Geometrija hiperbolične ravnine

delo diplomskega seminarja

Maja Blažej (Author), Janez Mrčun (Mentor), Jure Kališnik (Co-mentor)

Abstract

Peti aksiom evklidske geometrije pravi, da skozi dano točko T, ki ne leži na premici p, poteka natanko ena vzporednica k p skozi točko T. Če ta aksiom izpustimo, lahko za modele dobimo različne neevklidske geometrije. Mi bomo obravnavali hiperbolično geometrijo, kjer k vsaki premici lahko narišemo neskončno vzporednic skozi dano točko. Najprej bomo hiperbolično ravnino definirali, nato si bomo ogledali izometrije v hiperbolični ravnini in dokazali, da vse izometrije, ki ohranjajo orientacijo, lahko zapišemo v obliki Möbiusove transformacije. Pokazali bomo, da le-te tvorijo grupo izometrij v hiperbolični ravnini. Dokazali bomo tudi nekaj osnovnih izrekov hiperbolične trigonometrije.

Keywords

hiperbolična ravnina;izometrije hiperbolične ravnine;Möbiusova transformacija;hiperbolična trigonometrija;

Data

Language:	Slovenian
Year of publishing:	2020
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[M. Blažej]
UDC:	517.5
COBISS:	58540291
Views:	989
Downloads:	130
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Hyperbolic plane geometry
Secondary abstract:	The fifth axiom of Euclidean geometry says that for any given point T, that does not lie on a line p, there exists exactly one line through T that does not intersect p. If we disregard this axiom, we get different non-Euclidean geometries. We will investigate the hyperbolic geometry, where for each line an infinite number of parallel lines can be drawn through a given point. First, we will define the hyperbolic plane, then we will explore isometries of hyperbolic plane and prove that every isometry that preserves orientation can be written in the form of a Möbius transformation. We will show that isometries that preserve orientation form a group of isometries in the hyperbolic plane. In the end, we will prove some of the fundamental theorems of hyperbolic trigonometry.
Secondary keywords:	hyperbolic plane;isometries of hyperbolic plane;Möbius transformation;hyperbolic trigonometry;
Type (COBISS):	Final seminar paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	31 str.
ID:	12345779