Konformni modeli hiperboličke ravnine

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.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