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master's thesis
Konformni modeli hiperboličke ravnine
Zagreb: University of Zagreb, Faculty of Science, 2016. urn:nbn:hr:217:023588
Joha, Valentina
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Joha, V. (2016). Konformni modeli hiperboličke ravnine (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:023588

Joha, Valentina. "Konformni modeli hiperboličke ravnine." Master's thesis, University of Zagreb, Faculty of Science, 2016. https://urn.nsk.hr/urn:nbn:hr:217:023588

Joha, Valentina. "Konformni modeli hiperboličke ravnine." Master's thesis, University of Zagreb, Faculty of Science, 2016. https://urn.nsk.hr/urn:nbn:hr:217:023588

Joha, V. (2016). 'Konformni modeli hiperboličke ravnine', Master's thesis, University of Zagreb, Faculty of Science, accessed 23 December 2024, https://urn.nsk.hr/urn:nbn:hr:217:023588

Joha V. Konformni modeli hiperboličke ravnine [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2016 [cited 2024 December 23] Available at: https://urn.nsk.hr/urn:nbn:hr:217:023588

V. Joha, "Konformni modeli hiperboličke ravnine", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2016. Available at: https://urn.nsk.hr/urn:nbn:hr:217:023588

Metadata
TitleKonformni modeli hiperboličke ravnine
AuthorValentina Joha
MentorVedran Krčadinac (mentor)
Committee memberVedran Krčadinac (predsjednik povjerenstva)
Committee memberDragutin Svrtan (član povjerenstva)
Committee memberŽeljka Milin-Šipuš (član povjerenstva)
Committee memberIvica Nakić (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2016-11-29, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
Hiperbolička ravnina ima mnoga svojstva euklidske ravnine. Međutim, u njoj ne vrijedi peti euklidov postulat pa donosi i mnoge novosti u odnosu na euklidsku geometriju. Izabrali smo model gornje poluravnine \(\mathbb{H}\) u kojem smo istraživali svojstva ove geometrije. Glavni zadatak bio je na prirodan način uvesti metriku. Prije toga morali smo otkriti grupu Möbiusovih transformacija koje čuvaju incidenciju. Proučavali smo tranzitivnost i konformnost elemenata opće Möbiusove grupe. Uz to morali smo otkriti grupu koja čuva \(\mathbb{H}\). Iz kompleksne analize smo iskoristili duljine krivulja te krivuljne integrale. Otkrili smo formule za računanje hiperboličke duljine i udaljenosti te smo otkrili još neka metrička svojstva. Osim modela gornje poluravnine spomenuli smo i Poincaréov disk i opću konstrukciju hiperboličke ravnine pomoću kompleksne analize.
Abstract (english)
The hyperbolic plane has many properties of the usual Euclidean plane. However, since Euclid’s fifth postulate does not hold, it also has many strange features, different from Euclidean geometry. We study properties of the hyperbolic plane using the upper half-plane model \(\mathbb{H}\). Our main goal was to introduce the hyperbolic metric in a natural way. We consider the group of incidence-preserving Möbius transformations and study transitivity properties and conformality of the general Möbius group. After that we discover its subgroup preserving \(\mathbb{H}\). We use complex analysis to compute arc-length and path integrals, and obtain formulae for calculating hyperbolic length and distance. We discover some other metric properties of the hyperbolic plane. Besides the upper half-plane model, we also mention the Poincaré disk model and a general construction using complex analysis.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:023588
Study programmeTitle: Mathematics Education; specializations in: Mathematics Education
Course: Mathematics Education
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2017-03-22 13:32:24