Page 384 - Srednje skole - radovi

HIPERBOLIĈKA GEOMETRIJA. HYPERBOLIC GEOMETRY

Uĉenik:TAMARA ŠEKULARAC, Matematiĉka gimnazija, Beograd
Mentor: VESNA RAJŠIĆ, profesor matematike Elektrotehniĉka škola „Nikola Tesla”, Beograd
REZIME
Razvoj nauke podsticao je razvoj tehnike i doveo do napretka sveukupnog ĉoveĉanstva. Veliki broj problema koji se postavljao
pred tadašnji uĉeni svet i nauĉnike zahtevao je rešavanje mnogobrojnih problema u raznim prirodnim naukama, a meĊu njima i u
matematici i geometriji. Izuĉavanja u oblasti geometrije stara su hiljadama godina. Pokušaji razrešenja pitanja o petom euklidskom
postulatu doveli su do osnivanja potpuno nove geometrije koja je ne protivureĉna. Ova nova hiperboliĉka geometrija je taĉna

koliko i euklidska geometrija. Dokazi do kojih se izuĉavanjem ove geometrije došlo doprineli su proširenju postojećih znanja i
uvoĊenju potpuno novog pogleda na geometriju. Ova nova saznanja uticala su na potpuno novo, drugaĉije i kompleksnije
sagledavanje odnosa taĉke, prave i ravni u prostoru. Lobaĉevski je utvrdio da njegova neeuklidska geometrija ima neposrednu
primenu kod izraĉunavanja kosmiĉkih prostranstava. U okvirima obiĉnih zemaljskih dimenzija kod svih proraĉunavanja ljudi
koriste euklidsku geometriju kao geometriju koja je najjednostavnija i najrealnije odraţava stvarnost. Dostignuća u ovoj oblasti
govore o tome da se fiziĉki prostori prevelikih dimenzija ponašaju kao neeuklidski prostori. Za njihovo izuĉavanje potrebne su
neeuklidske geometrije odnosno geometrija Lobaĉevskog. Ideja o ovom istraţivanju o hiperboliĉkoj geometriji potekla je iz ţelje

da se sazna nešto više o mogućnostima matematiĉkog i geometrijskog rešavanja prostora, kao o neophodnom znanju bez koga se
ne mogu rešavati sloţeni prostorni problemi u raznim oblastima istraţivanja u domenu geometrije, fizike i astronomije.
Kljuĉne reĉi: hiperboliĉka geometrija, aksioma Lobaĉevskog, paralelnost, hiperparalelnost, modeli hiperboliĉke ravni, primena
hiperboliĉke geometrije
SUMMARY
The development of science encouraged the development of technology and led to the progress of mankind. A great number of

problems which scholars and scientists of that time were facing demanded a solution of numerous problems in various natural
sciences, among others, in mathematics and geometry. The studies in the field of geometry have been thousands of years old.
Attempts of resolving questions about the fifth Euclidian postulate led to the creation of entirely new geometry that is non-
contradictory. This new hyperbolic geometry is as accurate as the Euclidian geometry. The evidence obtained by studying this
type of geometry led to the expansion of existing knowledge and to the introduction of a completely new view of the geometry.
The new findings influenced a completely new, different and more complex understanding of the relations among dots, lines and
planes in space.Lobachevsky found that his new non-Euclidian geometry has direct application in the calculation of user spaces. In

terms of ordinary earthly dimensions in all calculations people use Euclidian geometry as the simplest geometry which reflects
reality in most realistic way. Achievements in this area suggest that physical spaces of too large dimensions behave as non-
Euclidian spaces. Their study requires non-Euclidian geometries, i.e. Lobachevskian geometry. The idea of this research on
hyperbolic geometry came from a desire to learn more about the possibilities of mathematical and geometrical solution of space,
as the knowledge necessary to solve complex spatial problems in various areas of research in the field of geometry, physics and
astronomy.

Key words: hyperbolic geometry, the axiom of Lobachevsky, parallelism, hyper-parallelism, models of the hyperbolic planes, the
application of hyperbolic geometry