Foundations of a Hyperbolic Neutrosophic Geometry: Extension of Euclidean Space Incorporating Uncertainty

Abstract

This paper addresses the need to extend classical Euclidean geometry toward a framework capable of incorporating uncertainty, recognizing that exact models are insufficient to describe phenomena in spaces where measurements are imprecise and structures exhibit negative curvature. This problem is crucial because hyperbolic geometry plays a central role in fields as diverse as cosmology, theoretical physics, and the analysis of complex networks, all of which are permeated by incomplete information and ambiguous data. Although there are approximations based on fuzzy or interval geometry, the literature lacks a formal approach that rigorously combines hyperbolic structure with neutrosophic logic, capable of explicitly handling truth, indeterminacy, and falsity. To overcome this limitation, a mathematical model is proposed that redefines points, distances, and triangles in terms of neutrosophic numbers, extending classical trigonometric axioms and laws to a hyperbolic context. The results show that the approach allows uncertainty to be captured, represented, and propagated in a structured manner, providing more faithful solutions than those offered by traditional methods. The contribution of this study is twofold: it establishes the theoretical foundations of hyperbolic neutrosophic geometry and lays the groundwork for practical applications in areas requiring accurate modeling of non-Euclidean spaces under uncertainty.