Research Article Shape Preserving Interpolation using Rational Cubic Spline

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1 Research Journal of Appled Scences, Engneerng and Technolog 8(2): 67-78, 4 DOI:.9026/rjaset.8.96 ISSN: ; e-issn: Mawell Scentfc Publcaton Corp. Submtted: Januar, 4 Accepted: Februar, 4 Publshed: Jul, 4 Research Artcle Shape Preservng Interpolaton usng Ratonal Cubc Splne Samsul Arffn Abdul Karm and 2 Kong Voon Pang Department of Fundamental and Appled Scences, Unverst Teknolog PETRONAS, Bandar Ser Iskandar, 370 Tronoh, Perak Darul Rdzuan, Malasa 2 School of Mathematcal Scences, Unverst Sans Malasa, 800 USM Mnden, Penang, Malasa Abstract: Ths stud proposes new C ratonal cubc splne nterpolant of the form cubc/quadratc wth three shape parameters to preserves the geometrc propertes of the gven data sets. Suffcent condtons for the postvt and data constraned modelng of the ratonal nterpolant are derved on one parameter whle the remanng two parameters can further be utlzed to change and modf the fnal shape of the curves. The suffcent condtons ensure the estence of postve and constraned ratonal nterpolant. Several numercal results wll be presented to test the capablt of the proposed ratonal nterpolant scheme. Comparsons wth the estng scheme also have been done. From all numercal results, the new ratonal cubc splne nterpolant gves satsfactor results. Kewords: Contnut, parameters, postvt preservng, ratonal cubc splne, shape preservng INTRODUCTION Shape preservng data nterpolaton and appromaton are mportant n computer graphcs, geometrc modelng and scentfc vsualzaton. There are man characterstcs of the geometrc for data sets. For eamples, the gven data sets mght be monotone, conve and postve. In postvt preservng, the proposed nterpolant (ether ratonal or polnomal) must be able to produce the nterpolatng curves and/or surfaces that wll preserves the characterstcs of the data namel postvt. There are man stuaton arse n our dal lfe nvolvng postvt. Notabl, the dstrbutons of wnd energ and ranfall measurement are alwas havng postve values. Thus, t s mportant that when the data s to be vsualzed for computer dspla, the resultant curves or surfaces must retan ts geometrc shape propertes (n our case t s postve and data constraned above an straght lne). There are man research papers concernng about postvt preservaton ether b usng cubc splne nterpolaton or ratonal splne nterpolaton. One of the earl fndng n postvt preservng b usng cubc splne nterpolaton can be found n Doughert et al. (989). The authors gve the suffcent condtons for the postvt of cubc and quntc splne polnomal. Meanwhle, Brodle and Butt (99) and Butt and Brodle (993) have used cubc polnomals for postvt and convet preservng b nsertng one or two etra notes n nterval n whch the postvt and/or convet of the curves are found. B havng nsertng etra knots, the computaton wll be ncreased, hence s not sutable to asssts the user n controllng the shape of the data sets. Besdes the use of cubc or quntc splne polnomal, several researchers have proposed ratonal splne nterpolant to preserves the postvt of the gven data sets. For eamples, Sarfraz (02), Sarfraz et al. (0), Hussan and Sarfraz (08) and Abbas et al. (2a) studed the use of ratonal cubc nterpolant (cubc numerator and cubc denomnator) for preservng the postve data. Meanwhle Sarfraz et al. () studed postvt preservng for curves and surfaces b utlzng ratonal cubc splne wth quadratc denomnator. In Hussan et al. (), the ratonal cubc splne wth quadratc denomnator have been used for postvt and convet preservng wth degree attaned s C 2. Hussan and Al (06) have dscussed the postvt preservng b usng ratonal cubc splne orgnall proposed b Tan et al. (0). Meanwhle Hussan and Hussan (06) have etended the results n Hussan and Al (06) to preserves the data above the straght lne and postve surfaces nterpolatng problem. Abbas et al. (2b) have proposed new ratonal cubc splne (cubc/quadratc) wth three parameters for monotonct preservng nterpolaton and those method has been etend to postvt preservng n Abbas et al. (3). Motvated b the works of Tan et al. (0), Hussan and Al (06), Hussan and Hussan (06) and Abbas et al. (2b), n ths stud, wth the authors wll proposed new ratonal cubc splne wth three parameters and the ratonal cubc splne of Tan et al. (0) s a specal case of our ratonal nterpolant. Numercal comparson also has been done wth estng shape preservng Correspondng Author: Samsul Arffn Abdul Karm, Department of Fundamental and Appled Scences, Unverst Teknolog PETRONAS, Bandar Ser Iskandar, 370 Tronoh, Perak Darul Rdzuan, Malasa Ths work s lcensed under a Creatve Commons Attrbuton 4.0 Internatonal Lcense (URL: 67

2 nterpolaton methods. From numercal results, t can be clearl seen that, our proposed ratonal cubc splne provdes good alternatve to the estng ratonal nterpolant scheme. The man scentfc contrbuton ths stud s as follows: In ths stud a new ratonal cubc splne (cubc/quadratc) wth three parameters has been used for postvt preservng whle n Hussan and Al (06), Hussan and Hussan (06) and Hussan et al. (), the ratonal cubc splne (cubc/quadratc) wth two parameters haven used for postvt preservng. The ratonal cubc splne reduces to the ratonal cubc splne of Tan et al. (0) when one the parameter s equal to zero,.e., γ = 0. Indeed, when γ = 0, our postvt-preservng and data constraned nterpolaton reduces to the works of Hussan and Al (06) and Hussan and Hussan (06), respectvel. The degree smoothness attaned s ths stud s C whereas n Abbas et al. (3) the degree smoothness attaned s C 2. Furthermore; the work n ths stud can easl be etended to produce C 2 ratonal cubc splne nterpolant. Numercal comparson between the proposed ratonal cubc schemes wth two estng methods also has been done. Smlar to the works b Hussan and Al (06), Hussan and Hussan (06) and Hussan et al. (), etc., our method also do not requred an etra knots. The cubc splne scheme b Butt and Brodle (993) and Brodle and Butt (99) requres one or two etra knots to be nserted n the nterval n whch the postvt of the gven data sets s not preserved. Our ratonal schemes s based from splne functon whle n Bashr and Al (3) and Ibraheem et al. (2) the nterpolant s based from trgonometrc splne. Our method works well and guarantee preserves the postvt of the data. Fnall, n ths stud two algorthms have been gven for the computer mplementaton whle there are no algorthms for computer mplementaton n Hussan and Al (06) and Hussan and Hussan (06). MATERIALS AND METHODS Ratonal cubc splne nterpolant: Ths secton wll ntroduce a new ratonal cubc splne nterpolant wth three parameters. Let us assume that, gven the set of data, Suppose {(, f ), = 0,,.,n} s a gven set of data ponts, where 0 < < < n. Let h = + -, = ( ) 0 θ. Res. J. Appl. Sc. Eng. Technol., 8(2): 67-78, 4 and a local varable, = ( ) For, +, = 0,,2,..., n : where 68 where, P ( θ) ( θ), = () Q s s P θ = A θ + Aθ θ + Aθ θ + Aθ, 2 2 = Q θ θ α θ θ 2 α β γ θ β. The followng condtons are requred to ensure that the ratonal functon n () has C contnut: = ( ) = ( ) ( ) s f, s f, + + s = d, s = d, + + (2) Now, b some algebrac manpulaton to the C condton n (2), the unknowns A, = 0,, 2, 3 are gven as follows: A = α f 0 A = 2 α β + α + γ f + α h d, A = 2 α β + β + γ f β h d, = β f + A where, s () () denotes dervatve wth respect to and d denotes the dervatve value whch s gven at the k not, = 0,, 2,., n. The parameters α, β >0, γ 0. The data dependent suffcent condtons on the parameters α and β wll be developed n order to preserves the postvt on the entre nterval [, + ], = 0,, 2,, n-. Some observaton to the new ratonal cubc splne nterpolant gven n () can be descrbed as follows: When α >0, β >0, γ = 0 the ratonal nterpolant n () reduce to the ratonal splne of the form cubc/quadratc b Tan et al. (0). When α = β =, γ = 0 the ratonal cubc nterpolant n () s a standard cubc Hermte splne gven as follow: ( )= ( ) (+2 ) + (3 2 ) + ( ) ( ) (3) When α 0, β 0 ratonal nterpolant n () converges to straght lne gven below: lm s α, β 0 ( θ) f θ f + = + (4) Furthermore the ratonal nterpolant n () can be rewrtten as: ( θ) + ( ) ( )( ) + ( + ) Q ( θ) () hθ θ α d θ β d θ s = f + θ f +.

3 Res. J. Appl. Sc. Eng. Technol., 8(2): 67-78, 4 The shape parameters α, β = 0,, 2,, n- are free parameter (ndependent) whle the postvt and monotonct constraned wll be derved from the other parameter γ (dependent). The two parameters α, β can be used to refne and modf the fnal shape of the nterpolatng curve. Ths wll be useful for shape control of nterpolatng curves. Determnaton of dervatves: For the scalar data sets, the dervatve parameters must be estmated b usng mathematcs formulaton. The orgnal dervatons to estmate the frst dervatve are gven b Delbourgo and Gregor (98) and Sarfraz et al. (997). Among those methods are Arthmetc Mean Method (AMM), Geometrc Mean Method (GMM) and Harmonc Mean Method (HMM). For our purpose n ths stud, the Arthmetc Mean Method (AMM) wll be used. Ths s because, AMM s a smple method and sutable for postve data. It also provdes ver vsual pleasng results as can be seen n Results and Dscusson Secton later. Below the mathematcal formula of AMM. At the end ponts 0 and n : Fg. : Default cubc splne polnomal (α = β =, γ = 0) for data n Table d d h0 = + ( ) h0 + h hn = + ( ) hn + hn 2 n n n n 2 (6) (7) Fg. 2: Default cubc splne polnomal (α = β =, γ = 0) for data n Table 4 Assumng that the strctl postve set of data (, f ), = 0,,, n are gven, so that: At the nteror ponts,, =, 2,..., n-, the values of d are gven as: h d = + h h + h (8) Postvt-preservng usng ratonal cubc splne nterpolant: The proposed ratonal cubc splne nterpolant (cubc/quadratc) wth three parameters n below secton does not alwas preserves the postvt of the postve data. The ordnar cubc splne nterpolaton and cubc Hermte splne also does not guarantee to preserve the postvt of the data sets and t wll destro the characterstcs of the data sets. These shape volatons can be seen clearl from Fg. and 2, respectvel. The user ma manpulate the values of the shape parameters α, β, γ, = 0,,, n- b tral and error bass n order to preserves the postvt of the data. But ths approach s reall tme consumng and obvousl t s not recommended to the user. Followng the same dea b Sarfraz (02), the automated choce of the shape parameter γ wll be derved from the other two parameters α, β and the data dependent suffcent condtons for postvt of the ratonal nterpolant defned b Eq. () wll be developed. 69 and, 0 < < < n (9) f >0, = 0,,., n () Now, the suffcent condtons for postvt of pecewse ratonal cubc splne (cubc/quadratc) wth C contnut wll be developed. The man dea s that, n order to preserve the postvt of s (), the sutable values of shape parameter γ n each correspondng nterval must be chosen and assgned properl. For all α, β >0 and γ 0 the denomnator Q (θ) >0, = 0,,, n-, therefore the postvt of ratonal nterpolant n () solel depends to the postvt of cubc splne polnomal P (θ), = 0,,, n-. The cubc polnomal P (θ), = 0,,, n- can be rewrtten as follows: where, 3 2 P θ = Bθ + Cθ + Dθ + E, B = α hd + β h d + 2αβ f + γ f 2 αβ f γ f, C = 2α hd β h d + α f 4αβ f 2γ f + β f + 2αβ f + γ f D = α hd 2α f + 2 αβ f + γ f, E = α f.

4 Res. J. Appl. Sc. Eng. Technol., 8(2): 67-78, 4 For strctl monotone data sets gven n (), the followng theorem gves the suffcent condtons for postvt preservng. Theorem : For a strctl postve data defned n (), the ratonal cubc nterpolant defned over the nterval [ 0, n ] s postve f n each subnterval [, + ], = 0,,, n- the followng suffcent condtons are satsfed: ( 2β ) ( 2α ) h d + + f h d+ + f+ γ > Ma 0, α, β f f+ () Proof: To prove Theorem, the followng Proposton from Schmdt and Hess (988) s requred. Proposton : (Postvt of Cubc polnomal). For the strct nequalt postve data n (), P (θ) >0 f and onl f: where, ( P ( 0 ), P ( ) ) R R2 (2) ( ) f 2αβ + β + γ + β h d + 3β f 3β f = < (6) h h P The nequalt () and (6) leads to the followng relatons: and, ( 2β ) h d f γ > α + + f ( 2α ) h d+ + f+ γ > β f+ (7) (8) Now, Eq. (7) and (8) can be combned to the followng suffcent condtons: ( 2β ) ( 2α ) h d+ + f h d+ + f+ γ > Ma 0, α, β f f+ Ths complete the proof of Theorem. Furthermore, (P' (0), P' ()) R 2 f: P 3P 0 3 R = ( a, b) : a>, b<, h h ( ) ( a, b) : 36 f f + a b ab 3 ( a b) 3 3( ) R2 = + f+ a fb h ab f+ a+ fb h ( f+ a fb ) h a b > 0 (3) (4) ( ) ( a b) f f a b ab ( a b), : φ( a, b) = + 3( f+ a fb)( 2h ab 3f+ a+ 3fb) h ( f+ a fb ) h a b 0 where, a= P 0, b= P. + (9) where, a = P' (0), b = P' () and P (0) = α f, P () = β f +. Now, b dfferentatng P (θ) from () wth respect to θ, the results are: 3α f + f 2α β + α + γ + α h d P ( 0) = h The constrants on the free parameters can be derved ether from Eq. (3) or (4). But Eq. (4) nvolves a lot of computaton, thus we wll develop the data dependent constrants for postvt preservng b usng Eq. (3) because t s more econom and less ntrcate as compared wth condton (4) and (9). The suffcent condton n () can be rewrtten as: and, f 2α β + β + γ + β h d + 3β f P ( ) = h Now from Proposton, t can be deduced that (P' (0), P' ()) R f: 3α f + f 2α β + α + γ + αh d 3α f P ( 0) = > () h h and, 70 ( 2β ) ( 2α ) h d + + f h d+ + f+ γ = λ +Ma 0, α, β, λ > 0. f f+ () Remark : The postvt-preservng b usng the ratonal splne wth γ = 0 have been dscussed n detals b Hussan and Al (06). One of the man dfferences between the proposed scheme wth the work of Hussan and Al (06) s that, our ratonal cubc splne nterpolant have two free parameters α >0, β >0 that can be used to refne the fnal shape of the curves meanwhle there s no free parameters n the work of Hussan and Al (06). It s mportant that, the method should have free parameters n order to modf and

5 Res. J. Appl. Sc. Eng. Technol., 8(2): 67-78, 4 alterng the fnal shape of the nterpolatng curves. Theorem 2 (Hussan and Hussan, 06) below gves the suffcent condton for postvt preservng b usng ratonal cubc splne of Tan et al. (0). It s the same as our proposed ratonal cubc wth γ = 0. Theorem 2: (Hussan and Al, 06). The ratonal cubc splne n () wth γ = 0 preserves postvt f and onl f the followng suffcent condtons are satsfed: α Ma h d 0, h d, Ma 0,. 2 f β + > + > (2) 2 f An algorthm to generate C postvt-preservng curves usng the results n Theorem s gven below. Algorthm :. Input the number of data ponts, n and data ponts n { f } = 0,. 2. For = 0,,, n estmate d usng Arthmetc Mean Method (AMM). 3. For = 0,,, n-: Calculate h and Choose an sutable values of α >0, β >0 Calculate the shape parameter γ usng () wth sutable choces of λ >0 Calculate the nner control ordnates A and A 2 4. For = 0,,, n- Construct the pecewse postve nterpolatng curves usng (). Repeat Step untl Step 4 for each tested data sets. CONSTRAINED DATA MODELING For data modelng constraned b usng the propose ratonal cubc splne, the result from postvt preservng n Secton 3 wll be generalzed n order to constraned the data that les above arbtrar straght lne = m + c. In other word, the suffcent condton for the ratonal nterpolant to be above straght lne wll be derved. In general, the standard cubc splne nterpolaton was unable to produce the nterpolatng curves that wll les above the gven straght lne. Fgure 3 and 4 show these eamples. Clearl there are some parts of the curves that le below the gven straght lne. Ths unwanted flaw must be recovered ncel and should be vsual pleasng enough for computer graphcs dsplas. The followng theorem gves the man results for data constraned modelng b usng the proposed ratonal cubc splne. Fg. 3: Default cubc splne polnomal (α = β =, γ = 0) for data n Table Fg. 4: Default cubc splne polnomal (α = β =, γ = 0) for data n Table 2 subnterval [, + ], = 0,,, n-, the free parameter γ, satsf the followng suffcent condton: ( ) α f h d+ b β f+ + h d+ + a γ > Ma 0,,, = 0,,..., n. f a f+ b (22) Proof: Assumng that, we are gven the set of data (, f ), = 0,,, n lng above the straght lne = m + c, such that: f > m + c, = 0,,..., n. (23) Here we follow the same dea for data constrant modelng as dscussed n detals b Shakh et al. () and Sarfraz et al. (3). The curve wll le above the straght = m + c. If the proposed ratonal cubc splne s () n () satsfes the followng condton: 0 s > m+ c,, n. (24) Theorem 3: The pecewse ratonal cubc splne nterpolant s () preserves the shape of the data that les above the gven straght lne = m + c, f n 7 Now, for each subnterval [, + ], = 0,,, n-, the relaton n Eq. (24) can be epressed as follows:

6 Res. J. Appl. Sc. Eng. Technol., 8(2): 67-78, 4 (2) B transformng the straght lne equaton nto parametrc form, (2) can be rewrtten as follows: (26) where, a = m + c, b = m + + c. Snce for all α, β, γ >0, Q (θ) >0, we can multpl the rght hand sde b denomnator n (26). B rearrange (26), the followng equaton s obtaned: Now, we onl consder the numerator n (27). Let: where, 0 α, ( 2α β α γ ) α ( 2 αβ + γ ) a αb, c3 = β ( f+ b), c = f a c = + + f + h d c = 2αβ + α + γ f β h d 2 α β + γ b aβ (27) (28) Now, U (θ) >0 f c j >0. Clearl c 0 >0, c 3 >0, due to the fact that f - a >0, f + - b >0, thus the suffcent condton wll be derved from the followng condtons: c > 0, ( 2α β + α + γ ) f + αh d ( 2α β + γ ) a αb > 0 (29) and, c 2 > 0, 2αβ + β+ γ f+ βh d+ 2α β+ γ b aβ > 0 (30) ( ) α f h d + b β f+ + h d+ + a γ > Ma 0,,, = 0,,..., n. f a f+ b (33) The result s obtaned as requre. The suffcent condtons n Eq. (22) can be rewrtten as follows: ( ) α f h d+ b β f+ + h d+ + a γ = υ +Ma 0,,, = 0,,..., n. f a f+ b (34) where υ >0. The suffcent condton n (34) wll guarantee the estence of ratonal cubc nterpolant that les above the straght lne. For the purpose of numercal comparson later, Theorem 4 (Hussan and Hussan, 06) below gves the suffcent condton for data constraned modelng b usng Hussan and Hussan (06) method. It s equvalent wth our proposed ratonal cubc wth γ = 0. Theorem 4: (Hussan and Hussan, 06). The ratonal cubc splne n () wth γ = 0 preserves the shape of the data les above the straght lne, f n subnterval [, + ], free parameters α and β satsf the followng suffcent condtons: f+ + h d+ + a f h d + b α > Ma 0,, β > Ma 0,. 2 2 ( f+ b) ( f a) (3) Algorthm 2 can be used n order to generate the nterpolatng curves that les above an straght lne provded that the free parameter γ must be chosen from (34) wth some sutable values of the other there parameters α, β and υ. Algorthm 2:. Input the number of data ponts, n, data ponts n, f, and straght lne equaton, = m + c. { } = 0 2. For = 0,,, n, estmate d usng Arthmetc Mean Method (AMM): 3. For = 0,,, n-: Equaton (29) and (30) provdes the followng relatons: and, α γ > β γ > ( f h d + b ) f a ( f + h d + a ) + + f + b (3) (32) The suffcent condtons n (3) and (32) can be rewrte as follows: 72 Calculate h and Transform the straght lne equaton nto parametrc form and calculate a and b Choose an sutable values of α >0, β >0 Calculate the shape parameter γ usng (34) wth sutable choces of υ >0 Calculate the nner control ordnates A and A 2 defned n () 4. For = 0,,, n- Construct the pecewse data constraned modelng nterpolatng curves usng ().

7 Res. J. Appl. Sc. Eng. Technol., 8(2): 67-78, 4 Table : A data from Hussan and Hussan (06) above = f d Table 2: A data from Hussan and Hussan (06) above = / f d Table 3: A postve data from Brodle and Butt (99) f d Table 4: A postve data from Sarfraz et al. (0) f d Remark 2: The case for constraned data modelng that les below arbtrar straght lne can be treated n the same manner. RESULTS AND DISCUSSION In ths subsecton the postvt preservng and data constraned modelng b usng the results obtaned n Theorem and Theorem 3 are eplored. Two sets of postve data taken from Brodle and Butt (99) and Sarfraz et al. (0) were used. Meanwhle Table and 2 show the data that les above the straght lne taken from Hussan and Hussan (06). Fgure and 2 shows the default cubc splne nterpolaton for data n Table 3 and 4 respectvel. Fgure and 6 show the eamples of postvt preservng b usng Hussan and Al (06). Fgure 7 to 9, show the shape preservng b usng our propose ratonal cubc splne scehemes wth varous choces of free parameters α, β wth λ = 0.. Meanwhle Fgure shows the postvt preservng b usng our propose scehemes (blue) and Hussan and Al (06) (red) for data n Table 3. Fgure to 3, show the postvt preservng b usng our propose ratonal cubc splne wth varous choces of shape parameters α, β wth λ = 0. Fgure 4 shows vsual pleasng postve nterpolatng curves wth α = 0., 2, 0., 0., 3., 0., β = 2, 8,,.,.4, 0.8 for n Table 4. Fgure shows shape preservng b usng our schemes (red) wth α = β = 0., =, 2,, and α 0 = β 0 = 2. and Hussan and Al (06) (blue) Fg. 6: Shape preservng usng Hussan and Al (06) for data n Table Fg. : Shape preservng usng Hussan and Al (06) for data n Table 3 73 Fg. 7: Shape preservng usng our proposed ratonal splne wth (α = β = ) for data n Table 3

8 Res. J. Appl. Sc. Eng. Technol., 8(2): 67-78, Fg. 8: Shape preservng usng our proposed ratonal splne wth (α = β = 0.0) for data n Table 3 (the ratonal splne approach to straght lne) Fg. : Shape preservng usng our proposed ratonal splne wth (α = β = ) for data n Table Fg. 9: Shape preservng usng our proposed ratonal splne wth (α = β = 0.) for data n Table 3 Fg. 2: Shape preservng usng our proposed ratonal splne wth (α = β = 0.) for data n Table Fg. : Shape preservng wth α = β = 2. (blue) and Hussan and Al (06) (red) for data n Table 3 Fg. 3: Shape preservng usng our proposed ratonal splne wth (α = β =.) for data n Table 4 Fgure 3 and 4, show the default cubc splne constraned nterpolaton for data n Table and 2 respectvel. For data constraned modelng, Fgure 6 and 7 show the shape preservng b usng Hussan and Hussan (06) method. Fgure 8 to show the eamples of shape preservng b usng our propose ratonal cubc wth varous choces of shape parameters 74 as ndcated n the respectve fgures. Fgure 2 shows the eamples on shape control analss for data n Table 2. It can be seen clearl that, for the fed values of α, β and keep changng the γ values, the ratonal cubc splne nterpolatng curves wll les above the gven straght lne. From Fg. 2, when α = β = and γ = 8 (shown n blue color) the curves les above the

9 Res. J. Appl. Sc. Eng. Technol., 8(2): 67-78, Fg. 4: Shape preservng usng our proposed ratonal splne for data n Table 4 (vsual pleasng) 2 Fg. 7: Shape preservng usng Hussan and Hussan (06) for data n Table Fg. : Shape preservng nterpolaton wth our proposed splne (red) and Hussan and Al (06) (blue) for data n Table 4 Fg. 8: Shape preservng usng our proposed ratonal splne wth (α = β =, υ = 0.2) for data n Table Fg. 6: Shape preservng usng Hussan and Hussan (06) for data n Table straght lne. Fgure 22 and 23 show the shape preservng b usng ratonal cubc splne wth υ = 0.2. Fgure 24 shows two curves both les above straght lne wth α = β = 0., γ = 8 (black) and α = β = 0., γ = (blue). Fnall Fg. 2 shows nterpolatng curves for Hussan and Hussan (06) (black) and our ratonal scheme (red) Fg. 9: Shape preservng usng our proposed ratonal splne wth (α = β = 0., υ = 0.2) for data n Table From all 2 fgures that shown n ths secton, t s clear that, the proposed ratonal cubc splne wth three parameters provdes greater fleblt n controllng the fnal shape of postve and constrant nterpolatng curves. The postve and constrant nterpolatng curves

10 Res. J. Appl. Sc. Eng. Technol., 8(2): 67-78, Fg. : Shape preservng usng our proposed ratonal splne wth (α =, β = 2, υ = 0.) for data n Table Fg. 23: Shape preservng usng our proposed ratonal splne wth (α = β = 0., υ = 0.2) for data n Table Fg. 2: Ratonal splne wth α = β = γ = (black), α = β =, γ = (red), α = β =, γ = 8 (blue) for data n Table 2 Fg. 24: Shape preservng usng our proposed ratonal splne wth α = β = 0., γ = 8 (black) and α = β = 0., γ = (blue) for data n Table Fg. 22: Shape preservng usng our proposed ratonal splne wth (α = β =, υ = 0.2) for data n Table 2 can be change b choosng dfferent values of two free parameters α, β and the suffcent condtons s derved on the other parameter γ. Theorem and Theorem 3 wll guarantee the estence of postve and constraned data nterpolaton curves respectvel for α, β >0, γ 0. Once we assgned the values of α, β, then b choosng some sutable values of λ >0 (for postvt) and υ >0 76 Fg. 2: Shape preservng splne wth α = β =, υ = 0.2 (red) and Hussan and Hussan (06) (black) for data n Table 2 (for data constraned), the value of γ can be calculated from Eq. () and (34), respectvel. Ths process can be repeated to obtan the desre nterpolatng curves as user wsh. From numercal comparson between our proposed ratonal cubc splne wth the works of () Hussan and

11 Al (06) for postvt preservng and (2) Hussan and Hussan (06) for data constraned modelng (les above arbtrar straght lne), t s clear that our proposed ratonal cubc splne gves a good results and t s comparable wth both methods. It also provdes good alternatve to the estng ratonal cubc splne for postvt and constraned data modelng. Free parameters α and β can be utlzed n order to change the fnal shape of the curves. Ths wll gves us an added value to the proposed ratonal cubc splne scheme. Furthermore, the proposed ratonal cubc splne (cubc/quadratc) wth three parameters can be etended to preserves the postve data wth C 2 contnut. The authors n the fnal process to complete the stud. Fnal remark: Even though the proposed ratonal cubc splne wth three parameters s a mnor varant of the work of Tan et al. (0), Hussan and Al (06) and Hussan and Hussan (06), the free parameters α and β provde the user n controllng the fnal shape of postve curves and data constraned curves. In the orgnal works of Hussan and Al (06) and Hussan and Hussan (06), there s no free parameter. CONCLUSION New ratonal cubc splne wth three parameters has been dscussed n detals n ths work. Frstl, the proposed ratonal nterpolant has been use for postvt preservng. Whle the suffcent condton for postvt has been derved on one of the parameter γ and the other two parameters α and β wll be actng as free parameter. The free parameter can be used to refne and modf the fnal shape of the nterpolatng postve curves. Secondl, the results for postvt preservng has been etended to stud the data constraned modelng b usng the proposed ratonal cubc splne. Theorem and 3 wll guarantee the estence of postve and constraned data nterpolaton curves respectvel. Furthermore when α, β >0 and γ = 0 the proposed scheme reduces to the work of Hussan and Al (06) and Hussan and Hussan (06) for postvt preservng and constraned data nterpolaton respectvel. One of the man advantages of the proposed scheme s that there ests two free parameters α and β compare to the work b Hussan and Al (06) and Hussan and Hussan (06) have no free parameter (s). Works on preservng the monotone data as well as the conve data are underwa and t s nterestng to use ths method for medcal mage processng. Another potental research stud s to use the ratonal cubc splne for surface nterpolatng for postve, monotonct and convet data. Ths wll be our man target for future research. ACKNOWLEDGMENT The frst author would lke to acknowledge Unverst Teknolog PETRONAS for the fnancal Res. J. Appl. Sc. Eng. Technol., 8(2): 67-78, 4 77 support receved n the form of a research grant: Short Term Internal Research Fundng (STIRF) No. 3/2. REFERENCES Abbas, M., A.A. Majd, M.N.Hj. Awang and J.M. Al, 2a. Shape preservng postve surface data vsualzaton b splne functons. Appl. Math. Sc., 6(6): Abbas, M., A.A. Majd and J.M. Al, 2b. Monotonct-preservng C 2 ratonal cubc splne for monotone data. Appl. Math. Comput., 29: Abbas, M., A.A. Majd, M.N.Hj. Awang and J.M. Al, 3. Postvt-preservng C 2 ratonal cubc splne nterpolaton. Sc. Asa, 39: Bashr, U. and J.M. Al, 3. Data vsualzaton usng ratonal trgonometrc splne. J. Appl. Math., 3:, Artcle ID Brodle, K.W. and S. Butt, 99. Preservng convet usng pecewse cubc nterpolaton. Comput. Graph., : -23. Butt, S. and K.W. Brodle, 993. Preservng postvt usng pecewse cubc nterpolaton. Comput. Graph., 7(): -64. Delbourgo, R. and J.A. Gregor, 98. The determnaton of dervatve parameters for a monotonc ratonal quadratc nterpolant. IMA J. Numer. Anal., : Doughert, R.L., A. Edelman and J.M. Hman, 989. Nonnegatvt-, monotonct-, or convetpreservng cubc and quntc hermte nterpolaton. Math. Comput., 2(86): Hussan, M.Z. and J.M. Al, 06. Postvt preservng pecewse ratonal cubc nterpolaton. Matematka, 22(2): Hussan, M.Z. and M. Hussan, 06. Vsualzaton of data subject to postve constrant. J. Inform. Comput. Sc., (3): Hussan, M.Z. and M. Sarfraz, 08. Postvtpreservng nterpolaton of postve data b ratonal cubcs. J. Comput. Appl. Math., 28: Hussan, M.Z., M. Sarfraz and T.S. Shakh,. Shape preservng ratonal cubc splne for postve and conve data. Egpt. Inform. J., 2: Ibraheem, F., M. Hussan, M.Z. Hussan and A.A. Bhatt, 2. Postve data vsualzaton usng trgonometrc functon. J. Appl. Math., 2: 9, Artcle ID 247. Sarfraz, M., 02. Vsualzaton of postve and conve data b a ratonal cubc splne nterpolaton. Inform. Sc., 46(-4): Sarfraz, M., M.A. Mulhem and F. Ashraf, 997. Preservng monotonc shape of the data usng pecewse ratonal cubc functons. Comput. Graph., 2(): -4.

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