This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Research ArticleLocal Convexity-Preserving 1198622 Rational Cubic Spline forConvex Data
Muhammad Abbas1 Ahmad Abd Majid2 and Jamaludin Md Ali2
1 Department of Mathematics University of Sargodha Sargodha 40100 Pakistan2 School of Mathematical Sciences Universiti Sains Malaysia 11800 George Town Penang Malaysia
Correspondence should be addressed to Muhammad Abbas mabbasuosedupk
Received 31 August 2013 Accepted 10 February 2014 Published 13 March 2014
Academic Editors S De Marchi and J Tan
Copyright copy 2014 Muhammad Abbas et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We present the smooth and visually pleasant display of 2D data when it is convex which is contribution towards the improvementsover existing methods This improvement can be used to get the more accurate results An attempt has been made in order todevelop the local convexity-preserving interpolant for convex data using1198622 rational cubic spline It involves three families of shapeparameters in its representation Data dependent sufficient constraints are imposed on single shape parameter to conserve theinherited shape feature of data Remaining two of these shape parameters are used for the modification of convex curve to get avisually pleasing curve according to industrial demandThe scheme is tested through several numerical examples showing that thescheme is local computationally economical and visually pleasing
1 Introduction
In computer graphics a designer in industries needs togenerate splines which can interpolate the data points in sucha way that they conserve the inherited shape characteristics(positivity monotonicity and convexity) of data Among theproperties that the spline for curves and surfaces need tosatisfy smoothness and shape preservation of given dataare mostly needed by all the designers Convexity is a sub-stantial shape characteristic of the data The significance ofthe convexity-preserving interpolation problems in industrycannot be denied A number of examples can be quoted inthis regard like the modelling of cars in automobile industryaeroplane and ship design A crumpled curve is an unwantedcharacteristic Human aesthetic sense demands convexity-preserving nice and smooth curves without wiggles [1] Con-vexity should also be upheld in many applications includingnonlinear programming problems occurring in engineer-ing telecommunication system design approximation offunctions parameter estimation and optimal control Thetraditional cubic spline schemes have been used for quite along time to deal with the problems of constructing smoothcurves that passes through given data points However
these splines sometimes fail to conserve the inherited shapecharacteristics because of unwanted oscillations that are notsuitable for design purpose
Some work [1ndash11 13] on shape preservation has beenpublished in recent years Abbas et al [2 4 5] discussedthe problem of local convexity-preserving data visualizationusing 119862
1 piecewise rational cubic and bicubic functionwith three shape parameters The authors derived the datadependent conditions for single shape parameter to get theconvexity preserving curve and remaining shape parameterswere used for the modification of convex curve to obtaina visually pleasing curve Brodlie and Butt [6] solved theproblem of shape preserving of convex data by using thecubic Hermite interpolation The authors inserted one ortwo extra knots in the interval where the shape of data wasnot conserved Costantini [7] solved the shape preserving ofboundary valued problems using polynomial spline interpo-lation with arbitrary constraints Duan et al [12] developedrational interpolation based on function values and alsodiscussed constrained control of the interpolating curvesThey obtained conditions on function values for constrainingthe interpolating curves to lie above below or between thegiven straight lines The authors assumed suitable values of
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 391568 10 pageshttpdxdoiorg1011552014391568
2 The Scientific World Journal
parameters to obtain 1198622 continuous curve and the method
works for only equally spaced dataFiorot and Tabka [8] used 119862
2 cubic polynomial spline toconserve the shape of convex or monotone data The authorsobtained the values of derivative parameters by solving threesystems of linear equations Hussain et al [9] addressed theproblem of shape preserving 119862
2 rational cubic spline forpositive and convex data Simple data dependent constraintswere derived for free parameters used in the description ofrational cubic function to achieve the desired shape of thedata The scheme provided a limited freedom to designerto obtain a visually pleasing display of the data LambertiandManni [10] presented and investigated the approximationorder of a global 1198622 shape preserving interpolating functionusing parametric cubic curves The tension parameters wereused to control the shape of curve The authors derived thenecessary and sufficient conditions for convexity whereasonly sufficient conditions for positivity and monotonicity ofdata Sarfraz et al [11] developed a 119862
2 rational cubic splinewith two families of free parameters for positive monotoneand convex curve Sufficient data dependent constraintswere made for free parameters to maintain the shape ofdata The scheme did not provide a liberty to designer forthe refinement of positivity monotonicity and convexity-preserving curves
Every developedmethod needs improvements or modifi-cations to meet the required conditions It can be used to getmore accurate results Many researchers can use new tech-niques to getmore accurate results which are the contributionfor the advancement of such results The technique used inthis paper is also a contribution to achieve the goal and hasmany prominent features over existing schemes
(i) In this work the degree of smoothness is 1198622 continu-ity while in [2 13] it is 1198621
(ii) In [6] the authors developed the scheme to achievethe desired shape of data by inserting extra knotsbetween any two knots in the interval while weconserve the shape of convex data by only imposingconstraints on free parameters without any extraknots
(iii) In [12] the authors developed schemes that work forequally spaced data while the proposed schemeworksfor both equally and unequally spaced data
(iv) The authors [14] assumed the certain function valuesand derivative values to control the shape of the datawhile in this paper data dependent constraints forthe free parameters in the description of rational cubicfunction are used to achieve the required shape of thedata
(v) The authors [8] achieved the values of derivativeparameters by solving the three systems of linearequations which is computationally expensive ascompared to methods developed in this paper wherethere exists only one tridiagonal system of linearequations for finding the values of derivative param-eters
(vi) Experimental and interpolation error analysis evi-dence suggests that the scheme is not only localin comparison with global scheme [10] and com-putationally economical but also produces smoothergraphical results as compared to [9 11]
(vii) In [11] the interpolant does not allow the designer tomodify the convex curve as per industrial demandsto obtain a visually pleasing curve while in this papertwo out of three shape parameters are left free fordesigner to refine the convexity preserving curve asdesired
(viii) The proposed curve scheme is unique in its represen-tation and applicable equally well for the data withderivatives or without derivatives
(ix) The proposed scheme is not concerned with anarbitrary degree it is a rational cubic spline in theform of cubicquadratic and by particular settingof shape parameters it reduces to a standard cubicHermite spline
This paper is organized as follows A1198622 piecewise rational
cubic function with three shape parameters is rewritten inSection 2 Local convexity-preserving rational cubic splineInterpolation is discussed in Section 3 Error estimation ofinterpolation is discussed in Section 5 Sufficient numericalexamples and discussion are given in Section 4 to prove theworth of the scheme The concluding remarks are presentedto end the paper
2 Rational Cubic Spline Function
Let (119909119894 119891119894) 119894 = 0 1 2 119899 be the given set of data points
such that 1199090lt 1199091lt 1199092lt sdot sdot sdot lt 119909
119899 A piecewise rational cubic
function [3] with three shape parameters in each subinterval119868119894= [119909119894 119909119894+1
(119906119894+ 2V119894+ 119908119894) minus V119894ℎ119894119889119894+1
)
times 1205792(1 minus 120579) + V
119894119891119894+1
1205793
119902119894(120579) = (1 minus 120579)
2119906119894+ 120579 (1 minus 120579) (119908
119894+ 119906119894+ V119894) + 1205792V119894
(8)
Remark 1 (see [3]) The system of linear equations definedin (5) is a strictly tridiagonal and has a unique solutionfor the derivatives parameters 119889
119894 119894 = 1 2 119899 minus 1 for all
119906119894 V119894gt 0 and 119908
119894ge 0 Moreover it is efficient to apply LU
decomposition method to solve the system for the values ofderivatives parameters 1198891015840
119894119904
Table 1 2D convex data set
119894 1 2 3 4 5 6 7119909119894
minus12 minus10 minus5 0 5 10 12119910119894
4 2 05 025 05 2 4
Remark 2 (see [3]) To make the rational cubic functionsmoother 1198622 continuity is applied at each knot The system(5) involves 119899 minus 1 linear equations while unknown derivativevalues are 119899 + 1 So two more equations are required forunique solution For this we impose end conditions at endknots as
1198781015840(1199090) = 1198890 119878
1015840(119909119899) = 119889119899 (9)
Remark 3 (see [3]) For the values of shape parameters set as119906119894= 1 V
119894= 1 and 119908
119894= 0 in each subinterval 119868
119894= [119909119894 119909119894+1
]119894 = 0 1 2 119899 minus 1 the rational cubic function reduces tostandard cubic Hermite spline [15]
3 Local Convexity-Preserving Rational CubicSpline Interpolation
In this section we discuss the solution of convexity-preserving problem by using 119862
2 rational cubic function withthree shape parameters For this problem we impose appro-priate constraints on single shape parameter to conservethe shape of convex data This requires some mathematicalarguments so that the required shape of data is achieved
Let (119909119894 119891119894) 119894 = 0 1 2 119899 be the given convex data
set This data set is said to be convex if
Δ119894lt Δ119894+1
119894 = 0 1 119899 minus 2 (10)
In similar way it is concave if
Δ119894gt Δ119894+1
119894 = 0 1 119899 minus 2 (11)
For strictly convex curves necessary condition for deriva-tive parameters to obtain the smoothness is
minus Δ119894) minus 119906119894(Δ119894minus 119889119894)
11987810158401015840
119894(119909) gt 0 if
5
sum
119896=0
(1 minus 120579)5minus119896
120579119896119862119896119894
gt 0 ℎ119894(119902119894(120579))3
gt 0
(16)
Since 119906119894gt 0 V119894gt 0 and119908
119894ge 0 it follows that ℎ
119894(119902119894(120579))3gt
05
sum
119896=0
(1 minus 120579)5minus119896
120579119896119862119896119894
gt 0 if 119862119896119894
gt 0 119896 = 0 1 2 3 4 5 (17)
Hence 119862119896119894
gt 0 119896 = 0 1 2 3 4 5 if the shape parameterssatisfy the following constraints
119906119894gt 0 V
119894gt 0
119908119894gt max0
V119894(119889119894+1
minus Δ119894)
(Δ119894minus 119889119894)
119906119894(Δ119894minus 119889119894)
(119889119894+1
minus Δ119894)
(18)
The above constraints can be rewritten as
119906119894gt 0 V
119894gt 0
119908119894= 120572119894+max0
V119894(119889119894+1
minus Δ119894)
(Δ119894minus 119889119894)
119906119894(Δ119894minus 119889119894)
(119889119894+1
minus Δ119894) 120572
119894gt 0
(19)
The above discussion can be summarized as follows
Theorem 4 The rational cubic function (7) conserves the1198622 convex curve of strictly convex data in each subinterval
[119909119894 119909119894+1
] if and only if the shape parameters 119906119894 V119894 and 119908
119894
satisfy (19)
4 Numerical Examples and Discussion
In this section the efficiency of the proposed convexity-preserving scheme through several numerical examples ispresented A comparison of 1198622 scheme with PCHIP (piece-wise cubic Hermite interpolating polynomial Built-in MAT-LAB program) and cubic Hermite spline scheme is also partof this section
Example 1 A convex data set is taken in Table 1 which isborrowed from [4] Figure 1(a) is drawn by cubic Hermitespline scheme [15] that does not conserve the local convexitythrough given convex data Figure 1(b) is generated by PCHIPthat does not look smooth because the function has onlyability to remove the undulations in shape preserving curvesOn the other hand Figures 1(c) and 1(d) are generatedby developed local convexity-preserving 119862
2 rational cubicfunction with different values of parameters 119906
119894and V
119894
The effect of shape parameters can be seen by noting thedifference in smoothness of the curves in Figures 1(b) 1(c)and 1(d) Numerical results of Figure 1(d) are determinedfrom developed scheme shown in Table 2
Example 2 A 2D convex data set is taken in Table 3 which isborrowed from [9] Figure 2(a) is generated by cubic Hermitespline scheme [15] that does not maintain the convexity ofgiven data Figure 2(b) is produced by PCHIP to conservethe convexity of convex data but it looks tight at some datapoints Figures 2(c) and 2(d) are generated by convexity-preserving rational cubic interpolant developed in Section 3A comparison of convexity-preserving curve through convexdata in these figures depicts the flaw of the cubic Hermitescheme and tightness of PCHIP Figure 2(d) looks morepleasant and smooth as compared to Figure 2(b) Numerical
Figure 2 Comparison of convexity- and nonconvexity-preserving curves
Table 5 A convex data set
119894 1 2 3 4 5 6 7119909119894
minus9 minus8 minus4 0 4 8 9119910119894
7 5 35 325 35 5 7
results of Figure 2(d) are determined by developed convexity-preserving 119862
2 rational cubic spline scheme shown in Table 4
Example 3 ThecubicHermite spline scheme [15] and PCHIPhave been used to draw Figures 3(a) and 3(b) respectivelythrough convex data given in Table 5 which is borrowed from[4]The efficiency of the schemedeveloped in Section 3 can beseen in Figures 3(c) and 3(d) A remarkable difference in thesmoothness with a pleasant graphical view is visible in thesefigures drawn by PCHIP and proposed rational cubic schemedue to the freedom granted to the designer on the values ofshape parameters Table 6 demonstrates the numerical resultscomputed from the proposed scheme of Figure 3(d)
Example 4 A convex data set is taken in Table 7 A noncon-vex curve from this given data is drawn in Figure 4(a) bycubic Hermite scheme [15] Figure 4(b) is produced by usingPCHIP to conserve the shape of curve but the visual modellooks tight as compared to proposed rational model whereasFigures 4(c) and 4(d) are generated by convexity-preservingrational cubic function developed in Section 3 A comparisonof convexity-preserving curve in these figures depicts the flawof the cubic Hermite spline scheme and tightness of PCHIPFigure 4(d) looks more pleasant and smooth as comparedto Figures 4(c) and 4(b) due to different values of shapeparameters The numerical results computed from proposedscheme of Figure 4(d) are shown in Table 8
5 Error Estimation
In this section the error of interpolation is calculated by usingthe following Theorem 5 which was developed by Abbas etal in [3] at some fixed values of free shape parameters 119906
max 1205851(119906119894 V119894 119908119894 120579) 0 le 120579 le 120579
lowast
max 1205852(119906119894 V119894 119908119894 120579) 120579
lowastle 120579 le 1
max 1205853(119906119894 V119894 119908119894 120579) 0 le 120579 le 1
(21)
where 1205851(119906119894 V119894 119908119894) 1205852(119906119894 V119894 119908119894) and 120585
3(119906119894 V119894 119908119894) are given in
[3]
Theorem 6 (see [3]) For any given positive values of shapeparameters 119906
119894 V119894 and 119908
119894 the optimal error 119888
119894in Theorem 5
satisfies 0 lt 119888119894le 00640
Proof See immediately Tables 9 10 11 and 12
6 Concluding Remarks
A 1198622 rational cubic function has been developed in this
paper for the smooth and attractive display of convex dataThree shape parameters were utilized for the descriptionof function to conserve the shape of convex data Simpledata dependent sufficient constraints were derived for singleshape parameter to insure convexity Remaining two shapeparameters have provided freedom to designer to modifythe shape of the curve by simply adjusting the values ofthe shape parameters No extra knots were inserted in theinterval where the interpolant loses the convexity The valuesof derivative parameters were achieved by solving the singlesystem of linear equations in comparison with [8] there existthree systems of linear equations for finding these valueswhich is computationally expensive and time-consuming
The Scientific World Journal 9
Table 8 Numerical results of Figure 4(d)
119894 1 2 3 4 5 6119889119894
minus0544 minus0452 0060 0136 0215 0283Δ119894
minus05 minus0048 0093 0166 025 mdash119906119894
15 15 15 15 15 mdashV119894
15 15 15 15 15 mdash119908119894
5108 3620 1811 3532 3638 mdash
Table 9 Values of 119888119894for several values of shape parameters 119906
processThe proposed scheme is not only1198622 smoother localand computationally economical but also visually pleasingas compared to schemes developed in [9 11] The proposedscheme works for both equally and unequally spaced datawhile the schemes developed in [12 14] work for only equallyspaced data
Table 11 Values of 119888119894with different values of shape parameters 119906
The authors declare that they have no conflict of interestsregarding the publication of this paper
10 The Scientific World Journal
References
[1] F Kuijt Convexity preserving interpolationmdashstationary nonlin-ear subdivision and splines [PhD thesis] University of TwenteEnschede The Netherlands 1998
[2] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo ISRN Mathematical Analysis vol 2012 Article ID174048 14 pages 2012
[3] M Abbas A A Majid and J M Ali ldquoMonotonicity-preserving1198622 rational cubic spline formonotone datardquoAppliedMathemat-
ics and Computation vol 219 no 6 pp 2885ndash2895 2012[4] M Abbas Shape preserving data visualization for curves and
surfaces using rational cubic functions [PhD thesis] Schoolof Mathematical Sciences Universiti Sains Malaysia PenangMalaysia 2012
[5] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo British Journal of Mathematics amp Computer Sciencevol 2 no 2 pp 72ndash93 2012
[6] KW Brodlie and S Butt ldquoPreserving convexity using piecewisecubic interpolationrdquoComputersampGraphics vol 15 no 1 pp 15ndash23 1991
[7] P Costantini ldquoBoundary-valued shape-preserving interpolat-ing splinesrdquo ACM Transactions on Mathematical Software vol23 no 2 pp 229ndash251 1997
[8] J-C Fiorot and J Tabka ldquoShape-preserving 1198622 cubic polyno-
[9] M Z Hussain M Sarfraz and T S Shaikh ldquoShape preservingrational cubic spline for positive and convex datardquo EgyptianInformatics Journal vol 12 no 3 pp 231ndash236 2011
[10] P Lamberti and C Manni ldquoShape-preserving 1198622 functional
interpolation via parametric cubicsrdquoNumerical Algorithms vol28 no 1ndash4 pp 229ndash254 2001
[11] M Sarfraz M Z Hussain T S Shaikh and R Iqbal ldquoDatavisualization using shape preserving 119862
2 rational splinerdquo inProceedings of the 15th International Conference on InformationVisualisation (IV rsquo11) pp 528ndash533 London UK July 2011
[12] Q Duan L Wang and E H Twizell ldquoA new 1198622 rational inter-
polation based on function values and constrained control ofthe interpolant curvesrdquo Applied Mathematics and Computationvol 161 no 1 pp 311ndash322 2005
[13] M Sarfraz M Z Hussain and M Hussain ldquoShape-preservingcurve interpolationrdquo International Journal of Computer Mathe-matics vol 89 no 1 pp 35ndash53 2012
[14] F Bao Q Sun and Q Duan ldquoPoint control of the interpolatingcurve with a rational cubic splinerdquo Journal of Visual Commu-nication and Image Representation vol 20 no 4 pp 275ndash2802009
[15] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Wellesley Mass USA 1993
parameters to obtain 1198622 continuous curve and the method
works for only equally spaced dataFiorot and Tabka [8] used 119862
2 cubic polynomial spline toconserve the shape of convex or monotone data The authorsobtained the values of derivative parameters by solving threesystems of linear equations Hussain et al [9] addressed theproblem of shape preserving 119862
2 rational cubic spline forpositive and convex data Simple data dependent constraintswere derived for free parameters used in the description ofrational cubic function to achieve the desired shape of thedata The scheme provided a limited freedom to designerto obtain a visually pleasing display of the data LambertiandManni [10] presented and investigated the approximationorder of a global 1198622 shape preserving interpolating functionusing parametric cubic curves The tension parameters wereused to control the shape of curve The authors derived thenecessary and sufficient conditions for convexity whereasonly sufficient conditions for positivity and monotonicity ofdata Sarfraz et al [11] developed a 119862
2 rational cubic splinewith two families of free parameters for positive monotoneand convex curve Sufficient data dependent constraintswere made for free parameters to maintain the shape ofdata The scheme did not provide a liberty to designer forthe refinement of positivity monotonicity and convexity-preserving curves
Every developedmethod needs improvements or modifi-cations to meet the required conditions It can be used to getmore accurate results Many researchers can use new tech-niques to getmore accurate results which are the contributionfor the advancement of such results The technique used inthis paper is also a contribution to achieve the goal and hasmany prominent features over existing schemes
(i) In this work the degree of smoothness is 1198622 continu-ity while in [2 13] it is 1198621
(ii) In [6] the authors developed the scheme to achievethe desired shape of data by inserting extra knotsbetween any two knots in the interval while weconserve the shape of convex data by only imposingconstraints on free parameters without any extraknots
(iii) In [12] the authors developed schemes that work forequally spaced data while the proposed schemeworksfor both equally and unequally spaced data
(iv) The authors [14] assumed the certain function valuesand derivative values to control the shape of the datawhile in this paper data dependent constraints forthe free parameters in the description of rational cubicfunction are used to achieve the required shape of thedata
(v) The authors [8] achieved the values of derivativeparameters by solving the three systems of linearequations which is computationally expensive ascompared to methods developed in this paper wherethere exists only one tridiagonal system of linearequations for finding the values of derivative param-eters
(vi) Experimental and interpolation error analysis evi-dence suggests that the scheme is not only localin comparison with global scheme [10] and com-putationally economical but also produces smoothergraphical results as compared to [9 11]
(vii) In [11] the interpolant does not allow the designer tomodify the convex curve as per industrial demandsto obtain a visually pleasing curve while in this papertwo out of three shape parameters are left free fordesigner to refine the convexity preserving curve asdesired
(viii) The proposed curve scheme is unique in its represen-tation and applicable equally well for the data withderivatives or without derivatives
(ix) The proposed scheme is not concerned with anarbitrary degree it is a rational cubic spline in theform of cubicquadratic and by particular settingof shape parameters it reduces to a standard cubicHermite spline
This paper is organized as follows A1198622 piecewise rational
cubic function with three shape parameters is rewritten inSection 2 Local convexity-preserving rational cubic splineInterpolation is discussed in Section 3 Error estimation ofinterpolation is discussed in Section 5 Sufficient numericalexamples and discussion are given in Section 4 to prove theworth of the scheme The concluding remarks are presentedto end the paper
2 Rational Cubic Spline Function
Let (119909119894 119891119894) 119894 = 0 1 2 119899 be the given set of data points
such that 1199090lt 1199091lt 1199092lt sdot sdot sdot lt 119909
119899 A piecewise rational cubic
function [3] with three shape parameters in each subinterval119868119894= [119909119894 119909119894+1
(119906119894+ 2V119894+ 119908119894) minus V119894ℎ119894119889119894+1
)
times 1205792(1 minus 120579) + V
119894119891119894+1
1205793
119902119894(120579) = (1 minus 120579)
2119906119894+ 120579 (1 minus 120579) (119908
119894+ 119906119894+ V119894) + 1205792V119894
(8)
Remark 1 (see [3]) The system of linear equations definedin (5) is a strictly tridiagonal and has a unique solutionfor the derivatives parameters 119889
119894 119894 = 1 2 119899 minus 1 for all
119906119894 V119894gt 0 and 119908
119894ge 0 Moreover it is efficient to apply LU
decomposition method to solve the system for the values ofderivatives parameters 1198891015840
119894119904
Table 1 2D convex data set
119894 1 2 3 4 5 6 7119909119894
minus12 minus10 minus5 0 5 10 12119910119894
4 2 05 025 05 2 4
Remark 2 (see [3]) To make the rational cubic functionsmoother 1198622 continuity is applied at each knot The system(5) involves 119899 minus 1 linear equations while unknown derivativevalues are 119899 + 1 So two more equations are required forunique solution For this we impose end conditions at endknots as
1198781015840(1199090) = 1198890 119878
1015840(119909119899) = 119889119899 (9)
Remark 3 (see [3]) For the values of shape parameters set as119906119894= 1 V
119894= 1 and 119908
119894= 0 in each subinterval 119868
119894= [119909119894 119909119894+1
]119894 = 0 1 2 119899 minus 1 the rational cubic function reduces tostandard cubic Hermite spline [15]
3 Local Convexity-Preserving Rational CubicSpline Interpolation
In this section we discuss the solution of convexity-preserving problem by using 119862
2 rational cubic function withthree shape parameters For this problem we impose appro-priate constraints on single shape parameter to conservethe shape of convex data This requires some mathematicalarguments so that the required shape of data is achieved
Let (119909119894 119891119894) 119894 = 0 1 2 119899 be the given convex data
set This data set is said to be convex if
Δ119894lt Δ119894+1
119894 = 0 1 119899 minus 2 (10)
In similar way it is concave if
Δ119894gt Δ119894+1
119894 = 0 1 119899 minus 2 (11)
For strictly convex curves necessary condition for deriva-tive parameters to obtain the smoothness is
minus Δ119894) minus 119906119894(Δ119894minus 119889119894)
11987810158401015840
119894(119909) gt 0 if
5
sum
119896=0
(1 minus 120579)5minus119896
120579119896119862119896119894
gt 0 ℎ119894(119902119894(120579))3
gt 0
(16)
Since 119906119894gt 0 V119894gt 0 and119908
119894ge 0 it follows that ℎ
119894(119902119894(120579))3gt
05
sum
119896=0
(1 minus 120579)5minus119896
120579119896119862119896119894
gt 0 if 119862119896119894
gt 0 119896 = 0 1 2 3 4 5 (17)
Hence 119862119896119894
gt 0 119896 = 0 1 2 3 4 5 if the shape parameterssatisfy the following constraints
119906119894gt 0 V
119894gt 0
119908119894gt max0
V119894(119889119894+1
minus Δ119894)
(Δ119894minus 119889119894)
119906119894(Δ119894minus 119889119894)
(119889119894+1
minus Δ119894)
(18)
The above constraints can be rewritten as
119906119894gt 0 V
119894gt 0
119908119894= 120572119894+max0
V119894(119889119894+1
minus Δ119894)
(Δ119894minus 119889119894)
119906119894(Δ119894minus 119889119894)
(119889119894+1
minus Δ119894) 120572
119894gt 0
(19)
The above discussion can be summarized as follows
Theorem 4 The rational cubic function (7) conserves the1198622 convex curve of strictly convex data in each subinterval
[119909119894 119909119894+1
] if and only if the shape parameters 119906119894 V119894 and 119908
119894
satisfy (19)
4 Numerical Examples and Discussion
In this section the efficiency of the proposed convexity-preserving scheme through several numerical examples ispresented A comparison of 1198622 scheme with PCHIP (piece-wise cubic Hermite interpolating polynomial Built-in MAT-LAB program) and cubic Hermite spline scheme is also partof this section
Example 1 A convex data set is taken in Table 1 which isborrowed from [4] Figure 1(a) is drawn by cubic Hermitespline scheme [15] that does not conserve the local convexitythrough given convex data Figure 1(b) is generated by PCHIPthat does not look smooth because the function has onlyability to remove the undulations in shape preserving curvesOn the other hand Figures 1(c) and 1(d) are generatedby developed local convexity-preserving 119862
2 rational cubicfunction with different values of parameters 119906
119894and V
119894
The effect of shape parameters can be seen by noting thedifference in smoothness of the curves in Figures 1(b) 1(c)and 1(d) Numerical results of Figure 1(d) are determinedfrom developed scheme shown in Table 2
Example 2 A 2D convex data set is taken in Table 3 which isborrowed from [9] Figure 2(a) is generated by cubic Hermitespline scheme [15] that does not maintain the convexity ofgiven data Figure 2(b) is produced by PCHIP to conservethe convexity of convex data but it looks tight at some datapoints Figures 2(c) and 2(d) are generated by convexity-preserving rational cubic interpolant developed in Section 3A comparison of convexity-preserving curve through convexdata in these figures depicts the flaw of the cubic Hermitescheme and tightness of PCHIP Figure 2(d) looks morepleasant and smooth as compared to Figure 2(b) Numerical
Figure 2 Comparison of convexity- and nonconvexity-preserving curves
Table 5 A convex data set
119894 1 2 3 4 5 6 7119909119894
minus9 minus8 minus4 0 4 8 9119910119894
7 5 35 325 35 5 7
results of Figure 2(d) are determined by developed convexity-preserving 119862
2 rational cubic spline scheme shown in Table 4
Example 3 ThecubicHermite spline scheme [15] and PCHIPhave been used to draw Figures 3(a) and 3(b) respectivelythrough convex data given in Table 5 which is borrowed from[4]The efficiency of the schemedeveloped in Section 3 can beseen in Figures 3(c) and 3(d) A remarkable difference in thesmoothness with a pleasant graphical view is visible in thesefigures drawn by PCHIP and proposed rational cubic schemedue to the freedom granted to the designer on the values ofshape parameters Table 6 demonstrates the numerical resultscomputed from the proposed scheme of Figure 3(d)
Example 4 A convex data set is taken in Table 7 A noncon-vex curve from this given data is drawn in Figure 4(a) bycubic Hermite scheme [15] Figure 4(b) is produced by usingPCHIP to conserve the shape of curve but the visual modellooks tight as compared to proposed rational model whereasFigures 4(c) and 4(d) are generated by convexity-preservingrational cubic function developed in Section 3 A comparisonof convexity-preserving curve in these figures depicts the flawof the cubic Hermite spline scheme and tightness of PCHIPFigure 4(d) looks more pleasant and smooth as comparedto Figures 4(c) and 4(b) due to different values of shapeparameters The numerical results computed from proposedscheme of Figure 4(d) are shown in Table 8
5 Error Estimation
In this section the error of interpolation is calculated by usingthe following Theorem 5 which was developed by Abbas etal in [3] at some fixed values of free shape parameters 119906
max 1205851(119906119894 V119894 119908119894 120579) 0 le 120579 le 120579
lowast
max 1205852(119906119894 V119894 119908119894 120579) 120579
lowastle 120579 le 1
max 1205853(119906119894 V119894 119908119894 120579) 0 le 120579 le 1
(21)
where 1205851(119906119894 V119894 119908119894) 1205852(119906119894 V119894 119908119894) and 120585
3(119906119894 V119894 119908119894) are given in
[3]
Theorem 6 (see [3]) For any given positive values of shapeparameters 119906
119894 V119894 and 119908
119894 the optimal error 119888
119894in Theorem 5
satisfies 0 lt 119888119894le 00640
Proof See immediately Tables 9 10 11 and 12
6 Concluding Remarks
A 1198622 rational cubic function has been developed in this
paper for the smooth and attractive display of convex dataThree shape parameters were utilized for the descriptionof function to conserve the shape of convex data Simpledata dependent sufficient constraints were derived for singleshape parameter to insure convexity Remaining two shapeparameters have provided freedom to designer to modifythe shape of the curve by simply adjusting the values ofthe shape parameters No extra knots were inserted in theinterval where the interpolant loses the convexity The valuesof derivative parameters were achieved by solving the singlesystem of linear equations in comparison with [8] there existthree systems of linear equations for finding these valueswhich is computationally expensive and time-consuming
The Scientific World Journal 9
Table 8 Numerical results of Figure 4(d)
119894 1 2 3 4 5 6119889119894
minus0544 minus0452 0060 0136 0215 0283Δ119894
minus05 minus0048 0093 0166 025 mdash119906119894
15 15 15 15 15 mdashV119894
15 15 15 15 15 mdash119908119894
5108 3620 1811 3532 3638 mdash
Table 9 Values of 119888119894for several values of shape parameters 119906
processThe proposed scheme is not only1198622 smoother localand computationally economical but also visually pleasingas compared to schemes developed in [9 11] The proposedscheme works for both equally and unequally spaced datawhile the schemes developed in [12 14] work for only equallyspaced data
Table 11 Values of 119888119894with different values of shape parameters 119906
The authors declare that they have no conflict of interestsregarding the publication of this paper
10 The Scientific World Journal
References
[1] F Kuijt Convexity preserving interpolationmdashstationary nonlin-ear subdivision and splines [PhD thesis] University of TwenteEnschede The Netherlands 1998
[2] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo ISRN Mathematical Analysis vol 2012 Article ID174048 14 pages 2012
[3] M Abbas A A Majid and J M Ali ldquoMonotonicity-preserving1198622 rational cubic spline formonotone datardquoAppliedMathemat-
ics and Computation vol 219 no 6 pp 2885ndash2895 2012[4] M Abbas Shape preserving data visualization for curves and
surfaces using rational cubic functions [PhD thesis] Schoolof Mathematical Sciences Universiti Sains Malaysia PenangMalaysia 2012
[5] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo British Journal of Mathematics amp Computer Sciencevol 2 no 2 pp 72ndash93 2012
[6] KW Brodlie and S Butt ldquoPreserving convexity using piecewisecubic interpolationrdquoComputersampGraphics vol 15 no 1 pp 15ndash23 1991
[7] P Costantini ldquoBoundary-valued shape-preserving interpolat-ing splinesrdquo ACM Transactions on Mathematical Software vol23 no 2 pp 229ndash251 1997
[8] J-C Fiorot and J Tabka ldquoShape-preserving 1198622 cubic polyno-
[9] M Z Hussain M Sarfraz and T S Shaikh ldquoShape preservingrational cubic spline for positive and convex datardquo EgyptianInformatics Journal vol 12 no 3 pp 231ndash236 2011
[10] P Lamberti and C Manni ldquoShape-preserving 1198622 functional
interpolation via parametric cubicsrdquoNumerical Algorithms vol28 no 1ndash4 pp 229ndash254 2001
[11] M Sarfraz M Z Hussain T S Shaikh and R Iqbal ldquoDatavisualization using shape preserving 119862
2 rational splinerdquo inProceedings of the 15th International Conference on InformationVisualisation (IV rsquo11) pp 528ndash533 London UK July 2011
[12] Q Duan L Wang and E H Twizell ldquoA new 1198622 rational inter-
polation based on function values and constrained control ofthe interpolant curvesrdquo Applied Mathematics and Computationvol 161 no 1 pp 311ndash322 2005
[13] M Sarfraz M Z Hussain and M Hussain ldquoShape-preservingcurve interpolationrdquo International Journal of Computer Mathe-matics vol 89 no 1 pp 35ndash53 2012
[14] F Bao Q Sun and Q Duan ldquoPoint control of the interpolatingcurve with a rational cubic splinerdquo Journal of Visual Commu-nication and Image Representation vol 20 no 4 pp 275ndash2802009
[15] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Wellesley Mass USA 1993
(119906119894+ 2V119894+ 119908119894) minus V119894ℎ119894119889119894+1
)
times 1205792(1 minus 120579) + V
119894119891119894+1
1205793
119902119894(120579) = (1 minus 120579)
2119906119894+ 120579 (1 minus 120579) (119908
119894+ 119906119894+ V119894) + 1205792V119894
(8)
Remark 1 (see [3]) The system of linear equations definedin (5) is a strictly tridiagonal and has a unique solutionfor the derivatives parameters 119889
119894 119894 = 1 2 119899 minus 1 for all
119906119894 V119894gt 0 and 119908
119894ge 0 Moreover it is efficient to apply LU
decomposition method to solve the system for the values ofderivatives parameters 1198891015840
119894119904
Table 1 2D convex data set
119894 1 2 3 4 5 6 7119909119894
minus12 minus10 minus5 0 5 10 12119910119894
4 2 05 025 05 2 4
Remark 2 (see [3]) To make the rational cubic functionsmoother 1198622 continuity is applied at each knot The system(5) involves 119899 minus 1 linear equations while unknown derivativevalues are 119899 + 1 So two more equations are required forunique solution For this we impose end conditions at endknots as
1198781015840(1199090) = 1198890 119878
1015840(119909119899) = 119889119899 (9)
Remark 3 (see [3]) For the values of shape parameters set as119906119894= 1 V
119894= 1 and 119908
119894= 0 in each subinterval 119868
119894= [119909119894 119909119894+1
]119894 = 0 1 2 119899 minus 1 the rational cubic function reduces tostandard cubic Hermite spline [15]
3 Local Convexity-Preserving Rational CubicSpline Interpolation
In this section we discuss the solution of convexity-preserving problem by using 119862
2 rational cubic function withthree shape parameters For this problem we impose appro-priate constraints on single shape parameter to conservethe shape of convex data This requires some mathematicalarguments so that the required shape of data is achieved
Let (119909119894 119891119894) 119894 = 0 1 2 119899 be the given convex data
set This data set is said to be convex if
Δ119894lt Δ119894+1
119894 = 0 1 119899 minus 2 (10)
In similar way it is concave if
Δ119894gt Δ119894+1
119894 = 0 1 119899 minus 2 (11)
For strictly convex curves necessary condition for deriva-tive parameters to obtain the smoothness is
minus Δ119894) minus 119906119894(Δ119894minus 119889119894)
11987810158401015840
119894(119909) gt 0 if
5
sum
119896=0
(1 minus 120579)5minus119896
120579119896119862119896119894
gt 0 ℎ119894(119902119894(120579))3
gt 0
(16)
Since 119906119894gt 0 V119894gt 0 and119908
119894ge 0 it follows that ℎ
119894(119902119894(120579))3gt
05
sum
119896=0
(1 minus 120579)5minus119896
120579119896119862119896119894
gt 0 if 119862119896119894
gt 0 119896 = 0 1 2 3 4 5 (17)
Hence 119862119896119894
gt 0 119896 = 0 1 2 3 4 5 if the shape parameterssatisfy the following constraints
119906119894gt 0 V
119894gt 0
119908119894gt max0
V119894(119889119894+1
minus Δ119894)
(Δ119894minus 119889119894)
119906119894(Δ119894minus 119889119894)
(119889119894+1
minus Δ119894)
(18)
The above constraints can be rewritten as
119906119894gt 0 V
119894gt 0
119908119894= 120572119894+max0
V119894(119889119894+1
minus Δ119894)
(Δ119894minus 119889119894)
119906119894(Δ119894minus 119889119894)
(119889119894+1
minus Δ119894) 120572
119894gt 0
(19)
The above discussion can be summarized as follows
Theorem 4 The rational cubic function (7) conserves the1198622 convex curve of strictly convex data in each subinterval
[119909119894 119909119894+1
] if and only if the shape parameters 119906119894 V119894 and 119908
119894
satisfy (19)
4 Numerical Examples and Discussion
In this section the efficiency of the proposed convexity-preserving scheme through several numerical examples ispresented A comparison of 1198622 scheme with PCHIP (piece-wise cubic Hermite interpolating polynomial Built-in MAT-LAB program) and cubic Hermite spline scheme is also partof this section
Example 1 A convex data set is taken in Table 1 which isborrowed from [4] Figure 1(a) is drawn by cubic Hermitespline scheme [15] that does not conserve the local convexitythrough given convex data Figure 1(b) is generated by PCHIPthat does not look smooth because the function has onlyability to remove the undulations in shape preserving curvesOn the other hand Figures 1(c) and 1(d) are generatedby developed local convexity-preserving 119862
2 rational cubicfunction with different values of parameters 119906
119894and V
119894
The effect of shape parameters can be seen by noting thedifference in smoothness of the curves in Figures 1(b) 1(c)and 1(d) Numerical results of Figure 1(d) are determinedfrom developed scheme shown in Table 2
Example 2 A 2D convex data set is taken in Table 3 which isborrowed from [9] Figure 2(a) is generated by cubic Hermitespline scheme [15] that does not maintain the convexity ofgiven data Figure 2(b) is produced by PCHIP to conservethe convexity of convex data but it looks tight at some datapoints Figures 2(c) and 2(d) are generated by convexity-preserving rational cubic interpolant developed in Section 3A comparison of convexity-preserving curve through convexdata in these figures depicts the flaw of the cubic Hermitescheme and tightness of PCHIP Figure 2(d) looks morepleasant and smooth as compared to Figure 2(b) Numerical
Figure 2 Comparison of convexity- and nonconvexity-preserving curves
Table 5 A convex data set
119894 1 2 3 4 5 6 7119909119894
minus9 minus8 minus4 0 4 8 9119910119894
7 5 35 325 35 5 7
results of Figure 2(d) are determined by developed convexity-preserving 119862
2 rational cubic spline scheme shown in Table 4
Example 3 ThecubicHermite spline scheme [15] and PCHIPhave been used to draw Figures 3(a) and 3(b) respectivelythrough convex data given in Table 5 which is borrowed from[4]The efficiency of the schemedeveloped in Section 3 can beseen in Figures 3(c) and 3(d) A remarkable difference in thesmoothness with a pleasant graphical view is visible in thesefigures drawn by PCHIP and proposed rational cubic schemedue to the freedom granted to the designer on the values ofshape parameters Table 6 demonstrates the numerical resultscomputed from the proposed scheme of Figure 3(d)
Example 4 A convex data set is taken in Table 7 A noncon-vex curve from this given data is drawn in Figure 4(a) bycubic Hermite scheme [15] Figure 4(b) is produced by usingPCHIP to conserve the shape of curve but the visual modellooks tight as compared to proposed rational model whereasFigures 4(c) and 4(d) are generated by convexity-preservingrational cubic function developed in Section 3 A comparisonof convexity-preserving curve in these figures depicts the flawof the cubic Hermite spline scheme and tightness of PCHIPFigure 4(d) looks more pleasant and smooth as comparedto Figures 4(c) and 4(b) due to different values of shapeparameters The numerical results computed from proposedscheme of Figure 4(d) are shown in Table 8
5 Error Estimation
In this section the error of interpolation is calculated by usingthe following Theorem 5 which was developed by Abbas etal in [3] at some fixed values of free shape parameters 119906
max 1205851(119906119894 V119894 119908119894 120579) 0 le 120579 le 120579
lowast
max 1205852(119906119894 V119894 119908119894 120579) 120579
lowastle 120579 le 1
max 1205853(119906119894 V119894 119908119894 120579) 0 le 120579 le 1
(21)
where 1205851(119906119894 V119894 119908119894) 1205852(119906119894 V119894 119908119894) and 120585
3(119906119894 V119894 119908119894) are given in
[3]
Theorem 6 (see [3]) For any given positive values of shapeparameters 119906
119894 V119894 and 119908
119894 the optimal error 119888
119894in Theorem 5
satisfies 0 lt 119888119894le 00640
Proof See immediately Tables 9 10 11 and 12
6 Concluding Remarks
A 1198622 rational cubic function has been developed in this
paper for the smooth and attractive display of convex dataThree shape parameters were utilized for the descriptionof function to conserve the shape of convex data Simpledata dependent sufficient constraints were derived for singleshape parameter to insure convexity Remaining two shapeparameters have provided freedom to designer to modifythe shape of the curve by simply adjusting the values ofthe shape parameters No extra knots were inserted in theinterval where the interpolant loses the convexity The valuesof derivative parameters were achieved by solving the singlesystem of linear equations in comparison with [8] there existthree systems of linear equations for finding these valueswhich is computationally expensive and time-consuming
The Scientific World Journal 9
Table 8 Numerical results of Figure 4(d)
119894 1 2 3 4 5 6119889119894
minus0544 minus0452 0060 0136 0215 0283Δ119894
minus05 minus0048 0093 0166 025 mdash119906119894
15 15 15 15 15 mdashV119894
15 15 15 15 15 mdash119908119894
5108 3620 1811 3532 3638 mdash
Table 9 Values of 119888119894for several values of shape parameters 119906
processThe proposed scheme is not only1198622 smoother localand computationally economical but also visually pleasingas compared to schemes developed in [9 11] The proposedscheme works for both equally and unequally spaced datawhile the schemes developed in [12 14] work for only equallyspaced data
Table 11 Values of 119888119894with different values of shape parameters 119906
The authors declare that they have no conflict of interestsregarding the publication of this paper
10 The Scientific World Journal
References
[1] F Kuijt Convexity preserving interpolationmdashstationary nonlin-ear subdivision and splines [PhD thesis] University of TwenteEnschede The Netherlands 1998
[2] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo ISRN Mathematical Analysis vol 2012 Article ID174048 14 pages 2012
[3] M Abbas A A Majid and J M Ali ldquoMonotonicity-preserving1198622 rational cubic spline formonotone datardquoAppliedMathemat-
ics and Computation vol 219 no 6 pp 2885ndash2895 2012[4] M Abbas Shape preserving data visualization for curves and
surfaces using rational cubic functions [PhD thesis] Schoolof Mathematical Sciences Universiti Sains Malaysia PenangMalaysia 2012
[5] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo British Journal of Mathematics amp Computer Sciencevol 2 no 2 pp 72ndash93 2012
[6] KW Brodlie and S Butt ldquoPreserving convexity using piecewisecubic interpolationrdquoComputersampGraphics vol 15 no 1 pp 15ndash23 1991
[7] P Costantini ldquoBoundary-valued shape-preserving interpolat-ing splinesrdquo ACM Transactions on Mathematical Software vol23 no 2 pp 229ndash251 1997
[8] J-C Fiorot and J Tabka ldquoShape-preserving 1198622 cubic polyno-
[9] M Z Hussain M Sarfraz and T S Shaikh ldquoShape preservingrational cubic spline for positive and convex datardquo EgyptianInformatics Journal vol 12 no 3 pp 231ndash236 2011
[10] P Lamberti and C Manni ldquoShape-preserving 1198622 functional
interpolation via parametric cubicsrdquoNumerical Algorithms vol28 no 1ndash4 pp 229ndash254 2001
[11] M Sarfraz M Z Hussain T S Shaikh and R Iqbal ldquoDatavisualization using shape preserving 119862
2 rational splinerdquo inProceedings of the 15th International Conference on InformationVisualisation (IV rsquo11) pp 528ndash533 London UK July 2011
[12] Q Duan L Wang and E H Twizell ldquoA new 1198622 rational inter-
polation based on function values and constrained control ofthe interpolant curvesrdquo Applied Mathematics and Computationvol 161 no 1 pp 311ndash322 2005
[13] M Sarfraz M Z Hussain and M Hussain ldquoShape-preservingcurve interpolationrdquo International Journal of Computer Mathe-matics vol 89 no 1 pp 35ndash53 2012
[14] F Bao Q Sun and Q Duan ldquoPoint control of the interpolatingcurve with a rational cubic splinerdquo Journal of Visual Commu-nication and Image Representation vol 20 no 4 pp 275ndash2802009
[15] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Wellesley Mass USA 1993
minus Δ119894) minus 119906119894(Δ119894minus 119889119894)
11987810158401015840
119894(119909) gt 0 if
5
sum
119896=0
(1 minus 120579)5minus119896
120579119896119862119896119894
gt 0 ℎ119894(119902119894(120579))3
gt 0
(16)
Since 119906119894gt 0 V119894gt 0 and119908
119894ge 0 it follows that ℎ
119894(119902119894(120579))3gt
05
sum
119896=0
(1 minus 120579)5minus119896
120579119896119862119896119894
gt 0 if 119862119896119894
gt 0 119896 = 0 1 2 3 4 5 (17)
Hence 119862119896119894
gt 0 119896 = 0 1 2 3 4 5 if the shape parameterssatisfy the following constraints
119906119894gt 0 V
119894gt 0
119908119894gt max0
V119894(119889119894+1
minus Δ119894)
(Δ119894minus 119889119894)
119906119894(Δ119894minus 119889119894)
(119889119894+1
minus Δ119894)
(18)
The above constraints can be rewritten as
119906119894gt 0 V
119894gt 0
119908119894= 120572119894+max0
V119894(119889119894+1
minus Δ119894)
(Δ119894minus 119889119894)
119906119894(Δ119894minus 119889119894)
(119889119894+1
minus Δ119894) 120572
119894gt 0
(19)
The above discussion can be summarized as follows
Theorem 4 The rational cubic function (7) conserves the1198622 convex curve of strictly convex data in each subinterval
[119909119894 119909119894+1
] if and only if the shape parameters 119906119894 V119894 and 119908
119894
satisfy (19)
4 Numerical Examples and Discussion
In this section the efficiency of the proposed convexity-preserving scheme through several numerical examples ispresented A comparison of 1198622 scheme with PCHIP (piece-wise cubic Hermite interpolating polynomial Built-in MAT-LAB program) and cubic Hermite spline scheme is also partof this section
Example 1 A convex data set is taken in Table 1 which isborrowed from [4] Figure 1(a) is drawn by cubic Hermitespline scheme [15] that does not conserve the local convexitythrough given convex data Figure 1(b) is generated by PCHIPthat does not look smooth because the function has onlyability to remove the undulations in shape preserving curvesOn the other hand Figures 1(c) and 1(d) are generatedby developed local convexity-preserving 119862
2 rational cubicfunction with different values of parameters 119906
119894and V
119894
The effect of shape parameters can be seen by noting thedifference in smoothness of the curves in Figures 1(b) 1(c)and 1(d) Numerical results of Figure 1(d) are determinedfrom developed scheme shown in Table 2
Example 2 A 2D convex data set is taken in Table 3 which isborrowed from [9] Figure 2(a) is generated by cubic Hermitespline scheme [15] that does not maintain the convexity ofgiven data Figure 2(b) is produced by PCHIP to conservethe convexity of convex data but it looks tight at some datapoints Figures 2(c) and 2(d) are generated by convexity-preserving rational cubic interpolant developed in Section 3A comparison of convexity-preserving curve through convexdata in these figures depicts the flaw of the cubic Hermitescheme and tightness of PCHIP Figure 2(d) looks morepleasant and smooth as compared to Figure 2(b) Numerical
Figure 2 Comparison of convexity- and nonconvexity-preserving curves
Table 5 A convex data set
119894 1 2 3 4 5 6 7119909119894
minus9 minus8 minus4 0 4 8 9119910119894
7 5 35 325 35 5 7
results of Figure 2(d) are determined by developed convexity-preserving 119862
2 rational cubic spline scheme shown in Table 4
Example 3 ThecubicHermite spline scheme [15] and PCHIPhave been used to draw Figures 3(a) and 3(b) respectivelythrough convex data given in Table 5 which is borrowed from[4]The efficiency of the schemedeveloped in Section 3 can beseen in Figures 3(c) and 3(d) A remarkable difference in thesmoothness with a pleasant graphical view is visible in thesefigures drawn by PCHIP and proposed rational cubic schemedue to the freedom granted to the designer on the values ofshape parameters Table 6 demonstrates the numerical resultscomputed from the proposed scheme of Figure 3(d)
Example 4 A convex data set is taken in Table 7 A noncon-vex curve from this given data is drawn in Figure 4(a) bycubic Hermite scheme [15] Figure 4(b) is produced by usingPCHIP to conserve the shape of curve but the visual modellooks tight as compared to proposed rational model whereasFigures 4(c) and 4(d) are generated by convexity-preservingrational cubic function developed in Section 3 A comparisonof convexity-preserving curve in these figures depicts the flawof the cubic Hermite spline scheme and tightness of PCHIPFigure 4(d) looks more pleasant and smooth as comparedto Figures 4(c) and 4(b) due to different values of shapeparameters The numerical results computed from proposedscheme of Figure 4(d) are shown in Table 8
5 Error Estimation
In this section the error of interpolation is calculated by usingthe following Theorem 5 which was developed by Abbas etal in [3] at some fixed values of free shape parameters 119906
max 1205851(119906119894 V119894 119908119894 120579) 0 le 120579 le 120579
lowast
max 1205852(119906119894 V119894 119908119894 120579) 120579
lowastle 120579 le 1
max 1205853(119906119894 V119894 119908119894 120579) 0 le 120579 le 1
(21)
where 1205851(119906119894 V119894 119908119894) 1205852(119906119894 V119894 119908119894) and 120585
3(119906119894 V119894 119908119894) are given in
[3]
Theorem 6 (see [3]) For any given positive values of shapeparameters 119906
119894 V119894 and 119908
119894 the optimal error 119888
119894in Theorem 5
satisfies 0 lt 119888119894le 00640
Proof See immediately Tables 9 10 11 and 12
6 Concluding Remarks
A 1198622 rational cubic function has been developed in this
paper for the smooth and attractive display of convex dataThree shape parameters were utilized for the descriptionof function to conserve the shape of convex data Simpledata dependent sufficient constraints were derived for singleshape parameter to insure convexity Remaining two shapeparameters have provided freedom to designer to modifythe shape of the curve by simply adjusting the values ofthe shape parameters No extra knots were inserted in theinterval where the interpolant loses the convexity The valuesof derivative parameters were achieved by solving the singlesystem of linear equations in comparison with [8] there existthree systems of linear equations for finding these valueswhich is computationally expensive and time-consuming
The Scientific World Journal 9
Table 8 Numerical results of Figure 4(d)
119894 1 2 3 4 5 6119889119894
minus0544 minus0452 0060 0136 0215 0283Δ119894
minus05 minus0048 0093 0166 025 mdash119906119894
15 15 15 15 15 mdashV119894
15 15 15 15 15 mdash119908119894
5108 3620 1811 3532 3638 mdash
Table 9 Values of 119888119894for several values of shape parameters 119906
processThe proposed scheme is not only1198622 smoother localand computationally economical but also visually pleasingas compared to schemes developed in [9 11] The proposedscheme works for both equally and unequally spaced datawhile the schemes developed in [12 14] work for only equallyspaced data
Table 11 Values of 119888119894with different values of shape parameters 119906
The authors declare that they have no conflict of interestsregarding the publication of this paper
10 The Scientific World Journal
References
[1] F Kuijt Convexity preserving interpolationmdashstationary nonlin-ear subdivision and splines [PhD thesis] University of TwenteEnschede The Netherlands 1998
[2] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo ISRN Mathematical Analysis vol 2012 Article ID174048 14 pages 2012
[3] M Abbas A A Majid and J M Ali ldquoMonotonicity-preserving1198622 rational cubic spline formonotone datardquoAppliedMathemat-
ics and Computation vol 219 no 6 pp 2885ndash2895 2012[4] M Abbas Shape preserving data visualization for curves and
surfaces using rational cubic functions [PhD thesis] Schoolof Mathematical Sciences Universiti Sains Malaysia PenangMalaysia 2012
[5] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo British Journal of Mathematics amp Computer Sciencevol 2 no 2 pp 72ndash93 2012
[6] KW Brodlie and S Butt ldquoPreserving convexity using piecewisecubic interpolationrdquoComputersampGraphics vol 15 no 1 pp 15ndash23 1991
[7] P Costantini ldquoBoundary-valued shape-preserving interpolat-ing splinesrdquo ACM Transactions on Mathematical Software vol23 no 2 pp 229ndash251 1997
[8] J-C Fiorot and J Tabka ldquoShape-preserving 1198622 cubic polyno-
[9] M Z Hussain M Sarfraz and T S Shaikh ldquoShape preservingrational cubic spline for positive and convex datardquo EgyptianInformatics Journal vol 12 no 3 pp 231ndash236 2011
[10] P Lamberti and C Manni ldquoShape-preserving 1198622 functional
interpolation via parametric cubicsrdquoNumerical Algorithms vol28 no 1ndash4 pp 229ndash254 2001
[11] M Sarfraz M Z Hussain T S Shaikh and R Iqbal ldquoDatavisualization using shape preserving 119862
2 rational splinerdquo inProceedings of the 15th International Conference on InformationVisualisation (IV rsquo11) pp 528ndash533 London UK July 2011
[12] Q Duan L Wang and E H Twizell ldquoA new 1198622 rational inter-
polation based on function values and constrained control ofthe interpolant curvesrdquo Applied Mathematics and Computationvol 161 no 1 pp 311ndash322 2005
[13] M Sarfraz M Z Hussain and M Hussain ldquoShape-preservingcurve interpolationrdquo International Journal of Computer Mathe-matics vol 89 no 1 pp 35ndash53 2012
[14] F Bao Q Sun and Q Duan ldquoPoint control of the interpolatingcurve with a rational cubic splinerdquo Journal of Visual Commu-nication and Image Representation vol 20 no 4 pp 275ndash2802009
[15] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Wellesley Mass USA 1993
Figure 2 Comparison of convexity- and nonconvexity-preserving curves
Table 5 A convex data set
119894 1 2 3 4 5 6 7119909119894
minus9 minus8 minus4 0 4 8 9119910119894
7 5 35 325 35 5 7
results of Figure 2(d) are determined by developed convexity-preserving 119862
2 rational cubic spline scheme shown in Table 4
Example 3 ThecubicHermite spline scheme [15] and PCHIPhave been used to draw Figures 3(a) and 3(b) respectivelythrough convex data given in Table 5 which is borrowed from[4]The efficiency of the schemedeveloped in Section 3 can beseen in Figures 3(c) and 3(d) A remarkable difference in thesmoothness with a pleasant graphical view is visible in thesefigures drawn by PCHIP and proposed rational cubic schemedue to the freedom granted to the designer on the values ofshape parameters Table 6 demonstrates the numerical resultscomputed from the proposed scheme of Figure 3(d)
Example 4 A convex data set is taken in Table 7 A noncon-vex curve from this given data is drawn in Figure 4(a) bycubic Hermite scheme [15] Figure 4(b) is produced by usingPCHIP to conserve the shape of curve but the visual modellooks tight as compared to proposed rational model whereasFigures 4(c) and 4(d) are generated by convexity-preservingrational cubic function developed in Section 3 A comparisonof convexity-preserving curve in these figures depicts the flawof the cubic Hermite spline scheme and tightness of PCHIPFigure 4(d) looks more pleasant and smooth as comparedto Figures 4(c) and 4(b) due to different values of shapeparameters The numerical results computed from proposedscheme of Figure 4(d) are shown in Table 8
5 Error Estimation
In this section the error of interpolation is calculated by usingthe following Theorem 5 which was developed by Abbas etal in [3] at some fixed values of free shape parameters 119906
max 1205851(119906119894 V119894 119908119894 120579) 0 le 120579 le 120579
lowast
max 1205852(119906119894 V119894 119908119894 120579) 120579
lowastle 120579 le 1
max 1205853(119906119894 V119894 119908119894 120579) 0 le 120579 le 1
(21)
where 1205851(119906119894 V119894 119908119894) 1205852(119906119894 V119894 119908119894) and 120585
3(119906119894 V119894 119908119894) are given in
[3]
Theorem 6 (see [3]) For any given positive values of shapeparameters 119906
119894 V119894 and 119908
119894 the optimal error 119888
119894in Theorem 5
satisfies 0 lt 119888119894le 00640
Proof See immediately Tables 9 10 11 and 12
6 Concluding Remarks
A 1198622 rational cubic function has been developed in this
paper for the smooth and attractive display of convex dataThree shape parameters were utilized for the descriptionof function to conserve the shape of convex data Simpledata dependent sufficient constraints were derived for singleshape parameter to insure convexity Remaining two shapeparameters have provided freedom to designer to modifythe shape of the curve by simply adjusting the values ofthe shape parameters No extra knots were inserted in theinterval where the interpolant loses the convexity The valuesof derivative parameters were achieved by solving the singlesystem of linear equations in comparison with [8] there existthree systems of linear equations for finding these valueswhich is computationally expensive and time-consuming
The Scientific World Journal 9
Table 8 Numerical results of Figure 4(d)
119894 1 2 3 4 5 6119889119894
minus0544 minus0452 0060 0136 0215 0283Δ119894
minus05 minus0048 0093 0166 025 mdash119906119894
15 15 15 15 15 mdashV119894
15 15 15 15 15 mdash119908119894
5108 3620 1811 3532 3638 mdash
Table 9 Values of 119888119894for several values of shape parameters 119906
processThe proposed scheme is not only1198622 smoother localand computationally economical but also visually pleasingas compared to schemes developed in [9 11] The proposedscheme works for both equally and unequally spaced datawhile the schemes developed in [12 14] work for only equallyspaced data
Table 11 Values of 119888119894with different values of shape parameters 119906
The authors declare that they have no conflict of interestsregarding the publication of this paper
10 The Scientific World Journal
References
[1] F Kuijt Convexity preserving interpolationmdashstationary nonlin-ear subdivision and splines [PhD thesis] University of TwenteEnschede The Netherlands 1998
[2] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo ISRN Mathematical Analysis vol 2012 Article ID174048 14 pages 2012
[3] M Abbas A A Majid and J M Ali ldquoMonotonicity-preserving1198622 rational cubic spline formonotone datardquoAppliedMathemat-
ics and Computation vol 219 no 6 pp 2885ndash2895 2012[4] M Abbas Shape preserving data visualization for curves and
surfaces using rational cubic functions [PhD thesis] Schoolof Mathematical Sciences Universiti Sains Malaysia PenangMalaysia 2012
[5] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo British Journal of Mathematics amp Computer Sciencevol 2 no 2 pp 72ndash93 2012
[6] KW Brodlie and S Butt ldquoPreserving convexity using piecewisecubic interpolationrdquoComputersampGraphics vol 15 no 1 pp 15ndash23 1991
[7] P Costantini ldquoBoundary-valued shape-preserving interpolat-ing splinesrdquo ACM Transactions on Mathematical Software vol23 no 2 pp 229ndash251 1997
[8] J-C Fiorot and J Tabka ldquoShape-preserving 1198622 cubic polyno-
[9] M Z Hussain M Sarfraz and T S Shaikh ldquoShape preservingrational cubic spline for positive and convex datardquo EgyptianInformatics Journal vol 12 no 3 pp 231ndash236 2011
[10] P Lamberti and C Manni ldquoShape-preserving 1198622 functional
interpolation via parametric cubicsrdquoNumerical Algorithms vol28 no 1ndash4 pp 229ndash254 2001
[11] M Sarfraz M Z Hussain T S Shaikh and R Iqbal ldquoDatavisualization using shape preserving 119862
2 rational splinerdquo inProceedings of the 15th International Conference on InformationVisualisation (IV rsquo11) pp 528ndash533 London UK July 2011
[12] Q Duan L Wang and E H Twizell ldquoA new 1198622 rational inter-
polation based on function values and constrained control ofthe interpolant curvesrdquo Applied Mathematics and Computationvol 161 no 1 pp 311ndash322 2005
[13] M Sarfraz M Z Hussain and M Hussain ldquoShape-preservingcurve interpolationrdquo International Journal of Computer Mathe-matics vol 89 no 1 pp 35ndash53 2012
[14] F Bao Q Sun and Q Duan ldquoPoint control of the interpolatingcurve with a rational cubic splinerdquo Journal of Visual Commu-nication and Image Representation vol 20 no 4 pp 275ndash2802009
[15] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Wellesley Mass USA 1993
Figure 2 Comparison of convexity- and nonconvexity-preserving curves
Table 5 A convex data set
119894 1 2 3 4 5 6 7119909119894
minus9 minus8 minus4 0 4 8 9119910119894
7 5 35 325 35 5 7
results of Figure 2(d) are determined by developed convexity-preserving 119862
2 rational cubic spline scheme shown in Table 4
Example 3 ThecubicHermite spline scheme [15] and PCHIPhave been used to draw Figures 3(a) and 3(b) respectivelythrough convex data given in Table 5 which is borrowed from[4]The efficiency of the schemedeveloped in Section 3 can beseen in Figures 3(c) and 3(d) A remarkable difference in thesmoothness with a pleasant graphical view is visible in thesefigures drawn by PCHIP and proposed rational cubic schemedue to the freedom granted to the designer on the values ofshape parameters Table 6 demonstrates the numerical resultscomputed from the proposed scheme of Figure 3(d)
Example 4 A convex data set is taken in Table 7 A noncon-vex curve from this given data is drawn in Figure 4(a) bycubic Hermite scheme [15] Figure 4(b) is produced by usingPCHIP to conserve the shape of curve but the visual modellooks tight as compared to proposed rational model whereasFigures 4(c) and 4(d) are generated by convexity-preservingrational cubic function developed in Section 3 A comparisonof convexity-preserving curve in these figures depicts the flawof the cubic Hermite spline scheme and tightness of PCHIPFigure 4(d) looks more pleasant and smooth as comparedto Figures 4(c) and 4(b) due to different values of shapeparameters The numerical results computed from proposedscheme of Figure 4(d) are shown in Table 8
5 Error Estimation
In this section the error of interpolation is calculated by usingthe following Theorem 5 which was developed by Abbas etal in [3] at some fixed values of free shape parameters 119906
max 1205851(119906119894 V119894 119908119894 120579) 0 le 120579 le 120579
lowast
max 1205852(119906119894 V119894 119908119894 120579) 120579
lowastle 120579 le 1
max 1205853(119906119894 V119894 119908119894 120579) 0 le 120579 le 1
(21)
where 1205851(119906119894 V119894 119908119894) 1205852(119906119894 V119894 119908119894) and 120585
3(119906119894 V119894 119908119894) are given in
[3]
Theorem 6 (see [3]) For any given positive values of shapeparameters 119906
119894 V119894 and 119908
119894 the optimal error 119888
119894in Theorem 5
satisfies 0 lt 119888119894le 00640
Proof See immediately Tables 9 10 11 and 12
6 Concluding Remarks
A 1198622 rational cubic function has been developed in this
paper for the smooth and attractive display of convex dataThree shape parameters were utilized for the descriptionof function to conserve the shape of convex data Simpledata dependent sufficient constraints were derived for singleshape parameter to insure convexity Remaining two shapeparameters have provided freedom to designer to modifythe shape of the curve by simply adjusting the values ofthe shape parameters No extra knots were inserted in theinterval where the interpolant loses the convexity The valuesof derivative parameters were achieved by solving the singlesystem of linear equations in comparison with [8] there existthree systems of linear equations for finding these valueswhich is computationally expensive and time-consuming
The Scientific World Journal 9
Table 8 Numerical results of Figure 4(d)
119894 1 2 3 4 5 6119889119894
minus0544 minus0452 0060 0136 0215 0283Δ119894
minus05 minus0048 0093 0166 025 mdash119906119894
15 15 15 15 15 mdashV119894
15 15 15 15 15 mdash119908119894
5108 3620 1811 3532 3638 mdash
Table 9 Values of 119888119894for several values of shape parameters 119906
processThe proposed scheme is not only1198622 smoother localand computationally economical but also visually pleasingas compared to schemes developed in [9 11] The proposedscheme works for both equally and unequally spaced datawhile the schemes developed in [12 14] work for only equallyspaced data
Table 11 Values of 119888119894with different values of shape parameters 119906
The authors declare that they have no conflict of interestsregarding the publication of this paper
10 The Scientific World Journal
References
[1] F Kuijt Convexity preserving interpolationmdashstationary nonlin-ear subdivision and splines [PhD thesis] University of TwenteEnschede The Netherlands 1998
[2] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo ISRN Mathematical Analysis vol 2012 Article ID174048 14 pages 2012
[3] M Abbas A A Majid and J M Ali ldquoMonotonicity-preserving1198622 rational cubic spline formonotone datardquoAppliedMathemat-
ics and Computation vol 219 no 6 pp 2885ndash2895 2012[4] M Abbas Shape preserving data visualization for curves and
surfaces using rational cubic functions [PhD thesis] Schoolof Mathematical Sciences Universiti Sains Malaysia PenangMalaysia 2012
[5] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo British Journal of Mathematics amp Computer Sciencevol 2 no 2 pp 72ndash93 2012
[6] KW Brodlie and S Butt ldquoPreserving convexity using piecewisecubic interpolationrdquoComputersampGraphics vol 15 no 1 pp 15ndash23 1991
[7] P Costantini ldquoBoundary-valued shape-preserving interpolat-ing splinesrdquo ACM Transactions on Mathematical Software vol23 no 2 pp 229ndash251 1997
[8] J-C Fiorot and J Tabka ldquoShape-preserving 1198622 cubic polyno-
[9] M Z Hussain M Sarfraz and T S Shaikh ldquoShape preservingrational cubic spline for positive and convex datardquo EgyptianInformatics Journal vol 12 no 3 pp 231ndash236 2011
[10] P Lamberti and C Manni ldquoShape-preserving 1198622 functional
interpolation via parametric cubicsrdquoNumerical Algorithms vol28 no 1ndash4 pp 229ndash254 2001
[11] M Sarfraz M Z Hussain T S Shaikh and R Iqbal ldquoDatavisualization using shape preserving 119862
2 rational splinerdquo inProceedings of the 15th International Conference on InformationVisualisation (IV rsquo11) pp 528ndash533 London UK July 2011
[12] Q Duan L Wang and E H Twizell ldquoA new 1198622 rational inter-
polation based on function values and constrained control ofthe interpolant curvesrdquo Applied Mathematics and Computationvol 161 no 1 pp 311ndash322 2005
[13] M Sarfraz M Z Hussain and M Hussain ldquoShape-preservingcurve interpolationrdquo International Journal of Computer Mathe-matics vol 89 no 1 pp 35ndash53 2012
[14] F Bao Q Sun and Q Duan ldquoPoint control of the interpolatingcurve with a rational cubic splinerdquo Journal of Visual Commu-nication and Image Representation vol 20 no 4 pp 275ndash2802009
[15] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Wellesley Mass USA 1993
max 1205851(119906119894 V119894 119908119894 120579) 0 le 120579 le 120579
lowast
max 1205852(119906119894 V119894 119908119894 120579) 120579
lowastle 120579 le 1
max 1205853(119906119894 V119894 119908119894 120579) 0 le 120579 le 1
(21)
where 1205851(119906119894 V119894 119908119894) 1205852(119906119894 V119894 119908119894) and 120585
3(119906119894 V119894 119908119894) are given in
[3]
Theorem 6 (see [3]) For any given positive values of shapeparameters 119906
119894 V119894 and 119908
119894 the optimal error 119888
119894in Theorem 5
satisfies 0 lt 119888119894le 00640
Proof See immediately Tables 9 10 11 and 12
6 Concluding Remarks
A 1198622 rational cubic function has been developed in this
paper for the smooth and attractive display of convex dataThree shape parameters were utilized for the descriptionof function to conserve the shape of convex data Simpledata dependent sufficient constraints were derived for singleshape parameter to insure convexity Remaining two shapeparameters have provided freedom to designer to modifythe shape of the curve by simply adjusting the values ofthe shape parameters No extra knots were inserted in theinterval where the interpolant loses the convexity The valuesof derivative parameters were achieved by solving the singlesystem of linear equations in comparison with [8] there existthree systems of linear equations for finding these valueswhich is computationally expensive and time-consuming
The Scientific World Journal 9
Table 8 Numerical results of Figure 4(d)
119894 1 2 3 4 5 6119889119894
minus0544 minus0452 0060 0136 0215 0283Δ119894
minus05 minus0048 0093 0166 025 mdash119906119894
15 15 15 15 15 mdashV119894
15 15 15 15 15 mdash119908119894
5108 3620 1811 3532 3638 mdash
Table 9 Values of 119888119894for several values of shape parameters 119906
processThe proposed scheme is not only1198622 smoother localand computationally economical but also visually pleasingas compared to schemes developed in [9 11] The proposedscheme works for both equally and unequally spaced datawhile the schemes developed in [12 14] work for only equallyspaced data
Table 11 Values of 119888119894with different values of shape parameters 119906
The authors declare that they have no conflict of interestsregarding the publication of this paper
10 The Scientific World Journal
References
[1] F Kuijt Convexity preserving interpolationmdashstationary nonlin-ear subdivision and splines [PhD thesis] University of TwenteEnschede The Netherlands 1998
[2] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo ISRN Mathematical Analysis vol 2012 Article ID174048 14 pages 2012
[3] M Abbas A A Majid and J M Ali ldquoMonotonicity-preserving1198622 rational cubic spline formonotone datardquoAppliedMathemat-
ics and Computation vol 219 no 6 pp 2885ndash2895 2012[4] M Abbas Shape preserving data visualization for curves and
surfaces using rational cubic functions [PhD thesis] Schoolof Mathematical Sciences Universiti Sains Malaysia PenangMalaysia 2012
[5] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo British Journal of Mathematics amp Computer Sciencevol 2 no 2 pp 72ndash93 2012
[6] KW Brodlie and S Butt ldquoPreserving convexity using piecewisecubic interpolationrdquoComputersampGraphics vol 15 no 1 pp 15ndash23 1991
[7] P Costantini ldquoBoundary-valued shape-preserving interpolat-ing splinesrdquo ACM Transactions on Mathematical Software vol23 no 2 pp 229ndash251 1997
[8] J-C Fiorot and J Tabka ldquoShape-preserving 1198622 cubic polyno-
[9] M Z Hussain M Sarfraz and T S Shaikh ldquoShape preservingrational cubic spline for positive and convex datardquo EgyptianInformatics Journal vol 12 no 3 pp 231ndash236 2011
[10] P Lamberti and C Manni ldquoShape-preserving 1198622 functional
interpolation via parametric cubicsrdquoNumerical Algorithms vol28 no 1ndash4 pp 229ndash254 2001
[11] M Sarfraz M Z Hussain T S Shaikh and R Iqbal ldquoDatavisualization using shape preserving 119862
2 rational splinerdquo inProceedings of the 15th International Conference on InformationVisualisation (IV rsquo11) pp 528ndash533 London UK July 2011
[12] Q Duan L Wang and E H Twizell ldquoA new 1198622 rational inter-
polation based on function values and constrained control ofthe interpolant curvesrdquo Applied Mathematics and Computationvol 161 no 1 pp 311ndash322 2005
[13] M Sarfraz M Z Hussain and M Hussain ldquoShape-preservingcurve interpolationrdquo International Journal of Computer Mathe-matics vol 89 no 1 pp 35ndash53 2012
[14] F Bao Q Sun and Q Duan ldquoPoint control of the interpolatingcurve with a rational cubic splinerdquo Journal of Visual Commu-nication and Image Representation vol 20 no 4 pp 275ndash2802009
[15] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Wellesley Mass USA 1993
max 1205851(119906119894 V119894 119908119894 120579) 0 le 120579 le 120579
lowast
max 1205852(119906119894 V119894 119908119894 120579) 120579
lowastle 120579 le 1
max 1205853(119906119894 V119894 119908119894 120579) 0 le 120579 le 1
(21)
where 1205851(119906119894 V119894 119908119894) 1205852(119906119894 V119894 119908119894) and 120585
3(119906119894 V119894 119908119894) are given in
[3]
Theorem 6 (see [3]) For any given positive values of shapeparameters 119906
119894 V119894 and 119908
119894 the optimal error 119888
119894in Theorem 5
satisfies 0 lt 119888119894le 00640
Proof See immediately Tables 9 10 11 and 12
6 Concluding Remarks
A 1198622 rational cubic function has been developed in this
paper for the smooth and attractive display of convex dataThree shape parameters were utilized for the descriptionof function to conserve the shape of convex data Simpledata dependent sufficient constraints were derived for singleshape parameter to insure convexity Remaining two shapeparameters have provided freedom to designer to modifythe shape of the curve by simply adjusting the values ofthe shape parameters No extra knots were inserted in theinterval where the interpolant loses the convexity The valuesof derivative parameters were achieved by solving the singlesystem of linear equations in comparison with [8] there existthree systems of linear equations for finding these valueswhich is computationally expensive and time-consuming
The Scientific World Journal 9
Table 8 Numerical results of Figure 4(d)
119894 1 2 3 4 5 6119889119894
minus0544 minus0452 0060 0136 0215 0283Δ119894
minus05 minus0048 0093 0166 025 mdash119906119894
15 15 15 15 15 mdashV119894
15 15 15 15 15 mdash119908119894
5108 3620 1811 3532 3638 mdash
Table 9 Values of 119888119894for several values of shape parameters 119906
processThe proposed scheme is not only1198622 smoother localand computationally economical but also visually pleasingas compared to schemes developed in [9 11] The proposedscheme works for both equally and unequally spaced datawhile the schemes developed in [12 14] work for only equallyspaced data
Table 11 Values of 119888119894with different values of shape parameters 119906
The authors declare that they have no conflict of interestsregarding the publication of this paper
10 The Scientific World Journal
References
[1] F Kuijt Convexity preserving interpolationmdashstationary nonlin-ear subdivision and splines [PhD thesis] University of TwenteEnschede The Netherlands 1998
[2] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo ISRN Mathematical Analysis vol 2012 Article ID174048 14 pages 2012
[3] M Abbas A A Majid and J M Ali ldquoMonotonicity-preserving1198622 rational cubic spline formonotone datardquoAppliedMathemat-
ics and Computation vol 219 no 6 pp 2885ndash2895 2012[4] M Abbas Shape preserving data visualization for curves and
surfaces using rational cubic functions [PhD thesis] Schoolof Mathematical Sciences Universiti Sains Malaysia PenangMalaysia 2012
[5] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo British Journal of Mathematics amp Computer Sciencevol 2 no 2 pp 72ndash93 2012
[6] KW Brodlie and S Butt ldquoPreserving convexity using piecewisecubic interpolationrdquoComputersampGraphics vol 15 no 1 pp 15ndash23 1991
[7] P Costantini ldquoBoundary-valued shape-preserving interpolat-ing splinesrdquo ACM Transactions on Mathematical Software vol23 no 2 pp 229ndash251 1997
[8] J-C Fiorot and J Tabka ldquoShape-preserving 1198622 cubic polyno-
[9] M Z Hussain M Sarfraz and T S Shaikh ldquoShape preservingrational cubic spline for positive and convex datardquo EgyptianInformatics Journal vol 12 no 3 pp 231ndash236 2011
[10] P Lamberti and C Manni ldquoShape-preserving 1198622 functional
interpolation via parametric cubicsrdquoNumerical Algorithms vol28 no 1ndash4 pp 229ndash254 2001
[11] M Sarfraz M Z Hussain T S Shaikh and R Iqbal ldquoDatavisualization using shape preserving 119862
2 rational splinerdquo inProceedings of the 15th International Conference on InformationVisualisation (IV rsquo11) pp 528ndash533 London UK July 2011
[12] Q Duan L Wang and E H Twizell ldquoA new 1198622 rational inter-
polation based on function values and constrained control ofthe interpolant curvesrdquo Applied Mathematics and Computationvol 161 no 1 pp 311ndash322 2005
[13] M Sarfraz M Z Hussain and M Hussain ldquoShape-preservingcurve interpolationrdquo International Journal of Computer Mathe-matics vol 89 no 1 pp 35ndash53 2012
[14] F Bao Q Sun and Q Duan ldquoPoint control of the interpolatingcurve with a rational cubic splinerdquo Journal of Visual Commu-nication and Image Representation vol 20 no 4 pp 275ndash2802009
[15] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Wellesley Mass USA 1993
processThe proposed scheme is not only1198622 smoother localand computationally economical but also visually pleasingas compared to schemes developed in [9 11] The proposedscheme works for both equally and unequally spaced datawhile the schemes developed in [12 14] work for only equallyspaced data
Table 11 Values of 119888119894with different values of shape parameters 119906
The authors declare that they have no conflict of interestsregarding the publication of this paper
10 The Scientific World Journal
References
[1] F Kuijt Convexity preserving interpolationmdashstationary nonlin-ear subdivision and splines [PhD thesis] University of TwenteEnschede The Netherlands 1998
[2] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo ISRN Mathematical Analysis vol 2012 Article ID174048 14 pages 2012
[3] M Abbas A A Majid and J M Ali ldquoMonotonicity-preserving1198622 rational cubic spline formonotone datardquoAppliedMathemat-
ics and Computation vol 219 no 6 pp 2885ndash2895 2012[4] M Abbas Shape preserving data visualization for curves and
surfaces using rational cubic functions [PhD thesis] Schoolof Mathematical Sciences Universiti Sains Malaysia PenangMalaysia 2012
[5] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo British Journal of Mathematics amp Computer Sciencevol 2 no 2 pp 72ndash93 2012
[6] KW Brodlie and S Butt ldquoPreserving convexity using piecewisecubic interpolationrdquoComputersampGraphics vol 15 no 1 pp 15ndash23 1991
[7] P Costantini ldquoBoundary-valued shape-preserving interpolat-ing splinesrdquo ACM Transactions on Mathematical Software vol23 no 2 pp 229ndash251 1997
[8] J-C Fiorot and J Tabka ldquoShape-preserving 1198622 cubic polyno-
[9] M Z Hussain M Sarfraz and T S Shaikh ldquoShape preservingrational cubic spline for positive and convex datardquo EgyptianInformatics Journal vol 12 no 3 pp 231ndash236 2011
[10] P Lamberti and C Manni ldquoShape-preserving 1198622 functional
interpolation via parametric cubicsrdquoNumerical Algorithms vol28 no 1ndash4 pp 229ndash254 2001
[11] M Sarfraz M Z Hussain T S Shaikh and R Iqbal ldquoDatavisualization using shape preserving 119862
2 rational splinerdquo inProceedings of the 15th International Conference on InformationVisualisation (IV rsquo11) pp 528ndash533 London UK July 2011
[12] Q Duan L Wang and E H Twizell ldquoA new 1198622 rational inter-
polation based on function values and constrained control ofthe interpolant curvesrdquo Applied Mathematics and Computationvol 161 no 1 pp 311ndash322 2005
[13] M Sarfraz M Z Hussain and M Hussain ldquoShape-preservingcurve interpolationrdquo International Journal of Computer Mathe-matics vol 89 no 1 pp 35ndash53 2012
[14] F Bao Q Sun and Q Duan ldquoPoint control of the interpolatingcurve with a rational cubic splinerdquo Journal of Visual Commu-nication and Image Representation vol 20 no 4 pp 275ndash2802009
[15] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Wellesley Mass USA 1993
[1] F Kuijt Convexity preserving interpolationmdashstationary nonlin-ear subdivision and splines [PhD thesis] University of TwenteEnschede The Netherlands 1998
[2] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo ISRN Mathematical Analysis vol 2012 Article ID174048 14 pages 2012
[3] M Abbas A A Majid and J M Ali ldquoMonotonicity-preserving1198622 rational cubic spline formonotone datardquoAppliedMathemat-
ics and Computation vol 219 no 6 pp 2885ndash2895 2012[4] M Abbas Shape preserving data visualization for curves and
surfaces using rational cubic functions [PhD thesis] Schoolof Mathematical Sciences Universiti Sains Malaysia PenangMalaysia 2012
[5] M Abbas A A Majid M N H Awang and J M AlildquoLocal convexity shape-preserving data visualization by splinefunctionrdquo British Journal of Mathematics amp Computer Sciencevol 2 no 2 pp 72ndash93 2012
[6] KW Brodlie and S Butt ldquoPreserving convexity using piecewisecubic interpolationrdquoComputersampGraphics vol 15 no 1 pp 15ndash23 1991
[7] P Costantini ldquoBoundary-valued shape-preserving interpolat-ing splinesrdquo ACM Transactions on Mathematical Software vol23 no 2 pp 229ndash251 1997
[8] J-C Fiorot and J Tabka ldquoShape-preserving 1198622 cubic polyno-
[9] M Z Hussain M Sarfraz and T S Shaikh ldquoShape preservingrational cubic spline for positive and convex datardquo EgyptianInformatics Journal vol 12 no 3 pp 231ndash236 2011
[10] P Lamberti and C Manni ldquoShape-preserving 1198622 functional
interpolation via parametric cubicsrdquoNumerical Algorithms vol28 no 1ndash4 pp 229ndash254 2001
[11] M Sarfraz M Z Hussain T S Shaikh and R Iqbal ldquoDatavisualization using shape preserving 119862
2 rational splinerdquo inProceedings of the 15th International Conference on InformationVisualisation (IV rsquo11) pp 528ndash533 London UK July 2011
[12] Q Duan L Wang and E H Twizell ldquoA new 1198622 rational inter-
polation based on function values and constrained control ofthe interpolant curvesrdquo Applied Mathematics and Computationvol 161 no 1 pp 311ndash322 2005
[13] M Sarfraz M Z Hussain and M Hussain ldquoShape-preservingcurve interpolationrdquo International Journal of Computer Mathe-matics vol 89 no 1 pp 35ndash53 2012
[14] F Bao Q Sun and Q Duan ldquoPoint control of the interpolatingcurve with a rational cubic splinerdquo Journal of Visual Commu-nication and Image Representation vol 20 no 4 pp 275ndash2802009
[15] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Wellesley Mass USA 1993