Um estudo sobre curvas e suas paralelas: proposta de ensino de geometria diferencial na educação básica utilizando a ferramenta TikZ/LaTeX

• Autor(a) principal: Prado, Danilo de Sousa
• Data de Publicação: 2021
• Tipo de documento: Dissertação
• Idioma: por
• Título da fonte: Repositório Institucional da UFG
• dARK ID: ark:/38995/00130000032xm
• Texto Completo: http://repositorio.bc.ufg.br/tede/handle/tede/11427
• topic: Geometria diferencial; Curvas planas parametrizadas; Curvas paralelas; TikZ; Differential geometry; Parameterized plane curves; Parallel curves; TikZ; CIENCIAS EXATAS E DA TERRA::MATEMATICA

Studying the planar and analytical geometry, it is common to come across terms like parallel lines and concentric circles. More rarely, however, one encounters terms like parallels of a parabola – or of any other curve, for that matter. This thesis treats elements from diferential geometry of parameterized planar curves: the parallels of a given curve. We start by establishing the concepts of vectors and parameterized curves, and the concept of derivative is also introduced, presenting the derivation rules of some types of real and vector functions. We present parametrizations of a number of curves, including ellipsis, cycloid as well as hypercicloids, which along with the parabola, represents the main examples presented. Within the concepts of differential geometry, we introduce geometrical properties of parameterized planar curves, such as Frenet system, curvature and evolute (the curve of traced by the center of curvatures of a given curve). Parallel curves, also known as offsets or wavefronts, was first studied by Leibniz, around 1692 (FAROUKI; NEFF, 1990). Within this historical context, we present a few applications, based on which we proceed with a more detailed analytical and geometrical study of their propeties. At the end, a didactic sequence is proposed for the application of the content treated here in a class of the 3rd year of high school. The methodological resource consists of a set of activities to be worked, exclusively, in the computer lab, which aims to include a computational tool (TikZ) that helps in the process of teaching and learning the parallel curves, and that provides students with the development of the ability to produce algorithms during the elaboration of the figures. The implementation of such a proposal is based on the National Common Curricular Base (BNCC), where it emphasizes the need for the development of computational thinking in students; and also in Ávila (1991), which considers the importance of teaching the subject of Calculus in secondary education.