Geometria hiperbólica : consistência do modelo de disco de Poincaré

• Ano de defesa: 2015
• Autor(a) principal: SOUZA, Carlos Bino de
• Orientador(a): SILVA, Thiago Dias Oliveira
• Banca de defesa: RODRIGUES, César Augusto, SILVA, Adriano Régis Melo Rodrigues da, KULESZA, Maité
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Universidade Federal Rural de Pernambuco
• Programa de Pós-Graduação: Programa de Pós-Graduação em Matemática (PROFMAT)
• Departamento: Departamento de Matemática
• País: Brasil
• Palavras-chave em Português: Poincaré; Disco de Poincaré; Geometria hiperbólica; Axiomas de Hilbert; Plano hiperbólico; Inversão; Disc Poincaré; Hyperbolic geometry; Axioms of Hilber; Hyperbolic plane; Inversion
• Área do conhecimento CNPq: CIENCIAS EXATAS E DA TERRA::MATEMATICA
• Link de acesso: http://www.tede2.ufrpe.br:8080/tede2/handle/tede2/6695

Euclid wrote a book in 13 volumes called Elements where systematized all the mathematical knowledge of his time. In this work, the 5 postulates of Euclidean geometry were presented. For several years, the 5th Postulate was frequently asked, this inquiries it was discovered that there are several other possible geometries, including hyperbolic geometry. Beltrimi proved that hyperbolic geometry is consistent if Euclidean geometry is consistent. Hilbert showed that Euclidean geometry is consistent if the arithmetic is consistent and presented an axiomatic system that capped the gaps in Euclid’s axiomatic system. Poincaré created a model, called the Poincaré disk, to represent the plan of hyperbolic geometry. The objective of this work is to show that the Poincaré disk model is consistent with reference Axioms Hilbert, replacing only the Axioms of Parallel to "On a point outside a line passes through the two parallel straight lines given", by constructions of Euclidean geometry.