Geometria esférica : uma abordagem axiomática dos conceitos fundamentais e da trigonometria

Abstract

This work presents a systematic study of Spherical Geometry, covering its axiomatic construction, fundamental properties, and an introduction to spherical trigonometry. It begins with a historical contextualization, from ancient civilizations to its mathematical consolidation. It discusses the concept of a line on the sphere via geodesics, distinguishing between intrinsic and extrinsic approaches, and formally establishes concepts such as great circles, antipodality, spherical distance, and convexity. The main focus is on the theory of spherical triangles, including properties, congruence criteria (such as the AAA case), and the relationship between the sum of angles and curvature. It also addresses the area of spherical triangles, culminating in the Girard-Euler Theorem, and presentes proofs of the Spherical Pythagorean Theorem and the Spherical Law of Sines. Finally, it compares plane and spherical geometry, presents the extrinsic viewpoint, and original results on antipodal and parallel circles. The work aims to offer a formal and accessible exposition of Non-Euclidean Geometries, highlighting their historical, educational, and modern geometrical relevance.