Modely euklidovských a neeuklidovských rovin

Abstract

This bachelor's thesis deals with the study of models of Euclidean and non-Euclidean planes. The introduction presents Hilbert's axiomatic system of geometry with a focus on the axioms of incidence. It then describes individual models of planes, namely the model of the elliptic plane, the model of the hyperbolic plane, and the model of the Euclidean plane, realized by means of a paraboloid. For each model, the thesis presents basic concepts and examines their properties, in particular the relationships between points and lines. It also verifies selected axioms in each model. The aim of the thesis is to show how changing the parallel axiom affects the properties of a plane and gives rise to different geometric models.