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ISRN Mathematical Analysis
VolumeΒ 2012Β (2012), Article IDΒ 174048, 14 pages
http://dx.doi.org/10.5402/2012/174048
Research Article

Local Convexity Shape-Preserving Data Visualization by Spline Function

1School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia
2Department of Mathematics, University of Sargodha, 40100 Sargodha, Pakistan
3School of Distance Education, Universiti Sains Malaysia, 11800 Penang, Malaysia

Received 16 November 2011; Accepted 21 December 2011

Academic Editor: R.Β Barrio

Copyright Β© 2012 Muhammad Abbas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The main purpose of this paper is the visualization of convex data that results in a smooth, pleasant, and interactive convexity-preserving curve. The rational cubic function with three free parameters is constructed to preserve the shape of convex data. The free parameters are arranged in a way that two of them are left free for user choice to refine the convex curve as desired, and the remaining one free parameter is constrained to preserve the convexity everywhere. Simple data-dependent constraints are derived on one free parameter, which guarantee to preserve the convexity of curve. Moreover, the scheme under discussion is, 𝐢1 flexible, simple, local, and economical as compared to existing schemes. The error bound for the rational cubic function is 𝑂(β„Ž3).

1. Introduction

Spline interpolation is a significant tool in computer graphics, computer-aided geometric design and engineering as well. Convexity is prevalent shape feature of data. Therefore, the need for convexity preserving interpolating curves and surfaces according to the given data becomes inevitable. The aspiration of this paper is to preserve the hereditary attribute that is the convexity of data. There are many applications of convexity preserving of data, for instance, in the design of telecommunication systems, nonlinear programming arising in engineering, approximation of functions, optimal control, and parameter estimation.

The problem of convexity-preserving interpolation has been considered by a number of authors [1–21] and references therein. Bao et al. [1] used function values and first derivatives of function to introduce a rational cubic spline (cubic/cubic). A method for value control, inflection-point control and convexity control of the interpolation at a point was developed to be used in practical curve design. Asaturyan et al. [3] constructed a six-degree piecewise polynomial interpolant for the space curves to satisfy the shape-preserving properties for collinear and coplanar data.

Brodlie and Butt [4] developed a piecewise rational cubic function to preserve the shape of convex data. In [4], the authors inserted extra knots in the interval where the interpolation loses the convexity of convex data which is the drawback of this scheme. Carnicer et al. [5] analyzed the convexity-preserving properties of rational BΓ©zier and non-uniform rational B-spline curves from a geometric point of view and also characterize totally positive systems of functions in terms of geometric convexity-preserving properties of the rational curves.

Clements [6] developed a 𝐢2 parametric rational cubic interpolant with tension parameter to preserve the convexity. Sufficient conditions were derived to preserve the convexity of the function on strictly left/right winding polygonal line segments. Costantini and Fontanella [8] preserved the convexity of data by semi-global method. The scheme has some research gaps like the degree of rectangular patches in the interpolant that was too large; the resulting surfaces were not visually pleasing and smooth.

Delbourgo and Gregory [9] developed an explicit representation of rational cubic function with one free parameter which can be used to preserve the convexity of convex data. Meng and Shi Long [11] also developed an explicit representation of rational cubic function with two free parameters which can be used to preserve the convexity of convex data. In the schemes [9, 11], there was no choice for user to refine the convexity curve as desired. The rational spline was represented in terms of first derivative values at the knots and provided an alternative to the spline under tension to preserve the shape of monotone and convex data by Gregory [10].

McAllister [12], Passow [13], and Roulier [14] considered the problem of interpolating monotonic and convex data in the sense of monotonicity and convexity preserving. They used a piecewise polynomial Bernstein-BΓ©zier function and introduce additional knots into their schemes. Such a scheme for quadratic spline interpolation was described by McAllister [12] and was further developed by Schumaker [15] using piecewise quadratic polynomial which was very economical, but the method generally inserts an extra knot in each interval to interpolate.

Sarfraz and Hussain [17] used the rational cubic function with two shape parameters to solve the problem of convexity preserving of convex data. Data-dependent sufficient constraints were derived to preserve the shape of convex data. Sarfraz [18] developed a piecewise rational cubic function with two families of parameters. In [18], the authors derived the sufficient conditions on shape parameters to preserve the physical shape properties of data. Sarfraz [19–21] used piecewise rational cubic interpolant in parametric context for shape preserving of plane curves and scalar curves with planar data. The schemes [17–21] are local, but, unfortunately, they have no flexibility in the convexity-preserving curves.

In this paper, we construct a rational cubic function with three free parameters. One of the free parameter is used as a constrained to preserve the convexity of convex data while the other two are left free for the user to modify the convex curve. Sufficient data-dependent constraints are derived. Our scheme has a number of attributes over the existing schemes.(i)In this paper, the shape-preserving of convex data is achieved by simply imposing the conditions subject to data on the shape parameters used in the description of rational cubic function. The proposed scheme works evenly good for both equally and unequally spaced data. In contrast [1] assumed certain function values and derivative values to control the shape of the data.(ii)In [12, 15], the smoothness of interpolant is 𝐢0while in this work the degree of smoothness is𝐢1.(iii)The developed scheme has been demonstrated through different numerical examples and observed that the scheme is not only local, computationally economical, and easy to compute, time saving but also visually pleasant as compared to existing schemes [17–21].(iv)In [9–11, 17–21], the schemes do not allow to user to refine the convex curve as desired while for more pleasing curve (and still having the convex shape preserved) an additional modification is required, and this task is more easily done in this paper by simply adjustment of free parameters in the rational cubic function interpolation on user choice.(v)In [17–21], the authors did not provide the error analysis of the interpolants while a very good𝑂(β„Ž3)error bound is achieved in this paper.(vi)In [4, 12–15], the authors developed the schemes to achieve the desired shape of data by inserting extra knots between any two knots in the interval while we preserve the shape of convex data by only imposing constraints on free parameters without any extra knots.

The remaining part of this paper is organized as follows. A rational cubic function is defined in Section 2. The error of the rational cubic interpolant is discussed in Section 3. The problem of shape preserving convexity curve is discussed in Section 4. Derivatives approximation method is given in Section 5. Some numerical results are given in Section 6. Finally, the conclusion of this work is discussed in Section 7.

2. Rational Cubic Spline Function

Let{(π‘₯𝑖,𝑓𝑖),𝑖=0,1,2,…,𝑛}be the given set of data points such asπ‘₯0<π‘₯1<π‘₯2<β‹―<π‘₯𝑛. The rational cubic function with three free parameters introduced by Abbas et al. [2], in each subinterval𝐼𝑖=[π‘₯𝑖,π‘₯𝑖+1],𝑖=0,1,2,…,π‘›βˆ’1,is defined as 𝑆𝑖𝑝(π‘₯)=𝑖(πœƒ)π‘žπ‘–(πœƒ),(2.1) with 𝑝𝑖(πœƒ)=𝑒𝑖𝑓𝑖(1βˆ’πœƒ)3+𝑀𝑖𝑓𝑖+π‘’π‘–β„Žπ‘–π‘‘π‘–ξ€Έπœƒ(1βˆ’πœƒ)2+𝑀𝑖𝑓𝑖+1βˆ’π‘£π‘–β„Žπ‘–π‘‘π‘–+1ξ€Έπœƒ2(1βˆ’πœƒ)+𝑣𝑖𝑓𝑖+1πœƒ3,π‘žπ‘–(πœƒ)=𝑒𝑖(1βˆ’πœƒ)3+π‘€π‘–πœƒ(1βˆ’πœƒ)+π‘£π‘–πœƒ3,(2.2) whereπœƒ=π‘₯βˆ’π‘₯𝑖/β„Žπ‘–,β„Žπ‘–=π‘₯𝑖+1βˆ’π‘₯𝑖, and𝑒𝑖,𝑣𝑖,𝑀𝑖are the positive free parameters. It is worth noting that when we use the values of these free parameters as𝑒𝑖=1,𝑣𝑖=1and𝑀𝑖=3,then the𝐢1 piecewise rational cubic function (2.1) reduces to standard cubic Hermite spline discussed in Schultz [16].

The piecewise rational cubic function has the following interpolatory conditions: 𝑆𝑖π‘₯𝑖=𝑓𝑖,𝑆𝑖π‘₯𝑖+1ξ€Έ=𝑓𝑖+1,𝑆′𝑖π‘₯𝑖=𝑑𝑖,π‘†ξ…žπ‘–ξ€·π‘₯𝑖+1ξ€Έ=𝑑𝑖+1,(2.3) whereπ‘†ξ…žπ‘–(π‘₯)denotes the derivative with respect to β€œx,” and𝑑𝑖 denotes the derivative values at knots.

3. Interpolation Error Analysis

The error analysis of piecewise rational cubic function (2.1) is estimated, without loss of generality, in the subinterval𝐼𝑖=[π‘₯𝑖,π‘₯𝑖+1]. It is to mention that the scheme constructed in Section 2 is local. We suppose that𝑓(π‘₯)∈𝐢3[π‘₯0,π‘₯𝑛], and𝑆𝑖(π‘₯)is the interpolation of function𝑓(π‘₯)over arbitrary subinterval𝐼𝑖=[π‘₯𝑖,π‘₯𝑖+1]. The Peano Kernel Theorem, Schultz [16] is used to obtain the error analysis of piecewise rational cubic interpolation in each subinterval𝐼𝑖=[π‘₯𝑖,π‘₯𝑖+1], and it is defined as 𝑅[𝑓]=𝑓(π‘₯)βˆ’π‘†π‘–1(π‘₯)=2ξ€œπ‘₯𝑖+1π‘₯𝑖𝑓(3)(𝜏)𝑅π‘₯ξ€Ί(π‘₯βˆ’πœ)2+ξ€»π‘‘πœ.(3.1) In each subinterval, the absolute value of error is ||𝑓(π‘₯)βˆ’π‘†π‘–||≀1(π‘₯)2‖‖𝑓(3)β€–β€–ξ€œ(𝜏)π‘₯𝑖+1π‘₯𝑖||𝑅π‘₯ξ€Ί(π‘₯βˆ’πœ)2+ξ€»||π‘‘πœ,(3.2) where𝑅π‘₯ξ€Ί(π‘₯βˆ’πœ)2+ξ€»=⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩(π‘₯βˆ’πœ)2βˆ’ξ‚€π‘€π‘–ξ€·π‘₯𝑖+1ξ€Έβˆ’πœ2βˆ’2β„Žπ‘–π‘£π‘–ξ€·π‘₯𝑖+1ξ€Έξ‚πœƒβˆ’πœ2(1βˆ’πœƒ)+𝑣𝑖π‘₯𝑖+1ξ€Έβˆ’πœ2πœƒ3π‘žπ‘–π‘₯(πœƒ)𝑖𝑀<𝜏<π‘₯,𝑖π‘₯𝑖+1ξ€Έβˆ’πœ2βˆ’2β„Žπ‘–π‘£π‘–ξ€·π‘₯𝑖+1ξ€Έξ‚πœƒβˆ’πœ2(1βˆ’πœƒ)+𝑣𝑖π‘₯𝑖+1ξ€Έβˆ’πœ2πœƒ3π‘žπ‘–(πœƒ)π‘₯<𝜏<π‘₯𝑖+1,=ξ‚»π‘Ž(𝜏,π‘₯)π‘₯𝑖<𝜏<π‘₯,𝑏(𝜏,π‘₯)π‘₯<𝜏<π‘₯𝑖+1,(3.3) where𝑅π‘₯[(π‘₯βˆ’πœ)2+] is called the Peano Kernel of integral. To derive the error analysis, first of all we need to examine the properties of the kernel functionsπ‘Ž(𝜏,π‘₯) and𝑏(𝜏,π‘₯), and then to find the values of following integrals: ξ€œπ‘₯𝑖+1π‘₯𝑖||𝑅π‘₯ξ€Ί(π‘₯βˆ’πœ)2+ξ€»||ξ€œπ‘‘πœ=π‘₯π‘₯𝑖||||ξ€œπ‘Ž(𝜏,π‘₯)π‘‘πœ+π‘₯𝑖+1π‘₯||||𝑏(𝜏,π‘₯)π‘‘πœ.(3.4) So, we calculate these values in two parts. The proof of Theorem will be completed by combining these two parts.

3.1. Part 1

By simple computation, the roots of π‘Ž(π‘₯,π‘₯)=(πœƒ2(1βˆ’πœƒ)2((π‘€π‘–βˆ’πœˆπ‘–)πœƒ+(2πœˆπ‘–βˆ’π‘€π‘–))β„Ž2𝑖)/π‘žπ‘–(πœƒ)in[0,1] are πœƒ=0,πœƒ=1andπœƒβˆ—=1βˆ’πœˆπ‘–/(π‘€π‘–βˆ’πœˆπ‘–). It is easy to show that whenπœƒβ‰€πœƒβˆ—, π‘Ž(π‘₯,π‘₯)≀0 andπœƒβ‰₯πœƒβˆ—,π‘Ž(π‘₯,π‘₯)β‰₯0. The roots of quadratic functionπ‘Ž(𝜏,π‘₯)=0 are πœβˆ—1=π‘₯βˆ’πœƒβ„Žπ‘–ξ€·πœƒξ€·π‘€π‘–βˆ’πœˆπ‘–ξ€Έξ€Έ+𝐴(1βˆ’πœƒ)𝑒𝑖+πœƒπ‘€π‘–,𝜏1βˆ—βˆ—=π‘₯βˆ’πœƒβ„Žπ‘–ξ€·πœƒξ€·π‘€π‘–βˆ’πœˆπ‘–ξ€Έξ€Έβˆ’π΄(1βˆ’πœƒ)𝑒𝑖+πœƒπ‘€π‘–,(3.5) where √𝐴=πœˆπ‘–((π‘€π‘–βˆ’2πœˆπ‘–)+3πœƒ)+𝑀𝑖(π‘€π‘–βˆ’4πœˆπ‘–)πœƒ.

So, when πœƒ>πœƒβˆ—,π‘₯𝑖<𝜏1βˆ—βˆ—<π‘₯ and whenπœƒ<πœƒβˆ—,𝜏1βˆ—βˆ—>π‘₯. Thus,πœƒ<πœƒβˆ—,π‘Ž(𝜏,π‘₯)<0forall𝜏∈[π‘₯𝑖,π‘₯], ξ€œπ‘₯π‘₯𝑖||||ξ€œπ‘Ž(𝜏,π‘₯)π‘‘πœ=π‘₯π‘₯𝑖=ξ€·πœˆ(βˆ’π‘Ž(𝜏,π‘₯))π‘‘πœπ‘–(3βˆ’πœƒ)βˆ’π‘€π‘–ξ€Έ(1βˆ’πœƒ)(1βˆ’πœƒ)3πœƒ2β„Ž3𝑖3π‘žπ‘–+𝑀(πœƒ)π‘–βˆ’3𝑣𝑖(1βˆ’πœƒ)πœƒ2β„Ž3𝑖3π‘žπ‘–+𝑣(πœƒ)π‘–πœƒ3β„Ž3𝑖3π‘žπ‘–βˆ’πœƒ(πœƒ)3β„Ž3𝑖3.(3.6) The value ofπ‘Ž(𝜏,π‘₯) varies from negative to positive on the root𝜏1βˆ—βˆ—whenπœƒ>πœƒβˆ—, ξ€œπ‘₯π‘₯𝑖||||ξ€œπ‘Ž(𝜏,π‘₯)π‘‘πœ=𝜏1βˆ—βˆ—π‘₯π‘–ξ€œ(βˆ’π‘Ž(𝜏,π‘₯))π‘‘πœ+π‘₯𝜏1βˆ—βˆ—=2π‘€π‘Ž(𝜏,π‘₯)π‘‘πœξ€·ξ€·π‘–βˆ’πœˆπ‘–ξ€Έξ€Έπœƒβˆ’π΄3πœƒ3β„Ž3𝑖3ξ€·(1βˆ’πœƒ)𝑒𝑖+πœƒπ‘€π‘–ξ€Έ3βˆ’πœƒ3β„Ž3𝑖3βˆ’2β„Ž3𝑖3π‘žπ‘–ξƒ¬πœƒπ‘€(πœƒ)(1βˆ’πœƒ)+ξ€·ξ€·π‘–βˆ’πœˆπ‘–ξ€Έξ€Έπœƒβˆ’π΄(1βˆ’πœƒ)𝑒𝑖+πœƒπ‘€π‘–ξƒ­3Γ—ξ€·(1βˆ’πœƒ)𝑀𝑖+πœƒπœˆπ‘–ξ€Έ+2β„Ž3π‘–πœˆπ‘–πœƒ2(1βˆ’πœƒ)π‘žπ‘–ξƒ¬πœƒπ‘€(πœƒ)(1βˆ’πœƒ)+ξ€·ξ€·π‘–βˆ’πœˆπ‘–ξ€Έξ€Έπœƒβˆ’π΄(1βˆ’πœƒ)𝑒𝑖+πœƒπ‘€π‘–ξƒ­2.(3.7)

3.2. Part 2

In this part, we discuss the properties of function𝑏(𝜏,π‘₯). Consider𝑏(𝜏,π‘₯),𝜏∈[π‘₯,π‘₯𝑖+1] as function of𝜏. The roots of function𝑏(𝜏,π‘₯) are similar asπ‘Ž(𝜏,π‘₯)in Section 3.1 at𝜏=π‘₯. It is easy to show that whenπœƒβ‰€πœƒβˆ—,𝑏(π‘₯,π‘₯)≀0andπœƒβ‰₯πœƒβˆ—, 𝑏(π‘₯,π‘₯)β‰₯0. The roots of quadratic function𝑏(𝜏,π‘₯)=0are πœβˆ—2=π‘₯𝑖+1,𝜏2βˆ—βˆ—=π‘₯𝑖+1βˆ’2(1βˆ’πœƒ)πœˆπ‘–β„Žπ‘–(1βˆ’πœƒ)𝑀𝑖+πœƒπœˆπ‘–.(3.8) The function𝑏(𝜏,π‘₯)varies from negative to positive on the root𝜏2βˆ—βˆ—when πœƒβ‰€πœƒβˆ—. Thus, ξ€œπ‘₯𝑖+1π‘₯||||ξ€œπ‘(𝜏,π‘₯)π‘‘πœ=𝜏2βˆ—βˆ—π‘₯ξ€œ(βˆ’π‘(𝜏,π‘₯))π‘‘πœ+π‘₯𝑖+1𝜏2βˆ—βˆ—=𝑏(𝜏,π‘₯)π‘‘πœ8πœƒ2(1βˆ’πœƒ)3πœˆπ‘–3β„Ž3𝑖3π‘žπ‘–ξ€·(πœƒ)(1βˆ’πœƒ)𝑒𝑖+πœƒπ‘€π‘–ξ€Έ2+β„Ž3π‘–πœƒ2(1βˆ’πœƒ)33π‘žπ‘–ξ€·π‘€(πœƒ)𝑖(1βˆ’πœƒ)βˆ’πœˆπ‘–ξ€Έ,(3βˆ’πœƒ)(3.9) whenπœƒβ‰₯πœƒβˆ—,ξ€œπ‘₯𝑖+1π‘₯||||ξ€œπ‘(𝜏,π‘₯)π‘‘πœ=π‘₯𝑖+1π‘₯=β„Žπ‘(𝜏,π‘₯)π‘‘πœ3π‘–πœƒ2(1βˆ’πœƒ)33π‘žπ‘–ξ€·πœˆ(πœƒ)𝑖(3βˆ’πœƒ)βˆ’π‘€π‘–(ξ€Έ.1βˆ’πœƒ)(3.10) Thus, from (3.6) and (3.9), it can be shown that when 0β‰€πœƒβ‰€πœƒβˆ—,||𝑓(π‘₯)βˆ’π‘†π‘–||≀1(π‘₯)2‖‖𝑓(3)β€–β€–ξ€œ(𝜏)π‘₯𝑖+1π‘₯𝑖||𝑅π‘₯ξ€Ί(π‘₯βˆ’πœ(π‘₯βˆ’πœ)2+ξ€»||β€–β€–π‘“π‘‘πœ=(3)β€–β€–β„Ž(𝜏)3𝑖𝑝1𝑒𝑖,𝑣𝑖,𝑀𝑖,πœƒ,(3.11) where𝑝1𝑒𝑖,πœˆπ‘–,𝑀𝑖=ξ€·πœˆ,πœƒπ‘–(3βˆ’πœƒ)βˆ’π‘€π‘–ξ€Έ(1βˆ’πœƒ)(1βˆ’πœƒ)3πœƒ26π‘žπ‘–+𝑀(πœƒ)π‘–βˆ’3πœˆπ‘–ξ€Έ(1βˆ’πœƒ)πœƒ26π‘žπ‘–+𝜈(πœƒ)π‘–πœƒ36π‘žπ‘–βˆ’πœƒ(πœƒ)36+8πœƒ2(1βˆ’πœƒ)3πœˆπ‘–36π‘žπ‘–ξ€·(πœƒ)(1βˆ’πœƒ)𝑒𝑖+πœƒπ‘€π‘–ξ€Έ2+πœƒ2(1βˆ’πœƒ)36π‘žπ‘–ξ€·π‘€(πœƒ)𝑖(1βˆ’πœƒ)βˆ’πœˆπ‘–ξ€Έ,(3βˆ’πœƒ)(3.12) and, from (3.7) and (3.10), it can be shown that when πœƒβˆ—β‰€πœƒβ‰€1,||𝑓(π‘₯)βˆ’π‘†π‘–||≀1(π‘₯)2‖‖𝑓(3)β€–β€–ξ€œ(𝜏)π‘₯𝑖+1π‘₯𝑖||𝑅π‘₯ξ€Ί(π‘₯βˆ’πœ)2+ξ€»||β€–β€–π‘“π‘‘πœ=(3)β€–β€–β„Ž(𝜏)3𝑖𝑝2𝑒𝑖,πœˆπ‘–,𝑀𝑖,πœƒ,(3.13) where𝑝2𝑒𝑖,πœˆπ‘–,𝑀𝑖=2𝑀,πœƒξ€·ξ€·π‘–βˆ’πœˆπ‘–ξ€Έξ€Έπœƒβˆ’π΄3πœƒ36ξ€·(1βˆ’πœƒ)𝑒𝑖+πœƒπ‘€π‘–ξ€Έ3βˆ’πœƒ36βˆ’26π‘žπ‘–ξƒ¬πœƒπ‘€(πœƒ)(1βˆ’πœƒ)+ξ€·ξ€·π‘–βˆ’πœˆπ‘–ξ€Έξ€Έπœƒβˆ’π΄(1βˆ’πœƒ)𝑒𝑖+πœƒπ‘€π‘–ξƒ­3Γ—ξ€·(1βˆ’πœƒ)𝑀𝑖+πœƒπœˆπ‘–ξ€Έ+πœˆπ‘–πœƒ2(1βˆ’πœƒ)π‘žπ‘–ξƒ¬πœƒπ‘€(πœƒ)(1βˆ’πœƒ)+ξ€·ξ€·π‘–βˆ’πœˆπ‘–ξ€Έξ€Έπœƒβˆ’π΄(1βˆ’πœƒ)𝑒𝑖+πœƒπ‘€π‘–ξƒ­2+πœƒ2(1βˆ’πœƒ)36π‘žπ‘–ξ€·πœˆ(πœƒ)𝑖(3βˆ’πœƒ)βˆ’π‘€π‘–ξ€Έ.(1βˆ’πœƒ)(3.14)

Theorem 3.1. For the positive free parameters𝑒𝑖,πœˆπ‘–,and𝑀𝑖, the error of interpolating rational cubic function𝑆𝑖(π‘₯), for 𝑓(π‘₯)∈𝐢3[π‘₯0,π‘₯𝑛], in each subinterval 𝐼𝑖=[π‘₯𝑖,π‘₯𝑖+1] is ||𝑓(π‘₯)βˆ’π‘†π‘–||≀1(π‘₯)2‖‖𝑓(3)β€–β€–ξ€œ(𝜏)π‘₯𝑖+1π‘₯𝑖||𝑅π‘₯ξ€Ί(π‘₯βˆ’πœ)2+ξ€»||β€–β€–π‘“π‘‘πœ=(3)β€–β€–β„Ž(𝜏)3𝑖𝑐𝑖,𝑐𝑖=max0β‰€πœƒβ‰€1𝑝𝑒𝑖,πœˆπ‘–,𝑀𝑖,,πœƒ(3.15) where 𝑝𝑒𝑖,πœˆπ‘–,𝑀𝑖=ξ‚»,πœƒmax𝑝1𝑒𝑖,πœˆπ‘–,𝑀𝑖,πœƒ,0β‰€πœƒβ‰€πœƒβˆ—max𝑝2𝑒𝑖,πœˆπ‘–,𝑀𝑖,πœƒ,πœƒβˆ—β‰€πœƒβ‰€1.(3.16)

Remark 3.2. It is interesting to note that the rational cubic interpolation (2.1) reduces to standard cubic Hermite interpolation when we adjust the values of parameters as𝑒𝑖=1,πœˆπ‘–=1and𝑀𝑖=3. In this special case, the functions𝑝1(𝑒𝑖,πœˆπ‘–,𝑀𝑖,πœƒ)and𝑝2(𝑒𝑖,πœˆπ‘–,𝑀𝑖,πœƒ) are 𝑝1𝑒𝑖,𝑣𝑖,𝑀𝑖=,πœƒ4πœƒ2(1βˆ’πœƒ)33(3βˆ’2πœƒ)21,0β‰€πœƒβ‰€2𝑝,(3.17)2𝑒𝑖,πœˆπ‘–,𝑀𝑖=,πœƒ4πœƒ3(1βˆ’πœƒ)23(1+2πœƒ)2,12β‰€πœƒβ‰€0,(3.18) respectively. Since𝑐𝑖=max{max0β‰€πœƒβ‰€0.5𝑝1(𝑒𝑖,πœˆπ‘–,𝑀𝑖,πœƒ),max0.5β‰€πœƒβ‰€0𝑝2(𝑒𝑖,πœˆπ‘–,𝑀𝑖,πœƒ)}=1/96. This is the standard result for standard cubic Hemite spline interpolation.

4. Shape Preserving 2D Convex Data Rational Cubic Spline Interpolation

The piecewise rational cubic function (2.1) does not guarantee to preserve the shape of convex data. So, it is required to assign suitable constraints on the free parameters by some mathematical treatment to preserve the convexity of convex data.

Theorem 4.1. The𝐢1piecewise rational cubic function (2.1) preserves the convexity of convex data if in each subinterval𝐼𝑖=[π‘₯𝑖,π‘₯𝑖+1],𝑖=0,1,2,…,𝑛, the free parameters satisfy the following sufficient conditions: 𝑀𝑖𝑑>max0,𝑖+1πœˆπ‘–ξ€·π‘‘π‘–+1βˆ’Ξ”π‘–ξ€Έ,𝑑𝑖+1πœˆπ‘–ξ€·Ξ”π‘–βˆ’π‘‘π‘–ξ€Έ,2π‘’π‘–πœˆπ‘–ξ€·π‘‘π‘–+1βˆ’Ξ”π‘–ξ€Έξ€·π‘‘π‘–+1πœˆπ‘–βˆ’Ξ”π‘–π‘’π‘–ξ€Έ,2π‘’π‘–πœˆπ‘–ξ€·Ξ”π‘–βˆ’π‘‘π‘–ξ€Έξ€·Ξ”π‘–πœˆπ‘–βˆ’π‘‘π‘–π‘’π‘–ξ€Έ,π‘’π‘–πœˆπ‘–ξ€·π‘‘π‘–+1βˆ’π‘‘π‘–ξ€ΈΞ”π‘–ξ€·π‘’π‘–+πœˆπ‘–ξ€Έξƒ°,𝑒𝑖,πœˆπ‘–>0.(4.1) The above constraints are rearranged as 𝑀𝑖=𝑙𝑖𝑑+max0,𝑖+1πœˆπ‘–ξ€·π‘‘π‘–+1βˆ’Ξ”π‘–ξ€Έ,𝑑𝑖+1πœˆπ‘–ξ€·Ξ”π‘–βˆ’π‘‘π‘–ξ€Έ,2π‘’π‘–πœˆπ‘–ξ€·π‘‘π‘–+1βˆ’Ξ”π‘–ξ€Έξ€·π‘‘π‘–+1πœˆπ‘–βˆ’Ξ”π‘–π‘’π‘–ξ€Έ,2π‘’π‘–πœˆπ‘–ξ€·Ξ”π‘–βˆ’π‘‘π‘–ξ€Έξ€·Ξ”π‘–πœˆπ‘–βˆ’π‘‘π‘–π‘’π‘–ξ€Έ,π‘’π‘–πœˆπ‘–ξ€·π‘‘π‘–+1βˆ’π‘‘π‘–ξ€ΈΞ”π‘–ξ€·π‘’π‘–+πœˆπ‘–ξ€Έξƒ°,𝑙𝑖β‰₯0,𝑒𝑖,πœˆπ‘–>0.(4.2)

Proof. Let{(π‘₯𝑖,𝑓𝑖),𝑖=0,1,2,…,𝑛} be the given set of convex data. For the strictly convex set of data, so Ξ”1<Ξ”2<Ξ”3<β‹―<Ξ”π‘›βˆ’1.(4.3) In similar way for the concave set of data, we have Ξ”1>Ξ”2>Ξ”3>β‹―>Ξ”π‘›βˆ’1.(4.4) Now, for a convex interpolation𝑆𝑖(π‘₯), necessary conditions on derivatives parameters𝑑𝑖 should be in the form such that 𝑑1<Ξ”1<β‹―<Ξ”π‘–βˆ’1<𝑑𝑖<Δ𝑖<β‹―<Ξ”π‘›βˆ’1<𝑑𝑛.(4.5) Similarly, for concave interpolation, 𝑑1>Ξ”1>β‹―>Ξ”π‘–βˆ’1>𝑑𝑖>Δ𝑖>β‹―>Ξ”π‘›βˆ’1>𝑑𝑛.(4.6) The necessary conditions for the convexity of data are Ξ”π‘–βˆ’π‘‘π‘–β‰₯0,𝑑𝑖+1βˆ’Ξ”π‘–β‰₯0.(4.7) Now a piecewise rational cubic interpolation𝑆𝑖(π‘₯) is convex if and only if𝑆𝑖(2)(π‘₯)β‰₯0,βˆ€π‘₯∈[π‘₯1,π‘₯𝑛], for π‘₯∈[π‘₯𝑖,π‘₯𝑖+1] after some simplification it can be shown that; 𝑆𝑖(2)(βˆ‘π‘₯)=8π‘˜=1πœƒπ‘˜βˆ’1(1βˆ’πœƒ)8βˆ’π‘˜πΆπ‘–π‘˜β„Žπ‘–ξ€·π‘žπ‘–ξ€Έ(πœƒ)3,(4.8) where 𝐢𝑖1=2𝜈2𝑖𝑀𝑖𝑑𝑖+1βˆ’Ξ”π‘–ξ€Έ+π‘‘π‘–π‘’π‘–βˆ’π‘‘π‘–+1πœˆπ‘–ξ€Έ,𝐢𝑖2=4𝐢𝑖1+6𝜈2𝑖𝑑𝑖+1πœˆπ‘–βˆ’Ξ”π‘–π‘’π‘–ξ€Έ,𝐢𝑖3=𝐢𝑖2βˆ’πΆπ‘–1ξ€Έ+6πœˆπ‘–ξ€½π‘€π‘–ξ€·π‘‘π‘–+1πœˆπ‘–βˆ’Ξ”π‘–π‘’π‘–ξ€Έβˆ’2π‘’π‘–πœˆπ‘–ξ€·π‘‘π‘–+1βˆ’Ξ”π‘–,𝐢𝑖4=𝐢𝑖3+𝐢𝑖1βˆ’πΆπ‘–2ξ€Έ+2𝑀𝑖𝑀𝑖Δ𝑖𝑒𝑖+πœˆπ‘–ξ€Έξ€Έβˆ’π‘’π‘–πœˆπ‘–ξ€·π‘‘π‘–+1βˆ’π‘‘π‘–ξ€Έξ€Ύ+14π‘’π‘–πœˆπ‘–ξ€·π‘‘π‘–+1πœˆπ‘–βˆ’π‘‘π‘–π‘’π‘–ξ€Έ,𝐢𝑖5=𝐢𝑖6+𝐢𝑖8βˆ’πΆπ‘–7ξ€Έ+2𝑀𝑖𝑀𝑖Δ𝑖𝑒𝑖+πœˆπ‘–ξ€Έξ€Έβˆ’π‘’π‘–πœˆπ‘–ξ€·π‘‘π‘–+1βˆ’π‘‘π‘–ξ€Έξ€Ύ+14π‘’π‘–πœˆπ‘–ξ€·π‘‘π‘–+1πœˆπ‘–βˆ’π‘‘π‘–π‘’π‘–ξ€Έ,𝐢𝑖6=𝐢𝑖7βˆ’πΆπ‘–8ξ€Έ+6π‘’π‘–ξ€½π‘€π‘–ξ€·Ξ”π‘–πœˆπ‘–βˆ’π‘‘π‘–π‘’π‘–ξ€Έβˆ’2π‘’π‘–πœˆπ‘–ξ€·Ξ”π‘–βˆ’π‘‘π‘–,𝐢𝑖7=4𝐢𝑖8+6𝑒2π‘–ξ€·Ξ”π‘–πœˆπ‘–βˆ’π‘‘π‘–π‘’π‘–ξ€Έ,𝐢𝑖8=2𝑒2π‘–ξ€·π‘€π‘–ξ€·Ξ”π‘–βˆ’π‘‘π‘–ξ€Έ+π‘‘π‘–π‘’π‘–βˆ’π‘‘π‘–+1πœˆπ‘–ξ€Έ.(4.9) AllπΆπ‘–π‘˜β€™s are the expression involving the parameters𝑑′𝑖𝑠,Ξ”ξ…žπ‘–π‘ ,π‘’ξ…žπ‘–π‘ ,π‘£ξ…žπ‘–π‘ ,and𝑀′𝑖𝑠.
A 𝐢1piecewise rational cubic interpolant (2.1) preserves the convexity of data if𝑆𝑖(2)(π‘₯)β‰₯0.
𝑆𝑖(2)(π‘₯)>0 if both βˆ‘8π‘˜=1πœƒπ‘˜βˆ’1(1βˆ’πœƒ)8βˆ’π‘˜πΆπ‘–π‘˜>0andβ„Žπ‘–(π‘žπ‘–(πœƒ))3>0.
Since𝑒𝑖,πœˆπ‘–,𝑀𝑖 are positive free parameters, soβ„Žπ‘–(π‘žπ‘–(πœƒ))3>0must be positive 8ξ“π‘˜=1πœƒπ‘˜βˆ’1(1βˆ’πœƒ)8βˆ’π‘˜πΆπ‘–π‘˜>0ifπΆπ‘–π‘˜>0,π‘˜=1,2,3,4,5,6,7,8.(4.10) Hence,πΆπ‘–π‘˜>0,π‘˜=1,2,3,4,5,6,7,8if we have the following sufficient conditions on parameter𝑀𝑖: 𝑀𝑖𝑑>max0,𝑖+1πœˆπ‘–ξ€·π‘‘π‘–+1βˆ’Ξ”π‘–ξ€Έ,𝑑𝑖+1πœˆπ‘–ξ€·Ξ”π‘–βˆ’π‘‘π‘–ξ€Έ,2π‘’π‘–πœˆπ‘–ξ€·π‘‘π‘–+1βˆ’Ξ”π‘–ξ€Έξ€·π‘‘π‘–+1πœˆπ‘–βˆ’Ξ”π‘–π‘’π‘–ξ€Έ,2π‘’π‘–πœˆπ‘–ξ€·Ξ”π‘–βˆ’π‘‘π‘–ξ€Έξ€·Ξ”π‘–πœˆπ‘–βˆ’π‘‘π‘–π‘’π‘–ξ€Έ,π‘’π‘–πœˆπ‘–ξ€·π‘‘π‘–+1βˆ’π‘‘π‘–ξ€ΈΞ”π‘–ξ€·π‘’π‘–+πœˆπ‘–ξ€Έξƒ°.(4.11) The above constraints are rearranged as 𝑀𝑖=𝑙𝑖𝑑+max0,𝑖+1πœˆπ‘–ξ€·π‘‘π‘–+1βˆ’Ξ”π‘–ξ€Έ,𝑑𝑖+1πœˆπ‘–ξ€·Ξ”π‘–βˆ’π‘‘π‘–ξ€Έ,2π‘’π‘–πœˆπ‘–ξ€·π‘‘π‘–+1βˆ’Ξ”π‘–ξ€Έξ€·π‘‘π‘–+1πœˆπ‘–βˆ’Ξ”π‘–π‘’π‘–ξ€Έ,2π‘’π‘–πœˆπ‘–ξ€·Ξ”π‘–βˆ’π‘‘π‘–ξ€Έξ€·Ξ”π‘–πœˆπ‘–βˆ’π‘‘π‘–π‘’π‘–ξ€Έ,π‘’π‘–πœˆπ‘–ξ€·π‘‘π‘–+1βˆ’π‘‘π‘–ξ€ΈΞ”π‘–ξ€·π‘’π‘–+πœˆπ‘–ξ€Έξƒ°,𝑙𝑖β‰₯0,(4.12) whereΔ𝑖=(𝑓𝑖+1βˆ’π‘“π‘–)/β„Žπ‘–.

5. Determination of Derivatives

Usually, the derivative values at the knots are not given. These values are derived either at the given data set{(π‘₯𝑖,𝑓𝑖),𝑖=0,1,2,…,𝑛} or by some other means. In this paper, these values are determined by following arithmetic mean method for data in such a way that the smoothness of the interpolant (2.1) is maintained.

5.1. Arithmetic Mean Method

This method is the three point difference approximation with 𝑑𝑖=⎧βŽͺ⎨βŽͺ⎩0ifΞ”π‘–βˆ’1=0orΞ”π‘–β„Ž=0,π‘–Ξ”π‘–βˆ’1+β„Žπ‘–βˆ’1Ξ”π‘–β„Žπ‘–+β„Žπ‘–βˆ’1otherwise,𝑖=2,3,β€¦π‘›βˆ’1,(5.1) and the end conditions are given as 𝑑1=⎧βŽͺ⎨βŽͺ⎩0ifΞ”1𝑑=0orsgn1ξ€Έξ€·Ξ”β‰ sgn1ξ€Έ,Ξ”1+ξ€·Ξ”1βˆ’Ξ”2ξ€Έβ„Ž1β„Ž1+β„Ž2𝑑otherwise,𝑛=⎧βŽͺ⎨βŽͺ⎩0ifΞ”π‘›βˆ’1𝑑=0orsgn𝑛Δ≠sgnπ‘›βˆ’1ξ€Έ,Ξ”π‘›βˆ’1+ξ€·Ξ”π‘›βˆ’1βˆ’Ξ”π‘›βˆ’2ξ€Έβ„Žπ‘›βˆ’1β„Žπ‘›βˆ’1+β„Žπ‘›βˆ’2otherwise.(5.2)

6. Numerical Examples

In this section, a numerical demonstration of convexity-preserving scheme given in Section 4 is presented.

Example 6.1. Consider convex data set taken in Table 1. Figure 1 is produced by cubic Hermite spline. We remark that Figure 1 does not preserve the shape of convex data. To overcome this flaw, Figure 2 is produced by the convexity-preserving rational cubic spline interpolation developed in Section 4 with the values of free parameters𝑒𝑖=0.02,πœˆπ‘–=0.02 to preserve the shape of convex data. Numerical results of Figure 2 are determined by developed convexity preserving rational cubic spline interpolation shown in Table 2.

tab1
Table 1: Convex data set.
tab2
Table 2: Numerical results of Figure 2.
174048.fig.001
Figure 1: Cubic Hermite spline scheme.
174048.fig.002
Figure 2: Convexity shape-preserving rational cubic interpolation.

Example 6.2. Consider convex data set taken in Table 3. Figure 3 is produced by cubic Hermite spline, and it is easy to see that Figure 3 does not preserve the shape of convex data. Figure 4 is produced by the convexity-preserving rational cubic spline interpolation developed in Section 4 with the values of free parameters𝑒𝑖=0.02,πœˆπ‘–=0.02 to preserve the shape of convex data. Numerical results of Figure 4 are determined by developed convexity preserving rational cubic spline interpolation shown in Table 4.

tab3
Table 3: Convex data set [11].
tab4
Table 4: Numerical results of Figure 4.
174048.fig.003
Figure 3: Cubic hermite spline scheme.
174048.fig.004
Figure 4: Convexity shape-preserving rational cubic Interpolation.

7. Conclusion

In this paper, we have constructed a𝐢1piecewise rational cubic function with three free parameters. Data-dependent constraints are derived to preserve the shape of convex data. Remaining two free parameters are left free for user’s choice to refine the convexity-preserving shape of the convex data as desired. No extra knots are inserted in the interval when the curve loses the convexity. The developed curve scheme has been tested through different numerical examples, and it is shown that the scheme is not only local and computationally economical but also visually pleasant.

Acknowledgments

The authors are highly obliged to the anonymous referees for the inspiring comments and the precious suggestions which improved our manuscript significantly. This work was fully supported by USM-RU-PRGS (1001/PMATHS/844031) from the Universiti Sains Malaysia and Malaysian Government is gratefully acknowledged. The first author does acknowledge University of Sargodha, Sargodha-Pakistan for the financial support.

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