Page 375 - Srednje skole - radovi

HIPERBOLIĈKA GEOMETRIJA. HYPERBOLICGEOMETRY

Uĉenik : Dušica Bralović I razred Matematiĉka gimnazija, Beograd

Mentor: Violeta Komnenović,profesor matematike Elektro tehniĉka škola „Nikola Tesla”, Beograd

Rezime

Hiperboliĉka geometrija (geometrija Lobaĉevskog ) je neeuklidska geometrija u kojoj vaţe aksiome
apsolutne geometrije,ali u kojoj je peti postulat Euklidske geometrije promenjen. Pored Lobaĉevskog do
istih rezultata došao je i Janoš Boljaj ((1802 – 1860), maĊarski matematiĉar),pa ovu geometriju zovemo još i
geometrija Boljaj-Lobaĉevskog. Sem hiperboliĉke i euklidske geometrije postoji još geometrija koje imaju
promenjeni peti postulat i ove geometrije pripadaju grupi neeuklidskih geometrija. U matematici moţe biti
ĉudno da postoje dve teorije koje su u suprotnosti (Euklidska geometrija i geometrija Lobaĉevskog), ali obe
teorije su neprotivreĉne i potpune.Za modernu matematiku najvaţnije je da su obe taĉne, pa nemoţemo ih
porediti već ih prihvatiti bez dokaza.
Pojavom Ajnštajnove teorije relativiteta ( A. Ajnštajn (1879 – 1955),nemaĉki fiziĉar) pokazalo se da je u
kosmiĉkom prostoru pogodnije da se koristi neeuklidska geometrija, pa i geometrija Lobaĉevskog.
U okviru nastave geometrije prvi put sam se srela sa pojomhiperboliĉke geometrije, oblast koja je izuzetno
interesantna i koja je pobudila moja interesovanja. U ovom radu detaljno su date aksiome hiperboliĉke
geometrije , dokazane su neke od teorema takoĊe objašnjene su ravan i prostor ove geometrije.
Razvoj geometrije ovim nikako nije završen. I danas su mnogi problemi u matematici, posebno iz oblasti
neeuklidskih geometrija još uvek otvoreni za rešavanje.

Kljuĉne reĉi:paralelnost,hiperparalelnost,hiperboliĉka ravan , hiperboliĉki prostor , Oricikl, Eliptiĉki
pramen.

Abstract

Hyperbolicgeometry (Lobachevskiangeometry) is a non-Euclideangeometrywhere the axioms of
absolutegeometryarevalid, but the fifth postulate of Euclideangeometry is changed. BesidesLobachevsky
same resultscameandJánosBolyai ((1802 - 1860), Hungarianmathematician), so thisgeometry is alsocalled
the Bolyai-Lobachevskian geometry. Apart from the hyperbolic and Euclidean geometry, there are
geometries that have altered the fifth postulate and these geometries belong to the non-Euclidean geometry.
In mathematicscan be surprising that exist two theories which are in conflict (Euclidean geometry,
andLobachevskigeometry), but in the same time the bothareunambiguousandcompletely. For
modernmathematic is essentialthattheyarebothcorrect, so we can not compare them just accept them without
proof.
With the advent of Einstein'sTheory of Relativity (A. Einstein (1879 - 1955), German physicist) has shown
that in the cosmic space convenient to use non-Euclideangeometry, and geometry Lobachevsky.

The idea of thisworkhadstemmedby the factthathyperbolicgeometry is abstract. In this work details are given
axioms of hyperbolic geometry, demonstrated some of the theorems are also explained in this plane and
space geometry.

The development of geometry by this is not complete yet. There are still a lot of problems in mathematics,
especially in the field of non-Euclidean geometry is still open for resolution.
Keywords: parallel, hyper parallel, hyperbolic plane, hyperbolic space, Oricycle, Elliptic curve.