Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2014-09-22T13:24:09Z No. of bitstreams: 2 Nogueira, Leonardo Bernardes.pdf: 4758026 bytes, checksum: 81be665ec243b277cb285cc686730f04 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2014-09-23T11:17:17Z (GMT) No. of bitstreams: 2 Nogueira, Leonardo Bernardes.pdf: 4758026 bytes, checksum: 81be665ec243b277cb285cc686730f04 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2014-09-23T11:17:17Z (GMT). No. of bitstreams: 2 Nogueira, Leonardo Bernardes.pdf: 4758026 bytes, checksum: 81be665ec243b277cb285cc686730f04 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2013-03-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This paper begins with a brief history about the development of vector spaces and linear transformations, then presents fundamental concepts for the study of Linear Algebra, with greater focus on linear operators in the R2 space. Through examples it explores a wide range of operators in R2 in order to show other applications of matrices in high school and prepares the ground for the presentation a version of Spectral Theorem for selfadjoint operators in R2, which says that for every operator self-adjoint T : E!E in finite dimensional vector space with inner product, exists an orthonormal basis fu1; : : : ;ung E formed by eigenvectors of T, and culminates with their applications on the study of conic sections, quadratic forms and equations of second degree in x and y; on the study of operators associated to quadratic forms, a version of Spectral Theorem could be called as The Main Axis Theorem albeit this nomenclature is not used in this paper. Thereby summarizing a study made by Lagrange in "Recherche d’arithmétique ", between 1773 and 1775, which he studied the property of numbers that are the sum of two squares. Thus he was led to study the effects of linear transformation with integer coefficients in a quadratic form in two variables. / Este trabalho inicia-se com um breve embasamento histórico sobre o desenvolvimento de espaços vetoriais e transformações lineares. Em seguida, apresenta conceitos fundamentais básicos, que formam uma linguagem mínima necessária para falar sobre Álgebra Linear, com enfoque maior nos operadores lineares do plano R2. Através de exemplos, explora-se um vasto conjunto de transformações no plano a fim de mostrar outras aplicações de matrizes no ensino médio e prepara o terreno para a apresentação do Teorema Espectral para operadores auto-adjuntos de R2. Este Teorema diz que para todo operador auto-adjunto T : E!E, num espaço vetorial de dimensão finita, munido de produto interno, existe uma base ortonormal fu1; : : : ;ung E formada por autovetores de T. O trabalho culmina com aplicações sobre o estudo das secções cônicas, formas quadráticas e equações do segundo grau em x e y, no qual o Teorema Espectral se traduz como Teorema dos Eixos Principais, embora essa nomenclatura não seja usada nesse trabalho (para um estudo mais aprofundado neste tema ver [3], [4], [5], [7]). Retomando assim um estudo feito por Joseph Louis Lagrange em "Recherche d’Arithmétique", entre 1773 e 1775, no qual estudou a propriedade de números que são a soma de dois quadrados. Assim, foi levado a estudar os efeitos das transformações lineares com coeficientes inteiros numa forma quadrática de duas variáveis.

This thesis is intended for teachers and students of high schools and universities. It consists collection of solved problems from plane analytical geometry including various solutions and their comparison. The thesis aims to increase the student knowledge of the topic and to provide different approaches to problems and working materials for lessons for teachers. Pictures for better understanding are added for more difficult problems. The practical part focusing on common mistakes and their elimination is included.

No description available.

One of the important parameters observed during extruded fibre fabrication is its diameter. The diameter can be measured with a single scanning camera assuming that the fibre section has a circular shape. As proved in practice, another important parameter is ovality, that is the rate of fibre flattening. This paper assumes that the fibre section shape is elliptical. In such a case, at least three different views on examined fibre are needed. Mathematical part of this paper is concerned with analytical description of fibre ovality measurement using two different approaches based on the knowledge of linear algebra, projective geometry and conic sections theory. Main goal of this paper is thus to use both mathematical theory and image analysis methods for ovality and diameter determination. Precise calcluation of such quantities is, however, conditioned on precise camera system calibration, which is described in the paper as well. Additionally, the work contains a brief mention of technical realization of ovality measurement and its possible difficulties.

In this dissertation we study basic metrical properties of 2-dimensional normed linear spaces, so-called (Minkowski or) normed planes. In the first chapter we introduce a notion of angular measure, and we investigate under what conditions certain angular measures in a Minkowski plane exist. We show that only the Euclidean angular measure has the property that in an isosceles triangle the base angles are of equal size. However, angular measures with the property that the angle between orthogonal vectors has a value of pi/2, i.e, a quarter of the full circle, exist in a wider variety of normed planes, depending on the type of orthogonality. Due to this we have a closer look at isosceles and Birkhoff orthogonality. Finally, we present results concerning angular bisectors. In the second chapter we pay attention to convex quadrilaterals. We give definitions of different types of rectangles and rhombi and analyse under what conditions they coincide. Combinations of defining properties of rectangles and rhombi will yield squares, and we will see that any two types of squares are equal if and only if the plane is Euclidean. Additionally, we define a ``new\'\' type of quadrilaterals, the so-called codises. Since codises and rectangles coincide in Radon planes, we will explain why it makes sense to distinguish these two notions. For this purpose we introduce the concept of associated parallelograms. Finally we will deal with metrically defined conics, i.e., with analogues of conic sections in normed planes. We define metric ellipses (hyperbolas) as loci of points that have constant sum (difference) of distances to two given points, the so-called foci. Also we define metric parabolas as loci of points whose distance to a given point equals the distance to a fixed line. We present connections between the shape of the unit ball B and the shape of conics. More precisely, we will see that straight segments and corner points of B cause, under certain conditions, that conics have straight segments and corner points, too. Afterwards we consider intersecting ellipses and hyperbolas with identical foci. We prove that in special Minkowski planes, namely in the subfamily of polygonal planes, confocal ellipses and hyperbolas intersect in a way called Birkhoff orthogonal, whenever the respective ellipse is large enough.

In this dissertation we study basic metrical properties of 2-dimensional normed linear spaces, so-called (Minkowski or) normed planes. In the first chapter we introduce a notion of angular measure, and we investigate under what conditions certain angular measures in a Minkowski plane exist. We show that only the Euclidean angular measure has the property that in an isosceles triangle the base angles are of equal size. However, angular measures with the property that the angle between orthogonal vectors has a value of pi/2, i.e, a quarter of the full circle, exist in a wider variety of normed planes, depending on the type of orthogonality. Due to this we have a closer look at isosceles and Birkhoff orthogonality. Finally, we present results concerning angular bisectors. In the second chapter we pay attention to convex quadrilaterals. We give definitions of different types of rectangles and rhombi and analyse under what conditions they coincide. Combinations of defining properties of rectangles and rhombi will yield squares, and we will see that any two types of squares are equal if and only if the plane is Euclidean. Additionally, we define a ``new\'\' type of quadrilaterals, the so-called codises. Since codises and rectangles coincide in Radon planes, we will explain why it makes sense to distinguish these two notions. For this purpose we introduce the concept of associated parallelograms. Finally we will deal with metrically defined conics, i.e., with analogues of conic sections in normed planes. We define metric ellipses (hyperbolas) as loci of points that have constant sum (difference) of distances to two given points, the so-called foci. Also we define metric parabolas as loci of points whose distance to a given point equals the distance to a fixed line. We present connections between the shape of the unit ball B and the shape of conics. More precisely, we will see that straight segments and corner points of B cause, under certain conditions, that conics have straight segments and corner points, too. Afterwards we consider intersecting ellipses and hyperbolas with identical foci. We prove that in special Minkowski planes, namely in the subfamily of polygonal planes, confocal ellipses and hyperbolas intersect in a way called Birkhoff orthogonal, whenever the respective ellipse is large enough.:1 Introduction 2 On angular measures 3 Types of convex quadrilaterals 4 On conic sections