Biquartic Hesitant Fuzzy Bézier Surface Approximation Model with Its Visualization

Abstract

Geometric modeling has evolved significantly since its inception, with pioneers like Pierre Bézier laying the foundation for curve modeling. Concurrently, fuzzy set theory, introduced by Lotfi A. Zadeh in 1965, addressed uncertainty in decision-making processes. Integrating fuzzy set theory with geometric modeling has led to advancements in handling imprecise data and uncertainty. This paper proposes a novel approach, the hesitant fuzzy Bézier surface (HFBS) approximation model, which combines geometric modeling with hesitant fuzzy sets to address uncertainty in surface approximation. The model utilizes hesitant fuzzy control net relations to construct HFBSs, enabling visualization of surfaces under varying degrees of uncertainty. A biquartic HFBS example is presented, demonstrating the model’s ability to handle hesitancy among experts’ opinions. The paper discusses the properties of HFBS and suggests its extension to interpolation models for broader applicability. Ultimately, the HFBS model offers an approach to geometric modeling in situation where uncertainty is inherent, which aim to handle complex data in real-world applications.