Klasifikacija cirkularnih krivulja 3. razreda u kvazihiperboličkoj ravnini

doctoral thesis

• Title (english): Classification of the circular curves of the third class in the quasi-hyperbolic plane
• Author: Helena Koncul
• Mentor: Ema Jurkin (mentor)
• Committee member: Sonja Gorjanc (predsjednik povjerenstva)
• Committee member: Ema Jurkin (član povjerenstva)
• Committee member: Vladimir Volenec (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-03-09, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

Cilj ovog rada je klasifikacija cirkularnih krivulja 3. razreda u kvazihiperboličkoj ravnini i ispitivanje mogućnosti njihove tvorbe pravčastom inverzijom, nožišnom transformacijom te projektivnom tvorbom. U uvodnom se poglavlju definira ju pojmovi i iznose tvrdnje o krivuljama u euklidskoj ravnini. Drugo poglavlje posvećeno je kvazihiperboličkoj ravnini, odnosno osnovnim pojmovima, krivuljama 2. razreda te pojmovima vezanim uz cirkularnost razredne krivulje u kvazihiperboličkoj ravnini. Krivulje 3. razreda klasificirane su prema stupnju cirkularnosti na 1-cirkularne, 2-cirkularne i potpuno cirkularne krivulje te prema tipu cirkularnosti i vrsti izotropnih pravaca krivulje na sveukupno 18 tipova cirkularnih krivulja 3. razreda. U preostala tri poglavlja obrađuje se po jedno preslikavanje odnosno način tvorbe krivulja 3. razreda: pravčasta inverzija, nožišna transformacija i projektivna tvorba. Za svako se preslikavanje prvo proučavaju njihova svojstva, a zatim se promatraju uvjeti tvorbe cirkularnih krivulja 3. razreda. Nadalje, za pravčastu inverziju i nožišnu transformaciju pokazana je međusobna veza te je dana konstrukcija dualne krivulje od krivulje dobivene s ta dva preslikavanja. Pri istraživanju korištena je sintetička i analitička metoda. Pokazano je da je projektivnom tvorbom moguće konstruirati svih 18 tipova cirkularnih krivulja 3. razreda, a da je nožišnom transformacijom moguće konstruirati 5 tipova, dok je za pravčastu inverziju pokazano da se ne može konstruirati samo jedan tip cirkularnih krivulja 3. razreda.

Abstract (english)

The main aim of this thesis is the classification of the circular curves of the 3rd class in the quasi-hyperbolic plane and the analysis of the possibilities of their construction obtained by the line inversion, pedal transformation and projective mapping. The theory of the curves in the Euclidean plane has been given in the first chapter. In the second chapter the quasi-hyperbolic plane has been studied i.e. the basics, the types of the 2nd class curves, and notions of the circular curves of class n in the quasi-hyperbolic plane are given. The circular curves of the 3rd class have been classified according to the degree of circularity into 1-circular, 2-circular or entirely circular curves and, within these types, according to the type of circularity and the type of the isotropic lines of the curve into 18 subtypes. In each of the remaining chapters one mapping is observed - line inversion, pedal transformation and projective mapping. First, the properties of every mapping are investigated and then the conditions of obtaining the circular curves of the 3rd class by these mappings are observed. Furthermore, the connection between line inversion and pedal transformation is given and the construction of the curve dual to the curve obtained by these two mappings is explained. The synthetic and analytical methods have been used in the studies. It is shown that every subtype of the circular curves of the 3rd class can be obtained by projective mapping and five subtypes can be obtained by pedal transformation, while only one subtype of the circular curves of the 3rd class can not be obtained by line inversion.

• Keywords: kvazihiperbolička ravnina; krivulja 3. razreda; cirkularna krivulja; pravčasta inverzija; nožišna transformacija; projektivna tvorba
• Keywords (english): quasi-hyperbolic plane; curve of the 3rd class; circular curve; line inversion; pedal transformation; projective mapping
• Language: croatian
• URN:NBN: urn:nbn:hr:217:075130
• 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: 2019-03-08 12:46:46