On general form of fuzzy lines and its application in fuzzy line fitting

Abstract

In this paper, fuzzy geometrical construction and characteristics of fuzzy lines are investigated. A general form of fuzzy lines is proposed. It is shown that a fuzzy line passing through a set of fuzzy points whose cores are collinear is unique. Slope and intercept of a fuzzy line, vertical and perpendicular distances from a fuzzy point to a fuzzy line are also studied. Sup-min composition of fuzzy sets and concepts of same and inverse points in fuzzy geometry are applied to define all the ideas. Proposed general form of fuzzy line is applied to fit a fuzzy line for a dataset of imprecise locations or fuzzy points. It is shown that the fitted fuzzy line has the minimum sum of square vertical distances between the given fuzzy points and the fitted fuzzy line. Proposed definitions and ideas are supported by several numerical and pictorial illustrations.

Keywords

Fuzzy set fuzzy number fuzzy point same and inverse points fuzzy line fuzzy distance fuzzy line fitting

1 Introduction

Classical geometrical ideas may be potentially very powerful to provide a mathematical framework in analyzing shapes or positions of observed objects [13]. However, classical geometry is not always enough to investigate all practical problems, since often observed objects are inherently imprecise [22]. When description of objects are vague, imprecise or inadequate, their precise mathematical representation may not be possible [31]. The need for an effective presentation of objects or shapes whose realizations are inherently imprecise has necessitated the formulation of fuzzy geometrical ideas. However, there have been a few works [3 , 46] on rigorous treatment of fuzzy geometry and topology of imprecise imagesubsets.

In the analysis of fuzzy geometrical shapes Buckley and Eslami [3] may be the first to analyze fuzzy geometry using sup-min composition of fuzzy sets. Very recently, Ghosh and Chakraborty [6, 15] defined few basic ideas of fuzzy plane geometry using newly introduced concepts of same and inverse points. It is shown in [15] that the ideas of same and inverse points enhanced the formulations of fuzzy geometry. In this paper, an attempt has been made to study general form of fuzzy line in detail. This paper mainly addresses the questions: what is a fuzzy line? How to construct a fuzzy line? And what is the mathematical form of a fuzzy line? As an application of the proposed fuzzy line, which essentially motivated us to study fuzzy line, we have shown how to fit a fuzzy line for a given dataset of imprecise locations. A brief literature of fuzzy line, which shows the need to study fuzzy line, is given below.

Ammar [1] proposed fuzzy line passing through two fuzzy points as a fuzzy set whose membership function is convex combination of membership values of those fuzzy points. In [40], a fuzzy cell complex structure is defined for modeling fuzzy lines. Fuzzy line in [40] is a particular case of that in [1]. In [24], fuzzy line, which is eventually a fuzzy line segment, has been defined as a collection of line segments with varied membership values. An idea of imprecise lines, which is eventually a set of lines, is given in [27]. Fuzzy line in [3], passing through two fuzzy points whose supports are circles of same radius, has support alike to epsilon-butterfly of [25]. In [8], Chaudhuri defined fuzzy line as a fuzzy set for which any α-level set is either empty or a straight line for all α in (0, 1]. Gupta and Ray [16] defined fuzzy line as collection of collinear crisp points with varied membership values. These definitions of fuzzy line may not be suitable in fuzzy plane geometry. Since, according to [16], α-cut of fuzzy line may be union of disjoint line segments and core of a fuzzy line may not be a crisp line. Moreover, fuzzy lines in [8, 16] can have empty core also. Thus, these methods of defining fuzzy lines may be violating the corresponding well-known definition of crisp lines. Some more geometrical properties of fuzzy line and its applications can also be found in [17 –30].

In the existing literature of fuzzy lines, it may be noted that though ideas on fuzzy line are proposed by various researchers, the study on construction of a fuzzy line has not been yet focused extensively. Only Buckley and Eslami studied some concepts to construct fuzzy lines [3].

Recently, Zadeh [46] has proposed that the counterpart of a crisp line, C, in Euclidean geometry, is a fuzzy line. Fuzzy line may be formed by fuzzy-transform of C, with C playing the role of the prototype of fuzzy line.

Jian et al. [21] have mentioned that in classical mathematics, a straight line can be seen as a locus of a point along a fixed direction. Similarly, a fuzzy line can be seen as a locus of a fuzzy point in fuzzy geometrical space.

Since a fuzzy line can be considered as locus of a fuzzy point along a particular direction, a fuzzy line may be visualized as an infinitely long hazy band consisting of a bunch of crisp lines with varied membership grades [15]. Its core must contain a crisp line and its membership function must smoothly decrease from core to the neighboring points. It is reported in [15] that spread of the support of a fuzzy line cannot suddenly get wider and wider imprecise range 1 (epsilon-butterfly shape [25]) as is observed in [3].

In the application field of fuzzy line, such as fuzzy regression line or fuzzy line fitting corresponding to a given dataset of imprecise locations, which is a topic of extensive application of fuzzy line, Kao and Chyu [22] have mentioned that existing ideas on fitting fuzzy regression line have a common characteristic of increasing spreads for fuzzy responses (dependent variable) as the independent variable increases its magnitude, but this is not suitable for general cases. In this paper, we propose an approach to fitting a fuzzy line. In the proposed fuzzy line fitting, we will see that fitted fuzzy line will not have increasing spread. Due to the successful applications of fuzzy regression line in various fuzzy systems (reported in [10]), substantially a large number of research work is focused to find fuzzy linear regression corresponding to crisp data or fuzzy data [19 –37].

Several conflict and deficiencies of the existing methodologies have been pointed out by Shapiro [38] and Wang and Tusar [42]. We feel that prior to find a fuzzy regression line, first we need to know what is a fuzzy line. Thus, this paper is intended to discuss what is a fuzzy line and how to construct a fuzzy line. Then, we attempted to fit a fuzzy line for a set of fuzzy points. This paper is organized as follows.

Section 2 provides some basic definitions and terminologies which are used throughout this paper. A detailed investigation on fuzzy lines in general form is given in the Section 3, which also includes definition of slope of a fuzzy line, intercepts of a fuzzy line, (vertical) distance between a fuzzy point and a fuzzy line and fuzzy half plane. Section 4 provides a procedure of fuzzy line fitting as an application of the proposed fuzzy line. In Section 5, future scope and conclusions of the proposed work are drawn.

2 Preliminaries

The basic definitions which are used here are adopted from [3] and [15]. Capital letters with over tilde bar, i.e., $\tilde{A}$ , $\tilde{B}$ , $\tilde{C}$ , … all represent fuzzy subsets of $ℝ^{n}$ . Membership function of a fuzzy set $\tilde{A}$ of $ℝ^{n}$ is represented by $μ (x | \tilde{A})$ , $x \in ℝ^{n}$ with $μ (ℝ^{n}) \subseteq [0, 1]$ .

Definition 2.1. (Fuzzy set) Let X be a classical set of objects, called universe, whose generic elements are denoted by x. Let X be evaluated with regard to a fuzzy statement. Then the set of ordered pairs $\tilde{A} = {(x, μ (x | \tilde{A})) : x \in X}$ , where μ : X → [0, 1], is called a fuzzy set in X. The function μ : X → [0, 1] evaluates membership value of x in the fuzzy set $\tilde{A}$ .

Definition 2.2. (α-cut of a fuzzy set [15]) For a fuzzy set $\tilde{A}$ of $ℝ^{n}$ , its α-cut is denoted by $\tilde{A} (α)$ and isdefined by: $\tilde{A} (α) = {\begin{matrix} {x : μ (x | \tilde{A}) \geq α} & if 0 < α \leq 1 \\ closure {x : μ (x | \tilde{A}) > 0} & if α = 0 . \end{matrix}$ The set ${x : μ (x | \tilde{A}) > 0}$ is called as support of the fuzzy set $\tilde{A}$ . The set $\tilde{A} (0)$ is often said as base of $\tilde{A}$ . The 1-cut (α = 1) of a fuzzy set is known as core. If the core is non-empty, the fuzzy set is called as a normal fuzzy set.

A fuzzy set is said to be convex if all of its α-cuts are convex. A normal and convex fuzzy set in $ℝ$ is known as a fuzzy number. Equivalently, we have the following definition of a fuzzy number.

Definition 2.3. (Fuzzy number [3]) A fuzzy set $\tilde{A}$ of $ℝ$ is called as a fuzzy number if its membership function μ has the following properties:

$μ (x | \tilde{A})$ is upper semi-continuous 2 ,

$μ (x | \tilde{A}) = 0$ outside some interval [a, d], and

there exist real numbers b and c so that a ≤ b ≤ c ≤ d and $μ (x | \tilde{A})$ is increasing on [a, b] and decreasing on [c, d], and $μ (x | \tilde{A}) = 1$ for each x in [b, c].

Due to upper semi-continuity of $μ (x | \tilde{A})$ , for a fuzzy number $\tilde{A}$ , the set ${x : μ (x | \tilde{A}) \geq t}$ is closed for all t in $ℝ$ . Therefore, the set $\tilde{A} (α)$ is a closed and bounded interval of $ℝ$ for all α in [0, 1]. A closed and bounded interval is a convex set in $ℝ$ . Thus, all the α-cuts of a fuzzy number are convex. Hence a fuzzy number is always a convex fuzzy set. Again due to the condition (ii), the fuzzy set $\tilde{A}$ is normal. Therefore, the Definition 2.3 can be taken as equivalent definition of a fuzzy number.

Definition 2.4. (LR-type fuzzy number [11]) A function L : [0, + ∞) → [0, 1] which is non-increasing and satisfies either of

L (0) =1 and L (1) =0, or

L (x) >0 for x in [0, + ∞) and L (+ ∞) =0

is called as a reference function of a fuzzy number.

A fuzzy number $\tilde{A}$ is called an LR-type fuzzy number if there exist two reference functions L and R, and two numbers α > 0 and β > 0 such that $μ (x | \tilde{A})$ can be written as: $μ (x | \tilde{A}) = {\begin{matrix} L (\frac{m - x}{α}) & if x \leq m \\ R (\frac{x - m}{β}) & if x \geq m . \end{matrix}$

The notation (m - α/m/m + β) _LR or (a/b/c) _LR is used to represent an LR-type fuzzy number. If L (x) = R (x) = max {0, 1 - |x|}, then an LR-type fuzzy number (a/b/c) _LR is called as a triangular fuzzy number. A triangular fuzzy number is denoted by (a/b/c). A new perception of fuzzy number along a line is given below.

Definition 2.5. (Fuzzy number along a line [15]) In defining fuzzy number, conventionally, real line $(ℝ)$ is taken as universal set. Instead of real line as the universal set, any line lx + my = n in $ℝ^{2}$ plane can be taken as follows. Let in $ℝ^{2}$ , x-axis represents the real line and $\tilde{p}$ be a fuzzy number. In x-axis, $\tilde{p}$ can be represented by $μ ((x, 0) | \tilde{p})$ = $μ (x | \tilde{p})$ ∀ $x \in ℝ$ . More explicitly: $μ ((x, y) | \tilde{p}) = {\begin{matrix} μ (x | \tilde{p}) & if y = 0 \\ 0 & elsewhere . \end{matrix}$ Let $T : ℝ^{2} \to ℝ^{2}$ be the transformation which includes rotation of axes with angle θ and translation of origin to $(\frac{\ln}{l^{2} + m^{2}}$ , $\frac{mn}{l^{2} + m^{2}})$ , which is the point of intersection of lx + my = n and mx - ly = 0, which is the line through origin and perpendicular to lx + my = n. This T can be expressed by T (x, y) = $(x cos θ - y sin θ + \frac{\ln}{l^{2} + m^{2}}, x sin θ + y cos θ + \frac{mn}{l^{2} + m^{2}})$ , which is a bijective transformation and transforms x-axis to lx + my = n. Now, the fuzzy number $\tilde{p}$ may be considered as a fuzzy number on the line lx + my = n defined in the following way: $μ ((u, v) | \tilde{p}) = {\begin{matrix} μ ((x, 0) | \tilde{p}) & if (u, v) = T (x, 0) \\ and lu + mv = n \\ 0 & elsewhere . \end{matrix}$

It is easy to observe that for each and every fuzzy number $\tilde{p}$ on the line lx + my = n there always exists a unique fuzzy number in the real line and vice versa.

Definition 2.6. (Fuzzy points [3]) A fuzzy point at (a, b) in $ℝ^{2}$ , written as $\tilde{P} (a, b)$ , is defined by its membership function:

$μ ((x, y) | \tilde{P} (a, b))$ is upper semi-continuous,

$μ ((x, y) | \tilde{P} (a, b)) = 1$ if and only if (x, y) = (a, b), and

$\tilde{P} (a, b) (α)$ is a compact, convex subset of $ℝ^{2}$ , for all α in [0, 1].

The notations ${\tilde{P}}_{1} (a, b)$ , ${\tilde{P}}_{2} (a, b)$ , ${\tilde{P}}_{3} (a, b)$ , … or ${\tilde{P}}_{1}$ , ${\tilde{P}}_{2}$ , ${\tilde{P}}_{3}$ , … are used to represent fuzzy points.

Definition 2.7. (Same and inverse points with respect to fuzzy points [15]) Let (x ₁, y ₁) and (x ₂, y ₂) be two points on support of two continuous fuzzy points $\tilde{P} (a, b)$ and $\tilde{P} (c, d)$ respectively. Let L ₁ be a line joining (x ₁, y ₁) and (a, b). As $\tilde{P} (a, b)$ is a fuzzy point, along L ₁ there exists a fuzzy number, ${\tilde{r}}_{1}$ say, on the support of $\tilde{P} (a, b)$ . Membership function of this fuzzy number ${\tilde{r}}_{1}$ can be written as: $μ ((x, y) | {\tilde{r}}_{1})$ $= μ ((x, y) | \tilde{P} (a, b))$ for (x, y) in L ₁, and 0 otherwise.

Similarly, along a line, L ₂ say, joining (x ₂, y ₂) and (c, d), there exists a fuzzy number, ${\tilde{r}}_{2}$ say, on the support of $\tilde{P} (c, d)$ . Now (x ₁, y ₁) , (x ₂, y ₂) are said to be same points with respect to $\tilde{P} (a, b)$ and $\tilde{P} (c, d)$ if:

(x ₁, y ₁) and (x ₂, y ₂) are same points with respect to ${\tilde{r}}_{1}$ and ${\tilde{r}}_{2}$ , and

L ₁, L ₂ have equal angle with the line joining (a, b) and (c, d).

The points (x ₁, y ₁) , (x ₂, y ₂) are said to be inverse points if (x ₁, y ₁) , (- x ₁, - y ₁) are same point with respect to

\tilde{P} (a, b)

and

- \tilde{P} (c, d)

Definition 2.8. (Fuzzy distance [15]) Fuzzy distance $\tilde{D}$ between two fuzzy points ${\tilde{P}}_{1}$ and ${\tilde{P}}_{2}$ is defined by its membership function as: $μ (d | \tilde{D}) =$ sup {α : d = d (u, v), $u \in {\tilde{P}}_{1} (0)$ and $v \in {\tilde{P}}_{2} (0)$ are inverse points with membership value α}, d (,) is the Euclidean distance metric.

Definition 2.9. (Fuzzy line segment [15]) Fuzzy line segment ${\tilde{\bar{L}}}_{P_{1} P_{2}}$ joining two fuzzy points ${\tilde{P}}_{1}$ , ${\tilde{P}}_{2}$ is defined by its membership function as: $μ ((x, y) | {\tilde{\bar{L}}}_{P_{1} P_{2}})$ = sup {α : (x, y) lies on the line joining same points $(x_{1}, y_{1}) \in {\tilde{P}}_{1} (0), (x_{2}, y_{2}) \in {\tilde{P}}_{2} (0)$ and $μ ((x_{1}, y_{1}) | {\tilde{P}}_{1})$ = $μ ((x_{2}, y_{2}) | {\tilde{P}}_{2}) = α}$ .

Now let us study fuzzy lines and their characteristics.

3 Fuzzy lines

Definition 3.1. (Fuzzy line passing through n (≥2) collinear fuzzy points [6]) Suppose ${\tilde{P}}_{1} (a_{1}, b_{1})$ , ${\tilde{P}}_{2} (a_{2}, b_{2})$ , …, ${\tilde{P}}_{n} (a_{n}, b_{n})$ are n collinear fuzzy points, i.e., (a ₁, b ₁), (a ₂, b ₂), …, (a _n, b _n) lie on the same line l say. Suppose the points (a _i, b _i) are ordered on the line l from left to right for i = 1, 2, …, n. To construct a fuzzy line, $\tilde{L}$ say, passing through all of these n fuzzy points, it may be considered that: at infinite distance apart, there exist two fuzzy points having core on l, one is on the left of ${\tilde{P}}_{1} (a_{1}, b_{1})$ and another is on the right of ${\tilde{P}}_{n} (a_{n}, b_{n})$ . Let these two points be ${\tilde{P}}_{1 \infty}$ , ${\tilde{P}}_{n \infty}$ respectively; ${\tilde{\bar{L}}}_{1 \infty}$ and ${\tilde{\bar{L}}}_{n \infty}$ be the line segments joining ${\tilde{P}}_{1 \infty}$ , ${\tilde{P}}_{1}$ and ${\tilde{P}}_{n}$ , ${\tilde{P}}_{n \infty}$ respectively. The line $\tilde{L}$ may be defined as: $\tilde{L} = {\tilde{\bar{L}}}_{1 \infty} ⋃ {\tilde{\bar{L}}}_{P_{1} P_{2}} ⋃ \dots ⋃ {\tilde{\bar{L}}}_{P_{n - 1} P_{n}} ⋃ {\tilde{\bar{L}}}_{n \infty} .$

Since support of a fuzzy point must be a compact set, i.e., closed and bounded set in $ℝ^{2}$ , all lines joining same points of ${\tilde{P}}_{1}$ and ${\tilde{P}}_{1 \infty}$ must be parallel to the line l. Similarly all the lines joining same points of ${\tilde{P}}_{n}$ and ${\tilde{P}}_{n \infty}$ always be parallel to the line l. Because, otherwise, the support sets ${\tilde{P}}_{1 \infty} (0)$ and ${\tilde{P}}_{n \infty} (0)$ will be unbounded and hence they cannot represent supports of fuzzy points. Therefore, slope of the fuzzy half lines (or infinite tails) ${\tilde{\bar{L}}}_{1 \infty}$ and ${\tilde{\bar{L}}}_{n \infty}$ must be equal to slope of the line l. Thus the fuzzy line may have figure like Fig. 1, where a fuzzy line passing through five collinear fuzzy points ${\tilde{P}}_{1}$ , ${\tilde{P}}_{2}$ , …, ${\tilde{P}}_{5}$ is depicted.

In the above definition of fuzzy line, fuzzy line is perceived as union of crisp lines or line segments with different membership values. However there are other ways too. For instance, a fuzzy line may be imagined as a collection of crisp points of different membership values or as group of fuzzy points [15]. Basically whatever be the way being followed to define a fuzzy line, it must be visualized as a straight infinitely long hazy band having one (deep) crisp straight line on its core with a smooth transition of membership values between the neighboring points of the core line [15]. To express this visualization mathematically, for each fuzzy line there must exists three functions f (x, y), g (x, y) and h (x, y) where f (x, y) =0 and h (x, y) =0 represent boundary curves of the support of the fuzzy line and g (x, y) =0 represents the straight line on the core. Obviously, g is a linear function and not necessarily f and h are linear. This concept leads to define general form of fuzzy lines and also give an equation to fuzzy lines.

Definition 3.2. (General form ( ${\tilde{L}}_{G}$ )) In general form of fuzzy line, it is described as collection of crisp points with different membership values. Those points must lie in between two (boundary) curves and there must be one crisp line between those curves. Let those boundary curves be f (x, y) =0 and h (x, y) =0 and the crisp line be g (x, y) ≡ y - mx - c = 0. Then the fuzzy line ${\tilde{L}}_{G}$ can be expressed by an equational form (f (x, y)/g (x, y)/h (x, y)) _LR = 0.

Here the notation (f/g/h) _LR = 0 means that membership grade of the points on ${\tilde{L}}_{G} (0)$ gradually increases (through the function L) from 0 to 1 as we move from f to g and gradually decreases (through the function R) from 1 to 0 as we move from g to h.

Let f and h be functions having the properties that at each point (x, y) ∈ ${\tilde{L}}_{G} (0)$ for which f (x, y) = 0 = h (x, y)

$\frac{\partial f}{\partial x}$ , $\frac{\partial f}{\partial y}$ , $\frac{\partial h}{\partial x}$ and $\frac{\partial h}{\partial y}$ exist and are continuous, and

$\frac{\partial f}{\partial y}$ and $\frac{\partial h}{\partial y}$ are non-vanishing.

Then from the well-known Implicit Function Theorem of calculus, there exist two functions p (x) and q (x) such that f (x, y) =0 ⇔y = p (x) and h (x, y) =0 ⇔y = q (x). Let us consider a straight line perpendicular to

{\tilde{L}}_{G} (1) \equiv y = mx + c

. As explained in [15], along x + my = k, which is perpendicular to

{\tilde{L}}_{G} (1)

, there must exist one fuzzy number on the support of

{\tilde{L}}_{G}

. In Fig. 1, the line x + my = k intersects y = p (x), y = mx + c and y = q (x) at F, G and H respectively. It is easy to note that the above said fuzzy number on

{\tilde{L}}_{G} (0)

and along x + my = k can be expressed as (F/G/H) _LR.

Apparently, $G = (\frac{k - mc}{1 + m^{2}}, \frac{c + mk}{1 + m^{2}})$ . Let F = (u ₁, v ₁) and H = (u ₂, v ₂). The points (u ₁, v ₁) and (u ₂, v ₂) can be obtained by solving ${\begin{matrix} u_{1} + m v_{1} = k \\ v_{1} = p (u_{1}) \end{matrix}$ and ${\begin{matrix} u_{2} + m v_{2} = k \\ v_{2} = q (u_{2}) \end{matrix}$ respectively.

Now membership value of any point $(u, v) \in {\tilde{L}}_{G} (0) ⋂ {(x, y) : x + my = k}$ can be expressed by: $μ ((u, v; k) | {\tilde{L}}_{G}) = {\begin{matrix} L (\frac{d ((u, v), G)}{d (F, G)}) & if u_{1} \leq u \leq \frac{k - mc}{1 + m^{2}}, \\ \frac{c + mk}{1 + m^{2}} \leq v \leq v_{1}, \\ u + mv = k \\ R (\frac{d (G, (u, v))}{d (G, H)}) & if \frac{k - mc}{1 + m^{2}} \leq u \leq u_{2}, \\ v_{2} \leq v \leq \frac{c + mk}{1 + m^{2}}, \\ u + mv = k \\ 0 & elsewhere . \end{matrix}$ Varying k $\in ℝ$ , membership value of any point in ${\tilde{L}}_{G}$ can be obtained.

Example 1. Let us consider the fuzzy line, passing through ${\tilde{P}}_{1} (1, 2)$ and ${\tilde{P}}_{2} (5, 7)$ where membershipfunction of them are right circular cone with ${\tilde{P}}_{1} (1, 2) (0) = {(x, y) : (x - 1)^{2} + (y - 2)^{2} \leq (\frac{1}{2})^{2}}$ and ${\tilde{P}}_{1} (5, 7) (0) = {(x, y) : (x - 5)^{2} + (y - 7)^{2} \leq (\frac{1}{2})^{2}}$ . This fuzzy line is depicted in Fig. 2.

In the Fig. 2, $\begin{matrix} A = (1, 2), \\ B = (5, 7), \\ AB : 5 x - 4 y + 3 = 0, \\ IJ : 5 x - 4 y + 6.202 = 0, and \\ CD : 5 x - 4 y - 0.202 = 0 . \end{matrix}$

The lines IJ and CD respectively determine the boundary functions f (x, y) and h (x, y) of support of the line. So, $\begin{matrix} f (x, y) \equiv 5 x - 4 y + 6.202 = 0, \\ g (x, y) \equiv 5 x - 4 y + 3 = 0, and \\ h (x, y) \equiv 5 x - 4 y - 0.202 = 0 . \end{matrix}$

Here ${\tilde{L}}_{G} (0)$ = {(x, y) : f (x, y) ≥0, h (x, y) ≤0} and at each point (x, y)∈ ${\tilde{L}}_{G} (0)$ for which 5x - 4y + 6.202 = 0 or 5x - 4y + 0.202 = 0

all of $\frac{\partial f}{\partial x}$ , $\frac{\partial f}{\partial y}$ , $\frac{\partial h}{\partial x}$ and $\frac{\partial h}{\partial y}$ are constant and hence continuous, and

$\frac{∂ f}{\partial y}$ and $\frac{\partial h}{\partial y}$ are non-vanishing.

So, ∃ p (x) and q (x) such that, f (x, y) ≡ y - p (x) =0 and h (x, y) ≡ y - q (x) =0. It is easy to observed that, $p (x) = \frac{1}{4} (5 x + 6.202)$ and $q (x) = \frac{1}{4} (5 x - 2.202)$ .

The equation of ${\tilde{L}}_{G}$ is (5x - 4y + 6.202 / 5x - 4y + 3 / 5x - 4y - 0.202) _LR = 0 with L (x) = R (x) = max {0, 1 - |x|}. Its membership function can be constructed as follows.

The line RS ≡ 4x + 5y - k = 0 is a generic line perpendicular to PQ or AB. Along the line RS, the fuzzy number (H/G/F) _LR exists in ${\tilde{L}}_{G}$ where H, F and G are points of intersection of RS with IJ, CD and PQ respectively. This fuzzy number is same with the following two fuzzy numbers according to a rigid translation 3 ,

the fuzzy number lie on ${\tilde{P}}_{1} (1, 2) (0)$ along MN,

the fuzzy number lie on ${\tilde{P}}_{2} (5, 7) (0)$ along UV.

The points F, G and H are (-0.756 + 0.098k, 0.605 + 0.122k), (-0.366 + 0.098k, 0.293 + 0.122k) and (0.025 + 0.098k, - 0.020 + 0.122k), respectively. Therefore, varying $k \in ℝ$ , the membership function of ${\tilde{L}}_{G}$ can be obtained as: $μ ((u, v; k) | {\tilde{l}}_{G}) = {\begin{matrix} 1 - 2 d (G, (u, v)) & if (u, v) \\ \in {\tilde{l}}_{G} (0) \cap \bar{FH} \\ 0 & otherwise . \end{matrix}$

Theorem 3.1. Fuzzy line (by the Definition 3.2) passing through fuzzy points ${\tilde{P}}_{1}$ , ${\tilde{P}}_{2}$ , …, ${\tilde{P}}_{n}$ is unique.

Proof. Let us suppose there exist two fuzzy lines ${\tilde{L}}_{G}$ and ${\tilde{L}}_{G}^{'}$ passing through ${\tilde{P}}_{1}$ , ${\tilde{P}}_{2}$ , …, ${\tilde{P}}_{n}$ . As ${\tilde{L}}_{G}$ and ${\tilde{L}}_{G}^{'}$ are distinct, there must exist one point $(x_{0}, y_{0}) \in {\tilde{L}}_{G}$ such that $(x_{0}, y_{0}) \notin {\tilde{L}}_{G}^{'}$ or $μ ((x_{0}, y_{0}) | {\tilde{L}}_{G}) \neq μ ((x_{0}, y_{0}) | {\tilde{L}}_{G}^{'})$ .

If $(x_{0}, y_{0}) \notin {\tilde{L}}_{G}^{'}$ , then it implies that there do not exist same points in two consecutive fuzzy points amongst ${\tilde{P}}_{1}$ , ${\tilde{P}}_{2}$ , …, ${\tilde{P}}_{n}$ such that (x ₀, y ₀) lies on the line segment joining those two same points. This is impossible, since $(x_{0}, y_{0}) \in {\tilde{L}}_{G}$ implies that there must exist two same points $(x_{1}, y_{1}) \in {\tilde{P}}_{i}$ and $(x_{2}, y_{2}) \in {\tilde{P}}_{i + 1}$ , for some i, such that (x ₀, y ₀) lies on the line segment joining (x ₁, y ₁) and (x ₂, y ₂). Therefore, the possibility that $(x_{0}, y_{0}) \notin {\tilde{L}}_{G}^{'}$ cannot occur.

Again $μ ((x_{0}, y_{0}) | {\tilde{L}}_{G})$ ≠ $μ ((x_{0}, y_{0}) | {\tilde{L}}_{G}^{'})$ also impossible, since membership value of (x ₀, y ₀) on both the fuzzy lines ${\tilde{L}}_{G}$ and ${\tilde{L}}_{G}^{'}$ are evaluated by the same Definition 3.2.

Hence, once a point (x ₀, y ₀) belongs to ${\tilde{L}}_{G}$ , then (x ₀, y ₀) must belong to ${\tilde{L}}_{G}^{'}$ with $μ ((x_{0}, y_{0}) | {\tilde{L}}_{G})$ ≠ $μ ((x_{0}, y_{0}) | {\tilde{L}}_{G}^{'})$ . Changing the role of ${\tilde{L}}_{G}$ and ${\tilde{L}}_{G}^{'}$ , we will get that ${\tilde{L}}_{G} = {\tilde{L}}_{G}^{'}$ . Hence the theorem follows. □

Theorem 3.2. Let $\tilde{L}$ be a fuzzy line and l be a line perpendicular to $\tilde{L} (1)$ . Along l, there always exists a fuzzy number on $\tilde{L}$ .

Proof. Let $\tilde{L}$ = ${(x, μ (x | \tilde{L})) : x \in ℝ^{2}}$ be the fuzzy line. The line l can also be presented by {(x, μ (x|l)) : $x \in ℝ^{2}}$ where μ (x|l) =1 for x ∈ l and zero otherwise. Obviously, the set {(x, μ(x| $\tilde{L}))$ $: x \in ℝ^{2}}$ ⋂ ${(x, μ (x | l)) : x \in ℝ^{2}}$ is a fuzzy set, whose μ is evaluated by the t-norm ‘min’. Let us denote this fuzzy set as $\tilde{A}$ . We have to prove $\tilde{A}$ is a fuzzy number along l.

Let 0 < α ≤ 1. The set $\tilde{A} (α)$ is a line segment or union of line segments which are subsets of l. We will prove that it is exactly a line segment and cannot be union of line segments. Let A _α and B _α be the points of intersection of l with f (x, y ; α) =0 and h (x, y ; α) =0 respectively, where f (x, y ; α) =0 and h (x, y ; α) =0 are boundary of the α-cut of $\tilde{L}$ . We will show $\tilde{A} (α)$ = $\bar{A_{α} B_{α}}$ . Let ∃ K ∈ $\bar{A_{α} B_{α}}$ but K ∉ $\tilde{A} (α)$ . So $μ (K | \tilde{L})$ <α. Therefore, $K \notin \tilde{L} (α)$ . But this is not possible, because all α-level sets of $\tilde{L}$ are closed convex sets and for 0 < α < β ≤ 1, $\tilde{L} (β)$ ⊆ $\tilde{L} (α)$ . Therefore $\tilde{A} (α) = \bar{A_{α} B_{α}}$ which is a compact convex set and for 0 < α < β ≤ 1 we get $\tilde{A} (β)$ ⊆ $\tilde{A} (α)$ . Thus, membership function of $\tilde{A}$ is upper semi-continuous and $\tilde{A} (0)$ is a compactset.

If G be the point of intersection of l and $\tilde{L} (1)$ , then $μ (G | \tilde{A})$ = min ${μ (G | l), μ (G | \tilde{L})} = 1$ . Hence the fuzzy set $\tilde{A}$ is a fuzzy number along the line l. □

Theorem 3.2 helps to think the fuzzy line $\tilde{L}$ as a collection of fuzzy numbers at each points of $\tilde{L} (1)$ or collection of fuzzy points at each point of $\tilde{L} (1)$ .

From the Theorem 3.2 it is obtained that along a line l, which is perpendicular to $\tilde{L}$ , a fuzzy number exists in $\tilde{L} (0)$ . However, spread of all of those fuzzy numbers may not be equal in general. If spreads on both the sides of core are equal, then the fuzzy lines may be called as symmetric and otherwise non-symmetric.

Definition 3.3. (Symmetric and non-symmetric fuzzy lines) If all the fuzzy numbers, situated on a fuzzy line $\tilde{L}$ and along the perpendicular lines of $\tilde{L} (1)$ , are having same spread, then $\tilde{L}$ is said to be a symmetric fuzzy line and otherwise non-symmetric.

Note 1. Symmetric fuzzy line may be visualized by pulling a fuzzy number (point) Symmetric fuzzy line may be visualized as a figure obtained by pulling a fuzzy number (point) continuously along the line $\tilde{L} (1)$ with the restriction that core of the fuzzy number (point) always retained on $\tilde{L} (1)$ . Thus, evaluating membership value of the points on a symmetric fuzzy line is quite easier since once a fuzzy number along a perpendicular line on the support of the fuzzy line becomes known to us, the entire fuzzy line can be captured.

We note that if a fuzzy line $\tilde{L}$ is perceived as group of fuzzy points, then boundary of the supports of the fuzzy points $\tilde{P}$ which belong to $\tilde{L}$ touch the boundary of $\tilde{L} (0)$ . For example, if a fuzzy line ${\tilde{L}}_{P_{1} P_{2}}$ passing through ${\tilde{P}}_{1}$ and ${\tilde{P}}_{2}$ is considered, the fuzzy points lie on ${\tilde{\bar{L}}}_{P_{1} P_{2}}$ are $\tilde{P} = λ {\tilde{P}}_{1} + (1 - λ) {\tilde{P}}_{2}$ (0 ≤ λ ≤ 1) whose supports touch the boundary curves of $\tilde{L} (0)$ . However, there exist several fuzzy points whose core lie on $\tilde{L} (1)$ and support of them are subset of $\tilde{L} (0)$ .

Up to this point, we observe that fuzzy lines are perceived as one of the following ways:

collection of crisp points,

collection of fuzzy points, or

collection of crisp line segments or half-lines or lines.

However any of the three considerations depends on the other two, since fuzzy points or crisp lines are collection of crisp points and intersection of two curves is a point or set of points. A unique characterization of fuzzy lines or the identification of a fuzzy set to be called as a fuzzy line is given in the following theorem.

Theorem 3.3. (Characterization theorem) A fuzzy set is a fuzzy line if and only if its core is a crisp straight line and along any line perpendicular to the core line there exist a fuzzy number on the support of the fuzzy line.

Proof. Forward part of the theorem is true from the Theorem 3.2. As we observe that, under the assumption of the theorem, taking union of all of the fuzzy numbers along the perpendicular lines to core line we can construct a fuzzy line in general form, so the converse part is true. Hence the result follows. □

Let us now define slope and intercepts of a fuzzyline.

Definition 3.4. (Slope of a fuzzy line) Let $\tilde{L}$ be a fuzzy line and $\bar{l}$ be a line segment in $\tilde{L} (0)$ . Let us consider that membership value of $\bar{l}$ on $\tilde{L}$ as $inf {μ (x | \tilde{L}) : x \in \bar{l}}$ . Slope of the fuzzy line may defined by its membership function as: $μ (m | \tilde{m}) = sup {μ (\bar{l} | \tilde{L}) : slope of \bar{l} is m} .$

Example 2. (Slope of a fuzzy line) Let $\tilde{L}$ be the fuzzy line passing through the fuzzy points ${\tilde{P}}_{1} (2, 3)$ and ${\tilde{P}}_{2} (7, 9)$ . Supports of ${\tilde{P}}_{1} (2, 3)$ and ${\tilde{P}}_{2} (7, 9)$ are ${(x, y) : (x - 2)^{2} + (y - 3)^{2} \leq \frac{1}{4}}$ and {(x, y) : (x - 7) ² + (y - 9) ² ≤ 1} respectively. Membership functions of ${\tilde{P}}_{1} (2, 3)$ and ${\tilde{P}}_{2} (7, 9)$ have shapes right circular cones with vertices at (2, 3) and (7, 9) respectively.

Fuzzy line $\tilde{L}$ is determined by ${\tilde{\bar{L}}}_{1 \infty} ⋃ {\tilde{\bar{L}}}_{12} ⋃ {\tilde{\bar{L}}}_{2 \infty}$ .

Any half line in the infinite fuzzy half lines ${\tilde{\bar{L}}}_{1 \infty}$ and ${\tilde{\bar{L}}}_{2 \infty}$ have slope 1.2.

Same points of ${\tilde{P}}_{1}$ and ${\tilde{P}}_{2}$ with membership value α ∈ [0, 1] are $S_{θ α}^{1} = (2 + \frac{1 - α}{2} cos θ, 3 + \frac{1 - α}{2} sin θ)$ and $S_{θ α}^{2} = (7 + (1 - α) cos θ, 9 + (1 - α) sin θ)$ respecti-vely, for each θ ∈ [0, 2π].

Slope of the line segment $\bar{S_{θ α}^{1} S_{θ α}^{2}}$ is $\frac{6 + \frac{1 - α}{2} sin θ}{5 + \frac{1 - α}{2} cos θ}$ .

If $\tilde{m}$ be the slope of the fuzzy line $\tilde{L}$ , then according to the Definition 3.4, membership value of $sup_{θ \in [0, 2 π]} \frac{6 + \frac{1 - α}{2} sin θ}{5 + \frac{1 - α}{2} cos θ}$ on $\tilde{m}$ is α.

Therefore, core of $\tilde{m}$ is 1.2 and for each α ∈ [0, 1], $\tilde{m} (α) = [\frac{6 - \frac{1 - α}{2} sin b (α)}{5 + \frac{1 - α}{2} cos b (α)}, \frac{6 + \frac{1 - α}{2} sin b (α)}{5 + \frac{1 - α}{2} cos b (α)}]$ where $b (α) = 2 {tan}^{- 1} \frac{\sqrt{244 - (1 - α)^{2}} + 12}{(1 - α) - 10}$ .

In the next y-intercept of a fuzzy line is defined and similarly x-intercept can be defined.

Definition 3.5. (y-intercept of a fuzzy line) y-intercept of a fuzzy line $\tilde{L}$ , $\tilde{c}$ say, may be defined by its membership function as: $μ (c | \tilde{c}) = {\begin{matrix} μ ((c, 0) | \tilde{L}) & if (c, 0) \in \tilde{L} (0) ⋂ (y - axis) \\ 0 & otherwise, \end{matrix}$ i.e., $\tilde{c} = \tilde{L} ⋂$ (y-axis).

Example 3. (y-intercept of a fuzzy line) Let us evaluate y-intercept of the fuzzy line $\tilde{L}$ passing through ${\tilde{P}}_{1} (2, 3)$ and ${\tilde{P}}_{2} (7, 9)$ in the Example 2. According to the Definition 3.5, y-intercept of $\tilde{L}$ is the triangular fuzzy number $(\frac{3}{5} - \frac{\sqrt{61}}{10} / \frac{3}{5} / \frac{3}{5} + \frac{\sqrt{61}}{10})$ .

Let us now move to study fuzzy distance between a fuzzy point an a fuzzy line.

Definition 3.6. (Distance and vertical distance between a fuzzy point and a fuzzy line) Let $\tilde{P}$ be a fuzzy point and $\tilde{L}$ be a fuzzy line. Distance between $\tilde{P}$ and $\tilde{L}$ may be defined as: $\begin{matrix} ⋃_{α \in [0, 1]} ({(d_{l}, α) : d_{l} = inf A_{α}} ⋃ {(d_{u}, α) : \\ d_{u} = sup A_{α}}), \end{matrix}$ where A _α = ${d ((x_{1}, y_{1}), (x_{2}, y_{2})) : (x_{1}, y_{1}) \in \tilde{L} (α)$ and $(x_{2}, y_{2}) \in \tilde{P} (α)}$ . The tuples (d _l, α) or (d _u, α) mean that ‘the distance d _l or d _u is adjoined with membership value α’.

Here the distance metric d ((x ₁, y ₁) , (x ₂, y ₂)) is the usual Euclidean distance metric. Instead of the Euclidean metric if we consider the function d′ ((x ₁, y ₁), (x ₂, y ₂)) = |y ₁ - y ₂| to measure the distance between (x ₁, y ₁) and (x ₂, y ₂), then the turns out fuzzy set $\begin{matrix} \tilde{D} & = & ⋃_{α \in [0, 1]} ({(d_{l}, α) : d_{l} = inf A_{α}} ⋃ {(d_{u}, α) : \\ d_{u} = sup A_{α}}) \end{matrix}$ may be called as vertical distance between the fuzzy point $\tilde{P}$ and the fuzzy line $\tilde{L}$ .

Theorem 3.4. $\tilde{D}$ , the vertical distance between a fuzzy point and a fuzzy line is a fuzzy number.

Proof. Similar to Theorem 4.1 in [15]. □

Example 4. Let $\tilde{L}$ be the line (y - 1/y - 2/y - 3) =0. Let $\tilde{P} (1, 6)$ be a fuzzy point whose μ is a right elliptical cone with base ${(x, y) : \frac{(x - 1)^{2}}{(1 / 2)^{2}} + \frac{(y - 6)^{2}}{(1 / 3)^{2}} \leq 1}$ , i.e., $μ ((x, y) | \tilde{P} (1, 6)) = {\begin{matrix} 1 - \sqrt{\frac{(x - 1)^{2}}{(1 / 2)^{2}} + \frac{(y - 6)^{2}}{(1 / 3)^{2}}} \\ if \frac{(x - 1)^{2}}{(1 / 2)^{2}} + \frac{(y - 6)^{2}}{(1 / 3)^{2}} \leq 1, \\ 0 elsewhere . \end{matrix}$ Therefore, $\tilde{L} (α)$ = {(x, y) :1 + α ≤ y ≤ 3 - α} and $\tilde{P} (1, 6) (α)$ = ${(x, y) : \frac{(x - 1)^{2}}{(1 / 2)^{2}}$ $+ \frac{(y - 6)^{2}}{(1 / 3)^{2}} \leq (1 - α)^{2}}$ .

So, A _α = $[\frac{8 + 4 α}{3}, \frac{16 - 4 α}{3}]$ and hence $μ (\frac{8 + 4 α}{3} | \tilde{D})$ = $μ (\frac{16 - 4 α}{3} | \tilde{D})$ = α. Precisely, vertical distance $\tilde{D}$ between $\tilde{P}$ and $\tilde{L}$ is $μ (x | \tilde{D}) = {\begin{matrix} \frac{3 x - 8}{4} & if \frac{8}{3} \leq x \leq 4, \\ \frac{16 - 3 x}{4} & if 4 \leq x \leq \frac{16}{3}, \\ 0 & otherwise . \end{matrix}$

Note 2. We note that the general form of a fuzzy line is intended to describe a general characterization of a fuzzy line. In [6], the authors have defined four different forms of fuzzy lines: a two-point form, a slope-intercept form, a point-slope form and an intercept form. Each of those four forms can define a fuzzy line when at least two of the attributes: two points, a point and slope, slope and intercept, or two intercepts are priory known. The main advantage of the formulation of the general form of fuzzy line is that it does not require any prior information about the fuzzy line. The general form essentially gives a general mathematical form to a fuzzy line. Again it can be easily noted that each of the four different forms of the fuzzy lines in [6] satisfies the Characterization Theorem 3.3 of a general fuzzy line. Thus all those four forms of fuzzy lines are particular cases of the general form.

In the next section, as an application of the proposed general form of fuzzy line, we will show how to fit a fuzzy line corresponding to a given data of imprecise locations.

4 An application: Fuzzy line fitting

Let us suppose a set of fuzzy points ${\tilde{P}}_{1}$ , ${\tilde{P}}_{2}$ , …, ${\tilde{P}}_{n}$ is given to us. We want to fit a fuzzy line for the given set of fuzzy points such that sum of vertical distances between the fuzzy points and the fitted fuzzy line is minimum.

In classical line fitting, we shift the given set of crisp points vertically such that the points become linear. There will have several possible lines which are obtained by shifting points to a linear fit. Out of all of these linear fits, which has smallest possible value for the sum of the squares of the vertical distances of the points is considered as best fitted line to the given set of crisp points. We will follow the same procedure to obtain fuzzy line fitting. At first, let us define what does vertical shifting of a fuzzy point mean.

Definition 4.1. (Vertical shifting of a fuzzy point) Let $\tilde{P} (a, b)$ be a fuzzy point and d be any real number. If the fuzzy point $\tilde{P} (a, b)$ is shifted d distance vertically, then the shifted fuzzy point $\tilde{P} (a, b + d)$ will be obtained by $μ ((x, y) | \tilde{P} (a, b + d))$ = $μ ((x, y - d) | \tilde{P} (a, b))$ .

Here d > 0 corresponds to upper shift and d ≤ 0 corresponds to lower shift. Instead of saying that $\tilde{P} (a, b)$ is shifted up to d distance vertically, we may also call $\tilde{P} (a, b)$ is rigidly shifted up to the point $\tilde{P} (a, b + d)$ . The word ‘rigid’ explains that only the points on the support $\tilde{P} (a, b) (0)$ are being shifted vertically without changing the membership values.

Example 5. Let $\tilde{P} (- 1, 0)$ be a fuzzy point whose membership function is given by $μ ((x, y) | {\tilde{P}}_{1} (- 1, 0)) = {\begin{matrix} 1 - 4 \sqrt{(x + 1)^{2} + y^{2}} \\ if (x + 1)^{2} + y^{2} \leq \frac{1}{4^{2}}, \\ 0 elsewhere, \end{matrix}$ and we want to find shifted fuzzy point of this fuzzy point up to distance d = 0.6. Shifted fuzzy point will be given by $\begin{matrix} μ ((x, y) | {\tilde{P}}_{1} (- 1, 0.6)) \\ = {\begin{matrix} 1 - 4 \sqrt{(x + 1)^{2} + (y - 0.6)^{2}} \\ if (x + 1)^{2} + (y - 0.6)^{2} \leq \frac{1}{4^{2}} \\ 0 elsewhere . \end{matrix} \end{matrix}$

Let (a ₁, b ₁), (a ₂, b ₂), …, (a _n, b _n) be the core of the given fuzzy points ${\tilde{P}}_{1}$ , ${\tilde{P}}_{2}$ , …, ${\tilde{P}}_{n}$ , respectively. Let $\tilde{L}$ be the best fitted fuzzy line corresponding to ${\tilde{P}}_{1}$ , ${\tilde{P}}_{2}$ , …, ${\tilde{P}}_{n}$ . For the core points, let y = mx + c be the best fitted line with least square vertical distances. Obviously, core line of the fuzzy line $\tilde{L}$ is the crisp line y = mx + c.

Now let us shift the fuzzy point ${\tilde{P}}_{i} (a_{i}, b_{i})$ up to the fuzzy point ${\tilde{P}}_{i}^{'} (a_{i}, {ma}_{i} + c)$ rigidly, for each i in {1, 2, …, n}.

The fuzzy line $\tilde{L}$ will be considered as the line through ${\tilde{P}}_{1}^{'} (a_{1}, {ma}_{1} + c)$ , ${\tilde{P}}_{2}^{'} (a_{2}, {ma}_{2} + c)$ , …, ${\tilde{P}}_{n}^{'} (a_{n}, {ma}_{n} + c)$ . Thus, the fitted fuzzy line $\tilde{L}$ for the fuzzy points ${\tilde{P}}_{1}$ , ${\tilde{P}}_{2}$ , …, ${\tilde{P}}_{n}$ is given by $\tilde{L} = {\tilde{\bar{L}}}_{1 \infty} ⋃ {\tilde{\bar{L}}}_{P_{1}^{'} P_{2}^{'}} ⋃ \dots ⋃ {\tilde{\bar{L}}}_{P_{n - 1}^{'} P_{n}^{'}} ⋃ {\tilde{\bar{L}}}_{n \infty} .$

Theorem 4.1. Let us suppose n fuzzy points ${\tilde{P}}_{1}$ , ${\tilde{P}}_{2}$ , …, ${\tilde{P}}_{n}$ are given. According to the above procedure fuzzy line $\tilde{L}$ is being fitted for these fuzzy points. Let us now calculate ${\tilde{D}}_{1}^{2} + {\tilde{D}}_{2}^{2} + \dots + {\tilde{D}}_{n}^{2}$ where ${\tilde{D}}_{i}$ is vertical fuzzy distance between ${\tilde{P}}_{i}$ and $\tilde{L}$ , i = 1, 2, …, n. This sum ${\tilde{D}}_{1}^{2} + {\tilde{D}}_{2}^{2} + \dots + {\tilde{D}}_{n}^{2}$ is minimum amongst all possible fuzzy linear fit ${\tilde{L}}_{G}$ passing through shifted fuzzy points ${\tilde{P}}_{1}^{'}$ , ${\tilde{P}}_{2}^{'}$ , …, ${\tilde{P}}_{n}^{'}$ .

Proof. The theorem trivially follows from the Theorem 3.1, since there is only one possible fuzzy line passing through shifted fuzzy points ${\tilde{P}}_{1}^{'}$ , ${\tilde{P}}_{2}^{'}$ , …, ${\tilde{P}}_{n}^{'}$ . Thus, the sum ${\tilde{D}}_{1}^{2} + {\tilde{D}}_{2}^{2} + \dots + {\tilde{D}}_{n}^{2}$ obviously will be minimum corresponding to the line $\tilde{L}$ , which is obtained by the above described procedure. □

Let equation of the fuzzy line $\tilde{L}$ be (y - f (x)/y - mx - c/y - g (x)) _LR = 0. We note that evaluating exact mathematical form of the functions f, g, L and R may not be always an easier task due to complex structure of the fuzzy points ${\tilde{P}}_{1}$ , ${\tilde{P}}_{2}$ , …, ${\tilde{P}}_{n}$ . According to Bector and Chandra [2] and Dubois and Prade [11, 12]in the application field rany fuzzy number can be approximated by triangular fuzzy number. Thus rwe may take L and R as linear functions. To approximate the functions f and g we may follow the following methodology.

4.1 Equation of the fitted fuzzy line

The fitted fuzzy line $\tilde{L} \equiv (y - f (x) / y - mx - c / y - h (x)) = 0$ is obtained by ${\tilde{\bar{L}}}_{1 \infty} ⋃ {\tilde{\bar{L}}}_{P_{1}^{'} P_{2}^{'}} ⋃ \dots ⋃ {\tilde{\bar{L}}}_{P_{n - 1}^{'} P_{n}^{'}} ⋃ {\tilde{\bar{L}}}_{n \infty} .$ Part of the upper curve ry = f (x) in the fuzzy line segment ${\tilde{\bar{L}}}_{P_{i}^{'} P_{i + 1}^{'}}$ can be approximated as the line segment joining the points $U_{i} = ({\bar{x}}_{i} r {\bar{y}}_{i}) \in {\tilde{P}}_{i}^{'} (0)$ and $U_{i + 1} = ({\bar{x}}_{i + 1} r {\bar{y}}_{i + 1}) \in P_{i + 1}^{'} (0)$ where U _i and U _i+1 have the highest y-coordinates in the support sets ${\tilde{P}}_{i}^{'} (0)$ and ${\tilde{P}}_{i + 1}^{'} (0)$ respectively ri.e. r ${\bar{y}}_{i} = max_{(x r y) \in {\tilde{P}}_{i}^{'} (0)} y$ and ${\bar{y}}_{i + 1} = max_{(x r y) \in {\tilde{P}}_{i + 1}^{'} (0)} y$ . There may be several possible points in ${\tilde{P}}_{i}^{'} (0)$ with y-coordinate as ${\bar{y}}_{i}$ .

If slope of the core line y = mx + c is non-negative rthen out of all these points with y-coordinate as ${\bar{y}}_{i}$ rwe will consider the point U _i as the point for whichx-coordinate is minimum. Therefore rco-ordinate of the point $U_{i} = ({\bar{x}}_{i} r {\bar{y}}_{i})$ will be obtained by ${\bar{y}}_{i} = max_{(x r y) \in {\tilde{P}}_{i}^{'} (0)} y$ and ${\bar{x}}_{i} = min_{(x r y) \in {\tilde{P}}_{i + 1}^{'} (0)} x$ .

If slope of the core line y = mx + c is negative rthen out of the points with y-coordinate as ${\bar{y}}_{i}$ rwe will consider the point U _i as the point for which x-coordinate is maximum. Therefore rco-ordinate of the point $U_{i} = ({\bar{x}}_{i} r {\bar{y}}_{i})$ will be obtained by ${\bar{y}}_{i} = max_{(x r y) \in {\tilde{P}}_{i}^{'} (0)} y$ and ${\bar{x}}_{i} = max_{(x r y) \in {\tilde{P}}_{i + 1}^{'} (0)} x$ .

Similarly rthe point $U_{i + 1} = ({\bar{x}}_{i + 1} r {\bar{y}}_{i + 1})$ will be obtained.

Varying i from 1 to n requation of the upper curve y = f (x) will be obtained by joining the line segments U ₁ U ₂ rU ₂ U ₃ r… rU _n-1 U _n and the half lines U ₁ U _l∞ (parallel to the core line and passing through U ₁) and U _n U _r∞ (parallel to the core line and passing through U _n). Mathematically ry = f (x) is $f (x) = {\begin{matrix} U_{l \infty} U_{1} : m (x - {\bar{x}}_{1}) + {\bar{y}}_{1} & if - \infty < x \leq {\bar{x}}_{1} \\ U_{1} U_{2} : \frac{{\bar{y}}_{2} - {\bar{y}}_{1}}{{\bar{x}}_{2} - {\bar{x}}_{1}} (x - {\bar{x}}_{2}) + {\bar{y}}_{2} & if {\bar{x}}_{1} \leq x \leq {\bar{x}}_{2} \\ U_{2} U_{3} : \frac{{\bar{y}}_{3} - {\bar{y}}_{2}}{{\bar{x}}_{3} - {\bar{x}}_{2}} (x - {\bar{x}}_{3}) + {\bar{y}}_{3} & if {\bar{x}}_{2} \leq x \leq {\bar{x}}_{3} \\ ⋮ \\ U_{n - 1} U_{n} : \frac{{\bar{y}}_{n} - {\bar{y}}_{n - 1}}{{\bar{x}}_{n} - {\bar{x}}_{n - 1}} (x - {\bar{x}}_{n}) + {\bar{y}}_{n} & if {\bar{x}}_{n} \leq x \leq {\bar{x}}_{n + 1} \\ U_{n} U_{r \infty} : m (x - {\bar{x}}_{n}) + {\bar{y}}_{n} & if {\bar{x}}_{n} \leq x < \infty \end{matrix}$

Part of the lower curve ry = h (x) in the fuzzy line segment ${\tilde{\bar{L}}}_{P_{i}^{'} P_{i + 1}^{'}}$ can be approximated as the line segment joining the points $L_{i} = ({\bar{x^{'}}}_{i} r {\bar{y^{'}}}_{i}) \in {\tilde{P}}_{i}^{'} (0)$ and $L_{i + 1} = ({\bar{x^{'}}}_{i + 1} r {\bar{y^{'}}}_{i + 1}) \in P_{i + 1}^{'} (0)$ where L _i and L _i+1 have lowest y-coordinates in the support sets ${\tilde{P}}_{i}^{'} (0)$ and ${\tilde{P}}_{i + 1}^{'} (0)$ respectively ri.e. r ${\bar{y^{'}}}_{i} = min_{(x r y) \in {\tilde{P}}_{i} (0)} y$ and ${\bar{y^{'}}}_{i + 1} = min_{(x r y) \in {\tilde{P}}_{i + 1} (0)} y$ .

If slope of the core line y = mx + c is non-negative rthen the point $L_{i} = ({\bar{x^{'}}}_{i} r {\bar{y^{'}}}_{i})$ will be obtained by ${\bar{y^{'}}}_{i} = min_{(x r y) \in {\tilde{P}}_{i}^{'} (0)} y$ and ${\bar{x^{'}}}_{i} = max_{(x r y) \in {\tilde{P}}_{i + 1}^{'} (0)} x$ .

If slope of the core line y = mx + c is negative rthen the point $L_{i} = ({\bar{x^{'}}}_{i} r {\bar{y^{'}}}_{i})$ will be obtained by ${\bar{y^{'}}}_{i} = min_{(x r y) \in {\tilde{P}}_{i}^{'} (0)} y$ and ${\bar{x^{'}}}_{i} = min_{(x r y) \in {\tilde{P}}_{i + 1}^{'} (0)} x$ .

Similarly rthe point $L_{i + 1} = ({\bar{x^{'}}}_{i + 1} r {\bar{y^{'}}}_{i + 1})$ will be obtained.

Varying i from 1 to n requation of the upper curve y = f (x) will be obtained by joining the line segments L ₁ L ₂ rL ₂ L ₃ r… rL _n-1 L _n and the half lines L ₁ L _l∞ (parallel to the core line and passing through L ₁) and L _n L _r∞ (parallel to the core line and passing through L _n). Mathematically ry = h (x) is $h (x) = {\begin{matrix} L_{l \infty} L_{1} : m (x - {\bar{x^{'}}}_{1}) + {\bar{y^{'}}}_{1} & if - \infty < x \leq {\bar{x^{'}}}_{1} \\ L_{1} L_{2} : \frac{{\bar{y^{'}}}_{2} - {\bar{y^{'}}}_{1}}{{\bar{x^{'}}}_{2} - {\bar{x^{'}}}_{1}} (x - {\bar{x^{'}}}_{2}) + {\bar{y^{'}}}_{2} & if {\bar{x^{'}}}_{1} \leq x \leq {\bar{x^{'}}}_{2} \\ L_{2} L_{3} : \frac{{\bar{y^{'}}}_{3} - {\bar{y^{'}}}_{2}}{{\bar{x^{'}}}_{3} - {\bar{x^{'}}}_{2}} (x - {\bar{x^{'}}}_{3}) + {\bar{y^{'}}}_{3} & if {\bar{x^{'}}}_{2} \leq x \leq {\bar{x^{'}}}_{3} \\ ⋮ \\ L_{n - 1} L_{n} : \frac{{\bar{y^{'}}}_{n} - {\bar{y^{'}}}_{n - 1}}{{\bar{x^{'}}}_{n} - {\bar{x^{'}}}_{n - 1}} (x - {\bar{x^{'}}}_{n}) + {\bar{y^{'}}}_{n} & if {\bar{x^{'}}}_{n} \leq x \leq {\bar{x^{'}}}_{n + 1} \\ L_{n} L_{r \infty} : m (x - {\bar{x^{'}}}_{n}) + {\bar{y^{'}}}_{n} & if {\bar{x^{'}}}_{n} \leq x < \infty \end{matrix}$

Thus rcorresponding to the fuzzy points ${\tilde{P}}_{1}$ r ${\tilde{P}}_{2}$ r… r ${\tilde{P}}_{n}$ rfitted fuzzy line is given by (y - f (x)/y - mx - c/y - h (x)) =0.

4.2 Numerical example

Let us fit a fuzzy line for the fuzzy points ${\tilde{P}}_{1} (- 1 r 0)$ r ${\tilde{P}}_{2} (0 r 1)$ r ${\tilde{P}}_{3} (1 r 4)$ and ${\tilde{P}}_{4} (2 r 1)$ rwhere membership function of these fuzzy points are: $\begin{matrix} μ ((x r y) | {\tilde{P}}_{1} (- 1 r 0)) \\ = {\begin{matrix} 1 - 4 \sqrt{(x + 1)^{2} + y^{2}} & if (x + 1)^{2} + y^{2} \leq \frac{1}{4^{2}} \\ 0 & elsewhere r \end{matrix} \end{matrix}$ $\begin{matrix} μ ((x r y) | {\tilde{P}}_{2} (0 r 1)) \\ = {\begin{matrix} 1 - \sqrt{4 x^{2} + (y - 1)^{2}} & if 4 x^{2} + (y - 1)^{2} \leq 1 \\ 0 & elsewhere r \end{matrix} \end{matrix}$ $\begin{matrix} μ ((x r y) | {\tilde{P}}_{3} (1 r 4)) \\ = {\begin{matrix} min {μ (x | \tilde{1}) r μ (y | \tilde{4})} & if \frac{1}{2} \leq x \leq \frac{3}{2} r \frac{7}{2} \leq y \leq \frac{9}{2} \\ 0 & elsewhere r \end{matrix} \end{matrix}$ where $\tilde{1}$ and $\tilde{4}$ are triangular fuzzy numbers (0.5/1/1.5) and (3.5/4/4.5) respectively rand $\begin{matrix} μ ((x r y) | {\tilde{P}}_{4} (2 r 1)) \\ = {\begin{matrix} 1 - 2 \sqrt{(x - 2)^{2} + (y - 1)^{2}} \\ if (x - 2)^{2} + (y - 1)^{2} \leq \frac{1}{2^{2}} \\ 0 elsewhere . \end{matrix} \end{matrix}$

These fuzzy points are depicted in the Fig. 3.

Best fitted line for the core points (-1 r 0) r(0 r 1) r(1 r 4) and (2 r 1) is y = 0.6x + 1.2.

Shifted fuzzy points for the given fuzzy points are given by

$\begin{matrix} μ ((x r y) | {\tilde{P}}_{1}^{'} (- 1 r 0.6)) \\ = {\begin{matrix} 1 - 4 \sqrt{(x + 1)^{2} + (y - 0.6)^{2}} \\ if (x + 1)^{2} + (y - 0.6)^{2} \leq \frac{1}{4^{2}} \\ 0 elsewhere r \end{matrix} \\ μ ((x r y) | {\tilde{P}}_{2}^{'} (0 r 1.2)) \\ = {\begin{matrix} 1 - \sqrt{4 x^{2} + (y - 1.2)^{2}} \\ if 4 x^{2} + (y - 1.2)^{2} \leq 1 \\ 0 elsewhere r \end{matrix} \\ μ ((x r y) | {\tilde{P}}_{3}^{'} (1 r 1.8)) \\ = {\begin{matrix} min {μ (x | \tilde{1}) r μ (y | \tilde{1.8})} \\ if \frac{1}{2} \leq x \leq \frac{3}{2} r 1.3 \leq y \leq 2.3 \\ 0 elsewhere r \end{matrix} \end{matrix}$ where $\tilde{1}$ and $\tilde{1.8}$ are triangular fuzzy numbers (0.5/1/1.5) and (1.3/1.8/2.3) respectively rand $\begin{matrix} μ ((x r y) | {\tilde{P}}_{4}^{'} (2 r 2.4)) \\ = {\begin{matrix} 1 - 2 \sqrt{(x - 2)^{2} + (y - 2.4)^{2}} \\ if (x - 2)^{2} + (y - 2.4)^{2} \leq \frac{1}{2^{2}} \\ 0 elsewhere . \end{matrix} \end{matrix}$

These shifted fuzzy points ${\tilde{P}}_{1}^{'} (- 1 r 0.6)$ r ${\tilde{P}}_{2}^{'} (0 r 1.2)$ r ${\tilde{P}}_{3}^{'} (1 r 1.8)$ and ${\tilde{P}}_{4}^{'} (2 r 2.4)$ and the fitted core line is depicted in the Fig. 4.

To form the upper curve y = f (x) rthe required points U ₁ rU ₂ rU ₃ and U ₄ are given by (-1 r 0.85) r(0 r 2.2) r(0.5 r 2.3) and (2 r 2.9) respectively. Thus rthe function y = f (x) is: $f (x) = {\begin{matrix} U_{l \infty} U_{1} : 0.6 + 1.45 & if - \infty < x \leq - 1 \\ U_{1} U_{2} : 1.35 x + 2.2 & if - 1 \leq x \leq 0 \\ U_{2} U_{3} : 0.2 x + 2.2 & if 0 \leq x \leq 0.5 \\ U_{3} U_{4} : 0.4 x + 2.1 & if 0.5 \leq x \leq 2 \\ U_{4} U_{r \infty} : 0.6 x + 4.1 & if 2 \leq x < \infty . \end{matrix}$

To form the lower curve y = h (x) rthe required points L ₁ rL ₂ rL ₃ and L ₄ are given by (-1 r 0.35) r(0 r 0.2) r(1.5 r 1.3) and (2 r 1.9) respectively. Thus rthe function y = h (x) is: $h (x) = {\begin{matrix} L_{l \infty} U_{1} : 0.6 + 0.95 & if - \infty < x \leq - 1 \\ L_{1} L_{2} : - 0.15 x + 0.2 & if - 1 \leq x \leq 0 \\ L_{2} L_{3} : 0.73 x + 0.2 & if 0 \leq x \leq 1.5 \\ L_{3} L_{4} : 1.2 x - 0.5 & if 1.5 \leq x \leq 2 \\ L_{4} L_{r \infty} : 0.6 x + 0.7 & if 2 \leq x < \infty . \end{matrix}$ Therefore rfitted fuzzy line corresponding to the fuzzy points ${\tilde{P}}_{1} (- 1 r 0)$ r ${\tilde{P}}_{2} (0 r 1)$ r ${\tilde{P}}_{3} (1 r 4)$ and ${\tilde{P}}_{4} (2 r 1)$ is (y - f (x)/y - 0.6x - 1.2/y - h (x)) =0. This fuzzy line is depicted in the Fig. 5.

5 Conclusion

In this paper we have attempted to formulate general form of fuzzy line. From the presented investigation, we obtain that formulation of fuzzy lines can be made in four different ways—first, group of crisp points with varied membership values, second, group of line segments or half lines with different membership values, third, group of fuzzy points, and last one is the group of fuzzy numbers along perpendicular lines on the core of the fuzzy line. From the proposed formulations of fuzzy lines, it is observed that α-cut of fuzzy lines are closed, connected and arc-wise connected subset of $ℝ^{2}$ but not necessarily convex. Fuzzy line is always normalized and more precisely its core contains a crisp line. Membership function of fuzzy line is upper semi-continuous—it follows from the fact that α-cut being closed. Along with the construction procedure of membership function of fuzzy lines, we have attempted to introduce its slope, intercept and some concepts of distance and vertical distance between a fuzzy point and a fuzzy line. An equational form of fuzzy line is also proposed. Application on fuzzy linear and affine spaces [28] of the proposed analysis of fuzzy lines may be considered as future research.

As an application of the proposed fuzzy line, we have studied fuzzy line fitting. Fitted fuzzy line corresponding to a set of imprecise locations may have hopeful application on fuzzy regression line, fuzzy linear inequality, trends of a fuzzy data, pattern recognition, etc. When an explicit list of n fuzzy numbers for independent variables and corresponding list of n fuzzy numbers for dependent variable is available, then corresponding to each value of dependent and independent variable, we must obtain a fuzzy point with rectangular support (see Example 1 of [3]). Thus, even though formulated fuzzy regression line has been attempted to find a linear pattern of a given fuzzy data, but the methodology can also be applied to find vague relationship between cause and effect variables especially in fuzzy rule base systems [7]. An explicit application on finding vague relationship between dependent and independent variable can be found from our future research.

Footnotes

For a fuzzy set, its imprecise range refers to variability/diversity (distance) of elements on the fuzzy set from its core.

A real valued function f on a metric space M is called upper semi-continuous if for each real α, the set {x ∈ M : f (x) ≥ α} is closed in M (see []).

A translation is said to be rigid if it preserves relative distances –that is to say that if P ₁ and Q ₁ are transformed to P ₂ and Q ₂ then the distance from P ₁ to Q ₁ is the same as that from P ₂ to Q ₂.

References

D. Chakraborty and D. Ghosh, Analytical fuzzy plane geometry II, Fuzzy Sets Syst, (in press) 2013. (https://dx-doi-org.web.bisu.edu.cn/10.1016/j.fss.2013.06.016)

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