Evolute konika u projektivno-metričkim ravninama

doctoral thesis

• Title (english): The evolute of conics in projective-metric planes
• Author: Ivana Božić
• Mentor: Helena Koncul (mentor)
• Committee member: Ema Jurkin (predsjednik povjerenstva)
• Committee member: Vedran Krčadinac (član povjerenstva)
• Committee member: Helena Koncul (član povjerenstva)
• Committee member: Zdenka Kolar-Begović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-07-12, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

Evoluta dane krivulje je geometrijsko mjesto središta zakrivljenosti te krivulje, a istovremeno to je i omotaljka njezinih normala. Od 1665. godine kada je matematičar Huygens uveo pojam evolute, do danas o evolutama konika u euklidskoj ravnini napisani su brojni radovi. U ovom doktorskom radu proučavaju se svojstva evoluta uz isticanje njihovih karakteristika vezanih za Plückerove formule (red, razred, broj dvostrukih točaka/tangenata, broj šiljaka, infleksionih točaka/pravaca) u sedam od ukupno devet projektivno-metričkih ravnina. Rad je podijeljen na pet poglavlja u kojima je napravljen opsežan pregled osnovnih pojmova i svojstava evoluta konika u euklidskoj, pseudoeuklidskoj, kvazihiperboličnoj, kvazieliptičnoj i projektivno proširenoj hiperboličnoj ravnini. U radu je detaljno proučeno kako položaj konike prema apsolutnoj figuri utječe na red i razred njezine evolute. Pri istraživanju u projektivnim modelima projektivno-metričkih ravnina koriste se metode sintetičke geometrije koje omogućuju konstruktivnu obradu krivulja u dinamičkim računalnim programima The Geometer’s Sketchpad i Geogebra. Pri analitičkom istraživanju evoluta koriste se programi Wolfram Mathematica i Demos. Cilj je rada sistematizacija činjenica vezanih za istaknute projektivno-metričke ravnine te znanstveni doprinos pri klasifikaciji evoluta i njihovoj konstruktivnoj obradi.

Abstract (english)

The locus of centers of osculating circles is called the evolute of a curve. The evolute is also the envelope of the family of normal lines to the curve. From 1665, when the mathematician Huygens introduced the notion of the evolute of a conic in the Euclidean plane, to the present day numerous papers have been written. In this doctoral thesis, the properties of evolutes are studied with emphasis on their characteristics related to Plücker formulas (order, class, number of double points/tangents, number of cups, inflection points/lines) in seven out of a total of nine projective-metric planes. The thesis is divided into five chapters in which an extensive review about basic concepts and properties of the evolute of a conic is made in the Euclidean plane, pseudo-Euclidean plane, quasi-hyperbolic plane, quasi-elliptic plane and projective extended hyperbolic plane. The thesis studies in detail how the position of the conic towards the absolute figure affects the order and class of its evolute. The syntehetic and analytical methods have been used in the studies. The computer programs used in the research are The Geometer’s Sketchpad, Geogebra, Wolfram Mathematica and Demos. The main aim of this thesis is to systematize the facts related to prominent projective-metric planes and scientific contribution to the classification of evolutes and their constructive processing.

• Keywords: evoluta; oskulacijska kružnica konike; euklidska ravnina; pseudoeuklidska ravnina; kvazieliptična ravnina; kvazihiperbolična ravnina; hiperbolična ravnina
• Keywords (english): The evolute; the osculating circle of a conic; the Euclidean plane; the pseudo-Euclidean plane; quasi-hyperbolic plane; quasi-elliptic plane; the hyperbolic plane
• Language: croatian
• URN:NBN: urn:nbn:hr:217:929004
• Promotion: 2022
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: v, 79 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-01-24 13:24:07