Razão cruzada: dos clássicos aos contemporâneos

Abstract

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.