Research Article Convexity-preserving using Rational Cubic Spline Interpolation

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1 Research Journal of Appled Scences Engneerng and Technolog (3): DOI:.19/rjaset..97 ISSN: -79; e-issn: Mawell Scentfc Publcaton Corp. Submtted: October 13 Accepted: December 13 Publshed: Jul 1 Research Artcle Convet-preservng usng Ratonal Cubc Splne Interpolaton 1 Samsul Arffn Abdul Karm and Kong Voon Pang 1 Department of Fundamental and Appled Scences Unverst Teknolog PETRONAS Bandar Ser Iskandar 317 Tronoh Perak Darul Rdzuan Malasa School of Mathematcal Scences Unverst Sans Malasa 11 USM Mnden Penang Malasa Abstract: Ths stud s a contnuaton of our prevous paper. The ratonal cubc splne wth three parameters has been used to preserves the convet of the data. The suffcent condton for ratonal nterpolant to be conve on entre subnterval wll be developed. The constrant wll be on one of the parameter wth data dependent meanwhle the other are free parameters and wll determne the fnal shape of the conve curves. 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: Convet-preservng parameters ratonal cubc splne suffcent condton INTRODUCTION Shape preservng nterpolaton s an mportant task when the user are requre to nterpolate all fnte sets of data whch preserves the orgnal geometrc characterstcs of the data. For eample f the gven data s conve the nterpolatng curves must be able to produce the conve curves on each subnterval. There are three mportant geometrc features of the data.e. convet monotonct and postvt. It s the frst s of nterest of ths stud. Convet alwas happens n man dscplnes and applcatons. For eample n nonlnear programmng whch arses n fnance and engneerng based problem. Normall the cost functon s conve or concave thus the splne nterpolant must be able to retan the convet of the data. Snce the ntroducton of the paper b Dodd and Rouler (193) and Rouler (197) man researcher have dscussed the convet preservng b usng varous tpe of splne nterpolant. The splne nterpolant can be ether polnomal or ratonal splne. Doughert et al. (199) have dscussed the cubc and quntc polnomal splne for postvt monotonct and convet preservng nterpolaton. Brodle and Butt (1991) have stud the convet preservng b usng cubc splne nterpolaton b nsertng etra knots n whch the convet of the data s not preserves. Lam (199) and Schumaker (193) have used quadratc polnomal splne to preserves the convet of the data. Ther methods also requre the ntroducton of etra knots. Sarfraz () and Sarfraz et al. (1) have dscussed the convet and postvt preservng b usng ratonal cubc splne (cubc/cubc). The derved the suffcent condtons for the ratonal nterpolant to be conve and postve. Abbas et al. (1) also dscuss the convet preservng b usng ratonal cubc splne (cubc/quadratc) wth three shape parameters. Hussan and Hussan (7) have eplored the use of ratonal cubc splne for preservng convet of 3D data. Gregor (19) has stud the use of ratonal cubc splne (cubc/quadratc) wth one shape parameter for convet preservng wth C contnut. Hussan et al. (11) have proposed shape preservng for postve and conve data b usng ratonal cubc splne of the form cubc/quadratc wth two shape parameters. Tan et al. () have dscussed the use of ratonal cubc splne (cubc/quadratc) wth two shape parameters. Ther ratonal functon slghtl s dfferent from the works of Hussan et al. (11). Hussan and Hussan () have utlzed ratonal cubc splne of Tan et al. () for convet problem for surfaces lng on rectangular grds. Delbourgo and Gregor (19a) also stud the use of ratonal cubc splne for monotone and convet preservng. Sarfraz et al. (13) have proposed new ratonal cubc splne of the form cubc/quadratc wth two parameters. The derved the suffcent condtons for the ratonal nterpolant to be postve monotone and conve on entre gven nterval. Besde the use of ratonal cubc splne wth cubc denomnator or quadratc denomnator there ests another ratonal nterpolant schemes that have lnear denomnator. For eamples Hussan et al. () and Zhang et al. (7) have stud the convet preservng b usng ratonal cubc Correspondng Author: Samsul Arffn Abdul Karm Department of Fundamental and Appled Scences Unverst Teknolog PETRONAS Bandar Ser Iskandar 317 Tronoh Perak Darul Rdzuan Malasa Ths work s lcensed under a Creatve Commons Attrbuton. Internatonal Lcense (URL: 31

2 Res. J. App. Sc. Eng. Technol. (3): splne wth lnear denomnator. Meanwhle Hussan et al. () and Karm and Pah (9) have preserves the convet data b usng ratonal quartc splne wth lnear denomnator. Recentl the convet preservng b usng trgonometrc splne also have attract man researchers. Pan and Wang (7) have used trgonometrc splne polnomal wth one shape parameter for convet preservng. Dube and Twar (1 13) also dscussed the convet preservng b usng C ratonal trgonometrc splne. Zhu et al. (1) preserves the convet of planar data b usng a new quartc trgonometrc Bézer polnomal. Delbourgo (199) also dscussed the convet preservng but wth ratonal splne nterpolant of the form quadratc numerator and lnear denomnator. Grener (1991) provdes a good surve on shape preservng methods. As ndcated n the above abstract ths stud s contnuaton of our prevous paper n Karm and Kong (13). The man dfferent s that nstead of applng the new ratonal cubc splne n postvt preservng (Karm and Kong 13) n ths stud the ratonal cubc splne wll be used for convet preservng of scalar data sets. Thus the man objectve of ths paper s to use ratonal cubc splne orgnall proposed b Karm and Kong (13) for convet preservng wth C 1 contnut. Furthermore we dentf several features of our ratonal cubc splne for convet preservng. It s summarzed below: Our method lkes the works of Sarfraz () Abbas et al. (1) Hussan and Hussan (7 ) Hussan et al. (11) and Sarfraz et al. (13) etc. do not requre the nsertng knots. Lam (199) and Schumaker (193) requres an etra knots n the nterval n whch the convet of the data s not preserves. Even though ther methods also works well for the tested data b nsertng etra knot the computaton to generate the curves or surfaces wll be ncreased. METHODOLOGY Ratonal cubc splne nterpolant: Ths secton wll ntroduce a new ratonal cubc splne nterpolant wth three parameters orgnall proposed b Karm and Kong (13). Suppose {( f ) = 1 n} s a gven set of data ponts where < 1 < < n. Let h = + 1 = ( ) and = ( ) where θ 1. For where = 1... n ( ) ( ) P ( θ) ( θ) = (1) Q s s In ths stud the ratonal cubc splne (cubc/quadratc) wth three parameters of Karm and Kong (13) has been used for convet preservng whle n Sarfraz () Abbas et al. (1) and Hussan and Hussan (7) the ratonal cubc (cubc/cubc) have been used for convet preservng. Whereas Hussan et al. (11) and Sarfraz et al. (13) the ratonal cubc splne (cubc/quadratc) wth two parameters have been used nstead. The ratonal cubc splne reproduces the ratonal cubc splne of Tan et al. () when one the parameter s equal to zero.e. γ = Indeed when γ = our convet-preservng scheme reduced to the work of Tan et al. (). The degree smoothness attaned s ths paper s C 1 whereas n Gregor (19) the degree smoothness attaned s C. B havng C contnut some nonlnear equatons need to be solved through some teraton methods. Numercal comparson between the proposed ratonal cubc schemes wth Tan et al. () and Frtch and Carlson (19) (pchp) and Doughert et al. (199) also has been done. Furthermore no dervatve modfcaton requre n our ratonal cubc scheme whle to preserves the convet of the data b usng the cubc and quntc splne b usng Doughert et al. (199) deas the frst dervatve must be modfed when an shape volaton are found. wth ( ) ( ) ( ) ( ) P θ = A 1 θ + Aθ 1 θ + Aθ 1 θ + Aθ ( ) ( ) ( )( ) = Q θ 1 θ α θ 1 θ α β γ θ β. A = α f ( α β α γ ) ( α β β γ ) A1 = + + f + αh d A = + + f β h d = β f + 1 A The ratonal cubc splne n (1) satsfes the followng C 1 condtons: ( ) = ( ) = ( 1 ) ( 1 ( ) ) ( ) s f s f s = d s = d where s (1) () denotes dervatve wth respect to and d denotes the dervatve value whch s gven at the k not = 1... n. The parameters α β γ >. The data dependent suffcent condtons on the parameters α and β wll be developed n order to preserves the postvt on the entre nterval + 1 = 1... n 1. When α = β = 1 γ = ratonal cubc nterpolant n (1) s just a standard C 1 cubc Hermte splne gven as follow: 313 ()

3 Res. J. App. Sc. Eng. Technol. (3): parameters d must be estmated b usng some mathematcal formulaton. In ths stud the Arthmetc Mean Method (AMM) has been used to estmate the frst dervatve for postvt preservng. The formula for AMM s gven as follows (Delbourgo and Gregor 19b; Sarfraz et al. 1997). The frst dervatves for = and = n are gven b: Fg. 1: Default cubc splne polnomal (α = β = 1 γ = ) for data n Table 1 3 d d h = + ( ) h + h1 1 hn 1 = + ( ). hn 1+ hn n n 1 n 1 n For = 1 n - 1 the values of d are gven as: () () h + h d = h + h () Fg. : Default cubc splne polnomal (α = β = 1 γ = ) for data n Table 1 Fg. 3: Default cubc splne polnomal (α = β = 1 γ = ) for data n Table 3 The shape parameters α and β = 1 n - 1 are free parameter (ndependent) whle the postvt constraned wll be derved from the other parameter γ (dependent). The two parameters α β can be used to refned and modf the fnal shape of the nterpolatng curve. Some observatons ncludng shape control analss were dscussed n detals b Karm and Kong (13). (3) Determnaton of dervatves: Normall when the scalar data are beng nterpolated the frst dervatve 31 Convet-preservng usng ratonal cubc splne nterpolant: The ratonal cubc splne nterpolant (cubc/quadratc) orgnall proposed b Karm and Kong (13) n Secton above cannot preserves the convet of the conve data. There mght be some part of the nterpolatng curves are not conve. In general the cubc splne nterpolaton wll be unable to produce the conve nterpolatng curves for conve data sets. Ths fact can be observed from Fg. 1 to 3 respectvel. As proposed b man authors for nstance Sarfraz () Abbas et al. (1) and Tan et al. () and others the automated choces of parameters that wll ensure the convet of the data must be fnd. Ths can be acheved b developed the suffcent condtons for convet of the ratonal nterpolaton defned b Eq. (1). Ths condton s data dependent and t wll be derved on one parameter γ whle the other two free parameters α and β can be used for the refne the nterpolatng conve functon. We begn the dervaton of suffcent condton for convet b gvng the defnton of the conve data sets. It started as follows. Gven a strctl conve data set ( f ) = 1 n so that < 1<... < 1< <... < n 1 < 1 <... < n and: d < < d <... < < d <... < < d (7) 1 1 n 1 n Equaton (7) s equvalent to: d < < d 1. + () (work on concavt preservng also can be treated n the same manner). Now ratonal cubc nterpolant s () s conve f and onl f s () () [ +1 ] = 1 n - 1. Now the frst and second-order dervatve of ratonal nterpolant s () defned b Eq. (1) are gven as follows:

4 Res. J. App. Sc. Eng. Technol. (3): s where and ( 1 ) ( ) ( 1 θ) j Bj θ j= = Q ( θ) = α 1 = α + α 1 β + γ j (9) ((( ) + ) ) ( ) B d B 1 d (( ) ) B = β 1+ β d α + γ B = β d B = (( + ) d ) + ( d + 1+ ) + ( ( 1+ γ ) + γ γ d βd+ 1 βd + β ( d+ 1)). α β 1 β γ γ β β α β s where ( ) ( ) 3 ( 1 θ) 3 j Cj θ j= = 3 h Q( θ) j ( ( ) ( + ) ( )) ( ) ( ) C = α γ d β d αβ d 1 C1 = α β d C = αβ d+ ( ) ( ) ( ) () C3 = β γ d+ α d β d +. Now f s () () then the ratonal nterpolant s () wll be conve for all n. For + 1 wth > and the denomnator n () s alwas postve.e. ( ) 3 Q θ > = 1... n 1. Thus the suffcent condton for convet s depend to the postvt of numerator n Eq. (). From calculus s () () s non-negatve f and onl f C j > j = 1 3. The necessar condtons for convet are d d +. The suffcent condton for convet of ratonal nterpolant n (1) can be determned from the condton: C > j = 1 3. It s obvous that C 1 > C >. For strctl conve data and from C > C 3 > wll gves us the followng condtons: C > f j ( ) ( d ) d ( d ) β γ + α β + 1 >. (1) Equaton (11) and (1) provdes the followng condtons gven n Eq. (13) and (1) respectvel: d γ β + 1 > α d γ α d. > β d+ (13) (1) Thus s () () > f Eq. (13) and (1) holds. The followng Theorem gves the suffcent condton for convet preservng b usng ratonal cubc splne (cubc/quadratc) nterpolant. Theorem 1: Gven a strctl conve data satsfng (7) or () there ests a class of conve ratonal (of the form cubc/quadratc) nterpolatng splne 1 s( ) C n nvolvng parameters α β and γ provded that f t satsf the followng convet suffcent condtons: α > β > γ 1 Ma β d + d 1... n 1. d α α d β > = + () Remark 1: If the data gves d = or d + 1 = then we ma set d = d +1 =. Ths wll results the ratonal nterpolant n (1) reproduce the lnear segment n that nterval: ( ) ( θ) θ + 1 s = 1 f + f. (1) Remark : The suffcent condton for convet preservng n (17) can be rewrtten as: γ 1 η +Ma β d + d. d α α d β = (17) + where η >. The followng s an algorthm that can be used to generate C 1 convet-preservng curves usng the results n Theorem 1. It s gven as follows. and ( ( d) ( d+ 1 ) ( d ) ) α γ β αβ > (11) C 3 > f 3 Algorthm for convet-preservng: Input: Data ponts ( f ) d shape parameters > >. Output: and pecewse convet nterpolatng curves:

5 Res. J. App. Sc. Eng. Technol. (3): Table 1: A conve data from Brodle and Butt (1991) 3 3 f d Table : A conve data from Sarfraz () 1 f.... d Table 3: A conve data from Tan () f d For = 1 n nput data ponts ( f ). For = 1 n estmate d usng Arthmetc Mean Method (AMM) 3. For = 1 n - 1 Calculate h and Choose an sutable values of > > Calculate the shape parameter usng (19) wth sutable choces of > Calculate the nner control ordnates A 1 and A. For = 1 n - 1 Construct the pecewse convet nterpolatng curves usng (1). Remark 3: When γ = the convet preservng s correspondng to the work b Tan et al. (). For the purpose of numercal comparson later we cte the suffcent condton for convet preservng usng ratonal cubc splne wth two parameters orgnall proposed b Tan et al. (). Theorem (Tan et al. ): The pecewse ratonal cubc functon n (1) wth γ = preserves the convet of the data f n subnterval the free parameters α β satsfes the followng suffcent condtons: d d+ 1 α = β = Ma. ( d+ ) ( d+ 1) (1) RESULTS AND DISCUSSION In order to llustrate the convet-preservng nterpolaton b usng the proposed ratonal cubc splne nterpolaton (cubc/quadratc) three sets of conve data were taken from Brodle and Butt (1991) Sarfraz () and Tan (). Fgure 1 to 3 shows the default cubc splne nterpolaton for data n Table 1 to 3 respectvel. Fgure to shows the convet preservng usng (Tan et al. ; Hussan and Hussan 7) for all tested data sets. Fgure 7 to 9 show the convet preservng usng our ratonal cubc splne for data n Table 1. Fgure to 1 show convet preservng for 3 Fg. : Shape preservng usng Tan et al. () for data n Table 1 The condtons n (1) can be obtaned b rearrange Eq. (13) and (1) wth γ =. In the orgnal work of Tan et al. () there s no free parameter (s) to refne the fnal shape of the conve curve; meanwhle n ths stud there ests two free parameters α and β that can be used to changes the shape of conve curves. Secton below gves the comparson between our proposed convet-preservng b usng ratonal cubc splne wth the works of Tan et al. () and Doughert et al. (199) respectvel. 31 Fg. : Shape preservng usng Tan et al. () for data n Table

6 Res. J. App. Sc. Eng. Technol. (3): Fg. : Shape preservng usng Tan et al. () for data n Table 3 Fg. 9: Shape preservng usng our proposed ratonal splne wth (α = β =. η =.) for data n Table 1 3 Fg. 7: Shape preservng usng our proposed ratonal splne wth (α = β =. η =.) for data n Table 1 Fg. : Shape preservng usng our proposed ratonal splne wth (α = β = 1 η =.) for data n Table 3 Fg. : Shape preservng usng our proposed ratonal splne wth (α = β = 1 η =.) for data n Table 1 data n Table. Fgure 13 to show the convet preservng usng our ratonal scheme for conve data n Table 3. Fgure 1 17 and 1 show the comparson between our ratonal scheme and Tan et al. () for data n Table 1 to 3 respectvel. Fgure 19 to 1 show the convet preservng b usng cubc splne nterpolaton from Doughert et al. (199). It s mplemented through pchp functon n MATLAB. Through careful nspecton there are some parts of conve curves b usng Doughert et al. (199) 317 Fg. 11: Shape preservng usng our proposed ratonal splne wth (α = β =. η =.) for data n Table methods that the curves tend to overshoot and not smooth and not ver vsual pleasng. For eample from Fg. 1 n the nterval ( 11) the th and th curve segments are not smooth and not vsual pleasng for computer graphcs vsualzaton. Whereas from Fg. 1 and the conve curves s smooth and ver vsual pleasng. One of the man dfferences between our ratonal cubc splne wth three parameters wth the

7 Res. J. App. Sc. Eng. Technol. (3): Fg. 1: Shape preservng usng our proposed ratonal splne wth (α = β =. η =.) for data n Table Fg. : Shape preservng usng our proposed ratonal splne wth (α = β =. η =.) for data n Table Fg. 13: Shape preservng usng our proposed ratonal splne wth (α = β = 1 η =.) for data n Table 3 Fg. 1: Shape preservng wth α = β =. η =. (blue) α = β = 1 η =. (black) and Tan et al. () (green) for data n Table 1 Fg. 1: Shape preservng usng our proposed ratonal splne wth (α = β =. η =.) for data n Table 3 works of Frtch and Carlson (19) and Doughert et al. (199) s that our scheme does not requre the modfcaton of the frst dervatve whereas Frtch and Carlson (19) and Doughert et al. (199) requre the frst dervatve modfcaton n order to produce the conve curves for conve data sets. Even though the 31 Fg. 17: Shape preservng wth α = β =. η =. (red) and Tan et al. () (blue) for data n Table ratonal cubc splne of Tan et al. () works well for all tested data sets t has no free parameter (s) meanwhle b usng our proposed ratonal cubc splne there are two free parameters α and β. B havng etra free parameters the user can changes the fnal shape of

8 Res. J. App. Sc. Eng. Technol. (3): Fg. 1: Shape preservng wth (α = β =. η =.) (red) and Tan et al. () (blue) for data n Table Fg. 1: Shape preservng nterpolaton usng cubc splne for data n Table 3 CONCLUSION Fg. 19: Shape preservng nterpolaton usng cubc splne for data n Table 1 Ths stud s devoted to the preservng the conve data sets b usng ratonal cubc splne nterpolant of the form cubc/quadratc wth three parameters orgnall proposed b Karm and Kong (13). The suffcent condtons for the convet have been derved completel b usng basc calculus approach. Those condtons are based to one parameter whle the other two parameters can be changed n order to obtan the man conve curves. These curves not onl preserve the convet of the data provded but t has C 1 contnut. Furthermore when γ = the proposed scheme s the same as the work b Tan et al. (). Clearl our ratonal cubc nterpolant wth three parameters gves better freedom to the user n controllng the convet of the curves b manpulatng the values of free parameters α and β compare to the work of Tan et al. () wthout an etra free parameter (s). Work on convet-preservng b usng C ratonal cubc splne nterpolant wth three parameters s underwa and t wll be reported n our forthcomng stud. Fg. : Shape preservng nterpolaton usng cubc splne for data n Table the conve nterpolatng curves accordng to the what desgn requres. Indeed our ratonal scheme s eas to use and overall t gves a satsfactor results and comparable wth the estng ratonal scheme such as Tan et al. (). 319 ACKNOWLEDGMENT The author would lke to acknowledge Unverst Teknolog PETRONAS for the fnancal support receved n the form of a research grant: Short Term Internal Research Fundng (STIRF) No. 3/1. REFERENCES Abbas M. A.A. Majd M.N.Hj. Awang and J.M. Al 1. Local convet shape-preservng data vsualzaton b splne functon. ISRN Math. Anal. 1: 1 Artcle ID 17.

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