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

] 119894 = 0 1 2 119899 minus 1 is defined as

119878 (119909) = 119878119894(119909) =

sum3

119894=0(1 minus 120579)

3minus119894120579119894120585119894

119902119894(120579)

(1)

with

1205850= 119906119894119891119894

1205851= 119891119894(2119906119894+ V119894+ 119908119894) + 119906119894ℎ119894119889119894

1205852= 119891119894+1

(119906119894+ 2V119894+ 119908119894) minus V119894ℎ119894119889119894+1

1205853= V119894119891119894+1

119902119894(120579) = (1 minus 120579)

2119906119894+ 120579 (1 minus 120579) (119908

119894+ 119906119894+ V119894) + 1205792V119894

(2)

where ℎ119894= 119909119894+1

minus 119909119894 120579 = (119909 minus 119909

119894)ℎ119894 120579 isin [0 1] and 119906

119894

V119894 and 119908

119894are the positive shape parameters that are used

to control the shape of interpolating curve and provide thedesigner liberty to refine the curve as desired Let 119889

119894denote

the derivative value at knots119909119894that is used for the smoothness

of curve Let 1198781015840(119909) and 119878

10158401015840(119909) denote the first and second

ordered derivatives with respect to 119909

The Scientific World Journal 3

The following interpolatory conditions are imposed forthe 1198622 continuity of the piecewise rational cubic function (1)

119878 (119909119894) = 119891119894 119878 (119909

119894+1) = 119891119894+1

1198781015840(119909119894) = 119889119894 119878

1015840(119909119894+1

) = 119889119894+1

11987810158401015840(119909119894+) = 11987810158401015840(119909119894minus) 119894 = 1 2 119899 minus 1

(3)

with

11987810158401015840(119909119894+)

=2 ((119906119894+ 2V119894+ 119908119894) Δ119894minus V119894119889119894+1

minus (119906119894+ V119894+ 119908119894) 119889119894)

ℎ119894119906119894

11987810158401015840(119909119894minus)

= (2 (119906119894minus1

119889119894minus1

+ (119906119894minus1

+ V119894minus1

+ 119908119894minus1

) 119889119894

minus (2119906119894minus1

+ V119894minus1

+ 119908119894minus1

) Δ119894minus1

))

times (ℎ119894minus1

V119894minus1

)minus1

(4)

From (3) the 1198622 interpolating conditions produce the

following system of linear equations

120572119894119889119894minus1

+ 120575119894119889119894+ 120574119894119889119894+1

= 120582119894 (5)

with

120572119894= 119906119894119906119894minus1

ℎ119894

120575119894= ℎ119894119906119894(119906119894minus1

+ V119894minus1

+ 119908119894minus1

) + ℎ119894minus1

V119894minus1

(119906119894+ V119894+ 119908119894)

120574119894= V119894V119894minus1

ℎ119894minus1

120582119894= V119894minus1

ℎ119894minus1

(119906119894+ 2V119894+ 119908119894) Δ119894

+ 119906119894ℎ119894(2119906119894minus1

+ V119894minus1

+ 119908119894minus1

) Δ119894minus1

(6)

where Δ119894= (119891119894+1

minus 119891119894)ℎ119894 The 119862

2 piecewise rational cubicfunction (1) is reformulated after using (2) as

119878 (119909119894) =

119901119894(120579)

119902119894(120579)

(7)

with

119901119894(120579) = 119906

119894119891119894(1 minus 120579)

3+ (119891119894(2119906119894+ V119894+ 119908119894) + 119906119894ℎ119894119889119894)

times 120579(1 minus 120579)2+ (119891119894+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

1198890lt Δ0lt 1198891lt sdot sdot sdot lt Δ

119894minus1lt 119889119894lt Δ119894lt sdot sdot sdot lt 119889

119899 (12)

For concave data we have

1198890gt Δ0gt 1198891gt sdot sdot sdot gt Δ

119894minus1gt 119889119894gt Δ119894gt sdot sdot sdot gt 119889

119899 (13)

Necessary conditions for convexity are

119906119894gt 0 V

119894gt 0 119908

119894ge 0

(Δ119894minus 119889119894) gt 0

(119889119894+1

minus Δ119894) gt 0

(14)

Now the 1198622 rational cubic function 119878

119894(119909) defined in (7)

is convex if and only if 11987810158401015840119894(119909) gt 0 such that

11987810158401015840

119894(119909) =

sum5

119896=0(1 minus 120579)

5minus119896120579119896119862119896119894

ℎ119894(119902119894(120579))3

(15)

4 The Scientific World Journal

Table 2 Numerical results of Figure 1(d)

119894 1 2 3 4 5 6 7119889119894

minus12 minus07743 minus01143 0 01143 07743 12Δ119894

minus10 minus03 minus005 005 03 10 mdash119906119894

15 15 15 15 15 15 mdashV119894

15 15 15 15 15 15 mdash119908119894

11165 10776 00598 00598 10776 11165 mdash

Table 3 A convex data set

119894 119909119894

119910119894

1 10 1002 15 703 175 504 20 255 25 106 30 067 50 048 100 109 105 3010 11 5011 12 90

with

1198620119894=21199062

119894(119906119894+ V119894) (Δ119894minus 119889119894) + 119908119894(Δ119894minus 119889119894) minus V119894(119889119894+1

minus Δ119894)

1198621119894

= 1198620119894

+ 21199062

119894(5V119894(Δ119894minus 119889119894) + V119894119889119894)

1198622119894

= 1198620119894

+ 6119906119894(V2119894(119889119894+1

minus Δ119894) + 2119906

119894V119894(Δ119894minus 119889119894))

1198623119894

= 1198625119894

+ 6V119894(1199062

119894(Δ119894minus 119889119894) + 2119906

119894V119894(119889119894+1

minus Δ119894))

1198624119894

= 1198625119894

+ 2V2119894(5119906119894(119889119894+1

minus Δ119894) + 119906119894119889119894+1

)

1198625119894

= 2V2119894(119906119894+ V119894) (119889119894+1

minus Δ119894)

+ 119908119894(119889119894+1

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

The Scientific World Journal 5

Table 4 Numerical results of Figure 2(d)

119894 119889119894

Δ119894

119906119894

V119894

119908119894

1 minus46667 minus60 175 175 189582 minus72864 minus80 175 175 235863 minus91942 minus100 175 175 00014 minus78005 minus30 175 175 108355 minus15996 minus08 175 175 028746 minus05057 minus01 175 175 100637 minus00737 012 175 175 143398 38472 40 175 175 167239 40102 40 175 175 1553710 39991 40 175 175 16001198907

11 40 mdash mdash mdash mdash

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(a) Cubic Hermite spline curve

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(b) PCHIP curve

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(c) Convexity-preserving rational cubic curve with 119906119894 = 025 V119894 = 025

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(d) Convexity-preserving rational cubic curve with 119906119894 = 15 V119894 = 15

Figure 1 Convexity- and nonconvexity-preserving curves

6 The Scientific World Journal

0 2 4 8 10 12

0

2

4

8

10

6

6

x-axis

y-axis

minus2

(a) Cubic Hermite curve

0 2 4 8 10 120

1

2

3

4

5

6

7

8

9

10

6x-axis

y-axis

(b) PCHIP curve

0 2 4 8 10 120

1

2

3

4

5

6

7

8

9

10

6x-axis

y-axis

(c) 1198622 convex rational cubic curve with 119906119894 = 025 V119894 = 025

0 2 4 8 10 120

1

2

3

4

5

6

7

8

9

10

6x-axis

y-axis

(d) 1198622 convex rational cubic curve with 119906119894 = 175 V119894 = 175

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

119894

The Scientific World Journal 7

Table 6 Numerical results of Figure 3(d)

119894 1 2 3 4 5 6 7119889119894

minus2325 minus1644 minus0132 0 0132 1644 2325Δ119894

minus2 minus0375 minus0062 0062 0375 2 mdash119906119894

15 15 15 15 15 15 mdashV119894

15 15 15 15 15 15 mdash119908119894

5503 7902 1181 1181 7902 5503 mdash

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(a) Cubic Hermite curve

0 5 10minus5

x-axisminus10

45

5

55

3

35

4

65

7

6

y-axis

(b) PCHIP curve

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(c) Local convexity-preserving 1198622 rational cubic curve with 119906119894 = 025V119894 = 025

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(d) Local convexity-preserving 1198622 rational cubic curve with 119906119894 = 15V119894 = 15

Figure 3 Convex and nonconvex curves through given convex data set

V119894and various values of constrained parameter 119908

119894 Here we

take different values of shape parameters than [3] for theeffectiveness of proposed interpolant

Theorem 5 (see [3]) The error of interpolating rational cubicfunction (7) for 119891(119909) isin 119862

3[1199090 119909119899] in each subinterval 119868

119894=

[119909119894 119909119894+1

] is

1003816100381610038161003816119891 (119909) minus 119878119894(119909)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119909119894+1

119909119894

10038161003816100381610038161003816119877119909[(119909 minus 120591)

2

+]10038161003816100381610038161003816119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

119888119894= max0le120579le1

120585 (119906119894 V119894 119908119894 120579)

(20)

8 The Scientific World Journal

0 2 4 6 8 10 123

32

34

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

(a) Cubic Hermite spline curve

35

4

45

5

0 2 4 6 8 10 12minus8 minus6 minus4 minus2

x-axis

y-axis

(b) PCHIP curve

0 2 4 6 8 10 12

34

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

32

(c) 1198622 rational cubic convex curve with 119906119894 = 05 V119894 = 05

0 2 4 6 8 10 1234

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

(d) 1198622 rational cubic convex curve with 119906119894 = 15 V119894 = 15

Figure 4 Comparison of convex and nonconvex curves through given 2D convex data

Table 7 2D convex data set

119894 1 2 3 4 5 6119909119894

minus8 minus7 22 7 10 12119910119894

45 4 355 4 45 5

where

120585 (119906119894 V119894 119908119894 120579) =

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

119894 V119894 and

119908119894

119894 119906119894

V119894

119908119894

119888119894

1 001 001 001 001802 001 001 01 004583 001 001 06 006004 001 001 09 006135 001 001 16 006256 001 001 60 006377 001 001 120 006388 001 001 1150 006399 001 001 5150 0064010 001 001 7000 0064011 001 001 10000 0064012 001 001 15150 0064013 001 001 20000 0064014 001 001 200000 00640

Table 10 Values of 119888119894for different values of shape parameters 119906

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 150 150 15 001042 150 150 01 001093 150 150 06 001354 150 150 09 001515 150 150 16 001856 150 150 60 003317 150 150 120 004298 150 150 1150 006089 150 150 5150 0063310 150 150 7000 0063511 150 150 10000 0063712 150 150 15150 0063913 150 150 20000 0064014 150 150 200000 00640

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

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 5050 5050 15 001042 5050 5050 01 001043 5050 5050 06 001054 5050 5050 09 001055 5050 5050 16 001066 5050 5050 60 001137 5050 5050 120 001228 5050 5050 1150 002579 5050 5050 5150 0046010 5050 5050 7000 0049611 5050 5050 10000 0053212 5050 5050 15150 0056413 5050 5050 20000 0058114 5050 5050 200000 00634

Table 12 Values of 119888119894for various values of shape parameters 119906

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 2000 2000 15 001042 2000 2000 01 001043 2000 2000 06 001044 2000 2000 09 001045 2000 2000 16 001046 2000 2000 60 001067 2000 2000 120 001088 2000 2000 1150 001499 2000 2000 5150 0027210 2000 2000 7000 0031311 2000 2000 10000 0036312 2000 2000 15150 0042113 2000 2000 20000 0045814 2000 2000 200000 00616

Conflict of Interests

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-

mial interpolating splinesrdquoMathematics of Computation vol 57no 195 pp 291ndash298 1991

[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

Submit your manuscripts athttpwwwhindawicom

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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

] 119894 = 0 1 2 119899 minus 1 is defined as

119878 (119909) = 119878119894(119909) =

sum3

119894=0(1 minus 120579)

3minus119894120579119894120585119894

119902119894(120579)

(1)

with

1205850= 119906119894119891119894

1205851= 119891119894(2119906119894+ V119894+ 119908119894) + 119906119894ℎ119894119889119894

1205852= 119891119894+1

(119906119894+ 2V119894+ 119908119894) minus V119894ℎ119894119889119894+1

1205853= V119894119891119894+1

119902119894(120579) = (1 minus 120579)

2119906119894+ 120579 (1 minus 120579) (119908

119894+ 119906119894+ V119894) + 1205792V119894

(2)

where ℎ119894= 119909119894+1

minus 119909119894 120579 = (119909 minus 119909

119894)ℎ119894 120579 isin [0 1] and 119906

119894

V119894 and 119908

119894are the positive shape parameters that are used

to control the shape of interpolating curve and provide thedesigner liberty to refine the curve as desired Let 119889

119894denote

the derivative value at knots119909119894that is used for the smoothness

of curve Let 1198781015840(119909) and 119878

10158401015840(119909) denote the first and second

ordered derivatives with respect to 119909

The Scientific World Journal 3

The following interpolatory conditions are imposed forthe 1198622 continuity of the piecewise rational cubic function (1)

119878 (119909119894) = 119891119894 119878 (119909

119894+1) = 119891119894+1

1198781015840(119909119894) = 119889119894 119878

1015840(119909119894+1

) = 119889119894+1

11987810158401015840(119909119894+) = 11987810158401015840(119909119894minus) 119894 = 1 2 119899 minus 1

(3)

with

11987810158401015840(119909119894+)

=2 ((119906119894+ 2V119894+ 119908119894) Δ119894minus V119894119889119894+1

minus (119906119894+ V119894+ 119908119894) 119889119894)

ℎ119894119906119894

11987810158401015840(119909119894minus)

= (2 (119906119894minus1

119889119894minus1

+ (119906119894minus1

+ V119894minus1

+ 119908119894minus1

) 119889119894

minus (2119906119894minus1

+ V119894minus1

+ 119908119894minus1

) Δ119894minus1

))

times (ℎ119894minus1

V119894minus1

)minus1

(4)

From (3) the 1198622 interpolating conditions produce the

following system of linear equations

120572119894119889119894minus1

+ 120575119894119889119894+ 120574119894119889119894+1

= 120582119894 (5)

with

120572119894= 119906119894119906119894minus1

ℎ119894

120575119894= ℎ119894119906119894(119906119894minus1

+ V119894minus1

+ 119908119894minus1

) + ℎ119894minus1

V119894minus1

(119906119894+ V119894+ 119908119894)

120574119894= V119894V119894minus1

ℎ119894minus1

120582119894= V119894minus1

ℎ119894minus1

(119906119894+ 2V119894+ 119908119894) Δ119894

+ 119906119894ℎ119894(2119906119894minus1

+ V119894minus1

+ 119908119894minus1

) Δ119894minus1

(6)

where Δ119894= (119891119894+1

minus 119891119894)ℎ119894 The 119862

2 piecewise rational cubicfunction (1) is reformulated after using (2) as

119878 (119909119894) =

119901119894(120579)

119902119894(120579)

(7)

with

119901119894(120579) = 119906

119894119891119894(1 minus 120579)

3+ (119891119894(2119906119894+ V119894+ 119908119894) + 119906119894ℎ119894119889119894)

times 120579(1 minus 120579)2+ (119891119894+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

1198890lt Δ0lt 1198891lt sdot sdot sdot lt Δ

119894minus1lt 119889119894lt Δ119894lt sdot sdot sdot lt 119889

119899 (12)

For concave data we have

1198890gt Δ0gt 1198891gt sdot sdot sdot gt Δ

119894minus1gt 119889119894gt Δ119894gt sdot sdot sdot gt 119889

119899 (13)

Necessary conditions for convexity are

119906119894gt 0 V

119894gt 0 119908

119894ge 0

(Δ119894minus 119889119894) gt 0

(119889119894+1

minus Δ119894) gt 0

(14)

Now the 1198622 rational cubic function 119878

119894(119909) defined in (7)

is convex if and only if 11987810158401015840119894(119909) gt 0 such that

11987810158401015840

119894(119909) =

sum5

119896=0(1 minus 120579)

5minus119896120579119896119862119896119894

ℎ119894(119902119894(120579))3

(15)

4 The Scientific World Journal

Table 2 Numerical results of Figure 1(d)

119894 1 2 3 4 5 6 7119889119894

minus12 minus07743 minus01143 0 01143 07743 12Δ119894

minus10 minus03 minus005 005 03 10 mdash119906119894

15 15 15 15 15 15 mdashV119894

15 15 15 15 15 15 mdash119908119894

11165 10776 00598 00598 10776 11165 mdash

Table 3 A convex data set

119894 119909119894

119910119894

1 10 1002 15 703 175 504 20 255 25 106 30 067 50 048 100 109 105 3010 11 5011 12 90

with

1198620119894=21199062

119894(119906119894+ V119894) (Δ119894minus 119889119894) + 119908119894(Δ119894minus 119889119894) minus V119894(119889119894+1

minus Δ119894)

1198621119894

= 1198620119894

+ 21199062

119894(5V119894(Δ119894minus 119889119894) + V119894119889119894)

1198622119894

= 1198620119894

+ 6119906119894(V2119894(119889119894+1

minus Δ119894) + 2119906

119894V119894(Δ119894minus 119889119894))

1198623119894

= 1198625119894

+ 6V119894(1199062

119894(Δ119894minus 119889119894) + 2119906

119894V119894(119889119894+1

minus Δ119894))

1198624119894

= 1198625119894

+ 2V2119894(5119906119894(119889119894+1

minus Δ119894) + 119906119894119889119894+1

)

1198625119894

= 2V2119894(119906119894+ V119894) (119889119894+1

minus Δ119894)

+ 119908119894(119889119894+1

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

The Scientific World Journal 5

Table 4 Numerical results of Figure 2(d)

119894 119889119894

Δ119894

119906119894

V119894

119908119894

1 minus46667 minus60 175 175 189582 minus72864 minus80 175 175 235863 minus91942 minus100 175 175 00014 minus78005 minus30 175 175 108355 minus15996 minus08 175 175 028746 minus05057 minus01 175 175 100637 minus00737 012 175 175 143398 38472 40 175 175 167239 40102 40 175 175 1553710 39991 40 175 175 16001198907

11 40 mdash mdash mdash mdash

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(a) Cubic Hermite spline curve

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(b) PCHIP curve

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(c) Convexity-preserving rational cubic curve with 119906119894 = 025 V119894 = 025

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(d) Convexity-preserving rational cubic curve with 119906119894 = 15 V119894 = 15

Figure 1 Convexity- and nonconvexity-preserving curves

6 The Scientific World Journal

0 2 4 8 10 12

0

2

4

8

10

6

6

x-axis

y-axis

minus2

(a) Cubic Hermite curve

0 2 4 8 10 120

1

2

3

4

5

6

7

8

9

10

6x-axis

y-axis

(b) PCHIP curve

0 2 4 8 10 120

1

2

3

4

5

6

7

8

9

10

6x-axis

y-axis

(c) 1198622 convex rational cubic curve with 119906119894 = 025 V119894 = 025

0 2 4 8 10 120

1

2

3

4

5

6

7

8

9

10

6x-axis

y-axis

(d) 1198622 convex rational cubic curve with 119906119894 = 175 V119894 = 175

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

119894

The Scientific World Journal 7

Table 6 Numerical results of Figure 3(d)

119894 1 2 3 4 5 6 7119889119894

minus2325 minus1644 minus0132 0 0132 1644 2325Δ119894

minus2 minus0375 minus0062 0062 0375 2 mdash119906119894

15 15 15 15 15 15 mdashV119894

15 15 15 15 15 15 mdash119908119894

5503 7902 1181 1181 7902 5503 mdash

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(a) Cubic Hermite curve

0 5 10minus5

x-axisminus10

45

5

55

3

35

4

65

7

6

y-axis

(b) PCHIP curve

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(c) Local convexity-preserving 1198622 rational cubic curve with 119906119894 = 025V119894 = 025

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(d) Local convexity-preserving 1198622 rational cubic curve with 119906119894 = 15V119894 = 15

Figure 3 Convex and nonconvex curves through given convex data set

V119894and various values of constrained parameter 119908

119894 Here we

take different values of shape parameters than [3] for theeffectiveness of proposed interpolant

Theorem 5 (see [3]) The error of interpolating rational cubicfunction (7) for 119891(119909) isin 119862

3[1199090 119909119899] in each subinterval 119868

119894=

[119909119894 119909119894+1

] is

1003816100381610038161003816119891 (119909) minus 119878119894(119909)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119909119894+1

119909119894

10038161003816100381610038161003816119877119909[(119909 minus 120591)

2

+]10038161003816100381610038161003816119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

119888119894= max0le120579le1

120585 (119906119894 V119894 119908119894 120579)

(20)

8 The Scientific World Journal

0 2 4 6 8 10 123

32

34

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

(a) Cubic Hermite spline curve

35

4

45

5

0 2 4 6 8 10 12minus8 minus6 minus4 minus2

x-axis

y-axis

(b) PCHIP curve

0 2 4 6 8 10 12

34

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

32

(c) 1198622 rational cubic convex curve with 119906119894 = 05 V119894 = 05

0 2 4 6 8 10 1234

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

(d) 1198622 rational cubic convex curve with 119906119894 = 15 V119894 = 15

Figure 4 Comparison of convex and nonconvex curves through given 2D convex data

Table 7 2D convex data set

119894 1 2 3 4 5 6119909119894

minus8 minus7 22 7 10 12119910119894

45 4 355 4 45 5

where

120585 (119906119894 V119894 119908119894 120579) =

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

119894 V119894 and

119908119894

119894 119906119894

V119894

119908119894

119888119894

1 001 001 001 001802 001 001 01 004583 001 001 06 006004 001 001 09 006135 001 001 16 006256 001 001 60 006377 001 001 120 006388 001 001 1150 006399 001 001 5150 0064010 001 001 7000 0064011 001 001 10000 0064012 001 001 15150 0064013 001 001 20000 0064014 001 001 200000 00640

Table 10 Values of 119888119894for different values of shape parameters 119906

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 150 150 15 001042 150 150 01 001093 150 150 06 001354 150 150 09 001515 150 150 16 001856 150 150 60 003317 150 150 120 004298 150 150 1150 006089 150 150 5150 0063310 150 150 7000 0063511 150 150 10000 0063712 150 150 15150 0063913 150 150 20000 0064014 150 150 200000 00640

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

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 5050 5050 15 001042 5050 5050 01 001043 5050 5050 06 001054 5050 5050 09 001055 5050 5050 16 001066 5050 5050 60 001137 5050 5050 120 001228 5050 5050 1150 002579 5050 5050 5150 0046010 5050 5050 7000 0049611 5050 5050 10000 0053212 5050 5050 15150 0056413 5050 5050 20000 0058114 5050 5050 200000 00634

Table 12 Values of 119888119894for various values of shape parameters 119906

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 2000 2000 15 001042 2000 2000 01 001043 2000 2000 06 001044 2000 2000 09 001045 2000 2000 16 001046 2000 2000 60 001067 2000 2000 120 001088 2000 2000 1150 001499 2000 2000 5150 0027210 2000 2000 7000 0031311 2000 2000 10000 0036312 2000 2000 15150 0042113 2000 2000 20000 0045814 2000 2000 200000 00616

Conflict of Interests

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-

mial interpolating splinesrdquoMathematics of Computation vol 57no 195 pp 291ndash298 1991

[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

Submit your manuscripts athttpwwwhindawicom

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Advances in

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The Scientific World Journal 3

The following interpolatory conditions are imposed forthe 1198622 continuity of the piecewise rational cubic function (1)

119878 (119909119894) = 119891119894 119878 (119909

119894+1) = 119891119894+1

1198781015840(119909119894) = 119889119894 119878

1015840(119909119894+1

) = 119889119894+1

11987810158401015840(119909119894+) = 11987810158401015840(119909119894minus) 119894 = 1 2 119899 minus 1

(3)

with

11987810158401015840(119909119894+)

=2 ((119906119894+ 2V119894+ 119908119894) Δ119894minus V119894119889119894+1

minus (119906119894+ V119894+ 119908119894) 119889119894)

ℎ119894119906119894

11987810158401015840(119909119894minus)

= (2 (119906119894minus1

119889119894minus1

+ (119906119894minus1

+ V119894minus1

+ 119908119894minus1

) 119889119894

minus (2119906119894minus1

+ V119894minus1

+ 119908119894minus1

) Δ119894minus1

))

times (ℎ119894minus1

V119894minus1

)minus1

(4)

From (3) the 1198622 interpolating conditions produce the

following system of linear equations

120572119894119889119894minus1

+ 120575119894119889119894+ 120574119894119889119894+1

= 120582119894 (5)

with

120572119894= 119906119894119906119894minus1

ℎ119894

120575119894= ℎ119894119906119894(119906119894minus1

+ V119894minus1

+ 119908119894minus1

) + ℎ119894minus1

V119894minus1

(119906119894+ V119894+ 119908119894)

120574119894= V119894V119894minus1

ℎ119894minus1

120582119894= V119894minus1

ℎ119894minus1

(119906119894+ 2V119894+ 119908119894) Δ119894

+ 119906119894ℎ119894(2119906119894minus1

+ V119894minus1

+ 119908119894minus1

) Δ119894minus1

(6)

where Δ119894= (119891119894+1

minus 119891119894)ℎ119894 The 119862

2 piecewise rational cubicfunction (1) is reformulated after using (2) as

119878 (119909119894) =

119901119894(120579)

119902119894(120579)

(7)

with

119901119894(120579) = 119906

119894119891119894(1 minus 120579)

3+ (119891119894(2119906119894+ V119894+ 119908119894) + 119906119894ℎ119894119889119894)

times 120579(1 minus 120579)2+ (119891119894+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

1198890lt Δ0lt 1198891lt sdot sdot sdot lt Δ

119894minus1lt 119889119894lt Δ119894lt sdot sdot sdot lt 119889

119899 (12)

For concave data we have

1198890gt Δ0gt 1198891gt sdot sdot sdot gt Δ

119894minus1gt 119889119894gt Δ119894gt sdot sdot sdot gt 119889

119899 (13)

Necessary conditions for convexity are

119906119894gt 0 V

119894gt 0 119908

119894ge 0

(Δ119894minus 119889119894) gt 0

(119889119894+1

minus Δ119894) gt 0

(14)

Now the 1198622 rational cubic function 119878

119894(119909) defined in (7)

is convex if and only if 11987810158401015840119894(119909) gt 0 such that

11987810158401015840

119894(119909) =

sum5

119896=0(1 minus 120579)

5minus119896120579119896119862119896119894

ℎ119894(119902119894(120579))3

(15)

4 The Scientific World Journal

Table 2 Numerical results of Figure 1(d)

119894 1 2 3 4 5 6 7119889119894

minus12 minus07743 minus01143 0 01143 07743 12Δ119894

minus10 minus03 minus005 005 03 10 mdash119906119894

15 15 15 15 15 15 mdashV119894

15 15 15 15 15 15 mdash119908119894

11165 10776 00598 00598 10776 11165 mdash

Table 3 A convex data set

119894 119909119894

119910119894

1 10 1002 15 703 175 504 20 255 25 106 30 067 50 048 100 109 105 3010 11 5011 12 90

with

1198620119894=21199062

119894(119906119894+ V119894) (Δ119894minus 119889119894) + 119908119894(Δ119894minus 119889119894) minus V119894(119889119894+1

minus Δ119894)

1198621119894

= 1198620119894

+ 21199062

119894(5V119894(Δ119894minus 119889119894) + V119894119889119894)

1198622119894

= 1198620119894

+ 6119906119894(V2119894(119889119894+1

minus Δ119894) + 2119906

119894V119894(Δ119894minus 119889119894))

1198623119894

= 1198625119894

+ 6V119894(1199062

119894(Δ119894minus 119889119894) + 2119906

119894V119894(119889119894+1

minus Δ119894))

1198624119894

= 1198625119894

+ 2V2119894(5119906119894(119889119894+1

minus Δ119894) + 119906119894119889119894+1

)

1198625119894

= 2V2119894(119906119894+ V119894) (119889119894+1

minus Δ119894)

+ 119908119894(119889119894+1

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

The Scientific World Journal 5

Table 4 Numerical results of Figure 2(d)

119894 119889119894

Δ119894

119906119894

V119894

119908119894

1 minus46667 minus60 175 175 189582 minus72864 minus80 175 175 235863 minus91942 minus100 175 175 00014 minus78005 minus30 175 175 108355 minus15996 minus08 175 175 028746 minus05057 minus01 175 175 100637 minus00737 012 175 175 143398 38472 40 175 175 167239 40102 40 175 175 1553710 39991 40 175 175 16001198907

11 40 mdash mdash mdash mdash

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(a) Cubic Hermite spline curve

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(b) PCHIP curve

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(c) Convexity-preserving rational cubic curve with 119906119894 = 025 V119894 = 025

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(d) Convexity-preserving rational cubic curve with 119906119894 = 15 V119894 = 15

Figure 1 Convexity- and nonconvexity-preserving curves

6 The Scientific World Journal

0 2 4 8 10 12

0

2

4

8

10

6

6

x-axis

y-axis

minus2

(a) Cubic Hermite curve

0 2 4 8 10 120

1

2

3

4

5

6

7

8

9

10

6x-axis

y-axis

(b) PCHIP curve

0 2 4 8 10 120

1

2

3

4

5

6

7

8

9

10

6x-axis

y-axis

(c) 1198622 convex rational cubic curve with 119906119894 = 025 V119894 = 025

0 2 4 8 10 120

1

2

3

4

5

6

7

8

9

10

6x-axis

y-axis

(d) 1198622 convex rational cubic curve with 119906119894 = 175 V119894 = 175

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

119894

The Scientific World Journal 7

Table 6 Numerical results of Figure 3(d)

119894 1 2 3 4 5 6 7119889119894

minus2325 minus1644 minus0132 0 0132 1644 2325Δ119894

minus2 minus0375 minus0062 0062 0375 2 mdash119906119894

15 15 15 15 15 15 mdashV119894

15 15 15 15 15 15 mdash119908119894

5503 7902 1181 1181 7902 5503 mdash

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(a) Cubic Hermite curve

0 5 10minus5

x-axisminus10

45

5

55

3

35

4

65

7

6

y-axis

(b) PCHIP curve

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(c) Local convexity-preserving 1198622 rational cubic curve with 119906119894 = 025V119894 = 025

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(d) Local convexity-preserving 1198622 rational cubic curve with 119906119894 = 15V119894 = 15

Figure 3 Convex and nonconvex curves through given convex data set

V119894and various values of constrained parameter 119908

119894 Here we

take different values of shape parameters than [3] for theeffectiveness of proposed interpolant

Theorem 5 (see [3]) The error of interpolating rational cubicfunction (7) for 119891(119909) isin 119862

3[1199090 119909119899] in each subinterval 119868

119894=

[119909119894 119909119894+1

] is

1003816100381610038161003816119891 (119909) minus 119878119894(119909)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119909119894+1

119909119894

10038161003816100381610038161003816119877119909[(119909 minus 120591)

2

+]10038161003816100381610038161003816119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

119888119894= max0le120579le1

120585 (119906119894 V119894 119908119894 120579)

(20)

8 The Scientific World Journal

0 2 4 6 8 10 123

32

34

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

(a) Cubic Hermite spline curve

35

4

45

5

0 2 4 6 8 10 12minus8 minus6 minus4 minus2

x-axis

y-axis

(b) PCHIP curve

0 2 4 6 8 10 12

34

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

32

(c) 1198622 rational cubic convex curve with 119906119894 = 05 V119894 = 05

0 2 4 6 8 10 1234

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

(d) 1198622 rational cubic convex curve with 119906119894 = 15 V119894 = 15

Figure 4 Comparison of convex and nonconvex curves through given 2D convex data

Table 7 2D convex data set

119894 1 2 3 4 5 6119909119894

minus8 minus7 22 7 10 12119910119894

45 4 355 4 45 5

where

120585 (119906119894 V119894 119908119894 120579) =

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

119894 V119894 and

119908119894

119894 119906119894

V119894

119908119894

119888119894

1 001 001 001 001802 001 001 01 004583 001 001 06 006004 001 001 09 006135 001 001 16 006256 001 001 60 006377 001 001 120 006388 001 001 1150 006399 001 001 5150 0064010 001 001 7000 0064011 001 001 10000 0064012 001 001 15150 0064013 001 001 20000 0064014 001 001 200000 00640

Table 10 Values of 119888119894for different values of shape parameters 119906

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 150 150 15 001042 150 150 01 001093 150 150 06 001354 150 150 09 001515 150 150 16 001856 150 150 60 003317 150 150 120 004298 150 150 1150 006089 150 150 5150 0063310 150 150 7000 0063511 150 150 10000 0063712 150 150 15150 0063913 150 150 20000 0064014 150 150 200000 00640

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

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 5050 5050 15 001042 5050 5050 01 001043 5050 5050 06 001054 5050 5050 09 001055 5050 5050 16 001066 5050 5050 60 001137 5050 5050 120 001228 5050 5050 1150 002579 5050 5050 5150 0046010 5050 5050 7000 0049611 5050 5050 10000 0053212 5050 5050 15150 0056413 5050 5050 20000 0058114 5050 5050 200000 00634

Table 12 Values of 119888119894for various values of shape parameters 119906

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 2000 2000 15 001042 2000 2000 01 001043 2000 2000 06 001044 2000 2000 09 001045 2000 2000 16 001046 2000 2000 60 001067 2000 2000 120 001088 2000 2000 1150 001499 2000 2000 5150 0027210 2000 2000 7000 0031311 2000 2000 10000 0036312 2000 2000 15150 0042113 2000 2000 20000 0045814 2000 2000 200000 00616

Conflict of Interests

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-

mial interpolating splinesrdquoMathematics of Computation vol 57no 195 pp 291ndash298 1991

[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

Submit your manuscripts athttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

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HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

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4 The Scientific World Journal

Table 2 Numerical results of Figure 1(d)

119894 1 2 3 4 5 6 7119889119894

minus12 minus07743 minus01143 0 01143 07743 12Δ119894

minus10 minus03 minus005 005 03 10 mdash119906119894

15 15 15 15 15 15 mdashV119894

15 15 15 15 15 15 mdash119908119894

11165 10776 00598 00598 10776 11165 mdash

Table 3 A convex data set

119894 119909119894

119910119894

1 10 1002 15 703 175 504 20 255 25 106 30 067 50 048 100 109 105 3010 11 5011 12 90

with

1198620119894=21199062

119894(119906119894+ V119894) (Δ119894minus 119889119894) + 119908119894(Δ119894minus 119889119894) minus V119894(119889119894+1

minus Δ119894)

1198621119894

= 1198620119894

+ 21199062

119894(5V119894(Δ119894minus 119889119894) + V119894119889119894)

1198622119894

= 1198620119894

+ 6119906119894(V2119894(119889119894+1

minus Δ119894) + 2119906

119894V119894(Δ119894minus 119889119894))

1198623119894

= 1198625119894

+ 6V119894(1199062

119894(Δ119894minus 119889119894) + 2119906

119894V119894(119889119894+1

minus Δ119894))

1198624119894

= 1198625119894

+ 2V2119894(5119906119894(119889119894+1

minus Δ119894) + 119906119894119889119894+1

)

1198625119894

= 2V2119894(119906119894+ V119894) (119889119894+1

minus Δ119894)

+ 119908119894(119889119894+1

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

The Scientific World Journal 5

Table 4 Numerical results of Figure 2(d)

119894 119889119894

Δ119894

119906119894

V119894

119908119894

1 minus46667 minus60 175 175 189582 minus72864 minus80 175 175 235863 minus91942 minus100 175 175 00014 minus78005 minus30 175 175 108355 minus15996 minus08 175 175 028746 minus05057 minus01 175 175 100637 minus00737 012 175 175 143398 38472 40 175 175 167239 40102 40 175 175 1553710 39991 40 175 175 16001198907

11 40 mdash mdash mdash mdash

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(a) Cubic Hermite spline curve

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(b) PCHIP curve

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(c) Convexity-preserving rational cubic curve with 119906119894 = 025 V119894 = 025

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(d) Convexity-preserving rational cubic curve with 119906119894 = 15 V119894 = 15

Figure 1 Convexity- and nonconvexity-preserving curves

6 The Scientific World Journal

0 2 4 8 10 12

0

2

4

8

10

6

6

x-axis

y-axis

minus2

(a) Cubic Hermite curve

0 2 4 8 10 120

1

2

3

4

5

6

7

8

9

10

6x-axis

y-axis

(b) PCHIP curve

0 2 4 8 10 120

1

2

3

4

5

6

7

8

9

10

6x-axis

y-axis

(c) 1198622 convex rational cubic curve with 119906119894 = 025 V119894 = 025

0 2 4 8 10 120

1

2

3

4

5

6

7

8

9

10

6x-axis

y-axis

(d) 1198622 convex rational cubic curve with 119906119894 = 175 V119894 = 175

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

119894

The Scientific World Journal 7

Table 6 Numerical results of Figure 3(d)

119894 1 2 3 4 5 6 7119889119894

minus2325 minus1644 minus0132 0 0132 1644 2325Δ119894

minus2 minus0375 minus0062 0062 0375 2 mdash119906119894

15 15 15 15 15 15 mdashV119894

15 15 15 15 15 15 mdash119908119894

5503 7902 1181 1181 7902 5503 mdash

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(a) Cubic Hermite curve

0 5 10minus5

x-axisminus10

45

5

55

3

35

4

65

7

6

y-axis

(b) PCHIP curve

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(c) Local convexity-preserving 1198622 rational cubic curve with 119906119894 = 025V119894 = 025

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(d) Local convexity-preserving 1198622 rational cubic curve with 119906119894 = 15V119894 = 15

Figure 3 Convex and nonconvex curves through given convex data set

V119894and various values of constrained parameter 119908

119894 Here we

take different values of shape parameters than [3] for theeffectiveness of proposed interpolant

Theorem 5 (see [3]) The error of interpolating rational cubicfunction (7) for 119891(119909) isin 119862

3[1199090 119909119899] in each subinterval 119868

119894=

[119909119894 119909119894+1

] is

1003816100381610038161003816119891 (119909) minus 119878119894(119909)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119909119894+1

119909119894

10038161003816100381610038161003816119877119909[(119909 minus 120591)

2

+]10038161003816100381610038161003816119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

119888119894= max0le120579le1

120585 (119906119894 V119894 119908119894 120579)

(20)

8 The Scientific World Journal

0 2 4 6 8 10 123

32

34

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

(a) Cubic Hermite spline curve

35

4

45

5

0 2 4 6 8 10 12minus8 minus6 minus4 minus2

x-axis

y-axis

(b) PCHIP curve

0 2 4 6 8 10 12

34

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

32

(c) 1198622 rational cubic convex curve with 119906119894 = 05 V119894 = 05

0 2 4 6 8 10 1234

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

(d) 1198622 rational cubic convex curve with 119906119894 = 15 V119894 = 15

Figure 4 Comparison of convex and nonconvex curves through given 2D convex data

Table 7 2D convex data set

119894 1 2 3 4 5 6119909119894

minus8 minus7 22 7 10 12119910119894

45 4 355 4 45 5

where

120585 (119906119894 V119894 119908119894 120579) =

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

119894 V119894 and

119908119894

119894 119906119894

V119894

119908119894

119888119894

1 001 001 001 001802 001 001 01 004583 001 001 06 006004 001 001 09 006135 001 001 16 006256 001 001 60 006377 001 001 120 006388 001 001 1150 006399 001 001 5150 0064010 001 001 7000 0064011 001 001 10000 0064012 001 001 15150 0064013 001 001 20000 0064014 001 001 200000 00640

Table 10 Values of 119888119894for different values of shape parameters 119906

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 150 150 15 001042 150 150 01 001093 150 150 06 001354 150 150 09 001515 150 150 16 001856 150 150 60 003317 150 150 120 004298 150 150 1150 006089 150 150 5150 0063310 150 150 7000 0063511 150 150 10000 0063712 150 150 15150 0063913 150 150 20000 0064014 150 150 200000 00640

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

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 5050 5050 15 001042 5050 5050 01 001043 5050 5050 06 001054 5050 5050 09 001055 5050 5050 16 001066 5050 5050 60 001137 5050 5050 120 001228 5050 5050 1150 002579 5050 5050 5150 0046010 5050 5050 7000 0049611 5050 5050 10000 0053212 5050 5050 15150 0056413 5050 5050 20000 0058114 5050 5050 200000 00634

Table 12 Values of 119888119894for various values of shape parameters 119906

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 2000 2000 15 001042 2000 2000 01 001043 2000 2000 06 001044 2000 2000 09 001045 2000 2000 16 001046 2000 2000 60 001067 2000 2000 120 001088 2000 2000 1150 001499 2000 2000 5150 0027210 2000 2000 7000 0031311 2000 2000 10000 0036312 2000 2000 15150 0042113 2000 2000 20000 0045814 2000 2000 200000 00616

Conflict of Interests

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-

mial interpolating splinesrdquoMathematics of Computation vol 57no 195 pp 291ndash298 1991

[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

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 5

Table 4 Numerical results of Figure 2(d)

119894 119889119894

Δ119894

119906119894

V119894

119908119894

1 minus46667 minus60 175 175 189582 minus72864 minus80 175 175 235863 minus91942 minus100 175 175 00014 minus78005 minus30 175 175 108355 minus15996 minus08 175 175 028746 minus05057 minus01 175 175 100637 minus00737 012 175 175 143398 38472 40 175 175 167239 40102 40 175 175 1553710 39991 40 175 175 16001198907

11 40 mdash mdash mdash mdash

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(a) Cubic Hermite spline curve

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(b) PCHIP curve

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(c) Convexity-preserving rational cubic curve with 119906119894 = 025 V119894 = 025

0 5 10 15minus15 minus10 minus5

0

05

1

15

2

25

3

35

4

x-axis

y-axis

(d) Convexity-preserving rational cubic curve with 119906119894 = 15 V119894 = 15

Figure 1 Convexity- and nonconvexity-preserving curves

6 The Scientific World Journal

0 2 4 8 10 12

0

2

4

8

10

6

6

x-axis

y-axis

minus2

(a) Cubic Hermite curve

0 2 4 8 10 120

1

2

3

4

5

6

7

8

9

10

6x-axis

y-axis

(b) PCHIP curve

0 2 4 8 10 120

1

2

3

4

5

6

7

8

9

10

6x-axis

y-axis

(c) 1198622 convex rational cubic curve with 119906119894 = 025 V119894 = 025

0 2 4 8 10 120

1

2

3

4

5

6

7

8

9

10

6x-axis

y-axis

(d) 1198622 convex rational cubic curve with 119906119894 = 175 V119894 = 175

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

119894

The Scientific World Journal 7

Table 6 Numerical results of Figure 3(d)

119894 1 2 3 4 5 6 7119889119894

minus2325 minus1644 minus0132 0 0132 1644 2325Δ119894

minus2 minus0375 minus0062 0062 0375 2 mdash119906119894

15 15 15 15 15 15 mdashV119894

15 15 15 15 15 15 mdash119908119894

5503 7902 1181 1181 7902 5503 mdash

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(a) Cubic Hermite curve

0 5 10minus5

x-axisminus10

45

5

55

3

35

4

65

7

6

y-axis

(b) PCHIP curve

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(c) Local convexity-preserving 1198622 rational cubic curve with 119906119894 = 025V119894 = 025

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(d) Local convexity-preserving 1198622 rational cubic curve with 119906119894 = 15V119894 = 15

Figure 3 Convex and nonconvex curves through given convex data set

V119894and various values of constrained parameter 119908

119894 Here we

take different values of shape parameters than [3] for theeffectiveness of proposed interpolant

Theorem 5 (see [3]) The error of interpolating rational cubicfunction (7) for 119891(119909) isin 119862

3[1199090 119909119899] in each subinterval 119868

119894=

[119909119894 119909119894+1

] is

1003816100381610038161003816119891 (119909) minus 119878119894(119909)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119909119894+1

119909119894

10038161003816100381610038161003816119877119909[(119909 minus 120591)

2

+]10038161003816100381610038161003816119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

119888119894= max0le120579le1

120585 (119906119894 V119894 119908119894 120579)

(20)

8 The Scientific World Journal

0 2 4 6 8 10 123

32

34

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

(a) Cubic Hermite spline curve

35

4

45

5

0 2 4 6 8 10 12minus8 minus6 minus4 minus2

x-axis

y-axis

(b) PCHIP curve

0 2 4 6 8 10 12

34

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

32

(c) 1198622 rational cubic convex curve with 119906119894 = 05 V119894 = 05

0 2 4 6 8 10 1234

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

(d) 1198622 rational cubic convex curve with 119906119894 = 15 V119894 = 15

Figure 4 Comparison of convex and nonconvex curves through given 2D convex data

Table 7 2D convex data set

119894 1 2 3 4 5 6119909119894

minus8 minus7 22 7 10 12119910119894

45 4 355 4 45 5

where

120585 (119906119894 V119894 119908119894 120579) =

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

119894 V119894 and

119908119894

119894 119906119894

V119894

119908119894

119888119894

1 001 001 001 001802 001 001 01 004583 001 001 06 006004 001 001 09 006135 001 001 16 006256 001 001 60 006377 001 001 120 006388 001 001 1150 006399 001 001 5150 0064010 001 001 7000 0064011 001 001 10000 0064012 001 001 15150 0064013 001 001 20000 0064014 001 001 200000 00640

Table 10 Values of 119888119894for different values of shape parameters 119906

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 150 150 15 001042 150 150 01 001093 150 150 06 001354 150 150 09 001515 150 150 16 001856 150 150 60 003317 150 150 120 004298 150 150 1150 006089 150 150 5150 0063310 150 150 7000 0063511 150 150 10000 0063712 150 150 15150 0063913 150 150 20000 0064014 150 150 200000 00640

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

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 5050 5050 15 001042 5050 5050 01 001043 5050 5050 06 001054 5050 5050 09 001055 5050 5050 16 001066 5050 5050 60 001137 5050 5050 120 001228 5050 5050 1150 002579 5050 5050 5150 0046010 5050 5050 7000 0049611 5050 5050 10000 0053212 5050 5050 15150 0056413 5050 5050 20000 0058114 5050 5050 200000 00634

Table 12 Values of 119888119894for various values of shape parameters 119906

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 2000 2000 15 001042 2000 2000 01 001043 2000 2000 06 001044 2000 2000 09 001045 2000 2000 16 001046 2000 2000 60 001067 2000 2000 120 001088 2000 2000 1150 001499 2000 2000 5150 0027210 2000 2000 7000 0031311 2000 2000 10000 0036312 2000 2000 15150 0042113 2000 2000 20000 0045814 2000 2000 200000 00616

Conflict of Interests

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-

mial interpolating splinesrdquoMathematics of Computation vol 57no 195 pp 291ndash298 1991

[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

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 The Scientific World Journal

0 2 4 8 10 12

0

2

4

8

10

6

6

x-axis

y-axis

minus2

(a) Cubic Hermite curve

0 2 4 8 10 120

1

2

3

4

5

6

7

8

9

10

6x-axis

y-axis

(b) PCHIP curve

0 2 4 8 10 120

1

2

3

4

5

6

7

8

9

10

6x-axis

y-axis

(c) 1198622 convex rational cubic curve with 119906119894 = 025 V119894 = 025

0 2 4 8 10 120

1

2

3

4

5

6

7

8

9

10

6x-axis

y-axis

(d) 1198622 convex rational cubic curve with 119906119894 = 175 V119894 = 175

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

119894

The Scientific World Journal 7

Table 6 Numerical results of Figure 3(d)

119894 1 2 3 4 5 6 7119889119894

minus2325 minus1644 minus0132 0 0132 1644 2325Δ119894

minus2 minus0375 minus0062 0062 0375 2 mdash119906119894

15 15 15 15 15 15 mdashV119894

15 15 15 15 15 15 mdash119908119894

5503 7902 1181 1181 7902 5503 mdash

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(a) Cubic Hermite curve

0 5 10minus5

x-axisminus10

45

5

55

3

35

4

65

7

6

y-axis

(b) PCHIP curve

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(c) Local convexity-preserving 1198622 rational cubic curve with 119906119894 = 025V119894 = 025

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(d) Local convexity-preserving 1198622 rational cubic curve with 119906119894 = 15V119894 = 15

Figure 3 Convex and nonconvex curves through given convex data set

V119894and various values of constrained parameter 119908

119894 Here we

take different values of shape parameters than [3] for theeffectiveness of proposed interpolant

Theorem 5 (see [3]) The error of interpolating rational cubicfunction (7) for 119891(119909) isin 119862

3[1199090 119909119899] in each subinterval 119868

119894=

[119909119894 119909119894+1

] is

1003816100381610038161003816119891 (119909) minus 119878119894(119909)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119909119894+1

119909119894

10038161003816100381610038161003816119877119909[(119909 minus 120591)

2

+]10038161003816100381610038161003816119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

119888119894= max0le120579le1

120585 (119906119894 V119894 119908119894 120579)

(20)

8 The Scientific World Journal

0 2 4 6 8 10 123

32

34

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

(a) Cubic Hermite spline curve

35

4

45

5

0 2 4 6 8 10 12minus8 minus6 minus4 minus2

x-axis

y-axis

(b) PCHIP curve

0 2 4 6 8 10 12

34

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

32

(c) 1198622 rational cubic convex curve with 119906119894 = 05 V119894 = 05

0 2 4 6 8 10 1234

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

(d) 1198622 rational cubic convex curve with 119906119894 = 15 V119894 = 15

Figure 4 Comparison of convex and nonconvex curves through given 2D convex data

Table 7 2D convex data set

119894 1 2 3 4 5 6119909119894

minus8 minus7 22 7 10 12119910119894

45 4 355 4 45 5

where

120585 (119906119894 V119894 119908119894 120579) =

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

119894 V119894 and

119908119894

119894 119906119894

V119894

119908119894

119888119894

1 001 001 001 001802 001 001 01 004583 001 001 06 006004 001 001 09 006135 001 001 16 006256 001 001 60 006377 001 001 120 006388 001 001 1150 006399 001 001 5150 0064010 001 001 7000 0064011 001 001 10000 0064012 001 001 15150 0064013 001 001 20000 0064014 001 001 200000 00640

Table 10 Values of 119888119894for different values of shape parameters 119906

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 150 150 15 001042 150 150 01 001093 150 150 06 001354 150 150 09 001515 150 150 16 001856 150 150 60 003317 150 150 120 004298 150 150 1150 006089 150 150 5150 0063310 150 150 7000 0063511 150 150 10000 0063712 150 150 15150 0063913 150 150 20000 0064014 150 150 200000 00640

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

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 5050 5050 15 001042 5050 5050 01 001043 5050 5050 06 001054 5050 5050 09 001055 5050 5050 16 001066 5050 5050 60 001137 5050 5050 120 001228 5050 5050 1150 002579 5050 5050 5150 0046010 5050 5050 7000 0049611 5050 5050 10000 0053212 5050 5050 15150 0056413 5050 5050 20000 0058114 5050 5050 200000 00634

Table 12 Values of 119888119894for various values of shape parameters 119906

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 2000 2000 15 001042 2000 2000 01 001043 2000 2000 06 001044 2000 2000 09 001045 2000 2000 16 001046 2000 2000 60 001067 2000 2000 120 001088 2000 2000 1150 001499 2000 2000 5150 0027210 2000 2000 7000 0031311 2000 2000 10000 0036312 2000 2000 15150 0042113 2000 2000 20000 0045814 2000 2000 200000 00616

Conflict of Interests

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-

mial interpolating splinesrdquoMathematics of Computation vol 57no 195 pp 291ndash298 1991

[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

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 7

Table 6 Numerical results of Figure 3(d)

119894 1 2 3 4 5 6 7119889119894

minus2325 minus1644 minus0132 0 0132 1644 2325Δ119894

minus2 minus0375 minus0062 0062 0375 2 mdash119906119894

15 15 15 15 15 15 mdashV119894

15 15 15 15 15 15 mdash119908119894

5503 7902 1181 1181 7902 5503 mdash

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(a) Cubic Hermite curve

0 5 10minus5

x-axisminus10

45

5

55

3

35

4

65

7

6

y-axis

(b) PCHIP curve

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(c) Local convexity-preserving 1198622 rational cubic curve with 119906119894 = 025V119894 = 025

0 5 10minus10 minus5

45

5

55

75

3

35

4

65

7

6

x-axis

y-axis

(d) Local convexity-preserving 1198622 rational cubic curve with 119906119894 = 15V119894 = 15

Figure 3 Convex and nonconvex curves through given convex data set

V119894and various values of constrained parameter 119908

119894 Here we

take different values of shape parameters than [3] for theeffectiveness of proposed interpolant

Theorem 5 (see [3]) The error of interpolating rational cubicfunction (7) for 119891(119909) isin 119862

3[1199090 119909119899] in each subinterval 119868

119894=

[119909119894 119909119894+1

] is

1003816100381610038161003816119891 (119909) minus 119878119894(119909)

1003816100381610038161003816 le1

2

10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817int

119909119894+1

119909119894

10038161003816100381610038161003816119877119909[(119909 minus 120591)

2

+]10038161003816100381610038161003816119889120591

=10038171003817100381710038171003817119891(3)

(120591)10038171003817100381710038171003817119888119894

119888119894= max0le120579le1

120585 (119906119894 V119894 119908119894 120579)

(20)

8 The Scientific World Journal

0 2 4 6 8 10 123

32

34

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

(a) Cubic Hermite spline curve

35

4

45

5

0 2 4 6 8 10 12minus8 minus6 minus4 minus2

x-axis

y-axis

(b) PCHIP curve

0 2 4 6 8 10 12

34

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

32

(c) 1198622 rational cubic convex curve with 119906119894 = 05 V119894 = 05

0 2 4 6 8 10 1234

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

(d) 1198622 rational cubic convex curve with 119906119894 = 15 V119894 = 15

Figure 4 Comparison of convex and nonconvex curves through given 2D convex data

Table 7 2D convex data set

119894 1 2 3 4 5 6119909119894

minus8 minus7 22 7 10 12119910119894

45 4 355 4 45 5

where

120585 (119906119894 V119894 119908119894 120579) =

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

119894 V119894 and

119908119894

119894 119906119894

V119894

119908119894

119888119894

1 001 001 001 001802 001 001 01 004583 001 001 06 006004 001 001 09 006135 001 001 16 006256 001 001 60 006377 001 001 120 006388 001 001 1150 006399 001 001 5150 0064010 001 001 7000 0064011 001 001 10000 0064012 001 001 15150 0064013 001 001 20000 0064014 001 001 200000 00640

Table 10 Values of 119888119894for different values of shape parameters 119906

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 150 150 15 001042 150 150 01 001093 150 150 06 001354 150 150 09 001515 150 150 16 001856 150 150 60 003317 150 150 120 004298 150 150 1150 006089 150 150 5150 0063310 150 150 7000 0063511 150 150 10000 0063712 150 150 15150 0063913 150 150 20000 0064014 150 150 200000 00640

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

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 5050 5050 15 001042 5050 5050 01 001043 5050 5050 06 001054 5050 5050 09 001055 5050 5050 16 001066 5050 5050 60 001137 5050 5050 120 001228 5050 5050 1150 002579 5050 5050 5150 0046010 5050 5050 7000 0049611 5050 5050 10000 0053212 5050 5050 15150 0056413 5050 5050 20000 0058114 5050 5050 200000 00634

Table 12 Values of 119888119894for various values of shape parameters 119906

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 2000 2000 15 001042 2000 2000 01 001043 2000 2000 06 001044 2000 2000 09 001045 2000 2000 16 001046 2000 2000 60 001067 2000 2000 120 001088 2000 2000 1150 001499 2000 2000 5150 0027210 2000 2000 7000 0031311 2000 2000 10000 0036312 2000 2000 15150 0042113 2000 2000 20000 0045814 2000 2000 200000 00616

Conflict of Interests

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-

mial interpolating splinesrdquoMathematics of Computation vol 57no 195 pp 291ndash298 1991

[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

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 The Scientific World Journal

0 2 4 6 8 10 123

32

34

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

(a) Cubic Hermite spline curve

35

4

45

5

0 2 4 6 8 10 12minus8 minus6 minus4 minus2

x-axis

y-axis

(b) PCHIP curve

0 2 4 6 8 10 12

34

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

32

(c) 1198622 rational cubic convex curve with 119906119894 = 05 V119894 = 05

0 2 4 6 8 10 1234

36

38

4

42

44

46

48

5

minus8 minus6 minus4 minus2

x-axis

y-axis

(d) 1198622 rational cubic convex curve with 119906119894 = 15 V119894 = 15

Figure 4 Comparison of convex and nonconvex curves through given 2D convex data

Table 7 2D convex data set

119894 1 2 3 4 5 6119909119894

minus8 minus7 22 7 10 12119910119894

45 4 355 4 45 5

where

120585 (119906119894 V119894 119908119894 120579) =

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

119894 V119894 and

119908119894

119894 119906119894

V119894

119908119894

119888119894

1 001 001 001 001802 001 001 01 004583 001 001 06 006004 001 001 09 006135 001 001 16 006256 001 001 60 006377 001 001 120 006388 001 001 1150 006399 001 001 5150 0064010 001 001 7000 0064011 001 001 10000 0064012 001 001 15150 0064013 001 001 20000 0064014 001 001 200000 00640

Table 10 Values of 119888119894for different values of shape parameters 119906

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 150 150 15 001042 150 150 01 001093 150 150 06 001354 150 150 09 001515 150 150 16 001856 150 150 60 003317 150 150 120 004298 150 150 1150 006089 150 150 5150 0063310 150 150 7000 0063511 150 150 10000 0063712 150 150 15150 0063913 150 150 20000 0064014 150 150 200000 00640

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

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 5050 5050 15 001042 5050 5050 01 001043 5050 5050 06 001054 5050 5050 09 001055 5050 5050 16 001066 5050 5050 60 001137 5050 5050 120 001228 5050 5050 1150 002579 5050 5050 5150 0046010 5050 5050 7000 0049611 5050 5050 10000 0053212 5050 5050 15150 0056413 5050 5050 20000 0058114 5050 5050 200000 00634

Table 12 Values of 119888119894for various values of shape parameters 119906

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 2000 2000 15 001042 2000 2000 01 001043 2000 2000 06 001044 2000 2000 09 001045 2000 2000 16 001046 2000 2000 60 001067 2000 2000 120 001088 2000 2000 1150 001499 2000 2000 5150 0027210 2000 2000 7000 0031311 2000 2000 10000 0036312 2000 2000 15150 0042113 2000 2000 20000 0045814 2000 2000 200000 00616

Conflict of Interests

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-

mial interpolating splinesrdquoMathematics of Computation vol 57no 195 pp 291ndash298 1991

[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

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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

119894 V119894 and

119908119894

119894 119906119894

V119894

119908119894

119888119894

1 001 001 001 001802 001 001 01 004583 001 001 06 006004 001 001 09 006135 001 001 16 006256 001 001 60 006377 001 001 120 006388 001 001 1150 006399 001 001 5150 0064010 001 001 7000 0064011 001 001 10000 0064012 001 001 15150 0064013 001 001 20000 0064014 001 001 200000 00640

Table 10 Values of 119888119894for different values of shape parameters 119906

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 150 150 15 001042 150 150 01 001093 150 150 06 001354 150 150 09 001515 150 150 16 001856 150 150 60 003317 150 150 120 004298 150 150 1150 006089 150 150 5150 0063310 150 150 7000 0063511 150 150 10000 0063712 150 150 15150 0063913 150 150 20000 0064014 150 150 200000 00640

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

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 5050 5050 15 001042 5050 5050 01 001043 5050 5050 06 001054 5050 5050 09 001055 5050 5050 16 001066 5050 5050 60 001137 5050 5050 120 001228 5050 5050 1150 002579 5050 5050 5150 0046010 5050 5050 7000 0049611 5050 5050 10000 0053212 5050 5050 15150 0056413 5050 5050 20000 0058114 5050 5050 200000 00634

Table 12 Values of 119888119894for various values of shape parameters 119906

119894 V119894

and 119908119894

119894 119906119894

V119894

119908119894

119888119894

1 2000 2000 15 001042 2000 2000 01 001043 2000 2000 06 001044 2000 2000 09 001045 2000 2000 16 001046 2000 2000 60 001067 2000 2000 120 001088 2000 2000 1150 001499 2000 2000 5150 0027210 2000 2000 7000 0031311 2000 2000 10000 0036312 2000 2000 15150 0042113 2000 2000 20000 0045814 2000 2000 200000 00616

Conflict of Interests

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-

mial interpolating splinesrdquoMathematics of Computation vol 57no 195 pp 291ndash298 1991

[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

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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-

mial interpolating splinesrdquoMathematics of Computation vol 57no 195 pp 291ndash298 1991

[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

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014