ON SIMILARITY OF FUZZY TRIANGLES

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No.4, October 2013

ON SIMILARITY OF FUZZY TRIANGLES Debdas Ghosh1 and Debjani Chakraborty2 1,2

Department of Mathematics, Indian Institute of Technology Kharagpur Kharagpur 721302, West Bengal, India

ABSTRACT This paper investigates fuzzy triangle and similarity of fuzzy triangles. Five rules to determine similarity of fuzzy triangles are studied. Extension principle and concept of same and inverse points in fuzzy geometry are used to analyze the proposed concepts.

KEYWORDS Fuzzy numbers, Fuzzy point, Same and inverse points, Fuzzy angle, Fuzzy triangle.

1. INTRODUCTION In the literature, fuzzy triangle in the plane has been defined in four different ways−first, by three fuzzy points as its vertices [1], second, by intersection of three intersecting fuzzy half planes [2], third, by approximation of crisp triangle [3, 4] and last, by blurring the boundary of a crisp triangle [5].

Membership value of a point in the fuzzy half plane defined in [2] depends on the perpendicular distance between the point and the boundary of the fuzzy half plane. As this perpendicular distance increases, membership value of the points increases. This concept of defining fuzzy half plane cannot converge to the definition of crisp half plane. Over and above, core of a fuzzy half plane must be a crisp half-plane. This also does not follow from the definition of fuzzy half plane, and hence, definition of fuzzy triangle therein may be questionable. In [2], boundary of α-cuts of a fuzzy triangle are equivalent triangles having same measure of vertex angles, and hence, it is shown that sine law of triangle holds for fuzzy triangle also.

Buckley and Eslami [1] defined fuzzy triangle by three fuzzy points as its vertices. To form a fuzzy triangle, three intersecting fuzzy line segments are being adjoined. This definition for fuzzy triangle may be acceptable in fuzzy environment.

In [6], Fuzzy triangle is defined as a fuzzy set whose α-cuts are similar triangles. Fuzzy triangle defined in [6] cannot be a fuzzy triangle and it is a fuzzy point [1] whose support is a triangular region.

In [3, 4], fuzzy triangle or f-triangle is studied as approximate triangle. It is reported that instead of drawing a triangle by ruler, any triangle drawn by free hand is a fuzzy triangle. Subsequently similarities of fuzzy triangles are also studied. But we note that core of this fuzzy triangle is not a crisp triangle.

In [5], fuzzy triangle is defined by blurring boundary of a crisp triangle using smooth unit step function and implicit functions. But in the obtained shape, its 1-level set contain all the points DOI : 10.5121/ijfls.2013.3401

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No.4, October 2013

which lie outside the considered crisp triangle instead of the points on the boundary. Recently, Zadeh [7] has mentioned that the counterpart of a crisp triangle, C, in Euclidian geometry, is a fuzzy triangle. Fuzzy triangle may be formed by fuzzy-transform of C, with C playing the role of the prototype of fuzzy-triangle. It is helpful to visualize a fuzzy triangle as a fuzzy transform of C which is drawn by a spray pen [7]. Here the fuzzy-transformation is a one-to-many function.

An overview on fuzzy geometrical concepts prior to the work of Buckley and Eslami is reported in [8]. Some simple construction of fuzzy geometrical concepts can also be obtained in [9].

In this paper we have attempted to find rules to determine similarity of fuzzy triangles. A definition of fuzzy triangle has also been presented. Constructed fuzzy triangle here is similar to that defined by Zadeh [7]. To define similarity of fuzzy triangles, the definition which is proposed at first reflects generalization of well-known S-S-S rule for similarity of crisp triangles. However, other rules, like S-A-S, A-A-A and A-A-S rules have also been presented. A new concept, namely V-V-V rule, to determine similarity of fuzzy triangles is introduced. Delineation of the paper is as follows.

Section 2 is covered by basic definitions and terminologies used in this paper. Fuzzy triangle has been studied in the section 3. Similarity of fuzzy triangles has been studied in the Section 4. A brief discussion about the work presented here and its future scope are added in the Section 5.

2. PRELIMINARIES The basic definitions adopted here are taken from [1] and [10] with slight alteration. Small or capital letters with over tilde bar, i.e., A% , B% , C% ,... and a% , b%, c%,... denote fuzzy subsets of n , n = 1, 2. n Membership function of a fuzzy set A% of is represented by µ ( x | A% ), x ∈ n with

µ(

n

) ⊆ [0,1], n = 1, 2.

Definition 2.1. ( α - cut of a fuzzy set [10]). For a fuzzy set A% of denoted by A% (α ) and is defined by:

n

, n = 1, 2, an α - cut of A% is

{x : µ ( x | A% ) ≥ α } if 0 < α ≤ 1 A% (α ) =  closure{x : µ ( x | A% ) > 0} if α = 0.

The set {x: µ ( x | A% ) > 0 } is called support of the fuzzy set A% .

To represent the construction of membership function of a fuzzy set A% , the notation V{x: x ∈ A% (0) } is frequently used, which means µ ( x | A% ) = sup{α : x ∈ A% (α )}.

Definition 2.2. (Fuzzy number [1]). A fuzzy set A% of membership function µ has the following properties:

is called a fuzzy number if its

(i) µ ( x | A% ) is upper semi continuous, (ii) µ ( x | A% ) = 0 , outside some interval [a, d], and

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No.4, October 2013

(iii) there exist b and c so that a ≤ b ≤ c ≤ d and µ ( x | A% ) is increasing on [a, b], decreasing on [c, d] and µ ( x | A% ) = 1 ∀ x ∈ [b, c].

The notation (a / c / d)fg is used to represent the above defined fuzzy number when b = c, where f(x) = µ ( x | A% ) for x in [a, b] and g(x) = µ ( x | A% ) for x in [b, d]. A special type of fuzzy number called triangular fuzzy number is denoted by (a / b / c). A fuzzy number is called triangular fuzzy number if the functions f and g, in (a / b / c)fg, are linear.

Definition 2.3. (Fuzzy number along a line [10]). In defining fuzzy number, conventionally, real line ( ) is taken as universal set. Instead of real line as universal set, any line of 2 plane can be taken as well. Let, in 2 , x-axis represents real line and p% be a fuzzy number. In x-axis p% can be represented by µ (( x,0) | p% ) = µ ( x | p% ) ∀ x ∈ . More explicitly:

 µ ( x | p% ) if y = 0 elsewhere. 0

µ (( x , y ) | p% ) = 

Let T be a (bijective) transformation which transform x-axis to ax + by = c. Now, p% may be considered as a fuzzy number in the line ax + by = c defined in the following way:  µ (( x , 0) | p% ) 0

µ (( u , v ) | p% ) = 

if ( u , v ) = T ( x , 0), au + bv = c elsewhere.

Definition 2.4. (Fuzzy points [1]). A fuzzy point at (a, b) ∈ 2 , written as P% (a, b) , is defined by its membership function which satisfies the following conditions: (i) µ (( x, y) | P% (a, b)) is upper semi-continuous, (ii) µ (( x, y ) | P% (a, b)) = 1 if and only if (x, y) = (a, b), and (iii) P% (a, b)(α ) is a compact, convex subset of 2 for any α ∈ [0,1].

Often the notations P%1 , P%2 , P%3 ,... are used to represent fuzzy points.

Definition 2.5. (Same points [10]). Let (x1, y1) and (x2, y2) be two points on the supports of two continuous fuzzy points P% (a, b) and P% (c, d ) respectively. Let L1 be a line joining (x1, y1) and (a,

b). As P% (a, b) is a fuzzy point, along L1 there exists a fuzzy number, r%1 say, on the support of P% ( a, b). Membership function of r%1 can be written as µ (( x, y ) | r%1 ) = µ (( x, y ) | P% (a, b)) for (x, y) in L1, and zero 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 the points (x1, y1) and (x2, y2) are said to be same point with respect to P% (a, b) and P% (c, d ) if: 3

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No.4, October 2013

(i) (x1, y1) and (x2, y2) are same points with respect to r%1 and r%2 , and (ii) L1, L2 have made the same angle with the line joining (a, b) and (c, d).

Definition 2.6. (Inverse points [10]). Let (x1, y1) and (x2, y2) be two points in the support of two continuous fuzzy points P% (a, b) and P% (c, d ) respectively. The points (x1, y1) and (x2, y2) are said

to be inverse points with respect to P% (a, b) and P% (c, d ) if (x1, y1), (-x2, - y2) are same points with respect to P% (a, b) and − P% (c, d ).

Definition 2.7 (Fuzzy distance [10]). Fuzzy distance ( D% = D% ( P%1 , P%2 ) ) between two fuzzy points P%1 and P%2 is defined by its membership function: µ (d | D% ) = sup{α : d = d (u , v ), where u ∈ P% (0), v ∈ P% (0) are inverse points and µ (u | P% ) = µ (v | P% ) = α }. Here d (u , v) is the Euclidean 1

2

1

2

distance metric.

Definition 2.8 (Fuzzy line segment [10]). Fuzzy line segment L%P1P2 joining two fuzzy points P%1

and P%2 is defined by its membership function as: µ (( x, y ) | L%P1P2 ) = sup {α : (x, y) lies on the line

segment

joining

same

points % % µ (( x1 , y 1 ) | P1 ) = µ (( x 2 , y 2 ) | P2 ) = α }.

(x1,

y1)

∈ P%1 (0) ,

(x2,

y2) ∈ P%2 (0)

with

Definition 2.9 (Angle between two fuzzy line segments [10]). Let P%1 , P%2 , P%3 ,... be three continuous

% and is defined by: µ (θ | Θ % ) = sup{α : θ is fuzzy points. Angle between L%P1P2 , L%P2 P3 is denoted by Θ angle between two line segments Luv and Lvw where u, v and v, w are same points of membership value α ; u ∈ P% (0), v ∈ P% (0), w ∈ P% (0)}.

1

2

3

In the next section, first let us present the formation of fuzzy triangle and measurements of its side lengths, vertex angles and area using the concepts of same points and inverse points.

2. FUZZY TRIANGLE Let us suppose that three distinct fuzzy points P%1 , P%2 , P%3 ,... are given and a fuzzy triangle ( ∆% P1 P2 P3 ) has to form. A construction procedure may be designed as follows. Considering three same points u, v and w in the support of P% , P% and P% respectively, let us constructs a triangle 1

2

3

∆ whose vertices are u, v and w. If µ (( x1 , y1 ) | P%1 ) = α , then obviously µ (( x2 , y2 ) | P%2 ) = µ (( x3 , y3 ) | P%3 ) = α . Thus the membership value of ∆ in ∆% P1 P2 P3 may be considered as α. Now ∆% P P P can be considered as union of all of these ∆ s. Then ∆% P P P is a group of crisp triangles 1 2 3

1 2 3

with different membership grades. Thus a formal definition of a fuzzy triangle may be given by its membership function as:

µ ( x | ∆% P1 P2 P3 ) = sup {α : x in ∆ , where ∆ is constructed by the same points u ∈ P%1 (0), v ∈ P%2 (0) and w ∈ P%3 (0) as vertices with µ (u | P%1 ) = µ (v | P%2 ) = µ ( w | P%3 ) = α }.

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No.4, October 2013

Remark 3.0.1 Fuzzy triangle defined in the above definition is exactly equal to L%P1P2 U L%P2 P3 U L%P3 P1 .

Example 1 Let us consider the fuzzy triangle, ∆% P1 P2 P3 , whose vertices are three fuzzy points P% (1, 2), P% (5,7) and P% (6,1) . Membership functions of these fuzzy points are right elliptical / 1

2

3

circular cone with supports

P%1 (1, 2)(0 ) = { ( x , y ) : ( x − 1) 2 / 4 + ( y − 2) 2 ≤ 1}, P%2 (5, 7 )(0) = { ( x , y ) : ( x − 5) 2 + ( y − 7 ) 2 ≤ 4} and P% (6,1)(0) = {( x , y ) : ( x − 6) 2 + ( y − 1) 2 ≤ 1}.

3

Let us now evaluate membership value of (2, 4) in the fuzzy triangle ∆% P1 P2 P3 .

The same points with membership value α in [0, 1] on P%1 (1, 2), P%2 (5,7) and P%3 (6,1) are [10]

Aα ,θ : (1 + 2(1 − α ) cos θ

4sin 2 θ + cos 2 θ , 2 + 2(1 − α )sin θ

4sin 2 θ + cos 2 θ ),

Bα ,θ : (5 + 2(1 − α ) cosθ ,7 + 2(1 − α )sin θ ) and Cα ,θ : (6 + (1 − α ) cosθ ,1 + (1 − α ) sinθ ) respectively, where θ in [0, 2π]. Apparently, there is a possibility that (2, 4) may lie on the line segment L%P1P2 , but (2, 4) cannot lie on the line segments L%P2 P3 and L%P3 P1 . The condition that (2, 4) lies on L%P1P2 or on the line segment Aα ,θ Bα ,θ (for some θ in [0, 2π] and α in [0, 1]) is: 4 − (7 + 2(1 − α ) sin θ ) 2 + 2 k (1 − α ) sin θ − (7 + 2(1 − α ) sin θ ) = , where k = 2 − (5 + 2(1 − α ) cos θ ) 1 + 2k (1 − α ) cos θ − (5 + 2(1 − α ) cos θ ) ⇒ α =1−

1 4 sin θ + cos 2 θ 2

3 = f (θ ), say.

(8 + 6 k ) sin θ − (10 + 6 k ) cos θ ◦

◦

Here f(θ) must lie in [0, 1], and hence admissible domain of f(θ) is Df = [63 , 222.66 ]. Maximum ◦ value of f(θ) over Df occurred at 157.32 and the value is 0.8352, which measures the possibility

of containment of (2,4) on L%P1P2 . ◦

Thus, the point (2, 4) lies on the triangle ∆Aα ,θ Bα ,θ Cα ,θ for α = 0.8352 and θ = 157.32 , i.e, A ≡ (0.7726, 2.1193), B = (4.7081, 7.1531) and C ≡ (5.8541, 1.0766).

Hence, µ ((2, 4)| ∆% P1 P2 P3 ) = 0.8352.

In the Figure 1, construction of a fuzzy triangle has been displayed. Different grey level sets represent different α-cuts of P%1 , P%2 and P%3 . Deeper grey shading represents higher value of α.

Totally black points P1, P2 and P3 are core of P% , P% and P% respectively. For ten different values 1

2

3

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No.4, October 2013

of α (α = 0.1, 0.2, ..., 0.9, 1.0), P%1 (α ) , P%2 (α ) and P%3 (α ) are shown. The lines A1B1, A2B2 and A3B3 are parallel and passing through P1, P2 and P3 respectively. Let us consider any α-cut of P%1 , P%2 and P%3 , say for example α = 0.4. We note that intersection points of the boundaries of P% ( 0.4 ) , P% ( 0.4 ) and P% ( 0.4 ) with the lines A1B1, A2B2 and A3B3 respectively are S1, S2 and S3 and 1

2

3

T1, T2 and T3. Due to the definition of same points, the points S1, S2 and S3 and T1, T2 and T3 are same points with respect to P%1 , P%2 and P%3 with membership value α = 0.4. According to the definition of fuzzy triangle, ∆% P1 P2 P3 is union of all triangles like ∆ S1S2S3, ∆ S1S2S3, etc. with membership value α = 0.4.

Figure 1: Construction of a fuzzy triangle

Definition 3.1 (Side lengths and vertex angles of a fuzzy triangle). Length of the side of the fuzzy triangle ∆% P1 P2 P3 is defined by fuzzy distance (Definition 2.7) between the vertices, i.e., D% ( P% , P% ), D% ( P% , P% ) and D% ( P% , P% ) respectively; let us denote them as p% , p% and p% respectively.

1

2

2

3

3

P1 P2

P2 P3

1

3

1

2

∆% P1 P2 P3 may be defined as the fuzzy angles (Definition 4.5 in [10]) % % % % % % % % % % P P , LP P ) and ∠( LP P , LP P ); and the notations ∠P , ∠P and ∠P respectively

The vertex angles of % ( L% , L% ), ∠ % ( L% ∠

2 3

3 1

3 1

1 2

2

3

1

% P% is situated opposite to the side may be used to represent them. It is to note that vertex angle ∠ i with length p% i , i = 1, 2, 3.

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No.4, October 2013

Area of a fuzzy triangle may be defined in the following way.

Definition 3.2 (Area of a fuzzy triangle). Fuzzy area ( ∆% ) may be defined by its membership function as: µ (∆ | ∆% ) = sup{α : ∆ is the area of the triangle constructed by the same points u ∈ P% ( 0 ) , v ∈ P% ( 0 ) and w ∈ P% ( 0 ) as its vertices with µ (u | P% ) = µ (v | P% ) = µ ( w | P% ) = α }.

1

2

3

1

2

3

The results of the following theorem gives information to get α-cuts, and hence membership function, of ∆% .

Theorem 3.1: ∆% is a fuzzy number and ∆% (α ) = { ∆ : ∆ is the area of the triangle constructed by the same points u ∈ P%1 (α ) , v ∈ P%2 (α ) and w ∈ P%3 (α ) as its vertices}.

Proof. Similar to Theorem 4.1 of [10] and is skipped.

Let us now take two fuzzy triangles and try to compare them. In comparing fuzzy triangles, few questions may arise naturally—how to find similarity of two fuzzy triangles? When can we say that two given fuzzy triangles are similar? Whether alike to crisp triangle the same condition that if three sides of the triangles are in a constant ratio will be applied to fuzzy triangles also or not?

Or any other conditions should be added? Answers of all of the questions are addressed in the following section.

3. SIMILARITIES OF FUZZY TRIANGLES In classical trigonometry, two triangles are said as similar if their shapes are alike but sizes are different. That is, a triangle and its enlarged (magnified) versions are similar. Let us now try to generalize this idea in finding similar fuzzy triangles. To do so, we observe that if we enlarge a fuzzy triangle, side lengths of the fuzzy triangle before and after its enlargement can be found easily. But unlike to classical triangles, after the enlargement of a fuzzy triangle, imprecision of its sides also gets enlarged. This happens due to presence of imprecision in the sides. So, in finding similarity of fuzzy triangles, imprecision of the sides are also to be accounted. To measure imprecision of the sides of a fuzzy triangle ∆% P P P say, let us consider one of its side say L% .

1 2 3

P1 P2

We consider a line l(x, y) perpendicular to L%P1P2 (1) at (x, y) in L%P1P2 . As L%P1P2 is a fuzzy line segment, along the line l(x, y) there must exists one fuzzy number [10] given by l(x, y) ∩ L% .

P1 P2

We denote this fuzzy number by l%3 ( x, y ) . Thus, corresponding to each (x, y), the function l% ( x, y ) always gives one fuzzy number. We will say the function defined by (x, y) → l% ( x, y ) as 3

3

the imprecision function of the side L%P1P2 . Similarly, there will be two more imprecision functions and L% respectively. It is noticeable that l% ( x, y ) and l% ( x, y ) corresponding to the sides L% 1

2

P2 P3

P3 P1

when a fuzzy triangle is enlarged, then all of its imprecision functions are magnified by some constant multiplication. Thus, two fuzzy triangles may be said as similar fuzzy triangles if all of their corresponding side lengths and corresponding imprecision functions are constant multiplication of the other.

Definition 4.1 (Similarity of fuzzy triangles). Let ∆% P1 P2 P3 and ∆% Q1Q2 Q3 are two fuzzy triangles. % , lq % for i =1, 2, 3 are their imprecision p%1 , p% 2 , p% 3 and q%1 , q%2 , q%3 being their side lengths and lp i i functions of the corresponding sides. Fuzzy triangles ∆% P P P and ∆% Q Q Q are said to be similar 1 2 3

1

2

3

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No.4, October 2013

if there exists k in R such that (i) (ii)

p% i = kq%i , for all i and % ( x , y ) = klq % ( x , y ) ∀ t ∈ [0, 1] for each i = 1, 2, 3, where t is taken as follows. lp i pi t pi t i qi t qi t

For i = 1, let us consider L%P3 P2 and ( x p1t , y p1t ) ∈ L%P2 P3 (1) . Then the point (xpt1 , ypt1 ) corresponds

to that t for which (xp1t,yp1t)= t P2 + (1 − t) P3, t ∈ [0, 1]. Similar relation will be applied for the sides L%P1 P2 , L%P1P3 , L% Q Q , L% Q Q and L%Q3Q1 .

1 2

3 2

If above two conditions hold true, then one of the fuzzy triangles ∆% P1 P2 P3 and ∆% Q1Q2 Q3 is enlarged or contracted version of another. For enlargement we will have |k|≥ 1 and for lessening we will have |k| < 1.

Remark 4.1.1 Here question may arise whether the constant k would be fuzzy. Answer is that k will be crisp always, since magnification of one fuzzy triangle can be done by some crisp constant multiplication of all the points on the support of the fuzzy triangle and conversely.

Theorem 4.1 Let us suppose P%i and Q% i for i = 1, 2, 3 are six fuzzy points. If two fuzzy triangles ∆% P1 P2 P3 and ∆% Q1Q2 Q3 are similar, then the crisp triangles joining the same points, with membership value α, of the corresponding vertices of the fuzzy triangles are also similar triangles for each α in [0, 1].

Proof: Here ∆% P1 P2 P3 and ∆% Q1Q2 Q3 are fuzzily similar triangles. Therefore, imprecision function and length of each side of ∆% P1 P2 P3 are some constant multiplication of the imprecision function and side length of the corresponding side of ∆% Q Q Q . So, one of the fuzzy triangles is enlarged 1

2

3

version of another. Once a fuzzy triangle is enlarged, its all the vertices also magnified by the same measure. Thus there exists some constant k R such that P%i = kQ% i .

∈

A fuzzy triangle can be viewed as a collection of crisp triangles whose vertices are same points of fuzzy vertices. Thus enlargement of fuzzy triangle means enlargement of corresponding crisp triangles on its support. Let (x1, y1), (x2,y2) and (x3,y3) are same points with membership value α in [0, 1] of the fuzzy points P%1 , P%2 and P%3 and ∆ pα is the crisp triangle in the support of ∆% P P P joining these three same points. Then, it is easy to observe that (kx1, ky1), (kx2, ky2) and 1 2 3

(kx3, ky3) are same points with membership value α of the fuzzy points Q%1 , Q% 2 and Q% 3 ; and the crisp triangle ∆ qα with vertices as (kx1, ky1), (kx2, ky2) and (kx3, ky3) will have membership value α in the fuzzy triangle ∆% Q Q Q . Hence the result follows.

1

2

3

It is worthy to mention that that in the definition of similarity of fuzzy triangles (Definition 4.1), side lengths of the fuzzy triangles are in a constant ratio and corresponding imprecision functions of the sides are also in a constant ratio. Thus, the definition essentially reflects S-S-S (Side-SideSide) rule to find similar crisp triangles, since for the core of two similar fuzzy triangles the Definition 4.1 reduces to S-S-S rule. However, there are other rules also to investigate similarity of crisp triangles. Let us now try to investigate similarity of fuzzy triangles by these rules, like SA-S (Side-Angle-Side), A-A-A (Angle-Angle-Angle) and A-A-S (Angle-Angle-Side) rules. To examine so, we need to find a construction procedure of fuzzy triangle when its two sides and one 8

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No.4, October 2013

angle are given. A construction procedure may be given as follows. Procedure is explained with an example.

Suppose we have to construct a fuzzy triangle whose two sides are 3̃ = (1/3/5) and ◦

2 2 = ( 2 / 2 2 / 3 2) and the angle between those two sides is ‘around 45 ’; the fuzzy point of intersection of those two sides is P̃ say whose membership function is a right circular cone with support {(x, y): x2 + y2 ≤ 1} and vertex at (0, 0). First let us construct the core of the fuzzy triangle. Taking x-axis as one of its side, we obtain that core is the triangle with vertices O: (0, 0), Q: (3, 0), R: (2, 2).

Now let us try to obtain a fuzzy point having core at (3, 0) and 3̃ distance apart from P̃ (0, 0). We note that there are infinitely many such fuzzy points. For example some of them are R̃ 1(3, 0) with 2 2 support {(x, y): (x − 3) + y ≤ 1} and membership function is a right circular cone, R̃ 0.5(3, 0) with 2 2 2 support {(x, y): (x − 3) + y /0.5 ≤ 1}and membership function is a right circular cone, and R̃ ε(3, 2 2 2 0) with support {(x, y): (x − 3) + y /ε ≤ 1}and membership function is a right circular cone where ε (0, 1]. The fuzzy number (2/3/4), R̃ say, along x-axis is itself a fuzzy point at (3, 0) which is 3̃ distance apart from P̃ (0, 0). The fuzzy point R̃ ε(3, 0) is shown in the Figure 2.

∈

Figure 2: Two fuzzy points Q̃ ε and R̃ ε having distance 3̃ and 2 2 respectively from P̃ (0, 0)

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No.4, October 2013

Similarly, there are several fuzzy points having core at (2, 2) and 2 2 distance apart from P̃(0, 2

2

2

0). For instance some of them are Q̃ 1(2, 2) with support {(x, y): (x + y − 4) / 2 + (x − y) ≤ 1} and 2 2 membership function is a right circular cone, Q̃ 0.5(2, 2) with support {(x, y): (x + y − 4) / 2 + (x − 2 2 y) /0.5 ≤ 1} and membership function is a right circular cone, and Q̃ ε(2, 2) with support {(x, y): (x 2 2 2 2 + y − 4) / 2 + (x − y) / ε ≤ 1} and membership function is a right circular cone where ε (0, 1].

The triangular fuzzy number ((1, 1) / (2, 2) / (3, 3)), Q̃ say, along the line y = x is itself a fuzzy

∈

point at (2, 2) which is 2 2 distance apart from P̃ (0, 0). Fuzzy point Q̃ ε(2, 2) is shown in the Figure 2.

Here for the fuzzy triangles ∆% PQε Rε with 0 < ε < 1, lengths of the sides L%PQε and L%PRε are 3̃ and ◦

2 2 respectively and angle between those sides is ‘around 45 ’. This shows that there are

infinitely many fuzzy triangles whose two side lengths are 3̃ and 2 2 and fuzzy angle between ◦ them is ‘around 45 ’. ◦

% Q PR . We know its value is ‘around 45 ’, but let us evaluate its Let us measure the angle ∠ ε ε % and R% with membership value 0 are appropriate value. The same points of the fuzzy points P% , Q ε

ε

(cos θ, sin θ), (2 +

1 2 2

[( 2 − ε ) co s θ + ( 2 + ε ) sin θ ], 2 + √ [( 2 + ε ) co s θ + ( 2 − ε ) s in θ ])

and

(3 + cos θ , ε sin θ ) respectively.

% Q PR will be given by [θ ε ,θ ε ] where Thus support of the angle ∠ ε ε min max ε θ min = min f (θ ), 0 ≤θ ≤ 2 π

ε

θ max = max f (θ ) and 0 ≤θ ≤ 2 π

1

√ [(2 + ε ) cos θ + (2 − ε ) sin θ ] − sin θ (ε − 1) sin θ 2 2 . − tan −1 1 3 2+ [(2 − ε ) cos θ + (2 + ε ) sin θ ] − cos θ 2 2

2+ f (θ ) = tan −1

ε ε % Q PR is Obviously, [θ min ,θ max ] is an ε dependent interval. Therefore, as ε varies, support of ∠ ε ε

◦

also varies. Hence if support of angle ‘around 45 ’ and its membership function is known % Q PR is exactly equals previously, there may not exist any fuzzy triangle ∆% Qε PRε for which ∠ ε ε ◦

to the given around 45 .

From this example, we note that for given two fuzzy numbers ã and b̃, a fuzzy angle θ̃ and a fuzzy point P̃ , a fuzzy triangle may not be found whose two side lengths are ã and b̃ and corresponding vertex and vertex angle are P̃ and θ̃ respectively. For example, in the above example if ã and b̃ ◦ ◦ ◦ would have been (1 / 2 / 3) and θ̃ = (35 /45 /55 ), then no fuzzy triangle can be found.

Thus, in general, for a given vertex, corresponding given vertex angle and two given side lengths 10

International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No.4, October 2013

cannot always determine a fuzzy triangle. So, S-A-S rule cannot always be generalized in fuzzy environment. Main difficulty comes in implementing the S-A-S rule is that only two sides and one angle cannot determine vertices of the fuzzy triangle uniquely.

Similarly, A-A-A rule cannot be generalized. The main reason of A-A-A rule cannot be generalized is that addition of three vertex angles may not always fixed. Yes, of course this value ◦

is ‘around 180 ’, but the spreads differ for different fuzzy triangles.

However in A-A-S rule we have the following theorem.

Theorem 4.2 If for two fuzzy triangles two vertex angles are equal, length of the side opposite to the another vertex angle of them are equal or they are constant multiplication of another, and the imprecision functions of those sides are constant multiplication of another, then the fuzzy triangles are similar.

Proof: Let us recall that fuzzy line segments are collection of crisp line segments with varied membership values. According to the assumption of the theorem, for the given sides of the fuzzy triangles, we note that corresponding to each and every crisp line segment lying on the given side of a fuzzy triangle there must exists one crisp line segment, with same membership value, on the corresponding given sides of the another fuzzy triangle. Now let us suppose the given fuzzy % P P P and ∆% Q Q Q ; for them the side lengths and imprecision functions are given triangles are ∆ 1 2 3 1 2 3

%

%

for the given side LP1P2 and LQ1Q2 respectively; the given vertex angles are Θ̃1 and Θ̃2; the angle Θ̃i is

the vertex angle ∠̃Pi and ∠̃Qi for i = 1, 2. Let l and l are two line segments with membership value α on % % the support of LP1P2 and LQ1Q2 respectively. Now let us consider two triangles ∆ pα and ∆ qα whose two sides are l and l and two vertex angle on the two extremities of l and l are θ ∈Θ̃1 and θ2 ∈Θ̃2 with membership values α ∈ [0, 1]. We observe that ∆ pα and ∆ qα are similar triangle. As ∆̃P P P = ∨α∈[0,1] % P P P , ∆% Q Q Q are similar. ∆ and ∆̃Q Q Q = ∨α∈[0,1] ∆ , the fuzzy triangles ∆ 1 2 3 1 2 3 pα

pα

qα

qα

pα

qα

1

1

pα

1

2

3

2

3

qα

In the foregoing paragraphs, study has been made for different rules to determine similarity of fuzzy triangles. Let us try to investigate in respect of their area. In particular, how the area of fuzzy triangles are changing when one fuzzy triangle is enlarged. It is to note that as fuzzy triangle is collection of crisp triangles with different membership value and these vertices of these crisp triangles are same points of the vertices of the fuzzy triangle, enlargement of the fuzzy triangles effectively means enlargement of the vertices of the fuzzy triangles.

Theorem 4.3. If P%i and Q% i are fuzzy points such that Q% i = k P%i , where k is some real constant, then

following results hold.

(i) If (x, y) belongs to ∆% P1 P2 P3 (1), then (kx, ky) lie on ∆% Q1Q2 Q3 (1) and l%pi ( x , y ) = kl%qi ( kx , ky ) % where l% and lqi for some i in{1, 2, 3} are imprecision function as defined in the Definition 4.1. pi

(ii) Fuzzy triangles ∆% P1 P2 P3 and ∆% Q1Q2 Q3 are similar. (iii) If ∆% p and ∆% q are area of the fuzzy triangles ∆% P1 P2 P3 and ∆% Q1Q2 Q3 respectively, then ∆% = k 2 ∆% . q

p

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No.4, October 2013

Proof: (i) First part that if (x, y) belongs to ∆% P1 P2 P3 (1), then (kx, ky) will lie on ∆% Q1Q2 Q3 (1) is trivially true. For the second part that l% ( x, y ) = kl% (kx, ky ) for i in{1, 2, 3}, we will prove that pi

qi

l%pi ( x, y ) = kl%qi (kx, ky ) for i =1 and similar will be the case for i = 2, 3. To prove l%p1 ( x, y ) = kl%q1 (kx, ky ) we let (x1,y1), (x2,y2) belong to P%2 with membership value α in [0, 1] and (x3,y3), (x4,y4) belong to P% with membership value α in [0, 1]. Let us also suppose that (x1,y1), (x3,y3) are 3

same points and (x2,y2), (x4,y4) are same points. Let l13 and l24 are line segments joining (x1,y1), (x3,y3) and (x2,y2), (x4,y4) respectively. We also suppose l13k and l24k are line segments joining k(x1, y1), k(x3, y3) and k(x2, y2), k(x4, y4) respectively. Without loss of generality let the line segments l13 and l24 lie on the different sides of the line joining P1, P2. For all the assumptions please refer to the Figure 3. We note that distance between the line segments l13k and l24k is k times of the distance between l13 and l24. Thus support of the imprecision functional value l% (kx, ky ) will be k times of p1

the support of the imprecision functional value l%q1 (kx, ky ) for each (x, y) in L%P1P2 . Hence the result follows. (ii) (iii)

This part is clear from the Theorem 4.1.

From the construction of ∆% P1 P2 P3 , it is the union of all of the crisp triangles ∆ having vertices as three same points (x1,y1), (x2,y2) and (x3,y3) (say) of the vertices of the fuzzy 2 triangle. We note that if ∆ is the area of the crisp triangle ∆, then k ∆ will be the area of the triangle having vertices as k(x1, y1), k(x2, y2) and k(x3,y3). €

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No.4, October 2013

% P P P and ∆% Q Q Q where Q% = k P% for k = 3.

Figure 3. Fuzzy triangles ∆ 1 2 3 1 2 3 i i

Remark 4.1.2: In this theorem we note that if three vertices of a fuzzy triangle are some constant multiplication (i.e., enlarged or contracted version) of the vertices of another fuzzy triangle, then both the fuzzy triangles will be similar. Thus we may implement this result to determine similarity of fuzzy triangles. This rule may be refereed as V-V-V (Vertex-Vertex-Vertex) rule for fuzzy similarity of fuzzy triangles. It is noteworthy that V-V-V rule also applicable for crisp triangles, since two triangles having vertices (x1,y1), (x2,y2), (x3,y3) and k(x1, y1), k(x2, y2), k(x3, y3) (for some non-zero constant k) are similar.

Remark 4.1.3: From the proof of the above theorem, we obtain that two fuzzy triangles are similar if and only if corresponding to each crisp triangle in the support of a fuzzy triangle there exists one crisp triangle on the support of the another fuzzy triangle with same membership value as the prior triangle and vice versa.

Remark 4.1.4 The result of the above theorem also gives that ‘areas of similar fuzzy triangles are (real) constant multiplication of other’.

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No.4, October 2013

5. CONCLUSION This paper proposed some concepts of similarity of fuzzy triangles. The sup-min composition of fuzzy sets and the concepts of same and inverse points are used in all the discussion. All the 2 proposed study of fuzzy triangle has been made in a coordinate reference frame of R to account the present imprecision in the fuzzy triangle very easily. In the Figure 1, corresponding to Definition 3 of fuzzy triangle, we observe that taking the crisp triangle ∆P1 P2 P3 as prototype, fuzzy triangle can be obtained by f-transformation (x, y) → l% ( x, y ) , l% ( x, y ) or l% ( x, y ) . Thus 1

2

3

the defined concept of fuzzy triangle is similar to Zadeh [7].

In similarity of fuzzy triangles, the proposed Definition 4.1 essentially reflects generalization of well-known S-S-S rule to study similarity of crisp triangles, since two fuzzy triangles are said as similar when corresponding side lengths of the fuzzy triangles are in a constant ratio and corresponding imprecision functions are also in a constant ratio. Other studied concepts on similarity tried to generalize well known S-A-S, A-A-S and A-A-A rule to investigate similarity of fuzzy triangles. Similarities are being discussed under a co-ordinate system or fuzzy geometrical reference frame [10]. In general we have observed that S-S-S and A-A-S rules can be generalized in the fuzzy environment, but S-A-S and A-A-A rules cannot be generalized. One new rule for similarity called V-V-V rule has been introduced. Subsequently change of area of two similar fuzzy triangles is also investigated.

We hope that result of Theorem 4.3 can have nice application in fuzzy particle / fuzzy rigid body dynamics. Since some constant multiplication of the fuzzy point effectively mean to have a translation of the fuzzy point; constant multiplication of vertices of fuzzy triangle essentially means to give a translation of the entire fuzzy triangle.

Here an important point is to note that as reported by Zadeh [11] – “formulation of a valid, general-purpose definition of similarity is a challenging problem”, we have not intended to propose a measurement of how much two fuzzy triangles are similar. Beg and Ashraf [12] also mentioned that a valid and general-purpose definition of similarity of fuzzy sets may not exist.

Thus only fuzzy similarity and construction procedure of fuzzy triangles are being focused in our study. Future research can focus on this similarity measure.

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J. J. Buckley and E. Eslami, Fuzzy plane geometry II: circles and polygons, Fuzzy Sets and Syst., 87 (1997) 79–85.

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B. M. Imran, M. M. Sufyan Beg, Elements of sketching with words, in: X. Hu, T. Y. Lin, V. V. Raghavan, J. W. G-Busse, Q. Liu, A. Z. Broder, IEEE International Conference on Granular Computing, San Jose, California, USA, IEEE Computer Society, 2010 pp. 241– 246.

B. M. Imran, M. M. S. Beg, Estimation of f-Similarity in f-Triangles using FIS, in: N. Meghanathan, N. Chaki and D. Nagamalai (eds.) CCSIT 2012, Part III. LNICST vol. 86 (Springer, Heidelberg, 2012) pp. 290–299.

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Authors

Debdas Ghosh is a Research Scholar in the Department of Mathematics, IIT Kharagpur, India. He received his BSc (in 2004) from Calcutta University and his MSc (in 2007) from IIT Kharagpur in Mathematics. His research interests are theory and application of fuzzy optimization and fuzzy geometry. He is a recipient of Prof. J. C. Bose memorial gold medal, institute silver medal and best project award in 2009 from IIT Kharagpur

Debjani Chakraborty is an Associate Professor in the Mathematics Department, IIT Kharagpur. She received her BSc (Maths Hons) in 1986 from Calcutta University and MSc (in 1989) and PhD (in 1995) from IIT Kharagpur. Her main area of research is theory and application of fuzzy logic in optimisation. She is a recipient of the Young Scientist Award 1997 in Mathematics from the Indian Science Congress Association.

She has published one book and more than 70 research papers. She has also been awarded Young Scientist Scheme from the Department of Science and Technology, Government of India in 1997 as an individual scientist. She is a nominated member of the Indian National Science Academy of Science, Allahabad.

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