ph. D.)--University of Hawaii at Manoa, 2004. / Includes bibliographical references (leaves 113-114). / Electronic reproduction. / Also available by subscription via World Wide Web / vii, 114 leaves, bound ill. 29 cm

Thesis (Ph.D.)--University of Delaware, 2008. / Principal faculty advisor: Qing Xiang, Dept. of Mathematical Sciences. Includes bibliographical references.

The discovery of a new truth in the older fields of mathematics is a rare event. Here an investigator may hope at best to secure greater elegance in method or notation, or to extend known results by some process of generalization. It is our purpose to make a study of conic sections in the spirit of the above remark, using the symbolism developed by Josiah Williard Gibbs.

Submitted by Edilaine Cláudia Lima da Silva (edilaine.clsilva@gmail.com) on 2018-09-20T14:32:59Z No. of bitstreams: 1 ESTUDO DE CONICAS E QUADRICAS-CONSTRUÇÕES COM O USO DO GEOGEBRA - VERSÃO FINAL - (EDILAINE CLAUDIA LIMA DA SILVA - OK).pdf: 29127272 bytes, checksum: acfd2392a1e7dcb2bd20c58c69c1c5b8 (MD5) / Approved for entry into archive by Elza Mitiko Sato null (elzasato@ibilce.unesp.br) on 2018-09-20T17:21:30Z (GMT) No. of bitstreams: 1 silva_ecl_me_sjrp.pdf: 2655426 bytes, checksum: a1fdb09f1acb62ebdade4446ec1f8083 (MD5) / Made available in DSpace on 2018-09-20T17:21:30Z (GMT). No. of bitstreams: 1 silva_ecl_me_sjrp.pdf: 2655426 bytes, checksum: a1fdb09f1acb62ebdade4446ec1f8083 (MD5) Previous issue date: 2018-08-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Este trabalho tem como propósito estudar cônicas e quádricas que podem ser representadas algebricamente por equações do segundo grau em duas e três variáveis, respectivamente. Em particular, a temática de cônicas foi objeto de estudo dos gregos bem antes do início da era cristã, muito embora sob uma perspectiva meramente geométrica. As cônicas e as superfícies de revolução obtidas a partir destas possuem inúmeras aplicações práticas em várias áreas do conhecimento humano, sendo, portanto, um conceito interdisciplinar. Vale salientar que os Parâmetros Curriculares Nacionais (PCN’s), sugerem a investigação de temas e eixos transversais que possam ser discutidas em várias disciplinas ao longo da vida escolar do estudante. Atividades didáticas que exploram os elementos fundamentais associados a cada uma das cônicas foram propostas para serem desenvolvidas junto aos estudantes do ensino médio. No intuito de se diferenciar das formas tradicionais de ensino, procura-se fazer uso das denominadas novas tecnologias, em especial do software de matemática dinâmica Geogebra, que é capaz de trabalhar conteúdos de geometria, álgebra, cálculo e estatística e, em particular, simular construções geométricas baseadas em régua e compasso. Os inúmeros recursos de visualização em 2D e 3D, aliados a animação de objetos matemáticos, permite que os jovens estudantes possam ter níveis de abstração e enxergar relações entre objetos no espaço difíceis de serem obtidas por meios convencionais. Ademais, essa ferramenta permite aos estudantes explorar, investigar, conjecturar e com isso despertar e estimular o interesse dos mesmos pela construção de seu saber matemático, tornando-os agentes nesse processo. / The purpose of this work is to study conics and quadrics that can be represented algebraically by equations of the second degree in two and three variables, respectively. In particular, the concepts of conics was studied by the Greeks before the beginning of the Christian era, albeit from a purely geometric perspective. The conics and surfaces of revolution obtained from these have numerous practical applications in several areas of human knowledge, being, therefore, an interdisciplinary concept. It should be noted that the National Curricular Parameters (PCN’s) suggest the investigation of themes and transversal axes that can be discussed in various disciplines throughout the student’s school life. Didactic activities that explore the fundamental elements associated to each of the conics were proposed to be developed with high school students. In order to differentiate itself from the traditional forms of teaching, we try to make use of the so-called new technologies, especially Geogebra dynamic mathematics software, which is able to work with geometry, algebra, calculus and statistics content and, in particular, simulate geometric constructions based on ruler and compass. This software has numerous 2D and 3D visualization capabilities, coupled with the animation of mathematical objects, enable young students to have levels of abstraction and to see relationships between objects in space that are difficult to obtain by conventional means. In addition, this tool allows students to explore, investigate, conjecture and thereby awaken and stimulate their interest in building their mathematical knowledge, making them agents in this process.

Esta pesquisa tem por objetivo investigar as aplicações das cônicas no processo de ensino e aprendizagem, dando uma ampla visão histórica e aplicada do conteúdo em questão. Serão dadas as definições, propriedades, equações e conceitos das cônicas. Classificaremos cônicas a partir de sua equação geral. Apresentaremos as esferas de Dandelin e daremos algumas aplicações das cônicas. / This research aims to investigate conic sections in the process of teaching and learning, providing a broad historic and applied view of the subject. It will be discussed definitions, properties, equations and concepts of the conic sections. We classify them from their general equation. We present the Dandelin spheres and provide applications of the conics.

Esta pesquisa tem por objetivo investigar as aplicações das cônicas no processo de ensino e aprendizagem, dando uma ampla visão histórica e aplicada do conteúdo em questão. Serão dadas as definições, propriedades, equações e conceitos das cônicas. Classificaremos cônicas a partir de sua equação geral. Apresentaremos as esferas de Dandelin e daremos algumas aplicações das cônicas. / This research aims to investigate conic sections in the process of teaching and learning, providing a broad historic and applied view of the subject. It will be discussed definitions, properties, equations and concepts of the conic sections. We classify them from their general equation. We present the Dandelin spheres and provide applications of the conics.

Conics and Geometry is a report that focuses on the development of new approaches in mathematics by breaking from the accepted norm of the time. The conics themselves have their beginning in this manner. The author uses three ancient problems in geometry to illustrate this trend. Doubling the cube, squaring the circle, and trisecting an angle have intrigued mathematicians for centuries. The author shows various approaches at solving these three problems: Hippias’ Quadratrix to trisect an angle and square the circle, Pappus’ hyperbola to trisect an angle, and Little and Harris’ simultaneous solution to all three problems. After presenting these approaches, the focus turns to the conic sections in the non-Euclidean geometry known as Taxicab geometry. / text

Thesis (M.S.)--Brigham Young University. Dept. of Mathematics, 2005. / Includes bibliographical references (p. 51).

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.