On similarity of fuzzy triangles

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

DOI : 10.5121/ijfls.2013.3401 1

ON SIMILARITY OF FUZZY TRIANGLES

Debdas Ghosh1 and Debjani Chakraborty

2

1,2Department 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


2

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 , 1,2.n

n =�

Membership function of a fuzzy set A% of n� is represented by ( | ), n

x A xµ ∈% � with

( ) [0,1], 1,2.nnµ ⊆ =�

Definition 2.1. ( -α cut of a fuzzy set [10]). For a fuzzy set A% of , 1,2,n

n =� an -α cut of A% is

denoted by ( )A α% and is defined by:

{ : ( | ) } 0 1( )

{ : ( | ) 0} 0.

x x A ifA

closure x x A if

µ α αα

µ α

≥ < ≤=

> =

%%

%

The set {x: ( | ) 0x Aµ >% } 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

(0)A∈ % } is frequently used, which means ( | ) sup{ : ( )}.x A x Aµ α α= ∈% %

Definition 2.2. (Fuzzy number [1]). A fuzzy set A% of � is called a fuzzy number if its

membership function µ has the following properties:

(i) ( | )x Aµ % is upper semi continuous,

(ii) ( | ) 0x Aµ =% , outside some interval [a, d], and


3

(iii) there exist b and c so that a b c d≤ ≤ ≤ and ( | )x Aµ % is increasing on [a, b], decreasing on

[c, d] and ( | ) 1x Aµ =% ∀ x ∈ [b, c].

The notation (a / c / d)fg is used to represent the above defined fuzzy number when b = c, where

f(x) = ( | )x Aµ % for x in [a, b] and g(x) ( | )x Aµ= % 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, in2

� , x-axis represents real line and p% be a fuzzy number. In x-axis p% can

be represented by (( ,0) | ) ( | ) .x p x p xµ µ= ∀ ∈% % � More explicitly:

( | ) 0(( , ) | )

0 .

x p if yx y p

elsewhere

µµ

==

%%

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:

(( , 0) | ) ( , ) ( , 0), (( , ) | )

0 .

x p if u v T x au bv cu v p

elsewhere

µµ

= + ==

%%

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) (( , ) | ( , ))x y P a bµ % is upper semi-continuous,

(ii) (( , ) | ( , )) 1x y P a bµ =% 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 1 2 3, , ,...P P P% % % 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, 1r% say, on the support

of ( , ).P a b% Membership function of 1r% can be written as 1(( , ) | ) (( , ) | ( , ))x y r x y P a bµ µ= %%

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, 2r% 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:


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(i) (x1, y1) and (x2, y2) are same points with respect to 1r%

and 2

r% , 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 (1 2( , )D D P P=% % % % ) between two fuzzy points

1P% and2P% is defined by its membership function: ( | ) sup{ : ( , ),d D d d u vµ α= =% where

1 2(0), (0)u P v P∈ ∈% % are inverse points and 1 2( | ) ( | ) }.u P v Pµ µ α= =% % Here ( , )d u v is the Euclidean

distance metric.

Definition 2.8 (Fuzzy line segment [10]). Fuzzy line segment 1 2P P

L%

joining two fuzzy points 1P%

and 2P% is defined by its membership function as: 1 2

(( , ) | )P P

x y Lµ =% sup {α : (x, y) lies on the

line segment joining same points (x1, y1) 1(0)P∈ % , (x2, y2) 2 (0)P∈ % with

1 1 1 2 2 2(( , ) | ) (( , ) | )x y P x y Pµ µ α= =% % }.

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

fuzzy points. Angle between 1 2 2 3

,P P P P

L L% %

is denoted by Θ% and is defined by: ( | ) sup{ :µ θ α θΘ =% is

angle between two line segments uvL and vwL where u, v and v, w are same points of membership

value 1 2 3; (0), (0), (0)}.u P v P w Pα ∈ ∈ ∈% % %

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

1 2 3, , ,...P P P% % % are given and a fuzzy triangle

(1 2 3PP P∆% ) has to form. A construction procedure may be designed as follows. Considering three

same points u, v and w in the support of 1 2,P P% % and

3P% respectively, let us constructs a triangle

∆ whose vertices are u, v and w. If1 1 1(( , ) | )x y Pµ α=% , then obviously

2 2 2(( , ) | )x y Pµ =%

3 3 3(( , ) | ) .x y Pµ α=% Thus the membership value of ∆ in 1 2 3PP P∆% may be considered as α. Now

1 2 3PP P∆% can be considered as union of all of these ∆ s. Then 1 2 3PP P∆% is a group of crisp triangles

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

its membership function as:

1 2 3( | )x PP Pµ ∆% = sup {α : x in ∆ , where ∆ is constructed by the same points 1 2(0), (0)u P v P∈ ∈% %

and 3 (0)w P∈ % as vertices with

1 2 3( | ) ( | ) ( | ) }.u P v P w Pµ µ µ α= = =% % %


5

Remark 3.0.1 Fuzzy triangle defined in the above definition is exactly equal to

1 2 2 3 3 1.P P P P P PL L L

% % %U U

Example 1 Let us consider the fuzzy triangle, 1 2 3PP P∆% , whose vertices are three fuzzy points

1(1,2),P% 2 (5,7)P% and

3 (6,1)P% . Membership functions of these fuzzy points are right elliptical /

circular cone with supports 2 2

1

2 2

2

2 2

3

(1, 2)(0) {( , ) : ( 1) / 4 ( 2) 1},

(5, 7 )(0) {( , ) : ( 5) ( 7 ) 4} and

(6,1)(0) {( , ) : ( 6) ( 1) 1}.

P x y x y

P x y x y

P x y x y

= − + − ≤

= − + − ≤

= − + − ≤

%

%

%

Let us now evaluate membership value of (2, 4) in the fuzzy triangle 1 2 3.PP P∆%

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

3 (6,1)P% are [10]

2 2 2 2

,

,

,

: (1 2(1 )cos 4sin cos ,2 2(1 )sin 4sin cos ),

: (5 2(1 )cos ,7 2(1 )sin ) and

: (6 (1 ) cos ,1 (1 ) sin )

A

B

C

α θ

α θ

α θ

α θ θ θ α θ θ θ

α θ α θ

α θ α θ

+ − + + − +

+ − + −

+ − + −

respectively, where θ in [0, 2π]. Apparently, there is a possibility that (2, 4) may lie on the line

segment 1 2P P

L%

, but (2, 4) cannot lie on the line segments

2 3P PL%

and 3 1P P

L%

. The condition that (2, 4)

lies on 1 2P P

L%

or on the line segment , ,A Bα θ α θ (for some θ in [0, 2π] and α in [0, 1]) is:

2 2

4 (7 2(1 ) sin ) 2 2 (1 ) sin (7 2(1 ) sin ) 1 ,

2 (5 2(1 ) cos ) 1 2 (1 ) cos (5 2(1 ) cos ) 4 sin cos

31 ( ), .

(8 6 )sin (10 6 )cos

kwhere k

k

f sayk k

α θ α θ α θα θ α θ α θ θ θ

α θθ 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 1 2

.P P

L%

Thus, the point (2, 4) lies on the triangle , , ,A B Cα θ α θ α 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)|1 2 3PP P∆% ) = 0.8352.

In the Figure 1, construction of a fuzzy triangle has been displayed. Different grey level sets

represent different α-cuts of 1 2,P P% % and 3P% . Deeper grey shading represents higher value of α.

Totally black points P1, P2 and P3 are core of 1 2,P P% % and

3P% respectively. For ten different values


6

of α (α = 0.1, 0.2, ..., 0.9, 1.0), ( ) ( ) ( )1 2 3, and P P Pα α 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

1 2,P P% % and3P% , say for example α = 0.4. We note that intersection points of the boundaries of

( ) ( ) ( )1 2 30.4 , 0.4 and 0.4P P P% % % with the lines A1B1, A2B2 and A3B3 respectively are S1, S2 and S3 and

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 1 2,P P% % and 3P% with membership value α = 0.4. According to the

definition of fuzzy triangle, 1 2 3PP P∆% is union of all triangles like ∆ S1S2S3, ∆ S1S2S3, etc. with

membership value 0.4.α =

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 1 2 3PP P∆% is defined by fuzzy distance (Definition 2.7) between the vertices, i.e.,

1 2 2 3 3 1( , ), ( , ) and ( , )D P P D P P D P P% % % % % % % % % respectively; let us denote them as 3 1 2, and p p p% % % respectively.

The vertex angles of 1 2 3PP P∆% may be defined as the fuzzy angles (Definition 4.5 in [10])

1 2 2 3 2 3 3 1 3 1 1 2( , ), ( , ) and ( , );P P P P P P P P P P P PL L L L L L∠ ∠ ∠% % % % % %% % % and the notations 2 3 1, and P P P∠ ∠ ∠% % %% % % respectively

may be used to represent them. It is to note that vertex angle iP∠ %% is situated opposite to the side

with lengthip% , i = 1, 2, 3.


7

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: ( | ) sup{ : µ α∆ ∆ = ∆% is the area of the triangle constructed by the same points

( ) ( ) ( )1 2 30 , 0 and 0u P v P w P∈ ∈ ∈% % % as its vertices with 1 2 3( | ) ( | ) ( | ) }.u P v P w Pµ µ µ α= = =% % %

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 ( ) ( ) ( )1 2 3, and u P v P w Pα α 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 1 2 3PP P∆% say, let us consider one of its side say

1 2P PL%

.

We consider a line l(x, y) perpendicular to 1 2

(1)P P

L%

at (x, y) in1 2P P

L%

. As1 2P P

L%

is a fuzzy line

segment, along the line l(x, y) there must exists one fuzzy number [10] given by l(x, y) ∩1 2

.P P

L%

We denote this fuzzy number by3 ( , )l x y% . Thus, corresponding to each (x, y), the function

3 ( , )l x y% always gives one fuzzy number. We will say the function defined by (x, y) →3 ( , )l x y% as

the imprecision function of the side1 2P P

L%

. Similarly, there will be two more imprecision functions

1( , )l x y% and 2 ( , )l x y% corresponding to the sides

2 3P PL%

and 3 1P PL

% respectively. It is noticeable that

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 1 2 3PP P∆% and 1 2 3Q Q Q∆% are two fuzzy triangles.

1 2 3, ,p p p% % % and 1 2 3, ,q q q% % % being their side lengths and ,i ilp lq% % for i =1, 2, 3 are their imprecision

functions of the corresponding sides. Fuzzy triangles 1 2 3PP P∆% and

1 2 3Q Q Q∆% are said to be similar


8

if there exists k in R such that

(i) i ip kq=% % , for all i and

(ii) ( , ) ( , )i i i ii p t p t i q t q tlp x y klq x y=% % ∀ t ∈ [0, 1] for each i = 1, 2, 3, where t is taken as follows.

For i = 1, let us consider 3 2P P

L%

and ( )1 1 2 3

( , ) 1p t p t P Px y L∈ % . Then the point 1 1

( , )pt ptx y corresponds

to that t for which (xp1t,yp1t)= t P2 + (1 − t) P3, t ∈ [0, 1]. Similar relation will be applied

for the sides 1 2

,P P

L%

1 3,P PL

%1 2

,Q Q

L%

3 13 2

and .Q Q Q Q

L L% %

If above two conditions hold true, then one of the fuzzy triangles 1 2 3PP P∆% and

1 2 3Q Q Q∆% 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 iP% and

iQ% for i = 1, 2, 3 are six fuzzy points. If two fuzzy triangles

1 2 3PP P∆% and 1 2 3Q Q Q∆% 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 1 2 3PP P∆% and 1 2 3Q Q Q∆% are fuzzily similar triangles. Therefore, imprecision function

and length of each side of 1 2 3PP P∆% are some constant multiplication of the imprecision function

and side length of the corresponding side of 1 2 3Q Q Q∆% . So, one of the fuzzy triangles is enlarged

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 thati iP kQ= %% .

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 1 2,P P% % and

3P% and pα∆ is the crisp triangle in the support of

1 2 3PP P∆% joining these three same points. Then, it is easy to observe that (kx1, ky1), (kx2, ky2) and

(kx3, ky3) are same points with membership value α of the fuzzy points 1 2,Q Q% % and

3Q% ; and the

crisp triangle qα∆ with vertices as (kx1, ky1), (kx2, ky2) and (kx3, ky3) will have membership value α

in the fuzzy triangle1 2 3Q Q Q∆% . Hence the result follows.

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-Side-

Side) 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 S-

A-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


9

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 3 = (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 P 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 3 distance apart from P(0, 0). We

note that there are infinitely many such fuzzy points. For example some of them are R1(3, 0) with

support {(x, y): (x − 3)2

+ y2

≤ 1} and membership function is a right circular cone, R0.5(3, 0) with

support {(x, y): (x − 3)2

+ y2

/0.52

≤ 1}and membership function is a right circular cone, and Rε(3,

0) with support {(x, y): (x − 3)2

+ y2

/ε2

≤ 1}and membership function is a right circular cone where

ε ∈ (0, 1]. The fuzzy number (2/3/4), R say, along x-axis is itself a fuzzy point at (3, 0) which is 3

distance apart from P(0, 0). The fuzzy point Rε(3, 0) is shown in the Figure 2.

Figure 2: Two fuzzy points Qε and Rε having distance 3 and�2 2 respectively from P(0, 0)


10

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

0). For instance some of them are Q1(2, 2) with support {(x, y): (x + y − 4)2

/ 22

+ (x − y)2

≤ 1} and

membership function is a right circular cone, Q0.5(2, 2) with support {(x, y): (x + y − 4)2

/ 22

+ (x −

y)2

/0.52

≤ 1} and membership function is a right circular cone, and Qε(2, 2) with support {(x, y): (x

+ y − 4)2

/ 22

+ (x − y)2

/ ε2

≤ 1} and membership function is a right circular cone where ε ∈ (0, 1].

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

point at (2, 2) which is �2 2 distance apart from P(0, 0). Fuzzy point Qε(2, 2) is shown in the

Figure 2.

Here for the fuzzy triangles PQ Rε ε∆% with 0 < ε < 1, lengths of the sides PQ

Lε

% and

PRL

ε

%are 3 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 3 and �2 2 and fuzzy angle between

them is ‘around 45◦

’.

Let us measure the angle Q PRε ε∠% . We know its value is ‘around 45◦

’, but let us evaluate its

appropriate value. The same points of the fuzzy points , QP ε%% and Rε

% with membership value 0 are

(cos θ, sin θ),

1

(2 [(2 ) cos (2 ) sin ], 2 [(2 ) cos (2 ) s in ]) 2 2

ε θ ε θ ε θ ε θ+ − + + + √ + + − and

(3 cos , sin )θ ε θ+ respectively.

Thus support of the angle Q PRε ε∠% will be given by min max[ , ]ε εθ θ where

min0 2

max0 2

1 1

min ( ),

max ( ) and

12 [(2 ) cos (2 ) sin ] sin

( 1)sin2 2( ) tan tan .

1 32 [(2 ) cos (2 ) sin ] cos

2 2

f

f

f

ε

θ π

ε

θ π

θ θ

θ θ

ε θ ε θ θε θ

θε θ ε θ θ

≤ ≤

≤ ≤

− −

=

=

+ √ + + − −−

= −+ − + + −

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

also varies. Hence if support of angle ‘around 45◦

’ and its membership function is known

previously, there may not exist any fuzzy triangle Q PRε ε∆% for which Q PRε ε∠% is exactly equals

to the given around 45◦

.

From this example, we note that for given two fuzzy numbers a and b, a fuzzy angle θ and a fuzzy

point P, a fuzzy triangle may not be found whose two side lengths are a and b and corresponding

vertex and vertex angle are P and θ respectively. For example, in the above example if a and b

would have been (1 / 2 / 3) and θ = (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


11

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

triangles are 1 2 3PP P∆% and

1 2 3Q Q Q∆% ; for them the side lengths and imprecision functions are given

for the given side1 2P PL

%and

1 2Q QL%

respectively; the given vertex angles are Θ1 and Θ2; the angle Θi is

the vertex angle ∠ Pi and ∠ Qi for i = 1, 2. Let lpα and lqα are two line segments with membership value α on

the support of1 2P P

L% and

1 2Q QL% respectively. Now let us consider two triangles pα∆ and qα∆ whose

two sides are lpα and lqα and two vertex angle on the two extremities of lpα and lqα are θ1 ∈Θ1 and θ2 ∈Θ2 with

membership values α ∈ [0, 1]. We observe that pα∆ and qα∆ are similar triangle. As ∆P1P2P3 = ∨α∈[0,1]

∆pα and ∆Q1Q2Q3 = ∨α∈[0,1] ∆qα, the fuzzy triangles 1 2 3 ,PP P∆% 1 2 3Q Q Q∆% are similar.

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 iP% and iQ% are fuzzy points such that iQ% = k iP% , where k is some real constant, then

following results hold.

(i) If (x, y) belongs to 1 2 3PP P∆% (1), then (kx, ky) lie on 1 2 3Q Q Q∆% (1) and ( , ) ( , )i ip ql x y kl kx ky=% %

where ipl

% and iql%

for some i in{1, 2, 3} are imprecision function as defined in the Definition 4.1.

(ii) Fuzzy triangles 1 2 3PP P∆% and

1 2 3Q Q Q∆% are similar.

(iii) If and p q∆ ∆% % are area of the fuzzy triangles 1 2 3PP P∆% and

1 2 3Q Q Q∆% respectively, then

2.q pk∆ = ∆% %


12

Proof: (i) First part that if (x, y) belongs to1 2 3PP P∆% (1), then (kx, ky) will lie on

1 2 3Q Q Q∆% (1) is

trivially true. For the second part that ( , ) ( , )i ip ql x y kl kx ky=% % for i in{1, 2, 3}, we will prove that

( , )ipl x y =% ( , )

iqkl kx ky% for i =1 and similar will be the case for i = 2, 3. To prove 1( , )pl x y =%

1( , )qkl kx ky% we let (x1,y1), (x2,y2) belong to 2P% with membership value α in [0, 1] and (x3,y3),

(x4,y4) belong to3P% with membership value α in [0, 1]. Let us also suppose that (x1,y1), (x3,y3) are

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 13

kl and

24

kl

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 13

kl and

24

kl is k times of the distance

between l13 and l24. Thus support of the imprecision functional value1( , )pl kx ky% will be k times of

the support of the imprecision functional value 1( , )ql kx ky% for each (x, y) in

1 2P PL%

. Hence the result

follows.

(ii) This part is clear from the Theorem 4.1.

(iii) From the construction of1 2 3PP P∆% , 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

triangle. We note that if ∆ is the area of the crisp triangle ∆, then k2

∆ will be the area of

the triangle having vertices as k(x1, y1), k(x2, y2) and k(x3,y3).

€


13

Figure 3. Fuzzy triangles 1 2 3PP P∆% and 1 2 3Q Q Q∆% where iQ% = k iP% for k = 3.

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’.


14

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

proposed study of fuzzy triangle has been made in a coordinate reference frame of R2

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 1 2 3PP P∆ as prototype,

fuzzy triangle can be obtained by f-transformation (x, y) → ( )1 , ,l x y% ( )2 , l x y% or ( )3 , l x y% . Thus

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.

REFERENCES

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15

Letters, 12 (1991) 531–535.

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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.