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master's thesis
Neeuklidska geometrija
Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2017. urn:nbn:hr:126:735699
Lukanović, Ivana
Josip Juraj Strossmayer University of Osijek
Department of Mathematics
Chair of Pure Mathematics and Mathematics Teaching
Algebra, Analysis and Number Theory Research Group

Cite this document

Lukanović, I. (2017). Neeuklidska geometrija (Master's thesis). Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics. Retrieved from https://urn.nsk.hr/urn:nbn:hr:126:735699

Lukanović, Ivana. "Neeuklidska geometrija." Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2017. https://urn.nsk.hr/urn:nbn:hr:126:735699

Lukanović, Ivana. "Neeuklidska geometrija." Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2017. https://urn.nsk.hr/urn:nbn:hr:126:735699

Lukanović, I. (2017). 'Neeuklidska geometrija', Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, accessed 14 March 2025, https://urn.nsk.hr/urn:nbn:hr:126:735699

Lukanović I. Neeuklidska geometrija [Master's thesis]. Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics; 2017 [cited 2025 March 14] Available at: https://urn.nsk.hr/urn:nbn:hr:126:735699

I. Lukanović, "Neeuklidska geometrija", Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, Osijek, 2017. Available at: https://urn.nsk.hr/urn:nbn:hr:126:735699

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Metadata
TitleNeeuklidska geometrija
Title (english)Noneuclidean Geometry
AuthorIvana Lukanović
MentorIvan Matić (mentor)
Committee memberZdenka Kolar-Begović (predsjednik povjerenstva)
Committee memberIvan Matić (član povjerenstva)
Committee memberLjerka Jukić Matić (član povjerenstva)
GranterJosip Juraj Strossmayer University of Osijek
Department of Mathematics
(Chair of Pure Mathematics and Mathematics Teaching)
(Algebra, Analysis and Number Theory Research Group)
Osijek
Defense date and country2017-07-07, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Geometry and Topology
Abstract
U ovom radu pobliže se upoznajemo s neeuklidskom geometrijom, odnosno jednim njenim dijelom - hiperboličkom geometrijom kao geometrijom suprotnom od euklidske. Prvo poglavlje prikazuje postulat o paralelama koji postaje glavnim zanimanjem brojnih matematičara kroz povijest. Sama neintuitivnost ovog postulata dovodi do brojnih neočekivanih rezultata te naposlijetku i do nastanka nove neeuklidske geometrije. Pokazuje se da je nova geometrija jednako valjana kao i euklidska, no zbog čvrstih... More uvjerenja da je euklidska geometrija ta u kojoj se ogledaju pravila našeg prostora, hiperbolička geometrija ne poprima pozornost koju zaslužuje. Neeuklidskim geometrijama smatramo sfernu, eliptičnu i hiperboličku geometriju, no zbog opširnosti, ovaj rad bavi se samo hiperboličkom geometrijom, koja je glavna tema drugog poglavlja. Budući je postulat o paralelama bio glavnim izvorom otkrića nove geometrije, drugo poglavlje započinje upravo definiranjem paralelnih pravaca te hiperparalelnih pravaca. Iz različite definicije paralelnosti pravaca dolazimo do rezultata da je zbroj kutova u trokutu manji od 180º . Ne samo da je taj zbroj manji, već on čak nije ni isti za sve trokute. Veći dio ovog poglavlja bavi se prikazom trigonometrijskih jednadžbi. Pokazana su i pravila za sinus i kosinus ekvivalentna onima iz euklidske geometrije. Jednadžba th α = cos α dovodi do jednadžbe tanπ(p))2=e−pk koja pokazuje da kada k teži prema ∞, kut paralelnosti iznosi π2 , što pokazuje da je euklidska geometrija poseban slučaj hiperboličke. Ovime je hiperbolička geometrija dobila dugo traženu primjenu jer se javlja mogućnost da je upravo ona geometrija u kojoj vrijede pravila našeg prostora. Less
Abstract (english)
This framework deals with noneuclidean geometry, respectively one of its parts - hyperbolic geometry as geometry contrary to the euclidean one. The focus of Section 1 lies on the parallel postulate, which began the main interest of many mathematicians through the history. The non intuitiveness of the postulate leads to many unexpected results and even to the foundation of new noneuclidean geometry. Although it can be shown that this new type of geometry is equally consistent as the ... More euclidean one, because of the strong beliefs that the euclidean one describes rules of the real world, hyperbolic geometry didn’t get the popularity which it deserves. Sphere, elliptical and hyperbolic geometries are considered to be noneuclidean geometries, but this framework deals only with hyperbolic geometry, which is the main subject of Section 2. Considering that the parallel postulate was the main reason for the foundation of the new geometry, Section 2 begins with a definition of parallel and hyperparallel lines. This different definition of line parallelism leads us to the result that the sum of angles in a triangle is less than 180º. Not only is it less, but it isn’t even the same for all triangles. The best part of this section deals with trigonometric equations. There are presented a sine and cosine formulas equivalent to the one in the euclidean geometry. From the result th α = cos α it follows tanπ(p))2=e−pk which shows that for k → ∞ angle of parallelism becomes π2 , meaning that the euclidean geometry appears as a special case of hyperbolic geometry. This way hyperbolic geometry obtained a long desired use because it opens up the possibility that the rules of the real world are reflected exactly the same in hyperbolic geometry. Less
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:126:735699
Study programmeTitle: Mathematics and Computer Science
Study programme type: university
Study level: integrated undergraduate and graduate
Academic / professional title: magistar/magistra edukacije matematike i informatike (magistar/magistra edukacije matematike i informatike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
RepositoryRepository of School of Applied Mathematics and Informatics
Created on2017-07-10 07:37:30
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