# Constructing the Fuzzy Hyperbola and Its Applications in Analytical Fuzzy Plane Geometry

Authors: Seçil Özekinci; Cansel Aycan
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we studied about a detailed analysis of fuzzy hyperbola. In the previous studies, some methods for fuzzy parabola are discussed (Ghosh and Chakraborty, 2019). To define the fuzzy hyperbola, it is necessary to modify the method applied for the fuzzy parabola. To obtain a conic, it is necessary to know at least five points on this curve. First of all, in this study, we examined how to detect these five fuzzy points. We have discussed in detail the impact of points in this examination on finding fuzzy membership degrees and determining the curve. We show the use of the algorithm for calculating the coefficients in the conic equation on the examples. We make detailed drawings of all the fuzzy hyperbolas found and depicted the geometric location of fuzzy points with different membership degrees on the graph. As can be seen from the figures in our study, the importance of membership degrees in fuzzy space is that it causes us to find different numbers of hyperbola curves for the five points we study with. In addition, finding the membership of a given point to the fuzzy hyperbola is possible by solving nonlinear equations under different angular approaches. This examination is shown in detail in this study, and the results in the examples are evaluated by geometric comments. The systems formed by the fuzzy hyperbola curves are found to have different areas of use, as presented in the Conclusion. Some of usage areas of fuzzy hyperbola are radar systems, scanning devices, photosynthesis, heat, and CO2 distribution of plants.


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## Body

## 1. Introduction

In the case of crisp sets, a given objectx may belong to a set A or not belong to this set, and these two options are denoted by x∈A or x∉A,A classic set may be described by the characteristic function χA that takes two values: first is 1 (x belongs to a set A), and second is 0 (x does not belong to set A)(1)χAx=1,x∈A,0,x∉A.Fuzzy sets are introduced and described using membership functions by Zadeh in 1965 [1]. As opposed to a crisp set, if A¯ is a fuzzy set, we write its membership function as μx|A¯, and μx|A¯ is in 0,1 for all x.Some basic definitions of fuzzy logic are given by Zadeh in study [2]. Likewise, basic properties for fuzzy plane geometry have been given by Buckley and Eslami in the study [3] may be the first to analyze fuzzy sets.Fuzzy points and the fuzzy distance between fuzzy points were defined by Buckley and Eslami in [3]. And they showed it is a (weak) fuzzy metric and fuzzy point, fuzzy line segment, fuzzy distance, and the angle between two fuzzy segments, and same and inverse points are defined by Ghosh and Chakraborty [4].Gebray and Reddy dealt with properties of fuzzy set, fuzzy metric, fuzzy field, fuzzy set field, and fuzzy magnitude [5].A fuzzy line passing through several fuzzy points whose cores are collinear is reviewed by Ghosh and Chakraborty in the study [6].It has been studied about geometric reasoning with extended objects and transferred to geography [7].Rosenfeld presented fuzzy geometry and fuzzy topology of image subsets [8].The fuzzy triangle as the intersection of three fuzzy half-planes and its geometric forms was discussed by Rosenfeld in the study [9].Zimmermann dealt with types of fuzzy sets, fuzzy measures, fuzzy functions, and applications of fuzzy set theory and gave basic definitions and theorems about fuzzy sets [10].Ashraf et al. approached the ordering of the fuzzy numbers in the decision and fuzzy distance [11].It is known that in analytical geometry to get a conic also a hyperbola, we need five points. In this perspective, Ghosh and Chakraborty introduced a fuzzy parabola in the study [12].When all these studies are examined, it is seen that only fuzzy circle and fuzzy parabola curves are studied from the conics. No study has got been done to construct a fuzzy hyperbola. Fuzzy systems are used in the planning of technological structures developing in the field of engineering nowadays. On the other hand, hyperbola curves are very common curves in the wave distribution of radars, in the most suitable matching of points in scanning machines, and even in the carbon dioxide distribution of plants. While we aimed to analyze how the fuzzy hyperbola could be defined, calculated, and graphed mathematically, we thought that it would be useful to work on combining the applications mentioned above. We examine and prove these calculations with the evaluation of previous studies.


## 2. Preliminaries

In this section, we will mention the needed mathematical specialties for fuzzy plane geometry.We will draw “a bar” over capital letters to denote a fuzzy subset ofRn,i.eA¯,B¯,X¯,Y¯,…, and we will write membership of fuzzy set A¯ as μx|A¯,x∈Rn and μRn is in 0,1.Definition 1.
(Fuzzy Set) LetX be a nonempty set and μ:X⟶0,1 be a mapping. A¯=α,μα:α∈X is called a fuzzy set in X with membership function μ [5].Definition 2.
For a fuzzy setA¯ of Rn, its α−cut is denoted by A¯α, and it is defined by(2)A¯α=x,μx|A¯≥α,if0<α≤1,Clousex,μx|A¯>0,ifα=0.
The support of the fuzzy setA¯ is defined as positive membership degree namely μx|A¯>0. These points that have been positive membership degrees also denoted by A¯0=x,μx|A¯>0, and additionally, the set A¯0 defined as the base of fuzzy set A¯. On the other hand, the core of fuzzy set A¯ is denoted by x,μx|A¯=1. When the core of fuzzy set is nonempty and α−cuts are convex, it can be said that the fuzzy set is convex and normal [4].Definition 3.
(Fuzzy points):A fuzzy point at a,b in R2, written as P¯a,b is defined by its membership function:(i)
μx,y|P¯a,b is upper semicontinuous(ii)
μx,y|P¯a,b=1 if and only if x,y=a,b(iii)
P¯a,bα is a compact, convex subset of R2 of all α in 0,1
The notationsP¯1a,b,P¯2a,b,P¯3a,b,… or P¯1,P¯2,P¯3,… are used to represent fuzzy points [3].Definition 4.
(Same points with respect to fuzzy points): Let take two pointsx1,y1 and x2,y2. Such that x1,y1 is support of fuzzy point P¯a,b and similarly x2,y2 is support of fuzzy point P¯c,d. Let L1 is a line joining x1,y1 and a,b. There exists a fuzzy number r¯1 for the fuzzy point P¯a,b, along the line L1. The membership function for r¯1 can be written as follows:(3)μx,y|r¯1=μx,y|P¯a,b,forx,yinL1,0,otherwise.
Similarly, along a line,L2 say, joining x2,y2 and c,d, there exists a fuzzy number, r¯2 say, on the support of P¯c,d. Now, x1,y1, x2,y2 are said to be same points with respect to P¯a,b and P¯c,d if(i)
x1,y1 and x2,y2 are same points with respect to r¯1 and r¯2(ii)
L1,L2 have equal angle with line joining a,b and c,d [4]


## 3. Fuzzy Hyperbola

In this section, we will develop a method for obtaining a fuzzy hyperbola. As it is known, the general conic equation has the following form:(4)ax2+bxy+cy2+dx+ey+f=0.If we divide both sides of this equation byaa≠0, this equation takes the following form:(5)x2+b′xy+c′y2+d′x+e′y+f′=0.Thus, the number of unknown coefficients, in the equation containing six terms, becomes five. Then, the common solution of the five equations, will be obtained by substituting five different pairs forx and y, will be sufficient to find these unknowns. So, five points in the plane are enough to write a conic equation. Namely, five points on the plane will denote a single conic.In this study, since we will define a hyperbola in fuzzy plane geometry, first of all, these five points must be points in the fuzzy space that ensure the necessary properties. It will also be seen that the hyperbola in fuzzy space is formed by different crisp hyperbolic curves. Their combination will form the fuzzy hyperbola. The hyperbola curve passing through the core of five fuzzy points will be called a crisp hyperbola and is denoted by CH. However, since these points are fuzzy points, their membership degrees may change. Differences in membership degrees affect the drawing of the resulting hyperbola curves. Therefore, we will calculate five different coefficients in the conic equation (4). Calculation of these coefficients is possible with five determinants. For this reason, five different curves emerge for the fuzzy hyperbola that we want to reach in our study. Therefore, in terms of the importance of the fuzzy membership degree, the definite hyperbola CH with membership degree one is taken. The other four curves are hyperbola, and the combination of all of them gives the fuzzy hyperbola and is denoted by FH. The system formed by these curves can also be considered as a curvilinear system or distribution in mathematical applications.Now, we will denote a method to create a fuzzy hyperbola in a fuzzy plane by taking five fuzzy points. These points will be the same points which we gave in Definition4 in the preliminaries section.Necessary explanations and proofs are presented below.Now, we get five fuzzy points whose cores lie on a crisp hyperbolaCH, and these points are denoted by H¯iai,bi,i=1,2,…,5. First, to create a fuzzy hyperbola that passing through these points, we construct a fuzzy hyperbolic segment, and we will denote as FH¯1,…,5 briefly, joining H¯1,H¯2,…,H¯5. Subsequently, we extend the four sides of the hyperbolic segment FH¯1,…,5 to infinity to construct the fuzzy hyperbola. We will denote fuzzy hyperbola as FH¯, briefly. While constructing conic equations in fuzzy space, it is sufficient to examine only finite segments in closed curves such as ellipses and circles. But for curves with points at infinity, such as parabola and hyperbola, it is not sufficient to study only with finite segments. For this reason, based on the method presented for finite segments, semi-infinite segments are defined with the idea that it can start from a point and extend to infinity, that is, the image set must have a point at infinity. First of all, a finite segment is created by selecting at least two points on each hyperbola curve as shown in Figure 1. Then, as proved in Proposition 1, the semi-infinite segments are developed so that they remain parallel to the crisp hyperbola. Thus, as a result, fuzzy hyperbola FH¯ is obtained. We will express this more clearly below.Figure 1
Construction of fuzzy hyperbola in the method.We get four fuzzy points with a core onCH which are an infinitely far from the side of H¯1a1,b1, H¯3a3,b3, H¯4a4,b4, and H¯5a5,b5. Let these four fuzzy points be H¯1∞,H¯3∞,H¯4∞, and H¯5∞, respectively, and FH¯1∞,FH¯3∞,FH¯4∞, and FH¯5∞ be the semi-infinite fuzzy hyperbolic segments combining H¯1 with H¯1∞,H¯3 with H¯3∞,H¯4 with H¯4∞, and H¯5 with H¯5∞. Thereafter, we can define the fuzzy hyperbola FH¯ with the combinations of the hyperbolic segment FH¯1,…,5 and the semi-infinite segments,(6)FH¯=FH¯1∞∪FH¯3∞∪FH¯1,…,5∪FH¯4∞∪FH¯5∞.


### 3.1. Construction ofFH¯1,…,5

In this section, we construct the segmentFH¯1,…,5 for the fuzzy hyperbola. This segment is defined as(7)FH¯1,…,5=∨α∈0,1FHα:WhereFHαis a crisp hyperbola that passesthrough five same points anH¯iai,bi,i=1,2,…,5with membership degreeα.By the aid of the membership function, the hyperbolic segmentFH¯1,…,5 can be defined by(8)μx,y|FH¯1,…,5=supα:Wherex,ylies onFHαthatpasses through five same points onH¯i,i=1,2,…,5with membership degreeα.This definition shows that the fuzzy hyperbolic-segmentFH¯1,…,5 is a collection of crisp points with different degrees of membership. The fuzzy hyperbola is the union of all crisp hyperbolas passing through five same points on the supports of H¯i,i=1,2,…,5.


### 3.2. Construction ofFH¯1∞, FH¯3∞ and FH¯4∞, FH¯5∞

In this section, we create semi-infinitive fuzzy hyperbola segmentFH¯1∞,FH¯3∞,FH¯4∞, and FH¯5∞. But for this, fuzzy points H¯1∞, H¯3∞,H¯4∞, and H¯5∞ must be obtained first. We present these fuzzy points in the following Proposition 1, and also, the meaning of the phrase “infinitely far” is given. The shape and position of these fuzzy points H¯i∞,i=1,3,4,5 are not important for the construction of semi-infinity fuzzy segments FH¯i∞,i=1,3,4,5. There are two things to note here as follows:(i)
The support of fuzzy pointsH¯i∞,i=1,3,4,5 should be taken as a compact set(ii)
The cores of fuzzy pointsH¯i∞,i=1,3,4,5 should lie on the core hyperbola CHFuzzy hyperbola segmentsFH¯i∞,i=1,3,4,5 can be formed by using fuzzy same points H¯i∞,i=1,3,4,5 on the hyperbola, and their fuzzy points extended to infinity. The important detail here is connecting the same points of H¯1 with H¯1∞, H¯3 with H¯3∞, H¯4 with H¯4∞, H¯5 with H¯5∞ segments with these connecting same points give us the fuzzy hyperbola. Another feature we will draw attention to in Proposition 1 is that these fuzzy hyperbola segments should be parallel to the core hyperbola FH¯1. We will show later that a core fuzzy hyperbola FH¯1 and a crisp fuzzy hyperbola CH are equivalent. We can say the membership degrees of the hyperbolic segments used here will be different. According to this, semi-infinity crisp hyperbolic segments should be parallel to the CH.Proposition 1.
A fuzzy hyperbolaFH¯ formed by the determined fuzzy points, while the semi-infinite crisp hyperbolic segments in FH¯i∞,i=1,3,4,5 must be parallel to the core hyperbola FH¯1.Proof.
Let us prove withFH1∞ that any of the semi-infinite crisp hyperbolic segments. Suppose that there exists a semi-infinite crisp hyperbolic segment FH1∞ that is not parallel to FH¯1.
As per to theFH¯1∞0, it is union of the semi-infinite hyperbolic segments combining same points of the fuzzy points H¯1 and H¯1∞. Hence, corresponding to FH1∞, there must be two same points x1,y1∈H¯10 and x1∞,y1∞∈H¯1∞0 which are two extremities of FH1∞.
BecauseFH1∞ is not parallel to FH¯1, there are two cases. First case is the intersection FH1∞ with FH¯1, and the second case is the distance between the point x1∞,y1∞ and the semi-infinite hyperbolic segment FH1∞ must be infinitely wide.
The first case evidently clearly implies thatx1,y1∈H¯10 and x1∞,y1∞∈H¯1∞0 are not same points because, in this case, x1,y1 and x1∞,y1∞ lie on two different sides of FH¯1. Therefore, it lies on two different sides of the line combining core points of H¯1 and H¯1∞.
The second case implies thatH¯1∞0 which support of H¯1∞ is unlimited, and therefore, H¯1∞ cannot be a fuzzy point.
Therefore, it is impossible in both cases. So, a contradiction occurs.
Then,FH1∞ must be parallel to FH¯1. Hence, whole the semi-infinite hyperbolic segments in support of FH¯1∞ must be parallel to core hyperbola FH¯1. As with FH¯1∞0, supports of FH¯3∞,FH¯4∞, and FH¯5∞ must be parallel to core hyperbola FH¯1. We can now write FH¯1∞,FH¯3∞,FH¯4∞, and FH¯5∞ using the following formulations:(9)FH¯1∞=∨FH1∞x,y:wherex,y∈H¯10andFH1∞x,yis a semiinfinite hyperbolic segment parallel toFH¯1with membershipdegreex,yonH¯1,FH¯3∞=∨FH3∞x,y:wherex,y∈H¯30andFH3∞x,yis a semiinfinite hyperbolic segment parallel toFH¯1with membershipdegreex,yonH¯3,FH¯4∞=∨FH4∞x,y:wherex,y∈H¯40andFH4∞x,yis a semiinfinite hyperbolic segment parallel toFH¯1with membershipdegreex,yonH¯4,FH¯5∞=∨FH5∞x,y:wherex,y∈H¯50andFH5∞x,yis a semiinfinite hyperbolic segment parallel toFH¯1with membershipdegreex,yonH¯5.
Let now depict the fuzzy hyperbola in detail with the membership degrees of the given fuzzy points by making geometric comments.
In Figure1, H¯1,H¯2,H¯3,H¯4, and H¯5 are five fuzzy points. The regions inside the square with center at A1, ellipse with center A2, circle with center A3, ellipse with center at A4, and square with center at A5 are the supports of the points H¯1,H¯2,H¯3,H¯4, and H¯5, respectively. The gray color transitions that inside the supports of the fuzzy points indicate different α-cuts. That is, the intensity of the gray levels on the support of the fuzzy points expresses the change of membership degrees. The regions that depicted in dark gray in the graph are formed by points with high membership degrees. Light gray regions on the graph are obtained as the membership degrees approach 0. Namely, the membership degrees of the centers of circle, squares, and ellipses are “1;” then, it decreases gradually to “0” on the periphery of the support of H¯i for each i=1,2,3,4,5.
In Figure1, Liθ’s are five lines that passes through Ai,i=1,2,3,4,5. The Liθ’s here have an angle θ with the positive x-axis. Since H¯iAiα, the α-cut of a fuzzy point is convex, and Ai is an interior point of H¯iAiα, and the line Liθ must intersect with the boundary of H¯iAiα at two points which we will call Qiθα and Riθα. Each Q1θα,Q2θα,Q3θα,Q4θα, and Q5θα with respect to the angles θ form a set of five same points with membership degree α. Likewise, R1θα,R2θα,R3θα,R4θα, and R5θα are the set of five same points with the same membership degree α.
LetFHθUα is the hyperbola passing through Qiθα’s and FHθLα be the hyperbola passing through the five points Riθα’s in Figure 1, i=1,2,3,4,5. Because of all the points, Qiθα and Riθα have “α” membership degree, and we have assigned a membership degree of α to the hyperbola FHθUα and FHθLα on FH¯,i=1,2,3,4,5.
To get various hyperbolas asFHθUα and FHθLα, we will change the angle θ in 0,2π and membership degree α in 0,1. By definition, the fuzzy hyperbolic segment FH¯1,…,5 is the combination of whole the hyperbolas FHθUα and FHθLα with membership degree α.
Namely, we say(10)FH¯1,…,5=∨θ∈0,2πα∈0,1FHθUα,FHθLα.
We take a hyperbolaFH in FH¯0. Let define the membership degree of on hyperbola FH in FH¯ with(11)μFH|FH¯=minx,y∈FHμx,y|FH¯.
Let us show how to obtain the membership degree ofFH which is any hyperbola in FH¯ with Theorem 1.Theorem 1.
Suppose thatFH be any hyperbola in FH¯ and five same points xi,yi∈H¯i0 with μxi,yi|FH¯=α for all i=1,2,3,4,5 exist such that FH is the hyperbola that passes through the five xi,yi’s and μFH|FH¯=α.Proof.
We examine the proof in two different cases that (i)μFH|FH¯≮α and (ii) μFH|FH¯≯α.(i)
By contrast, let assume thatμFH|FH¯<α. In that case, by the definition of μFH|FH¯, there exist x0,y0 in FH¯ such that x0,y0∈FH and μx0,y0|FH¯<α. Let say μx0,y0|FH¯=β. Since x0,y0∈FH and FH is a hyperbola which is combining the five same points, let the membership degree of these same points must be α.(12)μx0,y0FH¯=supψ:wherex,ylies on the hyperbolawhich is combining the five same pointswith membership degreeψ≥α.But this contradicts our acceptanceβ<α. So, μFH|FH¯≮α.(ii)
SinceμFH|FH¯=minα:wherex,ylies onFHandμFH|FH¯=α and all the points xi,yi, lie on FH, it is clear that μFH|FH¯≯α, i=1,2,3,4,5.
Therefore,μFH|FH¯=α is obtained.
We depict the complete fuzzy hyperbolaFH¯ in Figure 2. The region between the curves f0L and f0U is the support of the FH¯. CH is the core hyperbola CH≡FH¯1 which passing through the five core points Ai of the fuzzy points H¯i.
Let’s mention the line perpendicular toCH≡FH¯1 that we take as the CD line in Figure 2. Along the CD, there exists a LR type fuzzy number that we denoted by (F/G/H)LR. If we explain LR type fuzzy number like this, L and R are reference functions LandR:0,+∞⟶0,1 that does not decrease and satisfies two conditions Lx=L−x and L0=1. Where α and β are positive and A¯ is a fuzzy number, μx|A¯ can be written as follows:(13)μx|A¯=Lm−xα,ifx≤m,Rx−mβ,ifx≥m.
The notation (m−α/m/m+β)LR is used to represent an LR-type fuzzy number. That is, all fuzzy hyperbolas can be visualized as a three-dimensional figure (a subset of x,y×0,1) whose cross section across FH¯ is a fuzzy number such as (F/G/H)LR.
Let (F/G/H)LR is a fuzzy number on fuzzy hyperbola FH¯ and take a convex region on FH¯0 such that, except F and H, all points on the line segment FH are inner of the convex region. When we take a fuzzy point H¯ such that the membership function is μ((x,y)|H¯=μ((x, y)|(F/G/H)LR, if x,y∈FH, μx,y|H¯≤μx,y|FH¯. Only at G, μx,y|H¯=1. Membership degree decreases gradually to “0” that approach F or H.Figure 2
Fuzzy hyperbola (towards completing the fuzzy hyperbola in Figure1).


### 3.3. Construction of Membership Function

To get the membership function first, we have to consider the middle hyperbolic segmentFH¯1…5 and the semi-infinite tails FH¯i∞,i=1,2,3,4,5. The membership degree μx,y|FH¯1,…,5 might not always be simple to evaluate. Also, we consider the membership degree and the definition of fuzzy hyperbola in the following form:(14)μx,y|FH¯1…5=supα:wherex,ylies in a hyperbola thatpasses through five same points inH¯i,i=1,2,3,4,5with membership degreeα,μx,y|FH¯i∞=supα:wherex,ylies on a crisp hyperbolaFHαthatpasses through a point onH¯i0with membership degreeα,i=1,2,3,4,5.Also, obtaining the closed form of the membership function ofFH¯ is a really difficult task. As in the definition of μx,y|FH¯1,…,5, to obtain μx,y|FH¯, we must get five same points with membership degree α∈0,1. Thereafter, all possible values of α are identified for which x,y lies on the hyperbola that joins five same points with membership degrees. To obtain α membership degree, the nonlinear equation systems that obtained by calculating 5 ∗ 5 determinants as can be seen in the next section are solved. From the solution of the equation, there may be real values between 0 and 1. The supremum of all these real α values is the membership degree of μx,y|FH¯1,…,5. We refer the hyperbola whose membership degree is supremum.Now, we obtain a systematic procedure to identify the membership degree of the pointx0,y0 in a fuzzy hyperbola FH¯ which passes through five fuzzy points H¯i,i=1,2,3,4,5We show the expansion of the same points onH¯i’s as xiθα,yiθα,i=1,2,3,4,5 (0≤θ≤2π, α∈0,1).As a result, we will have to examine the existence of solution of nonlinear equations by giving various values toθ and determining the α membership degrees according to the angle θ.Let the angleθ=θ00≤θ≤2π and Sθ0’s are the sets of membership degrees that can be compatible with respect to the various angle θ0.We assume that the supremum of the setSθ0 as sθ0. It can be seen from the given examples that nonlinear equation systems may not have a solution for some θ0. Fuzzy hyperbola FH¯ is obtained by determining and giving appropriate values.Then, the membership degree ofx0,y0 in the FH¯ fuzzy hyperbola is given by(15)μx0,y0|FH¯=supθsθ0.The explanation of this part is given also in this section where the membership function is explained.Let us give the application of the procedure with the following examples.Example 1.
LetH¯11,0,H¯22,33,H¯3−3,62,H¯4−4/3,−7, and H¯55/2,−321/2 be five fuzzy points. Let us get the fuzzy hyperbola that passes through these points.
The equation of the core hyperbola through the points is(16)x,y:x2−y29=1.
In this example, we take care of that the points are in different regions on the curve. The membership functions of these selected five fuzzy points are cones with circular and elliptical bases similar to the display of the points in Figure1:(17)x,y:x−12+y2≤1circular,x,y:x−22+4y−332≤1eliptical,x,y:x+32+y−622≤1circular,x,y:4x+432+y+72≤1eliptical,x,y:x−522+y+32122≤1circular.
The vertices of the membership functions areH¯11,0,H¯22,33,H¯3−3,62,H¯4−4/3,−7, and H¯55/2,−321/2, respectively.
Now, forα∈0,1, we may find the same points with membership degree α on H¯1,H¯2,H¯3,H¯4, and H¯5 as follows:(18)Q1θα:x1θα,y1θα=1+1−αcosθ,1−αsinθ,Q2θα=x2θα,y2θα=2+1−α⋅cosθ1+3sin2θ,33+1−α⋅sinθ1+3sin2θ,Q3θα=x3θα,y3θα=−3+1−αcosθ,62+1−αsinθ,Q4θα=x4θα,y4θα=−43+1−α⋅cosθ1+8cos2θ,−7+1−α⋅sinθ1+8cos2θ,Q5θα=x5θα,y5θα=52+1−αcosθ,−3212+1−αsinθ.
The hyperbolaHθα that passes through Q1θα,Q2θα,Q3θα,Q4θα, and Q5θα in (18) can be determinant by the following equation:(19)aθαx2+2hθαxy+bθαy2+2gθαx+2fθαy+cθα=0,with hθα2>aθα⋅bθα where(20)aθα=2hθαkθα−x1θαy1θαy1θα2x1θαy1θα1−x2θαy2θαy2θα2x2θαy2θα1−x3θαy3θαy3θα2x3θαy3θα1−x4θαy4θαy4θα2x4θαy4θα1−x5θαy1θαy5θα2x5θαy5θα1,bθα=2hθαkθαx1θα2−x1θαy1θαx1θαy1θα1x2θα2−x2θαy2θαx2θαy2θα1x3θα2−x3θαy3θαx3θαy3θα1x4θα2−x4θαy4θαx4θαy4θα1x5θα2−x5θαy1θαx5θαy5θα1,gθα=hθαkθαx1θα2y1θα2−x1θαy1θαy1θα1x2θα2y2θα2−x2θαy2θαy2θα1x3θα2y3θα2−x3θαy3θαy3θα1x4θα2y4θα2−x4θαy4θαy4θα1x5θα2y5θα2−x5θαy5θαy5θα1,fθα=hθαkθαx1θα2y1θα2x1θα−x1θαy1θα1x2θα2y2θα2x2θα−x2θαy2θα1x3θα2y3θα2x3θα−x3θαy3θα1x4θα2y4θα2x4θα−x4θαy4θα1x5θα2y5θα2x5θα−x5θαy5θα1,cθα=2hθαkθαx1θα2y1θα2x1θαy1θα−x1θαy1θαx2θα2y2θα2x2θαy2θα−x2θαy2θαx3θα2y3θα2x3θαy3θα−x3θαy3θαx4θα2y4θα2x4θαy4θα−x4θαy4θαx5θα2y5θα2x5θαy5θα−x5θαy5θα,kθα=x1θα2y1θα2x1θαy1θα1x2θα2y2θα2x2θαy2θα1x3θα2y3θα2x3θαy3θα1x4θα2y4θα2x4θαy4θα1x5θα2y5θα2x5θαy5θα1.
These determinants are composed by writing column(21)−x1θαy1θα−x2θαy2θα−x3θαy3θα−x4θαy4θα−x5θαy5θα,instead of columns in determinant(22)x1θα2y1θα2x1θαy1θα1x2θα2y2θα2x2θαy2θα1x3θα2y3θα2x3θαy3θα1x4θα2y4θα2x4θαy4θα1x5θα2y5θα2x5θαy5θα1.
LetA,B,C,F,G,andK be the determinant values used to find the values of aθα,bθα,cθα,fθα,gθα,andkθα, respectively. So,(23)aθα=2hθαkθα⋅A,bθα=2hθαkθα⋅B,cθα=2hθαkθα⋅C,fθα=hθαkθα⋅F,gθα=hθαkθα⋅G,kθα=K,are obtained.
The fuzzy hyperbolaFH¯ that passes through H¯i’s, i=1,2,3,4,5 is the union of all possible hyperbola Hθα’s that lies between Q1θα and Q5θα’s.
That is,(24)FH¯1,…,5=∨α∈0,1∪θ∈0,2πx,y=aθαx2+2hθαxy+bθαy2+2gθαx+2fθαy+cθα=0.
The infinite tailsFH¯2∞,FH¯3∞,FH¯4∞, and FH¯5∞ are determined by the following membership functions:(25)μx,y|FH¯2∞=α,9x2−x2θα2=y2−y2θα2,θ=16.1°,0,elsewhere,μx,y|FH¯3∞=α,9x2−x3θα2=y2−y3θα2,θ=19.4°,0,elsewhere,μx,y|FH¯4∞=α,9x2−x4θα2=y2−y4θα2,θ=26.7°,0,elsewhere,μx,y|FH¯5∞=α,9x2−x5θα2=y2−y5θα2,θ=16.98°,0,elsewhere.
The anglesθ=16.1°,19.4°,26.7°, and 16.98° are, respectively, the angles of the normals to the core hyperbola at 2,33,−3,62,−4/3,−7, and 5/2,−321/2 with the positive x-axis.
Now, we choose a point on the fuzzy hyperbolaFH¯ satisfying (19). This point can be taken as 1.2,0.4. We can show below how to calculate the membership degree of this selected point. It can also be seen from the figures that a fuzzy hyperbola passing through five points is formed as a combination of curves. In order to use the most suitable curve, the membership degrees must be known. The value of membership degrees is important for fuzzy hyperbolas.
Now, we get the equation as follows:(26)aθα1.22+2hθα1.20.4+bθα0.42+2gθα1.2+2fθα0.4+cθα=0,which simplifies to(27)1,44aθα+0,96hθα+0,16bθα+2,4gθα+0,8fθα+cθα=0.
Now, let us examine the angular values thatL lines make with the x-axis.
First, we admit thatθ0=45° in equation (27). We calculate above determinant values for this angle and will find kθα.(28)kθα=K,K=1+1−αcos45°21−αsin45°21+1−αcos45°1−αsin45°12+1−αcos45°1+3sin245°233+1−αsin45°1+3sin245°2+1−αcos45°1+3sin245°33+1−αsin45°1+3sin245°1−3+1−αcos45°262+1−αsin45°2−3+1−αcos45°62+1−αsin45°1−43+1−αcos45°1+8cos245°2−7+1−αsin45°1+8cos245°2−43+1−αcos45°1+8cos245°−7+1−αsin45°1+8cos245°152+1−αcos45°2−3212+1−αsin45°252+1−αcos45°−3212+1−αsin45°1,K=2.337α3−73.417α2−657.057α−585.977.
Then, we calculateaθα,bθα,cθα,fθα, and gθα.
For the value ofA,(29)−1+1−αcos45°1−αcos45°1−αsin45°21+1−αcos45°1−αsin45°1−2+1−αcos45°1+3sin245°33+1−αcos45°1+3sin245°33+1−αsin45°1+3sin245°2+1−αcos45°1+3sin245°33+1−αsin45°1+3sin245°1−−3+1−αcos45°62+1−αsin45°62+1−αsin45°2−3+1−αcos45°62+1−αsin45°1−−43+1−αcos45°1+8cos245°−7+1−αsin45°1+8cos245°−7+1−αsin45°1+8cos245°2−43+1−αcos45°1+8cos245°−7+1−αsin45°1+8cos245°1−52+1−αcos45°−3212+1−αsin45°−3212+1−αsin45°252+1−αcos45°−3212+1−αsin45°1=−1.168α3+859.293α2−1587.371α−71752.51.
For the value ofB,(30)1+1−αcos45°2−1+1−αcos45°1−αcos45°1+1−αcos45°1−αsin45°12+1−αcos45°1+3sin245°2−2+1−αcos45°1+3sin245°33+1−αsin45°1+3sin245°2+1−αcos45°1+3sin245°33+1−αsin45°1+3sin245°1−3+1−αcos45°2−−3+1−αcos45°62+1−αsin45°−3+1−αcos45°62+1−αsin45°1−43+1−αcos45°1+8cos245°2−−43+1−αcos45°1+8cos245°−7+1−αsin45°1+8cos245°−43+1−αcos45°1+8cos245°−7+1−αsin45°1+8cos245°152+1−αcos45°2−52+1−αcos45°−3212+1−αsin45°52+1−αcos45°−3212+1−αsin45°1=−1.168α3−121.751α2+643.496α+7532.952.
For the value ofG,(31)1+1−αcos45°21−αsin45°2−1+1−αcos45°1−αcos45°1−αsin45°12+1−αcos45°1+3sin245°233+1−αsin45°1+3sin245°2−2+1−αcos45°1+3sin245°33+1−αcos45°1+3sin245°33+1−αsin45°1+3sin245°1−3+1−αcos45°262+1−αsin45°2−−3+1−αcos45°62+1−αsin45°62+1−αsin45°1−43+1−αcos45°1+8cos245°2−7+1−αsin45°1+8cos245°2−−43+1−αcos45°1+8cos245°−7+1−αsin45°1+8cos245°−7+1−αsin45°1+8cos245°152+1−αcos45°2−3212+1−αsin45°2−52+1−αcos45°−3212+1−αsin45°−3212+1−αsin45°1=1033.751α3−2942.633α2−86043.569α+87952.451.
For the value ofF,(32)1+1−αcos45°21−αsin45°21+1−αcos45°−1+1−αcos45°1−αsin45°12+1−αcos45°1+3sin245°233+1−αsin45°1+3sin245°22+1−αcos45°1+3sin245°−2+1−αcos45°1+3sin245°33+1−αsin45°1+3sin245°1−3+1−αcos45°262+1−αsin45°2−3+1−αcos45°−−3+1−αcos45°62+1−αsin45°1−43+1−αcos45°1+8cos245°2−7+1−αsin45°1+8cos245°2−43+1−αcos45°1+8cos245°−−43+1−αcos45°1+8cos245°−7+1−αsin45°1+8cos245°152+1−αcos45°2−3212+1−αsin45°252+1−αcos45°−52+1−αcos45°−3212+1−αsin45°1=−267.139α3+1601.891α2−13501.485α−14836.237.
For the value ofC,(33)1+1−αcos45°21−αsin45°21+1−αcos45°1−αsin45°−1+1−αcos45°1−αsin45°2+1−αcos45°1+3sin245°233+1−αsin45°1+3sin245°22+1−αcos45°1+3sin245°33+1−αsin45°1+3sin245°−2+1−αcos45°1+3sin245°33+1−αsin45°1+3sin245°−3+1−αcos45°262+1−αsin45°2−3+1−αcos45°62+1−αsin45°−−3+1−αcos45°62+1−αsin45°−43+1−αcos45°1+8cos245°2−7+1−αsin45°1+8cos245°2−43+1−αcos45°1+8cos245°−7+1−αsin45°1+8cos245°−−43+1−αcos45°1+8cos245°−7+1−αsin45°1+8cos245°52+1−αcos45°2−3212+1−αsin45°252+1−αcos45°−3212+1−αsin45°−52+1−αcos45°−3212+1−αsin45°=210.014α4−551.863α3−19423.306α2+25857.334α+66389.578.
Let substitute these values in equation (23), and we obtain the coefficients aθα,bθα,cθα,fθα,gθα, and cθα.
If we substitute this coefficient in the conic equation (27) and simplify the equation with hθαhθα≠0, we obtain the following equation:(34)420.029α4+1162.068α3−42262.096α2−147723.644α+127196.826.
By solving equation (34), the values of α are 0.717,10.197,−4.516, and −9.164.
As alpha represents the membership degree, it must be0,1. So, the appropriate alpha real number is θ=0.717 as it is supremum.
Thus, forθ0=45°, the set Sθ0 of all possible value of α is the set 0.717.
We changeθ0 through 0,2π and obtain the value of sθ0. Then, we get supremum of all sθ0’s.
It can be easily verified that the supremum value is0.717 for the point 1.2,0.4, and it is obtained for θ0=45°. In conclusion, we obtain the hyperbola conic passing through to the same points for θ0=45° and α=0.717. As a result, we find that the conic which passing through to the same points are 1.205,0.205∈H¯1, 2.129,5.325∈H¯2, −2.784,8.69∈H¯3, −1.203,−2.516∈H¯4, and 2.705,−6.668∈H¯5.
Then, we obtain the conic equation (35) which passing through to the same points:(35)−72372.62x2−159.62xy+7917.21y2+25766.96x−4541.24y+74675.32=0.
This hyperbola equation contains the point1.2,0.4. And, we have(36)μ1.2,0.4|FH¯=0.717,alpha cut of fuzzy hyperbola.
Let also give an example of a fuzzy hyperbola whose core hyperbola is noncentral.Example 2.
LetH¯16,217,H¯22,2,H¯30,25,H¯4−5,−102, and H¯54,−25 be five fuzzy points. Let’s get the fuzzy hyperbola that passes through these points.
In this example, we take core of that the points are in different regions on the curve. The membership functions of these selected five fuzzy points are cones with circular and elliptical basessimilar to the display of the points in Figure 1;
(37)x,y:x−62+y−2172≤1circular,x,y:x−22+9y−22≤1elliptical,x,y:x2+y−252≤1circular,x,y:9x+52+y+1022≤1elliptical,x,y:x−42+y+252≤1circular.
The vertices of the membership functions areH¯16,217,H¯22,2,H¯30,25,H¯4−5,−102, and H¯54,−25, respectively.
Now, forα∈0,1, we may find the same points with membership degree α on H¯1,H¯2,H¯3,H¯4, and H¯5 as follows:(38)Q1θα=x1θα,y1θα=6+1−αcosθ,217+1−αsinθ,Q2θα=x2θα,y2θα=2+1−α⋅cosθ1+8sin2θ,25+1−α⋅sinθ1+8sin2θ,Q3θα=x3θα,y3θα=1−αcosθ,25+1−αsinθ,Q4θα=x4θα,y4θα=−5+1−α⋅cosθ1+8cos2θ,−25+1−α⋅sinθ1+8cos2θ,Q5θα=x5θα,y5θα=4+1−αcosθ,−25+1−αsinθ.
The equation of the core hyperbola through the points is(39)y24−x−22=1.
The hyperbolaHθα that passes through Q1θα,Q2θα,Q3θα,Q4θα, and Q5θα can be determinant by the following equation:(40)aθαx2+2hθαxy+bθαy2+2gθαx+2fθαy+cθα=0,with hθα2>aθα⋅bθα. In this equation, the determinants used to find aθα,hθα,bθα,gθα,fθα, and cθα will be taken as in Example 1, and the membership of a point that we will choose is calculated.
LetA,B,C,F,G,andK be the determinant values used to find the values of aθα,bθα,cθα,fθα,gθα,andkθα, respectively. So,(41)aθα=2hθαkθα⋅A,bθα=2hθαkθα⋅B,cθα=2hθαkθα⋅C,fθα=hθαkθα⋅F,gθα=hθαkθα⋅G,kθα=K.
The fuzzy hyperbolaFH¯ that passes through H¯i’s, i=1,2,3,4,5 is the union of all possible hyperbola Hθα’s that lies between Q1θα and Q5θα’s.
That is,(42)FH¯1,…,5=∨α∈0,1∪θ∈0,2πx,y=aθαx2+2hθαxy+bθαy2+2gθαx+2fθαy+cθα=0.
The infinite tailsFH¯1∞,FH¯3∞,FH¯4∞, and FH¯5∞ are determined by the following membership functions:(43)μx,y|FH¯1∞=α,y2−y1θα2=4x−22+x1θα2,θ=27.2°,0,elsewhere,μx,y|FH¯3∞=α,y2−y3θα2=4x−22+x3θα2,θ=29.2°,0,elsewhere,μx,y|FH¯4∞=α,y2−y4θα2=4x−22+x4θα2,θ=26.7°,0,elsewhere,μx,y|FH¯5∞=α,y2−y5θα2=4x−22+x5θα2,θ=29.2°,0,elsewhere.
The anglesθ=27.2°,29.2°,26.7°, and 29.2° are, respectively, the angles of the normals to the core hyperbola at 6,217,0,215,−5,−102, and 4,−25 with the positive x-axis.
2.05,2.05 is a point on the fuzzy hyperbola FH¯. First, we adjust the set of hyperbolas Hθα’s which the point 2.05,2.05 lies. We will calculate the membership degree of this point.
Let replace point2.05,2.05 in (40). We need to identify the possible values of α.
Then, we get the equation as follows:(44)aθα2.052+2hθα22.052.05+bθα2.052+2gθα2.05+2fθα2.05+cθα=0,4.2025aθα+8.405hθα+4.2025bθα+4.1gθα+4.1fθα+cθα=0.First, we admit that θ0=60°. We will calculate determinant values for this angle. These determinant values are calculated as in Example 1. We substitute θ0=60° in x1θα,y1θα,x2θα,y2θα,x3θα,y3θα,x4θα,y4θα, and x5θα,y5θα for determinants and calculate the determinant values approximately. We find them and replace in (44); then, we obtain the following nonlinear equation which determinants are found:(45)−3.184α4−17.469α3+14514.314α2−204227.678α+204987.984=0.
By solving this equation, real alpha values are found.(46)α=0.842,42.001.
But, we get0.842 from 0 to 1 from these real two values.
Thus, forθ0=60°, the set Sθ0 of all possible values of α is the set 0.842.
We changeθ0 through 0,2π and obtain the value of sθ0. Finally, we get supremum of all sθ0’s. It can be easily verified that the supremum value is 0,842 for the point 2.05,2.05, and it is obtained for θ0=60°. In conclusion, we obtain the hyperbola conic passing through to the same points, for θ0=60° and α=0.842.
Substitutingα and θ in Equation (38), we get the same points as 6.079,8.383∈H˜1,2.029,2.051∈H˜2,0.079,4.608∈H˜3,−4.954,−14.063∈H˜4 and 4.079,−4335∈H˜5.
Then, we obtain the following conic (47) which passes through to the same points(47)348199.353x2−1624.867xy−86002.529y2−1445151.495x+27947.976y+1809949.889=0.
The conic equation contains the point2.05,2.05. And, we have(48)μ2.05,2.05|FE¯=0.842.


## 3.1. Construction ofFH¯1,…,5

In this section, we construct the segmentFH¯1,…,5 for the fuzzy hyperbola. This segment is defined as(7)FH¯1,…,5=∨α∈0,1FHα:WhereFHαis a crisp hyperbola that passesthrough five same points anH¯iai,bi,i=1,2,…,5with membership degreeα.By the aid of the membership function, the hyperbolic segmentFH¯1,…,5 can be defined by(8)μx,y|FH¯1,…,5=supα:Wherex,ylies onFHαthatpasses through five same points onH¯i,i=1,2,…,5with membership degreeα.This definition shows that the fuzzy hyperbolic-segmentFH¯1,…,5 is a collection of crisp points with different degrees of membership. The fuzzy hyperbola is the union of all crisp hyperbolas passing through five same points on the supports of H¯i,i=1,2,…,5.


## 3.2. Construction ofFH¯1∞, FH¯3∞ and FH¯4∞, FH¯5∞

In this section, we create semi-infinitive fuzzy hyperbola segmentFH¯1∞,FH¯3∞,FH¯4∞, and FH¯5∞. But for this, fuzzy points H¯1∞, H¯3∞,H¯4∞, and H¯5∞ must be obtained first. We present these fuzzy points in the following Proposition 1, and also, the meaning of the phrase “infinitely far” is given. The shape and position of these fuzzy points H¯i∞,i=1,3,4,5 are not important for the construction of semi-infinity fuzzy segments FH¯i∞,i=1,3,4,5. There are two things to note here as follows:(i)
The support of fuzzy pointsH¯i∞,i=1,3,4,5 should be taken as a compact set(ii)
The cores of fuzzy pointsH¯i∞,i=1,3,4,5 should lie on the core hyperbola CHFuzzy hyperbola segmentsFH¯i∞,i=1,3,4,5 can be formed by using fuzzy same points H¯i∞,i=1,3,4,5 on the hyperbola, and their fuzzy points extended to infinity. The important detail here is connecting the same points of H¯1 with H¯1∞, H¯3 with H¯3∞, H¯4 with H¯4∞, H¯5 with H¯5∞ segments with these connecting same points give us the fuzzy hyperbola. Another feature we will draw attention to in Proposition 1 is that these fuzzy hyperbola segments should be parallel to the core hyperbola FH¯1. We will show later that a core fuzzy hyperbola FH¯1 and a crisp fuzzy hyperbola CH are equivalent. We can say the membership degrees of the hyperbolic segments used here will be different. According to this, semi-infinity crisp hyperbolic segments should be parallel to the CH.Proposition 1.
A fuzzy hyperbolaFH¯ formed by the determined fuzzy points, while the semi-infinite crisp hyperbolic segments in FH¯i∞,i=1,3,4,5 must be parallel to the core hyperbola FH¯1.Proof.
Let us prove withFH1∞ that any of the semi-infinite crisp hyperbolic segments. Suppose that there exists a semi-infinite crisp hyperbolic segment FH1∞ that is not parallel to FH¯1.
As per to theFH¯1∞0, it is union of the semi-infinite hyperbolic segments combining same points of the fuzzy points H¯1 and H¯1∞. Hence, corresponding to FH1∞, there must be two same points x1,y1∈H¯10 and x1∞,y1∞∈H¯1∞0 which are two extremities of FH1∞.
BecauseFH1∞ is not parallel to FH¯1, there are two cases. First case is the intersection FH1∞ with FH¯1, and the second case is the distance between the point x1∞,y1∞ and the semi-infinite hyperbolic segment FH1∞ must be infinitely wide.
The first case evidently clearly implies thatx1,y1∈H¯10 and x1∞,y1∞∈H¯1∞0 are not same points because, in this case, x1,y1 and x1∞,y1∞ lie on two different sides of FH¯1. Therefore, it lies on two different sides of the line combining core points of H¯1 and H¯1∞.
The second case implies thatH¯1∞0 which support of H¯1∞ is unlimited, and therefore, H¯1∞ cannot be a fuzzy point.
Therefore, it is impossible in both cases. So, a contradiction occurs.
Then,FH1∞ must be parallel to FH¯1. Hence, whole the semi-infinite hyperbolic segments in support of FH¯1∞ must be parallel to core hyperbola FH¯1. As with FH¯1∞0, supports of FH¯3∞,FH¯4∞, and FH¯5∞ must be parallel to core hyperbola FH¯1. We can now write FH¯1∞,FH¯3∞,FH¯4∞, and FH¯5∞ using the following formulations:(9)FH¯1∞=∨FH1∞x,y:wherex,y∈H¯10andFH1∞x,yis a semiinfinite hyperbolic segment parallel toFH¯1with membershipdegreex,yonH¯1,FH¯3∞=∨FH3∞x,y:wherex,y∈H¯30andFH3∞x,yis a semiinfinite hyperbolic segment parallel toFH¯1with membershipdegreex,yonH¯3,FH¯4∞=∨FH4∞x,y:wherex,y∈H¯40andFH4∞x,yis a semiinfinite hyperbolic segment parallel toFH¯1with membershipdegreex,yonH¯4,FH¯5∞=∨FH5∞x,y:wherex,y∈H¯50andFH5∞x,yis a semiinfinite hyperbolic segment parallel toFH¯1with membershipdegreex,yonH¯5.
Let now depict the fuzzy hyperbola in detail with the membership degrees of the given fuzzy points by making geometric comments.
In Figure1, H¯1,H¯2,H¯3,H¯4, and H¯5 are five fuzzy points. The regions inside the square with center at A1, ellipse with center A2, circle with center A3, ellipse with center at A4, and square with center at A5 are the supports of the points H¯1,H¯2,H¯3,H¯4, and H¯5, respectively. The gray color transitions that inside the supports of the fuzzy points indicate different α-cuts. That is, the intensity of the gray levels on the support of the fuzzy points expresses the change of membership degrees. The regions that depicted in dark gray in the graph are formed by points with high membership degrees. Light gray regions on the graph are obtained as the membership degrees approach 0. Namely, the membership degrees of the centers of circle, squares, and ellipses are “1;” then, it decreases gradually to “0” on the periphery of the support of H¯i for each i=1,2,3,4,5.
In Figure1, Liθ’s are five lines that passes through Ai,i=1,2,3,4,5. The Liθ’s here have an angle θ with the positive x-axis. Since H¯iAiα, the α-cut of a fuzzy point is convex, and Ai is an interior point of H¯iAiα, and the line Liθ must intersect with the boundary of H¯iAiα at two points which we will call Qiθα and Riθα. Each Q1θα,Q2θα,Q3θα,Q4θα, and Q5θα with respect to the angles θ form a set of five same points with membership degree α. Likewise, R1θα,R2θα,R3θα,R4θα, and R5θα are the set of five same points with the same membership degree α.
LetFHθUα is the hyperbola passing through Qiθα’s and FHθLα be the hyperbola passing through the five points Riθα’s in Figure 1, i=1,2,3,4,5. Because of all the points, Qiθα and Riθα have “α” membership degree, and we have assigned a membership degree of α to the hyperbola FHθUα and FHθLα on FH¯,i=1,2,3,4,5.
To get various hyperbolas asFHθUα and FHθLα, we will change the angle θ in 0,2π and membership degree α in 0,1. By definition, the fuzzy hyperbolic segment FH¯1,…,5 is the combination of whole the hyperbolas FHθUα and FHθLα with membership degree α.
Namely, we say(10)FH¯1,…,5=∨θ∈0,2πα∈0,1FHθUα,FHθLα.
We take a hyperbolaFH in FH¯0. Let define the membership degree of on hyperbola FH in FH¯ with(11)μFH|FH¯=minx,y∈FHμx,y|FH¯.
Let us show how to obtain the membership degree ofFH which is any hyperbola in FH¯ with Theorem 1.Theorem 1.
Suppose thatFH be any hyperbola in FH¯ and five same points xi,yi∈H¯i0 with μxi,yi|FH¯=α for all i=1,2,3,4,5 exist such that FH is the hyperbola that passes through the five xi,yi’s and μFH|FH¯=α.Proof.
We examine the proof in two different cases that (i)μFH|FH¯≮α and (ii) μFH|FH¯≯α.(i)
By contrast, let assume thatμFH|FH¯<α. In that case, by the definition of μFH|FH¯, there exist x0,y0 in FH¯ such that x0,y0∈FH and μx0,y0|FH¯<α. Let say μx0,y0|FH¯=β. Since x0,y0∈FH and FH is a hyperbola which is combining the five same points, let the membership degree of these same points must be α.(12)μx0,y0FH¯=supψ:wherex,ylies on the hyperbolawhich is combining the five same pointswith membership degreeψ≥α.But this contradicts our acceptanceβ<α. So, μFH|FH¯≮α.(ii)
SinceμFH|FH¯=minα:wherex,ylies onFHandμFH|FH¯=α and all the points xi,yi, lie on FH, it is clear that μFH|FH¯≯α, i=1,2,3,4,5.
Therefore,μFH|FH¯=α is obtained.
We depict the complete fuzzy hyperbolaFH¯ in Figure 2. The region between the curves f0L and f0U is the support of the FH¯. CH is the core hyperbola CH≡FH¯1 which passing through the five core points Ai of the fuzzy points H¯i.
Let’s mention the line perpendicular toCH≡FH¯1 that we take as the CD line in Figure 2. Along the CD, there exists a LR type fuzzy number that we denoted by (F/G/H)LR. If we explain LR type fuzzy number like this, L and R are reference functions LandR:0,+∞⟶0,1 that does not decrease and satisfies two conditions Lx=L−x and L0=1. Where α and β are positive and A¯ is a fuzzy number, μx|A¯ can be written as follows:(13)μx|A¯=Lm−xα,ifx≤m,Rx−mβ,ifx≥m.
The notation (m−α/m/m+β)LR is used to represent an LR-type fuzzy number. That is, all fuzzy hyperbolas can be visualized as a three-dimensional figure (a subset of x,y×0,1) whose cross section across FH¯ is a fuzzy number such as (F/G/H)LR.
Let (F/G/H)LR is a fuzzy number on fuzzy hyperbola FH¯ and take a convex region on FH¯0 such that, except F and H, all points on the line segment FH are inner of the convex region. When we take a fuzzy point H¯ such that the membership function is μ((x,y)|H¯=μ((x, y)|(F/G/H)LR, if x,y∈FH, μx,y|H¯≤μx,y|FH¯. Only at G, μx,y|H¯=1. Membership degree decreases gradually to “0” that approach F or H.Figure 2
Fuzzy hyperbola (towards completing the fuzzy hyperbola in Figure1).


## 3.3. Construction of Membership Function

To get the membership function first, we have to consider the middle hyperbolic segmentFH¯1…5 and the semi-infinite tails FH¯i∞,i=1,2,3,4,5. The membership degree μx,y|FH¯1,…,5 might not always be simple to evaluate. Also, we consider the membership degree and the definition of fuzzy hyperbola in the following form:(14)μx,y|FH¯1…5=supα:wherex,ylies in a hyperbola thatpasses through five same points inH¯i,i=1,2,3,4,5with membership degreeα,μx,y|FH¯i∞=supα:wherex,ylies on a crisp hyperbolaFHαthatpasses through a point onH¯i0with membership degreeα,i=1,2,3,4,5.Also, obtaining the closed form of the membership function ofFH¯ is a really difficult task. As in the definition of μx,y|FH¯1,…,5, to obtain μx,y|FH¯, we must get five same points with membership degree α∈0,1. Thereafter, all possible values of α are identified for which x,y lies on the hyperbola that joins five same points with membership degrees. To obtain α membership degree, the nonlinear equation systems that obtained by calculating 5 ∗ 5 determinants as can be seen in the next section are solved. From the solution of the equation, there may be real values between 0 and 1. The supremum of all these real α values is the membership degree of μx,y|FH¯1,…,5. We refer the hyperbola whose membership degree is supremum.Now, we obtain a systematic procedure to identify the membership degree of the pointx0,y0 in a fuzzy hyperbola FH¯ which passes through five fuzzy points H¯i,i=1,2,3,4,5We show the expansion of the same points onH¯i’s as xiθα,yiθα,i=1,2,3,4,5 (0≤θ≤2π, α∈0,1).As a result, we will have to examine the existence of solution of nonlinear equations by giving various values toθ and determining the α membership degrees according to the angle θ.Let the angleθ=θ00≤θ≤2π and Sθ0’s are the sets of membership degrees that can be compatible with respect to the various angle θ0.We assume that the supremum of the setSθ0 as sθ0. It can be seen from the given examples that nonlinear equation systems may not have a solution for some θ0. Fuzzy hyperbola FH¯ is obtained by determining and giving appropriate values.Then, the membership degree ofx0,y0 in the FH¯ fuzzy hyperbola is given by(15)μx0,y0|FH¯=supθsθ0.The explanation of this part is given also in this section where the membership function is explained.Let us give the application of the procedure with the following examples.Example 1.
LetH¯11,0,H¯22,33,H¯3−3,62,H¯4−4/3,−7, and H¯55/2,−321/2 be five fuzzy points. Let us get the fuzzy hyperbola that passes through these points.
The equation of the core hyperbola through the points is(16)x,y:x2−y29=1.
In this example, we take care of that the points are in different regions on the curve. The membership functions of these selected five fuzzy points are cones with circular and elliptical bases similar to the display of the points in Figure1:(17)x,y:x−12+y2≤1circular,x,y:x−22+4y−332≤1eliptical,x,y:x+32+y−622≤1circular,x,y:4x+432+y+72≤1eliptical,x,y:x−522+y+32122≤1circular.
The vertices of the membership functions areH¯11,0,H¯22,33,H¯3−3,62,H¯4−4/3,−7, and H¯55/2,−321/2, respectively.
Now, forα∈0,1, we may find the same points with membership degree α on H¯1,H¯2,H¯3,H¯4, and H¯5 as follows:(18)Q1θα:x1θα,y1θα=1+1−αcosθ,1−αsinθ,Q2θα=x2θα,y2θα=2+1−α⋅cosθ1+3sin2θ,33+1−α⋅sinθ1+3sin2θ,Q3θα=x3θα,y3θα=−3+1−αcosθ,62+1−αsinθ,Q4θα=x4θα,y4θα=−43+1−α⋅cosθ1+8cos2θ,−7+1−α⋅sinθ1+8cos2θ,Q5θα=x5θα,y5θα=52+1−αcosθ,−3212+1−αsinθ.
The hyperbolaHθα that passes through Q1θα,Q2θα,Q3θα,Q4θα, and Q5θα in (18) can be determinant by the following equation:(19)aθαx2+2hθαxy+bθαy2+2gθαx+2fθαy+cθα=0,with hθα2>aθα⋅bθα where(20)aθα=2hθαkθα−x1θαy1θαy1θα2x1θαy1θα1−x2θαy2θαy2θα2x2θαy2θα1−x3θαy3θαy3θα2x3θαy3θα1−x4θαy4θαy4θα2x4θαy4θα1−x5θαy1θαy5θα2x5θαy5θα1,bθα=2hθαkθαx1θα2−x1θαy1θαx1θαy1θα1x2θα2−x2θαy2θαx2θαy2θα1x3θα2−x3θαy3θαx3θαy3θα1x4θα2−x4θαy4θαx4θαy4θα1x5θα2−x5θαy1θαx5θαy5θα1,gθα=hθαkθαx1θα2y1θα2−x1θαy1θαy1θα1x2θα2y2θα2−x2θαy2θαy2θα1x3θα2y3θα2−x3θαy3θαy3θα1x4θα2y4θα2−x4θαy4θαy4θα1x5θα2y5θα2−x5θαy5θαy5θα1,fθα=hθαkθαx1θα2y1θα2x1θα−x1θαy1θα1x2θα2y2θα2x2θα−x2θαy2θα1x3θα2y3θα2x3θα−x3θαy3θα1x4θα2y4θα2x4θα−x4θαy4θα1x5θα2y5θα2x5θα−x5θαy5θα1,cθα=2hθαkθαx1θα2y1θα2x1θαy1θα−x1θαy1θαx2θα2y2θα2x2θαy2θα−x2θαy2θαx3θα2y3θα2x3θαy3θα−x3θαy3θαx4θα2y4θα2x4θαy4θα−x4θαy4θαx5θα2y5θα2x5θαy5θα−x5θαy5θα,kθα=x1θα2y1θα2x1θαy1θα1x2θα2y2θα2x2θαy2θα1x3θα2y3θα2x3θαy3θα1x4θα2y4θα2x4θαy4θα1x5θα2y5θα2x5θαy5θα1.
These determinants are composed by writing column(21)−x1θαy1θα−x2θαy2θα−x3θαy3θα−x4θαy4θα−x5θαy5θα,instead of columns in determinant(22)x1θα2y1θα2x1θαy1θα1x2θα2y2θα2x2θαy2θα1x3θα2y3θα2x3θαy3θα1x4θα2y4θα2x4θαy4θα1x5θα2y5θα2x5θαy5θα1.
LetA,B,C,F,G,andK be the determinant values used to find the values of aθα,bθα,cθα,fθα,gθα,andkθα, respectively. So,(23)aθα=2hθαkθα⋅A,bθα=2hθαkθα⋅B,cθα=2hθαkθα⋅C,fθα=hθαkθα⋅F,gθα=hθαkθα⋅G,kθα=K,are obtained.
The fuzzy hyperbolaFH¯ that passes through H¯i’s, i=1,2,3,4,5 is the union of all possible hyperbola Hθα’s that lies between Q1θα and Q5θα’s.
That is,(24)FH¯1,…,5=∨α∈0,1∪θ∈0,2πx,y=aθαx2+2hθαxy+bθαy2+2gθαx+2fθαy+cθα=0.
The infinite tailsFH¯2∞,FH¯3∞,FH¯4∞, and FH¯5∞ are determined by the following membership functions:(25)μx,y|FH¯2∞=α,9x2−x2θα2=y2−y2θα2,θ=16.1°,0,elsewhere,μx,y|FH¯3∞=α,9x2−x3θα2=y2−y3θα2,θ=19.4°,0,elsewhere,μx,y|FH¯4∞=α,9x2−x4θα2=y2−y4θα2,θ=26.7°,0,elsewhere,μx,y|FH¯5∞=α,9x2−x5θα2=y2−y5θα2,θ=16.98°,0,elsewhere.
The anglesθ=16.1°,19.4°,26.7°, and 16.98° are, respectively, the angles of the normals to the core hyperbola at 2,33,−3,62,−4/3,−7, and 5/2,−321/2 with the positive x-axis.
Now, we choose a point on the fuzzy hyperbolaFH¯ satisfying (19). This point can be taken as 1.2,0.4. We can show below how to calculate the membership degree of this selected point. It can also be seen from the figures that a fuzzy hyperbola passing through five points is formed as a combination of curves. In order to use the most suitable curve, the membership degrees must be known. The value of membership degrees is important for fuzzy hyperbolas.
Now, we get the equation as follows:(26)aθα1.22+2hθα1.20.4+bθα0.42+2gθα1.2+2fθα0.4+cθα=0,which simplifies to(27)1,44aθα+0,96hθα+0,16bθα+2,4gθα+0,8fθα+cθα=0.
Now, let us examine the angular values thatL lines make with the x-axis.
First, we admit thatθ0=45° in equation (27). We calculate above determinant values for this angle and will find kθα.(28)kθα=K,K=1+1−αcos45°21−αsin45°21+1−αcos45°1−αsin45°12+1−αcos45°1+3sin245°233+1−αsin45°1+3sin245°2+1−αcos45°1+3sin245°33+1−αsin45°1+3sin245°1−3+1−αcos45°262+1−αsin45°2−3+1−αcos45°62+1−αsin45°1−43+1−αcos45°1+8cos245°2−7+1−αsin45°1+8cos245°2−43+1−αcos45°1+8cos245°−7+1−αsin45°1+8cos245°152+1−αcos45°2−3212+1−αsin45°252+1−αcos45°−3212+1−αsin45°1,K=2.337α3−73.417α2−657.057α−585.977.
Then, we calculateaθα,bθα,cθα,fθα, and gθα.
For the value ofA,(29)−1+1−αcos45°1−αcos45°1−αsin45°21+1−αcos45°1−αsin45°1−2+1−αcos45°1+3sin245°33+1−αcos45°1+3sin245°33+1−αsin45°1+3sin245°2+1−αcos45°1+3sin245°33+1−αsin45°1+3sin245°1−−3+1−αcos45°62+1−αsin45°62+1−αsin45°2−3+1−αcos45°62+1−αsin45°1−−43+1−αcos45°1+8cos245°−7+1−αsin45°1+8cos245°−7+1−αsin45°1+8cos245°2−43+1−αcos45°1+8cos245°−7+1−αsin45°1+8cos245°1−52+1−αcos45°−3212+1−αsin45°−3212+1−αsin45°252+1−αcos45°−3212+1−αsin45°1=−1.168α3+859.293α2−1587.371α−71752.51.
For the value ofB,(30)1+1−αcos45°2−1+1−αcos45°1−αcos45°1+1−αcos45°1−αsin45°12+1−αcos45°1+3sin245°2−2+1−αcos45°1+3sin245°33+1−αsin45°1+3sin245°2+1−αcos45°1+3sin245°33+1−αsin45°1+3sin245°1−3+1−αcos45°2−−3+1−αcos45°62+1−αsin45°−3+1−αcos45°62+1−αsin45°1−43+1−αcos45°1+8cos245°2−−43+1−αcos45°1+8cos245°−7+1−αsin45°1+8cos245°−43+1−αcos45°1+8cos245°−7+1−αsin45°1+8cos245°152+1−αcos45°2−52+1−αcos45°−3212+1−αsin45°52+1−αcos45°−3212+1−αsin45°1=−1.168α3−121.751α2+643.496α+7532.952.
For the value ofG,(31)1+1−αcos45°21−αsin45°2−1+1−αcos45°1−αcos45°1−αsin45°12+1−αcos45°1+3sin245°233+1−αsin45°1+3sin245°2−2+1−αcos45°1+3sin245°33+1−αcos45°1+3sin245°33+1−αsin45°1+3sin245°1−3+1−αcos45°262+1−αsin45°2−−3+1−αcos45°62+1−αsin45°62+1−αsin45°1−43+1−αcos45°1+8cos245°2−7+1−αsin45°1+8cos245°2−−43+1−αcos45°1+8cos245°−7+1−αsin45°1+8cos245°−7+1−αsin45°1+8cos245°152+1−αcos45°2−3212+1−αsin45°2−52+1−αcos45°−3212+1−αsin45°−3212+1−αsin45°1=1033.751α3−2942.633α2−86043.569α+87952.451.
For the value ofF,(32)1+1−αcos45°21−αsin45°21+1−αcos45°−1+1−αcos45°1−αsin45°12+1−αcos45°1+3sin245°233+1−αsin45°1+3sin245°22+1−αcos45°1+3sin245°−2+1−αcos45°1+3sin245°33+1−αsin45°1+3sin245°1−3+1−αcos45°262+1−αsin45°2−3+1−αcos45°−−3+1−αcos45°62+1−αsin45°1−43+1−αcos45°1+8cos245°2−7+1−αsin45°1+8cos245°2−43+1−αcos45°1+8cos245°−−43+1−αcos45°1+8cos245°−7+1−αsin45°1+8cos245°152+1−αcos45°2−3212+1−αsin45°252+1−αcos45°−52+1−αcos45°−3212+1−αsin45°1=−267.139α3+1601.891α2−13501.485α−14836.237.
For the value ofC,(33)1+1−αcos45°21−αsin45°21+1−αcos45°1−αsin45°−1+1−αcos45°1−αsin45°2+1−αcos45°1+3sin245°233+1−αsin45°1+3sin245°22+1−αcos45°1+3sin245°33+1−αsin45°1+3sin245°−2+1−αcos45°1+3sin245°33+1−αsin45°1+3sin245°−3+1−αcos45°262+1−αsin45°2−3+1−αcos45°62+1−αsin45°−−3+1−αcos45°62+1−αsin45°−43+1−αcos45°1+8cos245°2−7+1−αsin45°1+8cos245°2−43+1−αcos45°1+8cos245°−7+1−αsin45°1+8cos245°−−43+1−αcos45°1+8cos245°−7+1−αsin45°1+8cos245°52+1−αcos45°2−3212+1−αsin45°252+1−αcos45°−3212+1−αsin45°−52+1−αcos45°−3212+1−αsin45°=210.014α4−551.863α3−19423.306α2+25857.334α+66389.578.
Let substitute these values in equation (23), and we obtain the coefficients aθα,bθα,cθα,fθα,gθα, and cθα.
If we substitute this coefficient in the conic equation (27) and simplify the equation with hθαhθα≠0, we obtain the following equation:(34)420.029α4+1162.068α3−42262.096α2−147723.644α+127196.826.
By solving equation (34), the values of α are 0.717,10.197,−4.516, and −9.164.
As alpha represents the membership degree, it must be0,1. So, the appropriate alpha real number is θ=0.717 as it is supremum.
Thus, forθ0=45°, the set Sθ0 of all possible value of α is the set 0.717.
We changeθ0 through 0,2π and obtain the value of sθ0. Then, we get supremum of all sθ0’s.
It can be easily verified that the supremum value is0.717 for the point 1.2,0.4, and it is obtained for θ0=45°. In conclusion, we obtain the hyperbola conic passing through to the same points for θ0=45° and α=0.717. As a result, we find that the conic which passing through to the same points are 1.205,0.205∈H¯1, 2.129,5.325∈H¯2, −2.784,8.69∈H¯3, −1.203,−2.516∈H¯4, and 2.705,−6.668∈H¯5.
Then, we obtain the conic equation (35) which passing through to the same points:(35)−72372.62x2−159.62xy+7917.21y2+25766.96x−4541.24y+74675.32=0.
This hyperbola equation contains the point1.2,0.4. And, we have(36)μ1.2,0.4|FH¯=0.717,alpha cut of fuzzy hyperbola.
Let also give an example of a fuzzy hyperbola whose core hyperbola is noncentral.Example 2.
LetH¯16,217,H¯22,2,H¯30,25,H¯4−5,−102, and H¯54,−25 be five fuzzy points. Let’s get the fuzzy hyperbola that passes through these points.
In this example, we take core of that the points are in different regions on the curve. The membership functions of these selected five fuzzy points are cones with circular and elliptical basessimilar to the display of the points in Figure 1;
(37)x,y:x−62+y−2172≤1circular,x,y:x−22+9y−22≤1elliptical,x,y:x2+y−252≤1circular,x,y:9x+52+y+1022≤1elliptical,x,y:x−42+y+252≤1circular.
The vertices of the membership functions areH¯16,217,H¯22,2,H¯30,25,H¯4−5,−102, and H¯54,−25, respectively.
Now, forα∈0,1, we may find the same points with membership degree α on H¯1,H¯2,H¯3,H¯4, and H¯5 as follows:(38)Q1θα=x1θα,y1θα=6+1−αcosθ,217+1−αsinθ,Q2θα=x2θα,y2θα=2+1−α⋅cosθ1+8sin2θ,25+1−α⋅sinθ1+8sin2θ,Q3θα=x3θα,y3θα=1−αcosθ,25+1−αsinθ,Q4θα=x4θα,y4θα=−5+1−α⋅cosθ1+8cos2θ,−25+1−α⋅sinθ1+8cos2θ,Q5θα=x5θα,y5θα=4+1−αcosθ,−25+1−αsinθ.
The equation of the core hyperbola through the points is(39)y24−x−22=1.
The hyperbolaHθα that passes through Q1θα,Q2θα,Q3θα,Q4θα, and Q5θα can be determinant by the following equation:(40)aθαx2+2hθαxy+bθαy2+2gθαx+2fθαy+cθα=0,with hθα2>aθα⋅bθα. In this equation, the determinants used to find aθα,hθα,bθα,gθα,fθα, and cθα will be taken as in Example 1, and the membership of a point that we will choose is calculated.
LetA,B,C,F,G,andK be the determinant values used to find the values of aθα,bθα,cθα,fθα,gθα,andkθα, respectively. So,(41)aθα=2hθαkθα⋅A,bθα=2hθαkθα⋅B,cθα=2hθαkθα⋅C,fθα=hθαkθα⋅F,gθα=hθαkθα⋅G,kθα=K.
The fuzzy hyperbolaFH¯ that passes through H¯i’s, i=1,2,3,4,5 is the union of all possible hyperbola Hθα’s that lies between Q1θα and Q5θα’s.
That is,(42)FH¯1,…,5=∨α∈0,1∪θ∈0,2πx,y=aθαx2+2hθαxy+bθαy2+2gθαx+2fθαy+cθα=0.
The infinite tailsFH¯1∞,FH¯3∞,FH¯4∞, and FH¯5∞ are determined by the following membership functions:(43)μx,y|FH¯1∞=α,y2−y1θα2=4x−22+x1θα2,θ=27.2°,0,elsewhere,μx,y|FH¯3∞=α,y2−y3θα2=4x−22+x3θα2,θ=29.2°,0,elsewhere,μx,y|FH¯4∞=α,y2−y4θα2=4x−22+x4θα2,θ=26.7°,0,elsewhere,μx,y|FH¯5∞=α,y2−y5θα2=4x−22+x5θα2,θ=29.2°,0,elsewhere.
The anglesθ=27.2°,29.2°,26.7°, and 29.2° are, respectively, the angles of the normals to the core hyperbola at 6,217,0,215,−5,−102, and 4,−25 with the positive x-axis.
2.05,2.05 is a point on the fuzzy hyperbola FH¯. First, we adjust the set of hyperbolas Hθα’s which the point 2.05,2.05 lies. We will calculate the membership degree of this point.
Let replace point2.05,2.05 in (40). We need to identify the possible values of α.
Then, we get the equation as follows:(44)aθα2.052+2hθα22.052.05+bθα2.052+2gθα2.05+2fθα2.05+cθα=0,4.2025aθα+8.405hθα+4.2025bθα+4.1gθα+4.1fθα+cθα=0.First, we admit that θ0=60°. We will calculate determinant values for this angle. These determinant values are calculated as in Example 1. We substitute θ0=60° in x1θα,y1θα,x2θα,y2θα,x3θα,y3θα,x4θα,y4θα, and x5θα,y5θα for determinants and calculate the determinant values approximately. We find them and replace in (44); then, we obtain the following nonlinear equation which determinants are found:(45)−3.184α4−17.469α3+14514.314α2−204227.678α+204987.984=0.
By solving this equation, real alpha values are found.(46)α=0.842,42.001.
But, we get0.842 from 0 to 1 from these real two values.
Thus, forθ0=60°, the set Sθ0 of all possible values of α is the set 0.842.
We changeθ0 through 0,2π and obtain the value of sθ0. Finally, we get supremum of all sθ0’s. It can be easily verified that the supremum value is 0,842 for the point 2.05,2.05, and it is obtained for θ0=60°. In conclusion, we obtain the hyperbola conic passing through to the same points, for θ0=60° and α=0.842.
Substitutingα and θ in Equation (38), we get the same points as 6.079,8.383∈H˜1,2.029,2.051∈H˜2,0.079,4.608∈H˜3,−4.954,−14.063∈H˜4 and 4.079,−4335∈H˜5.
Then, we obtain the following conic (47) which passes through to the same points(47)348199.353x2−1624.867xy−86002.529y2−1445151.495x+27947.976y+1809949.889=0.
The conic equation contains the point2.05,2.05. And, we have(48)μ2.05,2.05|FE¯=0.842.


## 4. Conclusion

The concept of fuzzy hyperbola has been examined, and the basic properties of fuzzy hyperbola have been explained in this study in detail. The membership degrees of fuzzy points have a specific role in the graph of hyperbola. The needed explanations based on these roles were made on the drawn graphics. Equations of conics such as hyperbola and ellipse can be obtained by determining five points. Starting from these, we developed a method for obtaining the fuzzy hyperbola in the study. But we cannot do this with five random points. We used the points which are called as same points. In the study, we have presented a method by determining the necessary properties for selecting points. As seen in the figures, the fuzzy hyperbolas can be depicted with different curves. The interpretations of this method and graphics presented in Section3 are described in detail. The method used is proven by theorem and prepositions. Fuzzy space has an important place in the coding and programing of technological devices. On the other hand, hyperbolic curves are curvilinear structures that are frequently encountered in the photosynthesis, CO2, and heat distributions of plants in biology. In addition, the distributions of radar waves appear especially in broadcasts based on spectral radar and even neural network. It is used when scanning documents. When these areas are combined with each other, there is a need to create fuzzy hyperbola as a result of the mentioned technological structures taking up in fuzzy space. Mathematically, we aim to define and analyze the fuzzy hyperbola, thinking that it will be useful in the future. In this study, we have presented the fuzzy hyperbola in detail in a way that can guide the studies to be done in the above-mentioned areas.



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