Laguerreova geometrija

master's thesis

• Author: Jelena Martić
• Mentor: Vedran Krčadinac (mentor)
• Committee member: Vedran Krčadinac (predsjednik povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Committee member: Mea Bombardelli (član povjerenstva)
• Committee member: Juraj Šiftar (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-09-24, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu bavimo se Laguerreovom geometrijom. Navodimo više modela koji zadovoljavaju aksiome Laguerreove ravnine te ih posebno dokazujemo za svaki od modela. Aksiomi Laguerreove ravnine su: Aksiom 1. Tri u parovima neparalelne točke spojene su jedinstvenim ciklusom. Aksiom 2. Za svaki ciklus \(c\) i za svake dvije neparalelne točke \(PIc\) i \(Q \cancel{I} c\), postoji jedinstveni ciklus \(d\) kroz \(Q\) koji dodiruje ciklus \(c\) u točki \(P\). Aksiom 3. Za svaku točku \(P\) i ciklus \(c\), postoji jedinstvena točka na \(c\) paralelna s \(PIc\). Aksiom 4. Postoji ciklus koji sadrži najmanje tri, ali ne i sve točke. Prikazujemo neke od modela u koordinatnom sustavu. Dokazujemo da postoji izomorfizam među konačnim minimalnim modelima i isto tako među dva beskonačna modela. Na kraju pokazujemo kako možemo definirati Laguerreovu ravninu nad bilo kojim poljem.

Abstract (english)

In this thesis we are dealing with Laguerre’s geometry. We give a several models that satisfy the axioms of Laguerre’s plane. We prove the axioms for each of the given models. The axioms of Laguerre plane: Axiom 1. Three pairwise non-parallel points can be joined by a unique cycle. Axiom 2. For any cycle $c$ and any two non-parallel points \(PIc\) and \(Q \cancel{I} c\), there is exactly one cycle \(d\) through \(Q\) touching \(c\) in \(P\). Axiom 3. For any point \(P\) and any cycle \(c\) there is exactly one point on \(c\) parallel with \(P\). Axiom 4. There is a cycle containing at least three, but not all points. We demonstrate some of the models in coordinate system. We prove that there is an isomorphism between the minimal models and between two of the infinite models. At the end we indicate the way to define Lagurre’s plane over an arbitrary field.

• Keywords: aksiomi Laguerreove ravnine; izomorfizam među modelima
• Keywords (english): axioms of Laguerre’s plane; isomorphism between models
• Language: croatian
• URN:NBN: urn:nbn:hr:217:252218
• Study programme: Title: 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 resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-01-23 10:52:39