Fuzzy plane geometry II: Circles and polygons

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sets and systems ELSEVIER

Fuzzy Sets and Systems 87 (1997) 79-85

Fuzzy plane geometry II: Circles and polygons J.J. B u c k l e y a'*, E. E s l a m i b a Mathematics Department, University of Alabama at Birmingham, Birmingham, AL 35294, USA b Department of Mathematics and Computer Science, Kerman University, Kerman, Iran

Received April 1995; revised October 1995

Abstract

This paper continues our research in fuzzy plane geometry. Previously, we studied fuzzy points and fuzzy lines. Now we investigate fuzzy circles and fuzzy polygons. We show that the fuzzy area of a fuzzy circle, or a fuzzy polygon, is a fuzzy number. We also argue that the fuzzy perimeter of a fuzzy circle, or a fuzzy polygon, is a fuzzy number. (~) 1997 Elsevier Science B.V. Keywords: Fuzzy sets; Geometry

I. Introduction

We will first present the basic notations, and the definitions and results from [3], that will be needed in this paper. Then we review the previous literature on fuzzy plane geometry and discuss how it relates to our results in this paper. Let us introduce the notation that will be used in the rest of this paper. We will place a " b a r " over a capital letter to denote a fuzzy subset of R or R 2. So, X, Y, ,4,/~, C .... all represent fuzzy subsets o f R n, n = 1,2. Any fuzzy set is defined by its membership function. I f A is a fuzzy subset of R, we write its membership function as/~(x IA ), x in R, with #(x 1.4 ) in [0, 1] for all x. If/5 is a fuzzy subset o f R 2 we write/~((x, y ) I P ) for its membership function with (x, y) in R 2. The acut of any fuzzy set )( of R, written )((~), is defined as {x: #(x I)?) ~>c¢}, 0 < c~< 1. )((0) is the closure of

the union o f ) ( ( ~ ) , 0 < ~ < 1. Similar definitions for a-cuts of fuzzy subsets of R 2. We will adopt the definition of a real fuzzy number given in [7, 8]. N is a (real) fuzzy number if and only if 1. #(x [~') is upper semi-continuous; 2. p(x[N-) = 0 outside some interval [c,d]; and 3. there are real numbers a and b so that c<~a<~b<~d and # ( x [ N ) is increasing on [c,a], # ( x [ N ) is decreasing on [b,d], #(x Ibm)= 1 on [a,b].

It is well-known that .N(~) is a (bounded) closed interval for all c¢, when N is a fuzzy number. A special type of fuzzy number is a triangular fuzzy number. A triangular fuzzy number N is defined by three numbers a , b , c so that: (1) a < b < c; (2) the graph of y = #(x I N ) is a triangle with base [a, c] and vertex at (b, 1 ). We denote triangular fuzzy numbers as N = (a,b,c).

* Corresponding author. Tel.: +1-205-934-2154; fax:+l-205934-9025; e-mail: [email protected].

Now we need to review two concepts from [3], fuzzy point and distance, needed in this paper.

0165-0114/97/$17.00 (~ 1997 Elsevier Science B.V. All rights reserved PI1 S01 65-01 14(96)00295-3

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J.J. Buckley, E. EslamilFuzzy Sets and Systems 87 (1997) 79-85

A fuzzy point at (a,b) in R 2, written P(a,b), is defined by its membership function: 1. #( (x, y ) lP( a, b ) ) is upper semi-continuous; 2. # ( ( x , y ) l f f ( a , b ) ) = l if and only if (x,y) = (a, b); and 3. /5(~) is a compact, convex, subset of R 2 for all ~,0~<~<1. We will some times abbreviate/5(a, b) as just/5. Next we define the fuzzy distance between fuzzy points. Let d(u, v) be the usual Euclidean distance metric between points u and v in R 2. We define the fuzzy distance L) between two fuzzy points #(al,bl) and /5(a2, b2) in terms of its membership function/~(d I/5). Let f2(~)= {d(u, v): u is in/5(al, bl )(00 and v is in /5(a2,b2)(=)}, 0~
special cases of fuzzy rectangles and triangles. The last section briefly discusses directions of future research.

2. Fuzzy circles In this section we: (1) first decide on the definition of a fuzzy circle; (2) show how to get ~-cuts of a fuzzy circle; (3) consider some examples of fuzzy circles; (4) define the fuzzy area of a fuzzy circle and show that it is a fuzzy number; (5) define the fuzzy circumference of a fuzzy circle and prove that it is also a fuzzy number; and (6) look at some examples of fuzzy area and circumference of fuzzy circles.

2.1. Definitions We know that the equation X2 + ax + y2 + by = C defines a circle when 4c > a 2 + b 2. Our first method is to fuzzify this procedure. Method 1. Let A, B, C be fuzzy numbers. A fuzzy circle ~ is all pairs of fuzzy numbers (J7",17)which are solutions to

(R) 2 + J 2 + (17)2 + M = d,

(1)

where 4C' > (A)2 + (j~)2. However, we know [4] that Eq. (1) usually has no solution (using standard fuzzy arithmetic) for R and I7. Therefore, we will not employ Method 1 in defining a fuzzy circle. Another procedure in specifying a circle is to use the standard equation for a circle: (x - a) 2 + (y - b) 2 = c2. This leads us to our second method of defining a fuzzy circle. Method 2. Let A,/~, C be fuzzy numbers. A fuzzy circle c~ is all pairs of fuzzy numbers Og,17) which are solutions to (o? - ~i )2 + (f _ a)2 = (0)2.

(2)

Unfortunately, Eq. (2) also has few, if any, solutions for)? and 17[4]. We need to try another way in defining fuzzy circles [2, 5, 6].

J.J. Buckley, E. EslamilFuzzy Sets and Systems 87 (1997) 79-85

81

Method 3. Let A, B, C be fuzzy numbers. Let

2.3. Examples

f2(a) = {(x, y): (x - a) 2 -b (y - b) 2 = c 2,

Example 1. This will be a thick (or "fat") fuzzy cir-

a E ,4(a), b E/~(a), c E C(a)},

(3)

for 0 ~
(4)

We will adopt Method 3 to define a fuzzy circle. As we shall see this type of fuzzy circle will have desirable properties including its fuzzy area and circumference are fuzzy numbers (or real numbers as a special case of fuzzy numbers).

cle. Let A = (0/1/2) = / ~ = C, all triangular fuzzy numbers. The support of ~, ~ ( 0 ) = 12(0), is the rectangle [ - 2 , 4] x [ - 2 , 4] with rounded edges. In fact, all a-cuts of (~, 0 ~
Example 2. A "regular" fuzzy circle. By a "regular" fuzzy circle we mean that a-cuts, 0 ~
2.2. Alfa-cuts

2.4. Area

As the following theorem shows it is not too difficult to obtain a-cuts of fuzzy circles.

Definition 1. Let A, B, C be fuzzy numbers in the definition of fuzzy circle c~ (Method 3). Set

Theorem 1. cg(a) = f2(a), 0,.
~r~a(a ) = {0:[9 is the area o f ( x - a ) 2 q - ( y - b ) 2 = c 2,

Proof. We first argue that the a-cuts are the same for 0 < a-~< 1. Let v E O(a). Then #(v[Cg)f>a and I2(a) is a subset of c~(a). We now argue that cg(a) is a subset of f2(a). Let v E c~(a). Then p ( v l ~ ) / > a . Set #(v[C~) = ft. We consider two cases: (a) fl > a; and (b) fl = a. (a) We have assumed fl > a. There is a 7, a < 7 ~ 0 there is a positive integer N so that a - e < 7,,n>>.N. Now v in 12(7n) for all n implies that v is also in I2(a - e), any e > 0. If v = (x, y), then (x - a) 2 + (y - b) 2 = c 2 for some a in A ( a - e), b i n / l ( a - e), c in C(a - ~). This means that/~(a [A)~>a - ~, p(blJ~) ~>a - e, #(c [t~)>~a - e. But e > 0 was arbitrary. So p(a [.4 ) >1a, #(b ]/~)/> a, p(c [ C)/> a and a E .4 (a), b E B(a), c E C(a). Hence v = (x, y) is in f2(a). We have shown that qq(a) is a subset of 12(a). It follows that ~ ( 0 ) = t2(0) since ~(ct) = f2(a), 0 < a-..< 1. []

a E .4(a), b E/~(a), c E C(a)},

(5)

0 ~
~(01o) -- sup(a:

0 E f2a(a)}.

(6)

There are some degenerate cases we must consider where O becomes a (crisp) real number (a special case of a fuzzy number). O f course, if.4,/1, C are all real numbers, then c~ is a crisp circle and O is a real number. So now assume that at least one of the A,/1, is a fuzzy number which is not a real number. The only case where O can degenerate into a crisp number is when C is real. Let C = c, c > 0, a real number. Then O = nc 2 a real number. So, in the next theorem let us assume that C is not a real number.

Theorem 2. O ( a ) = f2a(a), all a, and 0 is a fuzzy number. Proof. The proof that ~ ( a ) = I2a(a) for all a is similar to the proof that cg(a) = I2(a) all a, in Theorem 1, and is omitted. We show that O is a fuzzy number.

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J.J. Buckley, E. EslamilFuzzy Sets and Systems 87 (1997) 79-85

(a) Since a-cuts of A,/~, C are compact intervals it follows that (2(a) is a bounded closed interval for all a. Let f2a(a ) = [l(a),r(a)], 0-..
2.5. Circumference Definition 2. Let A, B, C be fuzzy numbers in the definition of c~. Define

Oc(a) = {A: A is the circumference of ( x - a ) 2 + (y - b) 2

=

C2, a EA(a),

b E/~(a), c ~ 0(a)},

(7)

0 ~
(8)

In the following theorem assume that C is not a real number. If (~ is real, then $ is a real number.

Theorem 3. 6(a) = f2c(~) for all a and 5 is a fuzzy number. Proof. Similar to the proof of Theorem 2 and is omitted. []

2.6. Examples

and vertex at (4n, 0). The fuzzy circumference 5 is a triangular fuzzy number (2n/4n/6rt). Example 4. We continue Example 1 where A = = C --- (0/1/2). Alfa-cuts of C are [a, 2 - a], 0 ~
3. Fuzzy polygons In this section we: (1) first discuss what we mean by a regular n-sided polygon; (2) define (regular) n-sided fuzzy polygons; (3) compute a-cuts of fuzzy polygons; (4) define the fuzzy area and perimeter of a fuzzy polygon and show that they are fuzzy numbers; and (5) look at two special cases as fuzzy rectangles and triangles.

3.1. Regular polygons A polygon is a rectilinear figure with n sides, n >~3. Quite often one assumes that n > 4 but in this paper we will allow n = 3 and n = 4 so that triangles and rectangles can be considered polygons. A regular polygon will have a convex interior. Let us make this definition more precise. Definition 3. We will say distinct points vl . . . . . v. in R 2 are convex independent if and only if any vi does not belong to the convex hull (convex closure) of the rest of the vj, 1 <~j <~n, j ~ i. Let vl .... , vn be convex independent. Assume they are numbered counterclockwise. That is, as we travel from Vl to v2. . . . . v,-1 to v, we continually travel in a counterclockwise direction. Now connect adjacent vi with line segments. So we draw a line segment 112 from Vl to v2. . . . . lnl from v, to vl. The vi, together with the line segments, define a regular n-sided polygon. Such a polygon we shall abbreviate as a n-gon. The interior of a n-gon is convex. When n = 3 we get a triangle.

E x a m p l e 3 . Consider the "regular" fuzzy circle from Example 2. We had A = / ~ = I , C=(1/2/3) a triangular fuzzy number. Alfa-cuts of C are [_~ + 1,3 - a], 0 ~
3.2. Fuzzy polygon

O(a) = [~(a + 1) 2, ~(3 - a)2], $(a) = [2~(a + 1), 2n(3 - a ) ] , 0~<~<1. So, the fuzzy area 8 is a triangular-shaped fuzzy number with support [n, 9n]

Let v l , . . . , v, be n distinct points in the plane that define a n-gon (a regular n-sided polygon). Next let

J.£ Buckley, E. Eslamil Fuzzy Sets and Systems 87 (1997) 79-85

83 m

Pi be a fuzzy point at vi, 1 ~ i <~n. We now require the definition of a fuzzy line segment. Definition 4. Let /5 and 0 be two distinct fuzzy points. Define (see [3]) I2t(e) = {line segments: from a point in/5(ct) to a point in 0(e)}.

(9)

The fuzzy line segment Lpq, from/5 to Q, is p((x,y) IZpq) = sup{co (x,y) E t2t(ct)}.

3.3. Alfa-cuts (10)

As in the proof of Theorem 1 we can show that Lpq = Ql(ct) for all ~. m

Example 6. Let n = 4 with P1 a fuzzy point at (0, 0), P2 a fuzzy point at (1,0), P3 a fuzzy point at (1, 1), and P4 a fuzzy point at (0, 1). Let Pl be a right circular cone with base B1 = {(x,y): x 2 + y2 ~<(0.1)2} and vertex at (0, 0). The rest of the Pi, i > 1, will just be rigid translations of P1 to their position in the plane. Then ~ is strongly non-degenerate; in fact, we would call ) a fuzzy rectangle (Section 3.5).

n

Theorem 4. ~ ( a ) = Ui=l ffii(e)for all ~. Proofi We know that Li(ot) = Qi(o~) where f2i(ct) = {line segments: from P~.(~) to P/+I(~)},

m

Definition 5. Let L1..... Ln be fuzzy line segments from P1 to P2 ..... Pn to P1, respectively. Then a (regular) n-sided fuzzy polygon ) is

(13) 0 ~<~ ~ 1. In the above equation we replace i + 1 by 1 when i = n. So, we need to show

(11)

) = LJZT,.

~(a)

i=l

~-~

U~'~i(a),

0~<~
(14)

i=1

The membership function for ~ is # ( ( x , y ) [ ~ ) = max { # ( ( x , y ) lL--i)}.

(12)

l<~i<~n

If we have two P/, say P1 and P2, so that P2(0) is a subset of P1 (0), then we will obtain a degenerate fuzzy polygon. In a degenerate fuzzy polygon the support, ~(0), will not show all the (fuzzy) n vertices. Example 5. Let n = 3 and the fuzzy points are all right circular cones. P-~ has base B1 = {(x, y): X 2 + y2<(0.1)2} and vertex at (0,0), P22 has base B2 = {(x,y): (x - 1) 2 + y2 ~<(0.1)2} with vertex at (1,0), and P3 has base B3 = {(x, y): ( x - 1)2 +(y_0.5)2 ~<1} having vertex at (1,0.5). Clearly, P2(0)C P3(0) and Pl(0), P3(0) are disjoint. Therefore, L2(O)CP3(O) and ) is degenerate. Also, L--T(0)c L--a(0) so the support of ~ is simply L-~(0).

Definition 6. We will abbreviate a regular n-sided fuzzy polygon as a fuzzy n~gon. We will say a fuzzy n-gun is non-degenerate if P-~.(0) is not a subset of Pj(0), j # i, for all i = 1.... , n. We say the fuzzy ngon is strongly non-degenerate if the Pi(O), 1 <~i <~n, are pairwise disjoint.

We first argue that Eq. (14) is true for 0 < ct~<1. Let v belong to the union of the f2i(a). Then there is some value of i, say i = 1, so that v E f21(a). Then /~(vlL1)~>e which implies that (by Eq. (12)) /~(v[~)>~a and v belongs to ~(~). We have shown that the union of the f2i(e) is a subset of ~(e). Let v belong to ~ ( e ) and let # ( v [ ~ ) = ft. Then f l > ~ or fl=~x. Suppose fl=cc By Eq. (12) there is an i, say i-- 1, so that p(v IL1)=~. Hence v E t21(ct) and v belongs to the union of the Qi(a). Suppose/3 > c~. By Eq. (12) there is an i, say i = 1, so that p(v [Ll)=fl. Then V E ~"~l(fl). But since fl > ct we also have v in I21(~). Hence v belongs to the union of the Oi(a). We have shown that )(ct) is a subset of the union of the f2i(e). Since Eq. (14) holds for 0 < a~
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J.J. Buckley, E EslamilFuzzy Sets and Systems 87 (1997) 79-85

Define flu(a) = {areas: n-gons defined by vi E Pi(~), l <~i<<.n},

(15)

0 ~<~ ~<1. The area 8 of 3~ is #(01 8 ) = sup{a: 0 E f2a(a)}.

(16)

In a similar way we can define the fuzzy perimeter of a fuzzy n-gon. Definition 8. Let ~ be a strongly non-degenerate fuzzy n-gon defined by fuzzy points Pi, i <~i <~n. Set 12p(~) = {perimeter: n-gons defined by vi in Pi(~), l<.i<~n),

(17)

0 ~ ~ ~<1. Then the fuzzy perimeter 3 is #(A I $) = sup{a: A E ap(~)}.

(18)

numbers. Now, since we have a fuzzy triangle we may investigate the beginnings of fuzzy trigonometry (see also [12]). For the rest of this subsection we assume that the fuzzy triangle is strongly non-degenerate. Definition 9. Let P1, P2, P3 be the three fuzzy points that define the fuzzy triangle off-.Let L12, L23, L31 be the fuzzy line segments connecting P1 to P2, P2 to P3, and P3 to P1, respectively. Define O~(~) = {angle in rad between 112 and 131: 112 is a line segment from a point in PI(~) to a point in P2(a), 131 is a line segment from P3(a) to PI(~), and 112, 131 initiate from the same point in Pl(ct)}, (19) 0 ~
(20)

Theorem 5. 8(00 = f2a( O0 and ~( ~z) = f2p( a ) for all o~.

Theorem 6. /)(a) = I2~(a) for alia and f l is a fuzzy number.

Proof. Similar to the proofs of Theorems 1 and 2 and is omitted. []

Proof. Follows that of Theorems 1 and 2 and is omitted. []

If we allow ~ to be degenerate we still obtain 8(00 = f2a(~), $(~) = Op(OO, for all ~ but 8, 3 may not be fuzzy numbers.

In the definition of/1 assume that P~ is a fuzzy point at vi in R 2, i -- 1,2, 3 . / I depends on the vi but also on the "size" of the fuzzy points. That is, if we substitute fuzzy point P2 at v2 for P2 and P2(a) c P~(a) for all a n d / F is the resulting fuzzy angle, then we can get H~< H r. So we could write/) = / ) ( P I , P 2 , P 3 ) to show the dependence o f / ) on the fuzzy points. Next, let us consider elementary fuzzy right triangle trigonometry. Consider fuzzy points P1, P2, P3 at vl = (0, 0), v2 = (a,O), v3 = (a,b), a > 0, b > 0, respectively. The Pi, 1 ~< i ~<3 define a fuzzy fight triangle because a triangle through the v~, 1 ~
Example 7. Consider the degenerate fuzzy polygon in Example 5. We see, since some a-cuts of P2 are inside the corresponding a-cuts of P3, that g2a(a) will be the interval (0, r(a)], 0 ~
Definition I0. We will first consider fuzzy triangles and then fuzzy rectangles. 3.5.1. Fuzzy triangles When n = 3 a fuzzy n-gon is a fuzzy triangle. When it is strongly non-degenerate we saw in the previous subsection that its fuzzy area and perimeter are fuzzy

tan(/I) =/)(P2, P3 )/D(P1, P2),

(21 )

where/5(P, Q) is the fuzzy distance from fuzzy point /St° Q. tan(/)) is well-defined, and a fuzzy number, because: (I) zero does not belong to the support of

J.J. Buckley, E. EslamilFuzzy Sets and Systems 87 (1997) 79-85

/3(Pl,P2); and (2) D(P2,P3) and D(PI,P2) are both fuzzy numbers. Similarly, we can define sin(/)) and cos(/)), both fuzzy numbers. However, many properties of triangles and crisp trigonometry identities (like cos2(0) + sin2(0) = 1) do not generalize to fuzzy triangles (see also [12]). The following example shows that the Pythagorean theorem may not hold for fuzzy right triangles.

Let A = D(P1,P2) ..... /3 = D(P4,P1). We see that¢[(0) . . . . . /3(0) = [0.8, 1.2]. So (A-/})(0) = [0.64, 1.44] and (A+ ... +/3)(0) = [3.2,4.8]. If f2a(e) = {area: rectangle with vertices in Pi(e), 1~
d =/3(E,~),/} =/3(~,~), c =/3(P3,P1). Now L12 and L23 are the two sides of the fuzzy right triangle J- and L31 is the hypotenuse. The lengths of L12, L23, L3--~areA, B, C, respectively. But (~)2 + (/})2 # ((~)z because.,t(0) =/}(0) = [0.8, 1.2], C(0) = [x/2 - 0.2, x/2 + 0.2] so that the support of (.4)2 + (/})2 is [1.28, 2.88] but the support of (~)2 is [1.474, 2.606].

3.5.2. Fuzzy rectangles Let Vl = (x, y), v2 = (x+A, y), v3 = (x+A,y+6), and v4 = (x, y + 6), A > 0, 6 > 0 be four points in the plane. Next let Pi be fuzzy point at vi, 1 <<.i <<.4. The Pi define a fuzzy 4-gon. We assume it is strongly non-degenerate. We will call this fuzzy 4-gon a fuzzy rectangle. We have seen in Section 3.4 that we can define the area and perimeter of a fuzzy rectangle and they will be fuzzy numbers. However, the fuzzy area of the fuzzy rectangle may not be the product of the side lengths and the fuzzy perimeter may not equal the sum of the side lengths. LetA = D(P1,P2), B = /3(P2,P3), C = /3(P3,P4) and/3 = /3(Pa,P1 ). First of all, since the a-cuts of the fuzzy points may have different sizes, we may get .4 # C and/} # / 3 . So, let us assume the fuzzy points are all the "same", just centered at different points in the plane. The following example shows that still the area may not be the product of the side lengths and the perimeter may not equal to the sum of the side lengths.

Example 9. Let P1, P2, P3, P4 be fuzzy points at/)1 = (0,0), v2 =(0, 1), v3 =(1, 1), v4 =(1,0), respectively. We get a fuzzy square. Assume each Pi is a right circular cone with base a circle of radius 0.1 centered at vi, and vertex at/3i, 1 ~
(22)

0 ~ • ~<1, then the fuzzy area O is /~(010) = sup{a: 0 E ~'~a(e)}.

Example 8. Let P1, P2, P3 be fuzzy points at /31 = ( 0 , 0 ) , /32 = (1,0), /)3 = (1, 1), respectively. E a c h Pi will be a right circular cone, base a circle of radius 0.1 centered at/3i, with vertex at vi, i = 1,2,3. Let

85

(23)

Similarly, we define the fuzzy perimeter 6. We easily see that the left end point of O(0) is greater than 0.64 and the left end point of 6(0) is more than 3.2. Hence,

# ~ L ~#A + . . . +/3. 4. Future research Our next research project is to extend our results to Rn, n >~3. We will define and study fuzzy points and lines in R n, n i> 3. Then introduce fuzzy planes in R 3 and fuzzy hyperplanes in R ~, n > 3. Of course, we will look at fuzzy distance in R ", n/> 3, and the intersection of fuzzy lines and fuzzy hyperplanes, etc.

References [1] A. Bogomolny, On the perimeter and area of fuzzy sets, Fuzzy Sets and Systems 23 (1987) 257-269. [2] J.J. Buckley, Solving fuzzy equations, Fuzzy Sets and Systems 50 (1992) 1-14. [3] J.J. Buckley and E. Eslami, Fuzzy plane geometry I: Points and lines, Fuzzy Sets and Systems 86 (1997). [4] J.J. Buckley and Y. Qu, Solving linear and quadratic fuzzy equations, Fuzzy Sets and Systems 38 (1990) 43-59. [5] J.J. Buckley and Y. Qu, Solving systems of fuzzy linear equations, Fuzzy Sets and Systems 43 (1991) 33-43. [6] J.J. Buckley and Y. Qu, Solving fuzzy equations: A new solution concept, Fuzzy Sets and Systems 39 (1991) 291-301. [7] R. Goetschel and W. Voxman, Topological properties of fuzzy numbers, Fuzzy Sets and Systems 10 (1983) 87-99. [8] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems 18 (1986) 31-43. [9] A. Rosenfeld, The diameter of a fuzzy set, Fuzzy Sets and Systems 13 (1984) 241-246. [10] A. Rosenfeld, Fuzzy rectangles, Pattern Recognition Lett. 11 (1990) 677-679. [11] A. Rosenfeld, Fuzzy plane geometry: Triangles, Proc. 3rd IEEE lnternat. Conf. on Fuzzy Systems, Orlando, Vol. II 26-29 June (1994) 891-893. [12] A. Rosenfeld and S. Haber, The perimeter of a fuzzy set, Pattern Recognition 18 (1985) 125-130.