Analytical fuzzy space geometry II

Abstract

In the present paper, we attempt to construct space fuzzy lines as well as fuzzy planes in R3. We construct a space fuzzy line as a bi-infinite extension of a space fuzzy line segment. A particular case of space fuzzy lines, namely, symmetric fuzzy lines, is also investigated. Importantly, the concept of skew fuzzy lines and the shortest distance between skew fuzzy lines are discussed in R3. In addition, the fuzzy planes are proposed in three different forms: a three-point form, an intercept form, and a fuzzy plane passing through an S-type space fuzzy point and perpendicular to a given crisp direction. The construction of all the different forms of fuzzy planes depends on the known details about the fuzzy planes, such as three S-type space fuzzy points, or its three intercepts, or an S-type space fuzzy point and a crisp direction. In the sequel, a symmetric fuzzy plane is investigated. The study and construction of all the proposed ideas are done with the help of the same and inverse points of S-type space fuzzy points. Geometric properties of the proposed space fuzzy lines and all the proposed forms of fuzzy planes are also explored. Numerical examples support all the formulations and studies. Notably, we also provide the algorithms to find the membership grade of a point in(i)the space fuzzy line passing through two continuous space fuzzy points,(ii)the shortest distance between symmetric skew fuzzy lines,(iii)the three-point form of the fuzzy plane,(iv)the intercept form of the fuzzy plane, and(v)the fuzzy plane passing through an S-type space fuzzy point and perpendicular to a given crisp direction.