Poincaréjeva metrika hiperbolične ravnine

diplomsko delo

Petra Peklar (Author), Dušan Pagon (Mentor)

Abstract

V pričujočem diplomskem delu je na podlagi pol ravninskega Poincaréjevega modela hiperbolične geometrije definirana in raziskana Poincaréjeva metrika hiperbolične ravnine. V drugem poglavju so na podlagi obravnavanega modela definirani in raziskani osnovni pojmi in definicije hiperbolične ravnine, ki so utemeljeni s pomočjo elementov iz evklidske geometrije. V tretjem poglavju je sistematično definirana in raziskana Poincaréjeva hiperbolična funkcija dolžine, ki je ponazorjena s primerom definicije hiperboličnega obsega hiperboličnega kroga. Dokazani sta tudi pomembni posledici hiperboličnega aksioma o vzporednosti, povezani z vsoto notranjih kotov hiperboličnega trikotnika in relacijo skladnosti v hiperbolični geometriji. V četrtem poglavju je sistematično definirana in raziskana Poincaréjeva hiperbolična funkcija ploščine hiperboličnih večkotnikov, ki je ponazorjena s primerom definicije hiperbolične ploščine hiperboličnega kroga. Na primeru hiperbolične ploščine in obsega hiperboličnega kroga je utemeljena povezava med hiperbolično in evklidsko geometrijo, ki nastopi kot mejna vrednost hiperbolične geometrije.

Keywords

matematika;hiperbolična geometrija;polravnine;Poincaréjev model;dolžina;ploščina;diplomska dela;

Data

Language:	Slovenian
Year of publishing:	2009
Source:	Maribor
Typology:	2.11 - Undergraduate Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[P. Peklar]
UDC:	51(043.2)
COBISS:	16859912
Views:	2825
Downloads:	202
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	The Poincaré Metric of the Hyperbolic Plane
Secondary abstract:	This thesis studies and defines the Poincaré metric of the hyperbolic plane based on the Poincaré half-plane model in hyperbolic geometry. The second chapter describes defined and explored basic concepts and definitions of hyperbolic geometry, which are supported by Euclid's elements from Euclidean geometry. Systematically defined and explored Poincaré hyperbolic distance function presented in the third chapter is exemplified by the definition of a hyperbolic length in a hyperbolic circle. Also, two important consequences of the hyperbolic axiom of parallelism are proven and linked to the internal angle sum in a hyperbolic triangle and symmetric relation in hyperbolic geometry. The fourth chapter systematically defines and explores the Poincaré hyperbolic area function of hyperbolic polygons, which is exemplified by the definition of a hyperbolic area in a hyperbolic circle. A link between hyperbolic and Euclidean geometry, which acts as a boundary value for hyperbolic geometry, is based on an example of a hyperbolic area and perimeter of a hyperbolic circle.
Secondary keywords:	hyperbolic geometry;Poincaré half-plane model;Poincaré hyperbolic length function;Poincaré hyperbolic area function;
URN:	URN:SI:UM:
Type (COBISS):	Undergraduate thesis
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	65 f.
Keywords (UDC):	mathematics;natural sciences;naravoslovne vede;matematika;mathematics;matematika;
ID:	17763