The Extension Principle for Angular Domains and its Application to Spatial Relations

This paper delves into Zadeh's extension principle (EP) for angular domains, particularly in the context of spatial relations. We introduce modifications to the EP, specifically tailored for addressing ambiguities inherent in angular scenarios. The first enhancement employs a vector space to mitigate truth degree variations in operations with input variability. The second modification introduces a user-defined tolerance parameter, enabling the judicious consideration of inputs based on their alignment with existing data. This augmentation facilitates a more adaptive handling of contradictory or uncertain data, enhancing the EP's versatility in real-world scenarios. Through a set of synthetic case studies, we illustrate the efficacy of our extension in the interpretation of spatial relations. Furthermore, a practical example is presented to demonstrate the generation of a prototypical spatial relation. Overall, our proposed modifications provide the groundwork for applying complex operations using fuzzy arithmetic in cyclical, angular domains.