Analytical fuzzy space geometry I

In this paper, we introduce a few basic concepts of fuzzy space geometry in the three-dimensional Euclidean space. The ideas that we study here are space fuzzy points, distance between two space fuzzy points, and space fuzzy line segments. To represent a space fuzzy point, we introduce the idea of a reference function of three variables. Accordingly, we define an S-type space fuzzy point. The concepts of same points and inverse points with respect to two continuous space fuzzy points are studied to formulate the fuzzy space geometrical concepts. To formulate same and inverse points for space fuzzy points, we provide the concepts of a fuzzy number along a line in the space. With the help of the introduced three-variable reference function and fuzzy number along a line, explicit general expressions of same and inverse points for space fuzzy points are provided. Employing the concept of inverse points, we define a fuzzy distance between two space fuzzy points. Using the idea of the same points, addition and convex combination of two space fuzzy points are defined. A fuzzy line segment is formulated by a convex combination of two space fuzzy points. In the sequel, a concept of coincidence of two space fuzzy points is also provided. All the provided ideas are supported with numerical examples and necessary pictorial illustrations. Importantly, we also provide algorithms to find(i)the fuzzy distance between two space fuzzy points,(ii)the membership value of a number in the fuzzy distance between two space fuzzy points and(iii)the membership value of a point in the space fuzzy line segment.