Três modelos para a geometria hiperbólica

Abstract

The objective of this work was to study the hyperbolic geometry building a model that mimics the construction of the model of the geometry of the sphere in Euclidean space R3. For this, we begin by doing a summary of historical nature of the emergence of the non-Euclidean geometries (the hyperbolic geometry and the spherical geometry). Then, in Chapter 1 we presented the basics notions, de.nitions and results, relevant to theunderstanding of the next two chapters. In chapter 2, we study the spherical geometry determining the spherical metric (induced of the metric Euclidean of the space . on di¤erent coordinates of S2), the geodesics in the sphere and the isometries. Already in chapter 3, we study the hyperbolic geometry similarly to the chapter 2, this is, determining the hyperbolic metric (in the models of Minkowski (H2M), of the disk of Poincaré (H2D) and of the semiplane of Poincaré (H2S)) the geodesics of H2S and the isometries of H2S. We end this work with the Appendix A that approach some important results referents the hyperbolic metrics relations such as: the hyperbolic distance between two points ofC+, the versions of the Pythagorean theorem, laws of the sines and of the cosines to the hyperbolic geometry and area of a triangle hyperbolic.