Modeli hiperbolične geometrije in njihova vizualizacija : magistrsko delo

Albreht, Katja (author), Bauer, Andrej (mentor), Podpečan, Vid (co-mentor)

Magistrsko delo v prvem delu vsebuje predstavitev osnovnih lastnosti hiperbolične geometrije. Predstavljeni so vsi trije njeni glavni modeli in osnovne geometrijske značilnosti posameznega modela. Delo obravnava tudi vrste tlakovanj, nato pa na kratko povzema tlakovanje Poincaréjevega diska in opisuje izpeljavo tlakovanja polsfernega modela hiperbolične geometrije. V nadaljevanju pa je predstavljen potek modeliranja in tiskanja 3D modelov, od pravilnih poliedrov do tlakovane hiperbolične polsfere. Dodane so izpeljave predpisov projekcij, ki izvirajo iz senčenja tiskanega 3D modela tlakovane polsfere.

• Language: Slovenian
• Keywords: geometrija, hiperbolična geometrija, 3D model, vizualizacija
• Work type: Master's thesis/paper
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2024
• PID: 20.500.12556/RUL-153457
• COBISS.SI-ID: 179957251
• Publication date in RUL: 07.01.2024

Secondary language

• Language: English
• Title: Models of hyperbolic geometry and their visualisation

Abstract

The first part of the master's thesis includes the presentation of the basic characteristics of the hyperbolic geometry including all three main models and basic geometry characteristics of each model. The thesis also covers the types of tessellations and then briefly summarises tessellation of Poincare’s discus and describes the derivationof semispheric model tessellation of the hyperbolic geometry. Furthermore the course of modelling and 3D model printing is presented, spanning from regular polyhedra to tesselled hyperbolic semisphere. The derivation of projection regulations are added, which derive from shadowing of the 3D model of a tessellated semisphere.

• Keywords: geometry, hyperbolic geometry, 3D model, visualisation