Uma introdução histórica à geometria hiperbólica e aplicações trigonométricas

Abstract

This work analyzes the historical and mathematical evolution of geometry, focu-sing on the rupture of the Euclidean dogma and the consolidation of hyperbolic geometry as a foundation for modern physics. The research aims to describe the transition from Euclid’s axiomatic system to non-Euclidean models, demonstrating how the negation of the fifth postulate grounds new understandings of space. The method utilizes a bibliographic and documental review of a deductive nature, based principaly in Barbosa’s works to reconstruct the logical rigor of geometric demonstrations(BARBOSA, 2007). The discussion covers the failed attempts by mathematicians such as Ptolemy, Proclus, and Saccheri to prove the parallel postu-late, culminating in the independent discovery by Gauss, Bolyai, and Lobachevsky. The text details elementary properties of neutral and hyperbolic geometry, suchas Saccheri and Lambert quadrilaterals, generalized triangles, and hyperbolic tri-gonometry. The results highlight that hyperbolic geometry provides the necessarymathematical framework for Einstein’s Theory of Relativity.