Estudo das geometrias afim e projetiva sob o ponto de vista de Felix Klein

Abstract

This work aims to present an introduction to Afim and Projective Geometries and their main resources through groups of transformations. Thus, it defines its properties, presents the theorems and the results of each geometry. Also, features the possibility of integrating three areas of mathematics, Geometry, Linear Algebra and Algebra. To make this work, used a specific bibliography that, in addition to Affine and Projective Geometry, also deals with other geometries, from the point of view of Felix Klein, but for this work he chose to work with only two of these geometries, having as alternatives for futures studies. Some examples were built using Geogebra and Cabri 3D software to explore details through figures in order to obtain a better understanding of the content presented. From this work, some results are obtained, for example, Ceva's Theorem using Fundamental Theorem of Affine Geometry, and Desargues Theorem that use the Fundamental Projective Geometry Theorem, which being different from those proof in Euclidean Geometry . Thus, in addition to making an introduction to these geometries, characterizing their geometric and algebraic aspects, proceeding to the classication of transformation groups, it opens the way for the continuation of studies of other geometries with the same purpose, that is, proceeding to the classication of different geometries via algebraic aspects.