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Notes on “Fuzzy plane geometry I, II”
Fuzzy Sets and Systems 121 (2001) 545–547
www.elsevier.com/locate/fss
Notes on “Fuzzy plane geometry I, II” Xuehai Yuan ∗ , Zhengwei Shen Department of Mathematics, Liaoning Normal University, Dalian 116029, People’s Republic of China Received 15 September 1998; received in revised form 31 May 2000; accepted 9 June 2000
Abstract In this paper, we give membership functions of some concepts in Buckley and Eslami (Fuzzy Sets and Systems 86 (1997) 179 –187, 87 (1997) 79 –85) such as fuzzy distance, fuzzy lines, fuzzy circles, fuzzy area and fuzzy circumference of a fuzzy circle. By the use of formulas of those membership functions, one can de7ne general fuzzy concepts in plane geometry. c 2001 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy sets; Membership function; Geometry
1. Introduction Buckley and Eslami have studied fuzzy plane geometry and given some important concepts such as fuzzy points, fuzzy distance between fuzzy points, fuzzy lines, fuzzy circles, fuzzy area and fuzzy circumference of a fuzzy circle, etc. These are very good works. In this paper, we give membership function of those concepts in [1,2]. By the use of formulas of those membership functions, one can de7ne general fuzzy concepts in plane geometry. For the sake of simplicity, we directly cite symbols in [1,2]. 2. Membership functions of some concepts in [1; 2] ? 1 ; b1 ) and P(a ? 2 ; b2 ) be two Theorem 1. Let P(a ? 1 ; b1 )(); v ∈ fuzzy points; () = {d(u; v) | u ∈ P(a ∗
Corresponding author.
? 2 ; b2 )()}; 0661; then the following statement P(a are equivalent: (i) ? = sup{d | d ∈ ()};
(d | D)
(1)
(ii) ?
(d | D) ? 1 ; b1 )) ∧ (v | P(a ? 2 ; b2 ))): = ( (u | P(a d=d(u;v)
(1) Proof. (i)⇒(ii): d∈()⇒d =d(u; v) for some u ∈ ? 1 ; b1 )() and v ∈ P(a ? 2 ; b2 )() ⇒ d = d(u; v) and P(a ? 1 ; b1)) ∧ (v | P(a ? 2 ; b2)) for some u; v ∈R 6 (u| P(a ? 1 ; b1)) ∧ (v| P(a ? 2 ; b2))) ⇒ ⇒ 6 d=d(u; v) ( (u| P(a ? ? 1 ; b1 ))
(d|D) = sup{ |d∈()}6 d=d(u; v) ( (u | P(a ? 2 ; b2 ))). Assume (d | D)¡ ? ∧ (v | P(a d=d(u; v) ( (u | ? 2 ; b2 ))). Then there is a ∈ [0; 1] ? 1 ; b1 )) ∧ (v | P(a P(a
c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 0 ) 0 0 0 9 9 - 3
546
X. Yuan, Z. Shen / Fuzzy Sets and Systems 121 (2001) 545–547
such that ? ¡
(d | D)
¡
? 1 ; b1 )) ∧ (v | P(a ? 2 ; b2 ))) ( (u | P(a
d=d(u;v)
then there are u; v such that d = d(u; v) and ¡ ? 2 ; b2)), it follows that ? 1 ; b1)) ∧ (v | P(a
(u | P(a ? ? u ∈ P(a1 ; b1 )(), v ∈ P(a2 ; b2 )() and d = d(u; v), and ? consequently d ∈ () and (d | D)¿. This is a contradiction. So ? = ? 1 ; b1 )) ∧ (v | P(a ? 2 ; b2 ))):
(d | D) ( (u | P(a d=d(u;v)
(ii) ⇒ (i): Let (d | D? 1 ) = sup{d | d ∈ ()}, then ? 1 ; b1 )) ∧ (v | P(a ? 2 ; b2 )))
(d | D? 1 ) = ( (u | P(a d=d(u;v)
? = (d | D):
Similarly, we have the following theorems. Theorem 2. L? a are fuzzy lines of methods 3 – 6 in [1] (a = 11; 12; 2; 3) if and only if
((x; y) | L?11 ) ? ∧ (b | B) ? ? ∧ (c | C)); = ( (a | A)
(2)
ax+by=c
? ( (m | M? ) ∧ (b | B));
(3)
y=mx+b
Theorem 3. Let L? 11 be a fuzzy line of method 3 ? and in [1]. Let (− (a=b) | M? ) = (a | A? ) ∧ (b | B) ? If ((x; y) | L? 12 ) = ?1 ) = (b | B) ? ∧ (c | C).
((c=b) | B ? ? ? y=mx+b ( (m | M ) ∧ (b | B1 )); then L12 is a fuzzy ? ? line of method 4 in [1] and L11 = L12 . Theorem 4. (1) C is a fuzzy circle of method 3 in [2] if and only if
((x; y) | C) =
? ∧ (b | B) ? ? ∧ (c | C)); ( (a | A) (6)
(2) is a fuzzy area of a fuzzy circle C in [2] if and only if ? ∧ (b | B) ? ? ∧ (c | C));
( | ) = ( (a | A) (a;b;c)∈2
(7) where 2 = {(a; b; c) | is a area of circle (x − a)2 + (y − b)2 = c2 ; for any a; b; c ∈ R}. (3) ? is an fuzzy circumference of a fuzzy circle C in [2] if and only if ? = ? ∧ (b | B) ? ? ∧ (c | C));
( | ) ( (a | A) (a;b;c)∈3
(8)
((x; y) | L?2 ) =
? ∧ (m | M? )); ( ((u; v) | K)
(4)
where 3 = {(a; b; c) | is a circumference of circle (x − a)2 + (y − b)2 = c2 ; for any a; b; c ∈ R}.
(5)
Note 2. One can de7ne general fuzzy concepts in plane geometry by the use of formulae (1) – (8). For ? C? be three fuzzy subsets of R, by example, let A;? B; the use of formula (6), we can de7ne a fuzzy circle as follows:
y−v=m(x−u)
((x; y) | L?3 ) =
v2 − v1 2 = ; (ui ; vi ) ∈ R (i = 1; 2) : u2 − u 1
(x−a)2 +(y−b)2 =c2
Note 1. We have seen from Theorem 1 that we are able to de7ne fuzzy distance between fuzzy points ? 1 ; b1 ) and P(a ? 2 ; b2 ) by formula (1). P(a
((x; y) | L?12 ) =
where y − v1 1 = ((u1 ; v1 ); (u2 ; v2 )) x − u1
( ((u1 ; v1 ) | P? 1 )
((u1 ;v1 );(u2 ;v2 ))∈1
∧ ((u2 ; v2 ) | P? 2 ));
X. Yuan, Z. Shen / Fuzzy Sets and Systems 121 (2001) 545–547
De nition. Let C be a fuzzy subset of R2 . If
((x; y) |C) =
? ∧ (b | B) ? ? ∧ (c | C)); ( (a | A)
(x−a)2 +(y−b)2 =c2
then C is called a fuzzy circle of R2 . We can see from this de7nition that (i) If A? = {a}, B? = {b} and C? = {c} are three sets of single element, then C = {(x; y) | (x −a)2 +(y −b)2 = c2 } is a circle. (ii) If A? = {a0 }, B? = {b0 } and C? is a fuzzy subset ? = 1 for some c0 ∈ R, then of R satisfying (c0 | C) ?
((x; y) | C) =
(c | C); (x−a0 )2 +(y−b0 )2 =c2
then C is a fuzzy circle with fuzzy boundary.
547
References [1] J.J. Buckley, E. Eslami, Fuzzy plane geometry I: points and lines, Fuzzy Sets and Systems 86 (1997) 179–187. [2] J.J. Buckley, E. Eslami, Fuzzy plane geometry II: circles and polygons, Fuzzy Sets and Systems 87 (1997) 79–85.
www.elsevier.com/locate/fss
Notes on “Fuzzy plane geometry I, II” Xuehai Yuan ∗ , Zhengwei Shen Department of Mathematics, Liaoning Normal University, Dalian 116029, People’s Republic of China Received 15 September 1998; received in revised form 31 May 2000; accepted 9 June 2000
Abstract In this paper, we give membership functions of some concepts in Buckley and Eslami (Fuzzy Sets and Systems 86 (1997) 179 –187, 87 (1997) 79 –85) such as fuzzy distance, fuzzy lines, fuzzy circles, fuzzy area and fuzzy circumference of a fuzzy circle. By the use of formulas of those membership functions, one can de7ne general fuzzy concepts in plane geometry. c 2001 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy sets; Membership function; Geometry
1. Introduction Buckley and Eslami have studied fuzzy plane geometry and given some important concepts such as fuzzy points, fuzzy distance between fuzzy points, fuzzy lines, fuzzy circles, fuzzy area and fuzzy circumference of a fuzzy circle, etc. These are very good works. In this paper, we give membership function of those concepts in [1,2]. By the use of formulas of those membership functions, one can de7ne general fuzzy concepts in plane geometry. For the sake of simplicity, we directly cite symbols in [1,2]. 2. Membership functions of some concepts in [1; 2] ? 1 ; b1 ) and P(a ? 2 ; b2 ) be two Theorem 1. Let P(a ? 1 ; b1 )(); v ∈ fuzzy points; () = {d(u; v) | u ∈ P(a ∗
Corresponding author.
? 2 ; b2 )()}; 0661; then the following statement P(a are equivalent: (i) ? = sup{d | d ∈ ()};
(d | D)
(1)
(ii) ?
(d | D) ? 1 ; b1 )) ∧ (v | P(a ? 2 ; b2 ))): = ( (u | P(a d=d(u;v)
(1) Proof. (i)⇒(ii): d∈()⇒d =d(u; v) for some u ∈ ? 1 ; b1 )() and v ∈ P(a ? 2 ; b2 )() ⇒ d = d(u; v) and P(a ? 1 ; b1)) ∧ (v | P(a ? 2 ; b2)) for some u; v ∈R 6 (u| P(a ? 1 ; b1)) ∧ (v| P(a ? 2 ; b2))) ⇒ ⇒ 6 d=d(u; v) ( (u| P(a ? ? 1 ; b1 ))
(d|D) = sup{ |d∈()}6 d=d(u; v) ( (u | P(a ? 2 ; b2 ))). Assume (d | D)¡ ? ∧ (v | P(a d=d(u; v) ( (u | ? 2 ; b2 ))). Then there is a ∈ [0; 1] ? 1 ; b1 )) ∧ (v | P(a P(a
c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 0 ) 0 0 0 9 9 - 3
546
X. Yuan, Z. Shen / Fuzzy Sets and Systems 121 (2001) 545–547
such that ? ¡
(d | D)
¡
? 1 ; b1 )) ∧ (v | P(a ? 2 ; b2 ))) ( (u | P(a
d=d(u;v)
then there are u; v such that d = d(u; v) and ¡ ? 2 ; b2)), it follows that ? 1 ; b1)) ∧ (v | P(a
(u | P(a ? ? u ∈ P(a1 ; b1 )(), v ∈ P(a2 ; b2 )() and d = d(u; v), and ? consequently d ∈ () and (d | D)¿. This is a contradiction. So ? = ? 1 ; b1 )) ∧ (v | P(a ? 2 ; b2 ))):
(d | D) ( (u | P(a d=d(u;v)
(ii) ⇒ (i): Let (d | D? 1 ) = sup{d | d ∈ ()}, then ? 1 ; b1 )) ∧ (v | P(a ? 2 ; b2 )))
(d | D? 1 ) = ( (u | P(a d=d(u;v)
? = (d | D):
Similarly, we have the following theorems. Theorem 2. L? a are fuzzy lines of methods 3 – 6 in [1] (a = 11; 12; 2; 3) if and only if
((x; y) | L?11 ) ? ∧ (b | B) ? ? ∧ (c | C)); = ( (a | A)
(2)
ax+by=c
? ( (m | M? ) ∧ (b | B));
(3)
y=mx+b
Theorem 3. Let L? 11 be a fuzzy line of method 3 ? and in [1]. Let (− (a=b) | M? ) = (a | A? ) ∧ (b | B) ? If ((x; y) | L? 12 ) = ?1 ) = (b | B) ? ∧ (c | C).
((c=b) | B ? ? ? y=mx+b ( (m | M ) ∧ (b | B1 )); then L12 is a fuzzy ? ? line of method 4 in [1] and L11 = L12 . Theorem 4. (1) C is a fuzzy circle of method 3 in [2] if and only if
((x; y) | C) =
? ∧ (b | B) ? ? ∧ (c | C)); ( (a | A) (6)
(2) is a fuzzy area of a fuzzy circle C in [2] if and only if ? ∧ (b | B) ? ? ∧ (c | C));
( | ) = ( (a | A) (a;b;c)∈2
(7) where 2 = {(a; b; c) | is a area of circle (x − a)2 + (y − b)2 = c2 ; for any a; b; c ∈ R}. (3) ? is an fuzzy circumference of a fuzzy circle C in [2] if and only if ? = ? ∧ (b | B) ? ? ∧ (c | C));
( | ) ( (a | A) (a;b;c)∈3
(8)
((x; y) | L?2 ) =
? ∧ (m | M? )); ( ((u; v) | K)
(4)
where 3 = {(a; b; c) | is a circumference of circle (x − a)2 + (y − b)2 = c2 ; for any a; b; c ∈ R}.
(5)
Note 2. One can de7ne general fuzzy concepts in plane geometry by the use of formulae (1) – (8). For ? C? be three fuzzy subsets of R, by example, let A;? B; the use of formula (6), we can de7ne a fuzzy circle as follows:
y−v=m(x−u)
((x; y) | L?3 ) =
v2 − v1 2 = ; (ui ; vi ) ∈ R (i = 1; 2) : u2 − u 1
(x−a)2 +(y−b)2 =c2
Note 1. We have seen from Theorem 1 that we are able to de7ne fuzzy distance between fuzzy points ? 1 ; b1 ) and P(a ? 2 ; b2 ) by formula (1). P(a
((x; y) | L?12 ) =
where y − v1 1 = ((u1 ; v1 ); (u2 ; v2 )) x − u1
( ((u1 ; v1 ) | P? 1 )
((u1 ;v1 );(u2 ;v2 ))∈1
∧ ((u2 ; v2 ) | P? 2 ));
X. Yuan, Z. Shen / Fuzzy Sets and Systems 121 (2001) 545–547
De nition. Let C be a fuzzy subset of R2 . If
((x; y) |C) =
? ∧ (b | B) ? ? ∧ (c | C)); ( (a | A)
(x−a)2 +(y−b)2 =c2
then C is called a fuzzy circle of R2 . We can see from this de7nition that (i) If A? = {a}, B? = {b} and C? = {c} are three sets of single element, then C = {(x; y) | (x −a)2 +(y −b)2 = c2 } is a circle. (ii) If A? = {a0 }, B? = {b0 } and C? is a fuzzy subset ? = 1 for some c0 ∈ R, then of R satisfying (c0 | C) ?
((x; y) | C) =
(c | C); (x−a0 )2 +(y−b0 )2 =c2
then C is a fuzzy circle with fuzzy boundary.
547
References [1] J.J. Buckley, E. Eslami, Fuzzy plane geometry I: points and lines, Fuzzy Sets and Systems 86 (1997) 179–187. [2] J.J. Buckley, E. Eslami, Fuzzy plane geometry II: circles and polygons, Fuzzy Sets and Systems 87 (1997) 79–85.