A new paradigm of fuzzy control point in space curve

Transcription

1 MATEMATIKA, 2016, Volume 32, Number 2, c Penerbt UTM Press All rghts reserved A new paradgm of fuzzy control pont n space curve 1 Abd Fatah Wahab, 2 Mohd Sallehuddn Husan and 3 Mohammad Izat Emr Zulkfly 1,2,3 School of Informatcs and Appled Mathematcs Unverst Malaysa Terengganu, 21300, Kuala Terengganu, Terengganu, Malaysa e-mal: 1 fatah@umtedumy, 2 salleh 85s@yahoocom, 3 emr zul@yahoocom Abstract The control pont s the most mportant element n the producton of splne curve or surface model Ths s because any changes of control ponts n the splne model affect the shape of the resultng curve or surface Wahab and colleagues have ntroduced fuzzy control ponts to solve the problem of uncertanty prevalng n the splne modelng However, based on ths concept, ths paper wll dscusses a new type of fuzzy control pont that can generates a splne space curve model n 3-dmensonal Ths s because the generated control pont s a 3-dmensonal that satsfes the basc concepts of fuzzy set was ntroduced by Zadeh However, ths paper only takng a B-Splne model as a numercal example n the dscussed model Keywords Space curve; fuzzy set; splne; control pont 2010 Mathematcs Subject Classfcaton 03E72, 93C42, 94D05 1 Introducton Fuzzy control pont s a geometrc coeffcent that was hybrdzed wth fuzzy approach to produce fuzzy geometrc modelng The fuzzy control pont was blended wth the splne bass functons to produce Fuzzy Splne Model such as Fuzzy Bezer, Fuzzy B-Splne and Fuzzy NURBS Non-Unform Ratonal B-Splnes Idea of fuzzy control pont has been ntroduced by Wahab et al to resolve the uncertanty prevalng n geometrc modelng Many prevous studes related to fuzzy geometry have been dscussed to solve the problem of uncertanty n ndustral desgn and Geographc Informaton System GIS [1-3] However, ths paper wll dscuss a new paradgm of fuzzy control pont whch generates space curve n 3-dmensonal space that meets the characterstcs of fuzzy set was ntroduced by Zadeh n 1965 The resultng curve looks lke floatng n a 3-dmensonal space that can be seen n varous planes or axes 2 Prelmnares In ths secton, we brefly revew the concept of fuzzy sets [4] and [5], the α cuts of fuzzy sets [6], the fuzzy number [7] and the fuzzy control ponts [8-12] 21 Fuzzy set In 1965, Zadeh proposed the theory of fuzzy sets [4] The fundamental of fuzzy set theory wll be used n solvng the problem of uncertanty data Defnton 1 Let X be a nonempty set A fuzzy set A denoted à n X s characterzed by a membershp functon µ A : X [0, 1] and µ A x s called the degree of membershp

2 154 Abd Fatah Wahab, Mohd Sallehuddn Husan and Mohammad Izat Emr Zulkfly element x n à for every x X Fuzzy set s also a set of ordered pars and can be wrtten as à = {x, µ A x : x X} [11] If X = {x 1, x 2,, x n } s a fnte set and à X, then à can be wrtten as à = {x 1, µ A x 1,x 2, µ A x 2,, x n, µ A x n } X 1 Chen and Chang [13] represented the trapezodal fuzzy set shown n Fgure 1 by quadruple a 1, a 2, a 3, a 4 where A = a 1, a 2, a 3, a 4 If a 2 = a 3, the trapezodal fuzzy set shown n Fgure 1 becomes a trangular fuzzy set, as shown n Fgure 2, where A = a 1, a 2, a 3, a 4 = a 1, a 3, a 3, a 4 Fgure 1: A trapezodal fuzzy set A Fgure 2: A trangular fuzzy set A Defnton 2 Let A be a fuzzy set n the unverse of dscourse X and let µ A be the membershp functon of the fuzzy set A The α cut A α of the fuzzy seta s an nterval n the unverse of dscoursex defned as follows [14]: A α = {x x X and µ A x α} 2

3 A new paradgm of fuzzy control pont n space curve 155 where α 0, 1] A fuzzy set n the unverse of dscourse X s convex f and only f for all x 1, x 2 n X, µ A λx λx 2 mnµ A x 1,µ A x 2 3 where λ 0, 1] If x X, such that µ A x = 1, then the fuzzy set s called a normal fuzzy set A fuzzy number s a fuzzy set n the unverse of dscourse that s both convex and normal Defnton 3 Let X, Y R be unversal sets, then s called a fuzzy relaton on X Y [15] R = { x, y, µ R x, y x, y X Y } 4 Defnton 4 Let X, Y R and A = {x, µx x X} and B = {y, µy y Y } are A B two fuzzy sets Then { x, } R = y, µ x, y x, y X Y s a fuzzy relaton on A and R B f µ µx x, y X Y and µ µy x, y X Y [15] R A R B Defnton 5 Let X, Y R and M = {x, µ M x x X} and N = {x, µ N y y Y } represent two fuzzy data Then the fuzzy relaton between both fuzzy data s gven by C = { x, y, µ C x, y x, y X Y } [15] Defnton 6 Set D = { D } n a space ψ D consdered as uncertanty data fuzzy data f and only f for every D wth = 0, 1, 2,, n, ts grade membershp that defned by functon f D = α 0, 1] for D [a, b, f ncreasng 1 for D µ D D = [b, c], atpeak 5 g D = β 0, 1] for D c, d], f decreasng 0 for D < a ord > d, f D = g D = 0 { } n { } Then, D = D 0 = D 0, D 0, D 0, D 1, D 1, D 1,, D n, D n, D n are 3-tuple set nterval forms of fuzzy data If α 0, 1], then D α = D =0 { } ε, D, D + δ n ψ D α wth D [b, c] If f D = g D, then ɛ = δ, and f f D g D, then ɛ δ Defnton 7 Fuzzy set P n a space of S sad set of fuzzy control pont f and only f for every α level set was chosen, there exst ponted nterval whch s P = P, P, P n S wth every P s crsp pont and membershp functon µ P : S 0, 1] whch s defned as µ P P = 1 [13], [14] and [15] 0 f P / S 0 f P / S µ P P = c 0, 1 f P S S 1 f P and µ P P = k 0, 1 f P S 1 f P S 6

4 156 Abd Fatah Wahab, Mohd Sallehuddn Husan and Mohammad Izat Emr Zulkfly wth µ P P and µ P P are left membershp grade value and rght membershp grade value respectvely and generally wrtten as { } P = P : = 0, 1, 2,, n 7 for every P = P, P, P wth P, P and P are left fuzzy control pont, crsp control pont and rght control pont respectvely [8], [9], [16] and [17] 3 Fuzzy system of lnear equaton The n n fuzzy system of lnear equatons wth m degree polynomal parametrc form may be wrtten as ã 11 x 1 + ã 12 x ã 1n x n = b 1 ã 21 x 1 + ã 22 x ã 2n x n = b 2 ã n1 x 1 + ã n2 x ã nn x n = b n [Ã] { b} In matrx, notaton the above system may be wrtten as {X} =, where the coeffcent matrx = ã kj = a kj, a kj, a kj [Ã], 1 k, j n s a fuzzy n n matrx { b} = bk = b k, b k, b k, 1 k s a column vector of fuzzy number and {X} = {x j} s a vector of crsp unknown For postve nteger m, all ã kj and b k are fuzzy numbers wth m degree polynomal [Ã] form } The above system {X} = { b, can be wrtten as n ã kj x j = b k for k = 1, 2,, n 9 j=1 As per the parametrc form we may wrte Equaton 9 as n a kj, a kj, a kj x j = b k, b k, b k for k = 1, 2,, n 10 j=1 4 A new paradgm of fuzzy control pont As dscussed above, ths paper only focuses on polynomal n splne model Wahab et al redefned a fuzzy control pont as n [10-12] In ths secton, we propose a new paradgm of fuzzy control pont coeffcent whch s control pont n the model wll produce a curve model n 3-dmensonal space Defnton 8 Let P α be a fuzzy control pont n 3 dmensonal space Pα = { x, ỹ, z x, ỹ X Y, z 0, 1] } where x, ỹ s fuzzy data pont, D { } D = D mn α, D, D max α for = 0, 1, 2,, n and α 0, 1] 8

5 A new paradgm of fuzzy control pont n space curve 157 mn If z = µ x,y and D α = mnx, y,µ D x mn, y, D = x 1, y 1,µ D x 1, y 1 mak and D α = maxx, y,µ D x max, y respectvely, then mnx, y,µ D x mn, y for left fuzzy number f ncreasng, P = x1, y 1,µ D x 1, y 1 α = 1 forcrsp numberatpeak, 11 maxx, y,µ D x max, y for rght fuzzy numberf decreasng By Defnton 1, a new type of fuzzy control pont can be produced as below that satsfy the concept of fuzzy set P = D, µ D 12 P So, Equaton 12 can be rewrte as P = x, ỹ, µ x, ỹ P 5 Dscusson and numercal example Fuzzy B-Splne curve was ntroduced by usng fuzzy number concept whch s used n modelng the uncertanty data pont for curve modelng Fuzzy B-Splne curve had been dscussed n [10], [11] and [12] Therefore, fuzzy B-Splne can be defned as BS t = k+n 1 =1 13 P B,n t 14 where P are fuzzy control pont and B,n t are B-Splne basc functon wth crsp knot sequences t 1, t 2, t m=d+n+1 where d represents the degree of B-Splne functon and n represents the numbers of control ponts Fgure 3 shows the curve of a Fuzzy B-Splne Model n the space generated by B-Splne bass functons and fuzzy control pont that s characterzed by three dmensonal A new type of Fuzzy Control Pont x, ỹ, µ x, ỹ meets the basc theory of fuzzy set whch P were ntroduced by Zadeh as n Defnton 1 Fuzzy control pont P has a degree of expertse n the z-axs to generate a new B- Splne curve whch looked as f floatng n a 3-dmensonal space The changes of the degree of membershp n each control ponts P 0α, P 1α, P 2α, P 3α, P 4α wll change the poston of curve as shown n Fgure 3 In ths example, we use the control ponts wth membershp value P 02, P 05 and P 07 to reflect the changes that occur on the curve Fgure 3 b and c each show Fuzzy B-Splne curves can be seen from the plan vew z-axs and the front vew x-axs 6 Concluson Ths paper dscussed a new paradgm of fuzzy control pont that generates fuzzy splne space curve Ths new fuzzy control pont was redefned and satsfes the fuzzy set theory

6 158 Abd Fatah Wahab, Mohd Sallehuddn Husan and Mohammad Izat Emr Zulkfly Fgure 3: B-Splne space curve n a 3-dmesonal x, y, z-axs; b top vew z-axs and c front vew x-axs ntroduced by Zadeh The proposed deas can be extended to other splne model such as Bezer and NURBS model Acknowledgments On ths occason, we would lke to gve our sncere apprecaton to the Research Management Centre RMC, Unverst Malaysa Terengganu and the Mnstry of Educaton Malaysa for research grant ERGS Vote to conduct ths research References [1] Anle, A M, Falcdeno, B, Gallo, G, Spagnuolo, M and Spnello, S Modelng uncertan data wth fuzzy B-Splnes Fuzzy Sets System : [2] Gallo, G, Spagnuolo, M and Spnello, S Ranfall estmaton from sparse data wth fuzzy B-Splnes Journal of Geographc Informaton and Decson Analyss : [3] Gallo, G, Spagnuolo, S and Spnello, S Fuzzy B-Splnes: A surface model encapsulatng uncertanty Graphcal Models : [4] Zadeh, L A A fuzzy-set-theoretc nterpretaton of lngustc hedges Journal of Cybernetcs : 4-34 [5] Zadeh, L A The concept of a lngustc varable and ts applcaton to approxmate reasonng I Informaton Scences : [6] Dubos, D and Prade, H Fuzzy Sets and Systems: Theory and Applcatons New York: Academc Press 1980 [7] Zmmermann, H J Fuzzy Set Theory and Its Applcatons New York: Kluwer Academc 1980

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