A study on fuzzy sphere and fuzzy cone

Abstract

In this paper, we investigate three different methods to construct a fuzzy sphere. These methods depend on the entities available to form a fuzzy sphere. In the first method, we formulate a fuzzy sphere when the position of the center and the value of the radius are given imprecisely. In the second method, a fuzzy sphere is constructed when the diameter of a fuzzy sphere is known. In the last method, we construct a fuzzy sphere that passes through four given S-type space fuzzy points whose core points are not co-planar. Also, a rotation invariance property of a fuzzy sphere is investigated. In addition, we show that the intersection of a fuzzy sphere with a crisp plane is a fuzzy circle. In a sequel, the idea of a fuzzy cone is developed. The properties concerned with the intersection of a fuzzy cone with a crisp plane are studied sequentially. All the studies are supported with geometrical interpretations and numerical examples. Moreover, we also develop the algorithms to get the membership grade of a number (i) in the diameter form of the fuzzy sphere and (ii) in the four-point form of the fuzzy sphere. © 2024 Elsevier Inc.