Breve estudo sobre a geometria hiperbólica.

Abstract

The present work is a monograph in which we discuss, in order not to exhaust the theme, about Hyperbolic Geometry and disseminate it. Therefore, we seek to analyze this geometry in three dimensions: historical, mathematical and utilitarian. In the historical part we bring historical fragments about the problem involving the 5th Postulate of Euclid, which during its investigation throughout history, led to the emergence of non-Euclidean Geometries. Regarding Mathematics, we develop results present only in this theory, as well as results that it “inherits” from Euclidean Geometry, such as the congruence of triangles and Pasch’s postulate. Still, we present a table in which we list some of the peculiarities of each Geometry mentioned. Further- more, we present some models through which Hyperbolic Geometry is represented. Finally, with regard to utility, we discuss some possibilities and applications of the Hyperbolic Geometric theory, which are present in Mathematics itself, as well as in other branches of knowledge.