Razão cruzada: dos clássicos aos contemporâneos

Detalhes bibliográficos
Ano de defesa: 2016
Autor(a) principal: Hellen Lima de Paula
Orientador(a): Não Informado pela instituição
Banca de defesa: Não Informado pela instituição
Tipo de documento: Dissertação
Tipo de acesso: Acesso aberto
Idioma: por
Instituição de defesa: Universidade Federal de Minas Gerais
Brasil
ICX - DEPARTAMENTO DE MATEMÁTICA
Programa de Pós-Graduação em Matemática
UFMG
Programa de Pós-Graduação: Não Informado pela instituição
Departamento: Não Informado pela instituição
País: Não Informado pela instituição
Palavras-chave em Português:
Link de acesso: http://hdl.handle.net/1843/48964
https://orcid.org/0000-0002-4731-0159
Resumo: Inspired by What is a cross-ratio?, published by Fran¢ois Labourie in Notices of AMS - American Mathematical Society, our goal is to go through several contexts in which the concept of cross-ratio is used. Starting with the onedimensional case, in the context of classical Euclidean Geometry, following to the fundamental theorems of Projective Geometry and arriving at the complex plane (and Riemann sphere) and real Hyperbolic Geometry. The cross-ratio is preserved by the fractional linear transformations, or Möbius transformations, and is essentially the only projective invariant of a quadruple of collinear points, which justies its importance for Projective Geometry. In the Cayley-Klein model of real Hyperbolic Geometry, the distance between points is expressed in terms of the cross-ratio. We also approach the concept of complex cross-ratio, presented by Korányi and Riemann, which is a generalization of classical cross-ratio and an important geometric invariant of a quadruple of points at the boundary of the complex hyperbolic plane. In the context of dynamics the concept of the derivative of Schwarz is presented as an innitesimal version of the cross-ratio, which allows us to evaluate the variation, under a dened transformation in the projective line, of the cross-ratio of innitely close points.