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Analytical fuzzy space geometry I
Fuzzy Sets and Systems ( IF 3.2 ) Pub Date : 2020-10-15 , DOI: 10.1016/j.fss.2020.10.001
Debdas Ghosh , Diksha Gupta , Tanmoy Som

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.



中文翻译:

解析模糊空间几何I

在本文中,我们介绍了三维欧几里得空间中模糊空间几何的一些基本概念。我们在这里研究的思想是空间模糊点两个空间模糊点之间的距离空间模糊线段。为了表示空间模糊点,我们引入了三变量参考函数的思想。因此,我们定义了一个S型空间模糊点。研究关于两个连续空间模糊点的同点反点概念,形成模糊空间几何概念。为了为空间模糊点制定同点和反点,我们提供了一个概念沿着空间中的一条线的模糊数。借助引入的三变量参考函数和沿线模糊数,给出了空间模糊点同反点的显式一般表达式。利用反点的概念,我们定义了两个空间模糊点之间的模糊距离。利用相同点的思想,定义了两个空间模糊点的加法和凸组合。模糊线段由两个空间模糊点的凸组合形成。在后续中,还提供了两个空间模糊点重合的概念。所有提供的想法都得到了数值例子和必要的插图的支持。重要的是,我们还提供算法来查找

(一世)

两个空间模糊点之间的模糊距离,

(二)

一个数在两个空间模糊点之间的模糊距离中的隶属度值和

(三)

空间模糊线段中某个点的隶属度值。

更新日期:2020-10-15
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