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Mikučić Crnković, Vedrana; Novak, Ivona
Math.e, 12/2012, Volume: 22, Issue: 1Paper
Nastanak neeuklidske geometrije počinje razmatranjem Euklidovog petoga aksioma (aksioma o paralelama) i zaključkom da on nije zavisan o prethodna četiri. N. I. Lobačevski negirao je Euklidov peti aksiom i sagradio novi geometrijski sustav. Matematičari tadašnjega vremena daljnjim proučavanjem došli su do zaključka da se geometrija Lobačevskog može promatrati na hiperboloidu u R3, zbog čega i dobiva naziv hiperbolička geometrija. Osim modela na hiperboloidu, u članku su prikazana još četiri modela hiperboličke ravnine: Kleinov model, model hemisfere, Poincareov model te model gornje poluravnine. U svakom od navedenih modela opisan je međusobni odnos dvaju pravaca. Osim toga, opisana je bijektivna korespondencija između pojedinih modela. Radi lakšeg predočavanja pojedinih modela hiperboličke ravnine, članak je upotpunjen slikama izrađenim u programskom paketu Mathematica (http://www.wolfram.com/) te programima NonEuclid (http://www.cs.unm.edu/~joel/NonEuclid/NonEuclid.html) i Interactive Non-Euclidean Geometry (http://fis.cie.uma.es/~ccriado/Interactive%20Non.htm).
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