Klasični rezultati globalne teorije ravninskih krivulja

master's thesis

• Author: Mateo Grabovac
• Mentor: Ilja Gogić (mentor)
• Committee member: Ilja Gogić (predsjednik povjerenstva)
• Committee member: Vedran Krčadinac (član povjerenstva)
• Committee member: Igor Ciganović (član povjerenstva)
• Committee member: Boris Guljaš (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Izoperimetrijska nejednakost, Hopfov teorem o indeksu rotacije i Teorem o četiri tjemena su jedni od glavnih rezultata koje su matematičari otkrili proučavajući globalna svojstva ravninskih krivulja u okvirima diferencijalne geometrije. U ovom diplomskom radu dajemo prikaz osnova teorije ravninskih krivulja te dokazujemo tri navedena teorema. Glavni dio ovog rada je podijeljen na šest poglavlja. U prvom poglavlju definiramo kruta i neprava euklidska gibanja, dok u drugom poglavlju definiramo krivulje i prikazujemo neka njihova osnovna svojstva. U trećem poglavlju koncentriramo se samo na ravninske krivulje i za njih definiramo pojmove orijentirane zakrivljenosti, kutne funkcije i indeksa rotacije. U istom poglavlju dokazujemo i fundamentalni teorem za ravninske krivulje. U četvrtom i petom poglavlju redom dokazujemo Hopfov teorem o indeksu rotacije i Izoperimetrijsku nejednakost, dok u zadnjem poglavlju uvodimo pojam tjemena krivulje i konačno dokazujemo Teorem o četiri tjemena.

Abstract (english)

Isoperimetric inequality, Hopf’s winding theorem and the Four vertex theorem are one of the main results discovered by mathematicians while studying the global properties of plane curves in the framework of differential geometry. In this graduate thesis we present basics of the theory of plane curves and prove the three aforementioned theorems. The main part of this paper is divided into six chapters. In the first chapter we define rigid and non-rigid Euclidean motions, while in the second chapter we define curves and present some of their basic properties. In the third chapter, we focus only on the plane curves and for them we define the concepts of oriented curvature, angular function and winding number. In the same chapter we also prove the fundamental theorem for plane curves. In the fourth and fifth chapter we prove the Hopf’s winding theorem (Hopf’s Umlaufsatz) and the Isoperimetric inequality, respectively, while in the last chapter we introduce the concept of a vertex of a curve and finally prove the Four vertex theorem.

• Keywords: ravninske krivulje; diferencijalna geometrija; izoperimetrijska nejednakost; Hopfov teorem o indeksu rotacije; teorem o četiri tjemena; fundamentalni teorem za ravninske krivulje
• Keywords (english): plane curves; differential geometry; isoperimetric inequality; Hopf’s winding theorem; Four vertex theorem; fundamental theorem for plane curves
• Language: croatian
• URN:NBN: urn:nbn:hr:217:560233
• 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: 2020-01-30 11:07:10