El plano hiperbólico : historia y fundamentos

Abstract

This thesis deals with three models of two-dimensional Hyperbolic Geometry, giving a description of two of them (the Poincaré disk model and the Klein-Beltrami model) and studying thoroughly the Poincaré half-plane model. The first part is a review of the history of Geometry in order to reach the two-dimensional Hyperbolic Geometry. It is explained here how until the 19th century it was thought that there was only one Geometry, the Euclidean Geometry. This is the Geometry we use to view or shape our physical space and whose origin can be found in a treatise written by the Greek mathematician Euclid around 300 b.C, Elements. This treatise collects an axiomatic system known as Euclid Postulates that lead to the Euclidean Geometry. In the first decades of the 19th century, Gauss, Bolyai and Lobachevsky were the first to find, independently, the existence of non-Euclidean Geometries, using Euclid's postulates. The first who achieves this is Lobachevsky and he creates what we known today as Hyperbolic Geometry. Eugenio Beltrami is the first to demonstrate that this Geometry is consistent and to prove this he sets out three models. These models, along with other contributions from Felix Klein and Henri Poincaré, are studied here. The second part of this thesis will describe the three models mentioned above. Firstly, properties of the Poincaré disk model and the Klein-Beltrami model will be analised, such as distances and relative positions of lines, as well as an equivalence between models. Secondly and in addition to lengths, distances, relative positions and areas of triangles, the concept of isometry for the Poincaré half-plane model will be defined, studying some of them as translations, rotations and reflections.