El modelo sl2(R) del plano hiperbólico

Abstract

Among the best known models of the hyperbolic plane are the Poincaré models of the half-plane and the disk and that of the hyperboloid. In this article a new model H of the hyperbolic plane is studied, based on the vector space sl₂(R) of the matrices in M₂(R) with trace zero (section 1), where the scalar and exterior products of two matrices are defined, and so is a trilinear form of three matrices –volumen form– through the scalar product of the first by the exterior product of the other two. Such volumen form defines an orientation in H, where each point in H is given by one matrix in sl₂(R) with determinant 1 and each oriented geodesic has a normal vector given by one matrix with determinant -1; the opposite matrix defines the normal vector of the same geodesic with the opposite orientation. The above model and such tools are somehow present in Iversen [1] and, with another notation, can also be found in Fenchel [2]. Here we have tried to restrict the definitions to what is essential, so that the results acquire the greatest possible generality.
This model of the hyperbolic plane and the Poincare’s models are isomorphic, and the projections of H on the half-plane and the disk allow to make all the representations in any of these.

Next to the previous definitions and results, in chapter 2, we analyse the relative position of two geodesics –secant, parallel or ultraparallel–; then, in chapter 3, we study the isomorphisms between the three models of the hyperbolic plane mentioned above, and so does the representation of hyperbolic circumferences and triangles, being next defined the power of a point with respect to a circle and obtained the formulae for the radical axis of two circles and the radical centre of three circles, as well as those corresponding to the elements of a triangle. The majority of the contents can be found in [3]. The representations have been made using Scientific Notebook and Geogebra [4].
This article is addressed to those interested in the study of the geometry of the hyperbolic plane unfamiliar with the sl₂(R) model.