Hyperbolic Geometry and Neutrosophic Trigonometry: A Unified Approach to Uncertainty Management in Geometric Spaces

Abstract

This paper addresses the challenge of modeling uncertainty in hyperbolic geometry, where non-Euclidean properties, such as angle sums in triangles smaller than 180°, complicate traditional trigonometric calculations. The relevance of this topic lies in its application in fields such as cosmology, robotics, and signal processing, where curved spaces and imprecision are common. Although the literature has explored fuzzy geometry and trigonometry under uncertainty, it lacks a framework that integrates negative curvature with explicit indeterminacy. To overcome this gap, a methodology is proposed that combines hyperbolic geometry with neutrosophic trigonometry, using neutrosophic sets (T, I, F) to represent angles and calculate hyperbolic functions. The results show that this integration accurately captures uncertainty in hyperbolic triangles, offering trigonometric values that are robust against imprecise measurements. This approach not only enriches geometric theory by introducing a flexible model for non-Euclidean spaces, but also provides practical tools for applications such as navigation in curved environments and the analysis of noisy signals. Thus, the study advances mathematical knowledge and facilitates innovative solutions in interdisciplinary contexts.