THE AXIOMATIC SYSTEM OF LOBACHEVSKIAN GEOMETRY ON THE PLANE AND ITS MATHEMATICAL CONSEQUENCES
Keywords:
Lobachevskian geometry, hyperbolic plane, non-Euclidean geometryAbstract
This paper explores the foundational axioms of Lobachevskian (hyperbolic) geometry on the Euclidean plane and derives significant geometric consequences from them. By comparing the hyperbolic axiom of parallelism with Euclidean postulates, we analyze the structure of space where the Euclidean fifth postulate fails. Using rigorous mathematical formulations, we discuss the implications on angle sums in triangles, parallel line behavior, and the construction of hyperbolic models. The findings highlight the consistency and significance of Lobachevskian geometry in the broader context of non-Euclidean geometry.
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