Local Convexity Shape-Preserving Data Visualization by Spline Function

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, 𝐶¹ flexible, simple, local, and economical as compared to existing schemes. The error bound for the rational cubic function is 𝑂(ℎ³).

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 Bézier 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 𝐶² 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-Bézier 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 𝐶⁰while in this work the degree of smoothness is𝐶¹.(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(ℎ³)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{(𝑥_𝑖,𝑓_𝑖),𝑖=0,1,2,…,𝑛}be the given set of data points such as𝑥₀<𝑥₁<𝑥₂<⋯<𝑥_𝑛. The rational cubic function with three free parameters introduced by Abbas et al. [2], in each subinterval𝐼_𝑖=[𝑥_𝑖,𝑥_𝑖+1],𝑖=0,1,2,…,𝑛−1,is defined as 𝑆_𝑖(𝑥)=𝑝^𝑖(𝜃)𝑞_𝑖(𝜃),(2.1) with 𝑝_𝑖(𝜃)=𝑢_𝑖𝑓_𝑖(1−𝜃)³+𝑤_𝑖𝑓𝑞_𝑖𝑖+𝑢_𝑖ℎ_𝑖𝑑_𝑖𝜃(1−𝜃)²+𝑤_𝑖𝑓_𝑖+1−𝑣_𝑖ℎ_𝑖𝑑_𝑖+1𝜃²(1−𝜃)+𝑣_𝑖𝑓_𝑖+1𝜃³,(𝜃)=𝑢_𝑖(1−𝜃)³+𝑤_𝑖𝜃(1−𝜃)+𝑣_𝑖𝜃³,(2.2) where𝜃=𝑥−𝑥_𝑖/ℎ_𝑖,ℎ_𝑖=𝑥_𝑖+1−𝑥_𝑖, and𝑢_𝑖,𝑣_𝑖,𝑤_𝑖are the positive free parameters. It is worth noting that when we use the values of these free parameters as𝑢_𝑖=1,𝑣_𝑖=1and𝑤_𝑖=3,then the𝐶¹ 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: 𝑆_𝑖𝑥_𝑖=𝑓_𝑖,𝑆_𝑖𝑥_𝑖+1=𝑓_𝑖+1,𝑆^′_𝑖𝑥_𝑖=𝑑_𝑖,𝑆^𝑖𝑥_𝑖+1=𝑑_𝑖+1,(2.3) where𝑆^𝑖(𝑥)denotes the derivative with respect to “x,” and𝑑_𝑖 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 subinterval𝐼_𝑖=[𝑥_𝑖,𝑥_𝑖+1]. It is to mention that the scheme constructed in Section 2 is local. We suppose that𝑓(𝑥)∈𝐶³[𝑥₀,𝑥_𝑛], and𝑆_𝑖(𝑥)is the interpolation of function𝑓(𝑥)over arbitrary subinterval𝐼_𝑖=[𝑥_𝑖,𝑥_𝑖+1]. The Peano Kernel Theorem, Schultz [16] is used to obtain the error analysis of piecewise rational cubic interpolation in each subinterval𝐼_𝑖=[𝑥_𝑖,𝑥_𝑖+1], and it is defined as 𝑅[𝑓]=𝑓(𝑥)−𝑆_𝑖(𝑥)=12^𝑥𝑖+1_𝑥𝑖𝑓⁽³⁾(𝜏)𝑅_𝑥(𝑥−𝜏)²⁺𝑑𝜏.(3.1) In each subinterval, the absolute value of error is ||𝑓(𝑥)−𝑆_𝑖(𝑥)||≤12‖‖𝑓⁽³⁾(𝜏)‖‖^𝑥𝑖+1_𝑥𝑖||𝑅_𝑥(𝑥−𝜏)²⁺||𝑑𝜏,(3.2) where𝑅_𝑥=(⎧⎪⎪⎨⎪⎪⎩𝑥(−𝑥𝜏−)²⁺𝜏)₂−𝑤^𝑖𝑥_𝑖+1−𝜏²−2ℎ_𝑖𝑣_𝑖𝑥_𝑖+1−𝜏𝜃²(1−𝜃)+𝑣_𝑖𝑥_𝑖+1−𝜏²𝜃³𝑞_𝑖(𝜃)𝑥^𝑖𝑤<𝜏<𝑥,_𝑖𝑥_𝑖+1−𝜏²−2ℎ_𝑖𝑣_𝑖𝑥_𝑖+1−𝜏𝜃²(1−𝜃)+𝑣_𝑖𝑥_𝑖+1−𝜏²𝜃³𝑞_𝑖(𝜃)𝑥<𝜏<𝑥^𝑖+1=𝑎(𝜏,𝑥)𝑥_𝑖,𝑏(𝜏,𝑥)𝑥<<𝜏𝜏<<𝑥𝑥,_𝑖+1,(3.3) where𝑅_𝑥[(𝑥−𝜏)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 functions𝑎(𝜏,𝑥) and𝑏(𝜏,𝑥), and then to find the values of following integrals: ^𝑥𝑖+1_𝑥𝑖||𝑅_𝑥(𝑥−𝜏)²⁺||𝑑𝜏=^𝑥𝑥_𝑖||𝑎(𝜏,𝑥)||𝑑𝜏+^𝑥𝑖+1_𝑥||𝑏(𝜏,𝑥)||𝑑𝜏.(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 𝑎(𝑥,𝑥)=(𝜃²(1−𝜃)²((𝑤_𝑖−𝜈_𝑖)𝜃+(2𝜈_𝑖−𝑤_𝑖))ℎ^2𝑖)/𝑞_𝑖(𝜃)in[0,1] are 𝜃=0,𝜃=1and𝜃^∗=1−𝜈_𝑖/(𝑤_𝑖−𝜈_𝑖). It is easy to show that when𝜃≤𝜃^∗, 𝑎(𝑥,𝑥)≤0 and𝜃≥𝜃^∗,𝑎(𝑥,𝑥)≥0. The roots of quadratic function𝑎(𝜏,𝑥)=0 are 𝜏^∗1=𝑥−𝜃ℎ^𝑖𝜃𝑤_𝑖−𝜈_𝑖+𝐴(1−𝜃)𝑢_𝑖+𝜃𝑤_𝑖,𝜏^1∗∗=𝑥−𝜃ℎ^𝑖𝜃𝑤_𝑖−𝜈_𝑖−𝐴(1−𝜃)𝑢_𝑖+𝜃𝑤_𝑖,(3.5) where 𝐴=√𝜈_𝑖((𝑤_𝑖−2𝜈_𝑖)+3𝜃)+𝑤_𝑖(𝑤_𝑖−4𝜈_𝑖)𝜃.

So, when 𝜃>𝜃^∗,𝑥_𝑖<𝜏^1∗∗<𝑥 and when𝜃<𝜃^∗,𝜏^1∗∗>𝑥. Thus,𝜃<𝜃^∗,𝑎(𝜏,𝑥)<0forall𝜏∈[𝑥_𝑖,𝑥], ^𝑥𝑥_𝑖||𝑎(𝜏,𝑥)||𝑑𝜏==𝜈^𝑥𝑥_𝑖(−𝑎(𝜏,𝑥))𝑑𝜏_𝑖(3−𝜃)−𝑤_𝑖(1−𝜃)(1−𝜃)³𝜃²ℎ^3𝑖3𝑞_𝑖(𝜃)+𝑤^𝑖−3𝑣_𝑖(1−𝜃)𝜃²ℎ^3𝑖3𝑞_𝑖(𝜃)+𝑣^𝑖𝜃³ℎ^3𝑖3𝑞_𝑖(𝜃)−𝜃³ℎ^3𝑖3.(3.6) The value of𝑎(𝜏,𝑥) varies from negative to positive on the root𝜏^1∗∗when𝜃>𝜃^∗, ^𝑥𝑥_𝑖||𝑎(𝜏,𝑥)||𝑑𝜏==2^𝜏1∗∗_𝑥𝑖𝑤(−_𝑖𝑎(𝜏,𝑥))𝑑𝜏+^𝑥𝜏_1∗∗𝑎(𝜏,𝑥)𝑑𝜏−𝜈_𝑖𝜃−𝐴³𝜃³ℎ^3𝑖3(1−𝜃)𝑢_𝑖+𝜃𝑤_𝑖³−𝜃³ℎ^3𝑖3−3𝑞2_𝑖ℎ(^3𝑖𝜃)(1−𝜃)+𝜃𝑤^𝑖−𝜈_𝑖𝜃−𝐴(1−𝜃)𝑢_𝑖+𝜃𝑤_𝑖³×(1−𝜃)𝑤_𝑖+𝜃𝜈_𝑖+2ℎ^3𝑖𝜈_𝑖𝜃²(1−𝜃)𝑞_𝑖(𝜃)(1−𝜃)+𝜃𝑤^𝑖−𝜈_𝑖𝜃−𝐴(1−𝜃)𝑢_𝑖+𝜃𝑤_𝑖².(3.7)

3.2. Part 2

In this part, we discuss the properties of function𝑏(𝜏,𝑥). Consider𝑏(𝜏,𝑥),𝜏∈[𝑥,𝑥_𝑖+1] as function of𝜏. The roots of function𝑏(𝜏,𝑥) are similar as𝑎(𝜏,𝑥)in Section 3.1 at𝜏=𝑥. It is easy to show that when𝜃≤𝜃^∗,𝑏(𝑥,𝑥)≤0and𝜃≥𝜃^∗, 𝑏(𝑥,𝑥)≥0. The roots of quadratic function𝑏(𝜏,𝑥)=0are 𝜏^∗2=𝑥_𝑖+1,𝜏^2∗∗=𝑥_𝑖+1−2(1−𝜃)𝜈^𝑖ℎ_𝑖(1−𝜃)𝑤_𝑖+𝜃𝜈_𝑖.(3.8) The function𝑏(𝜏,𝑥)varies from negative to positive on the root𝜏^2∗∗when 𝜃≤𝜃^∗. Thus, ^𝑥𝑖+1_𝑥||𝑏(𝜏,𝑥)||𝑑𝜏==^𝜏2∗∗_𝑥(8−𝜃𝑏₂(𝜏,𝑥))𝑑𝜏+^𝑥𝑖+1_𝜏2∗∗𝑏(𝜏,𝑥)𝑑𝜏(1−𝜃)³𝜈_𝑖3ℎ^3𝑖3𝑞_𝑖(𝜃)(1−𝜃)𝑢_𝑖+𝜃𝑤_𝑖²+ℎ^3𝑖𝜃²(1−𝜃)³3𝑞_𝑖(𝜃)𝑤^𝑖(1−𝜃)−𝜈_𝑖(3−𝜃),(3.9) when𝜃≥𝜃^∗,^𝑥𝑖+1_𝑥||𝑏(𝜏,𝑥)||𝑑𝜏==ℎ^𝑥𝑖+1_𝑥𝑏(𝜏,𝑥)𝑑𝜏_3𝑖𝜃²(1−𝜃)³3𝑞_𝑖(𝜃)𝜈^𝑖(3−𝜃)−𝑤_𝑖(1−𝜃).(3.10) Thus, from (3.6) and (3.9), it can be shown that when 0≤𝜃≤𝜃^∗,||𝑓(𝑥)−𝑆_𝑖(𝑥)||≤12‖‖𝑓⁽³⁾(𝜏)‖‖^𝑥𝑖+1_𝑥𝑖||𝑅_𝑥(𝑥−𝜏(𝑥−𝜏)²⁺||𝑑𝜏=‖‖𝑓⁽³⁾(𝜏)‖‖ℎ^3𝑖𝑝₁𝑢_𝑖,𝑣_𝑖,𝑤_𝑖,𝜃,(3.11) where𝑝₁𝑢_𝑖,𝜈_𝑖,𝑤_𝑖,𝜃=𝜈^𝑖(3−𝜃)−𝑤_𝑖(1−𝜃)(1−𝜃)³𝜃²6𝑞_𝑖(𝜃)+𝑤^𝑖−3𝜈_𝑖(1−𝜃)𝜃²6𝑞_𝑖(𝜃)+𝜈^𝑖𝜃³+8𝜃₂6𝑞_𝑖(𝜃)−𝜃6³(1−𝜃)³𝜈_𝑖36𝑞_𝑖(𝜃)(1−𝜃)𝑢_𝑖+𝜃𝑤_𝑖²+𝜃²(1−𝜃)³6𝑞_𝑖(𝜃)𝑤^𝑖(1−𝜃)−𝜈_𝑖(3−𝜃),(3.12) and, from (3.7) and (3.10), it can be shown that when 𝜃^∗≤𝜃≤1,||𝑓(𝑥)−𝑆_𝑖(𝑥)||≤12‖‖𝑓⁽³⁾(𝜏)‖‖^𝑥𝑖+1_𝑥𝑖||𝑅_𝑥(𝑥−𝜏)²⁺||𝑑𝜏=‖‖𝑓⁽³⁾(𝜏)‖‖ℎ^3𝑖𝑝₂𝑢_𝑖,𝜈_𝑖,𝑤_𝑖,𝜃,(3.13) where𝑝₂𝑢_𝑖,𝜈_𝑖,𝑤_𝑖,𝜃=2𝑤^𝑖−𝜈_𝑖𝜃−𝐴³𝜃³6(1−𝜃)𝑢_𝑖+𝜃𝑤_𝑖³−𝜃³6−6𝑞_𝑖2(𝜃)(1−𝜃)+𝜃𝑤^𝑖−𝜈_𝑖𝜃−𝐴(1−𝜃)𝑢_𝑖+𝜃𝑤_𝑖³×(1−𝜃)𝑤_𝑖+𝜃𝜈_𝑖+𝜈^𝑖𝜃²(1−𝜃)𝑞_𝑖(𝜃)(1−𝜃)+𝜃𝑤^𝑖−𝜈_𝑖𝜃−𝐴(1−𝜃)𝑢_𝑖+𝜃𝑤_𝑖²+𝜃²(1−𝜃)³6𝑞_𝑖(𝜃)𝜈^𝑖(3−𝜃)−𝑤_𝑖(1−𝜃).(3.14)

Theorem 3.1. For the positive free parameters𝑢_𝑖,𝜈_𝑖,and𝑤_𝑖, the error of interpolating rational cubic function𝑆_𝑖(𝑥), for 𝑓(𝑥)∈𝐶³[𝑥₀,𝑥_𝑛], in each subinterval 𝐼_𝑖=[𝑥_𝑖,𝑥_𝑖+1] is ||𝑓(𝑥)−𝑆_𝑖(𝑥)||≤12‖‖𝑓⁽³⁾(𝜏)‖‖^𝑥𝑖+1_𝑥𝑖||𝑅_𝑥(𝑥−𝜏)²⁺||𝑑𝜏=‖‖𝑓⁽³⁾(𝜏)‖‖ℎ^3𝑖𝑐_𝑖,𝑐_𝑖=max_0≤𝜃≤1𝑝𝑢_𝑖,𝜈_𝑖,𝑤_𝑖,𝜃,(3.15) where 𝑝𝑢_𝑖,𝜈_𝑖,𝑤_𝑖,𝜃=max𝑝¹𝑢_𝑖,𝜈_𝑖,𝑤_𝑖,𝜃,0≤𝜃≤𝜃^∗max𝑝₂𝑢_𝑖,𝜈_𝑖,𝑤_𝑖,𝜃,𝜃^∗≤𝜃≤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 as𝑢_𝑖=1,𝜈_𝑖=1and𝑤_𝑖=3. In this special case, the functions𝑝₁(𝑢_𝑖,𝜈_𝑖,𝑤_𝑖,𝜃)and𝑝₂(𝑢_𝑖,𝜈_𝑖,𝑤_𝑖,𝜃) are 𝑝₁𝑢_𝑖,𝑣_𝑖,𝑤_𝑖,𝜃=4𝜃²(1−𝜃)³3(3−2𝜃)²,0≤𝜃≤1𝑝2,(3.17)₂𝑢_𝑖,𝜈_𝑖,𝑤_𝑖,𝜃=4𝜃³(1−𝜃)²3(1+2𝜃)²,12≤𝜃≤0,(3.18) respectively. Since𝑐_𝑖=max{max_{0≤𝜃≤0.5}𝑝₁(𝑢_𝑖,𝜈_𝑖,𝑤_𝑖,𝜃),max_{0.5≤𝜃≤0}𝑝₂(𝑢_𝑖,𝜈_𝑖,𝑤_𝑖,𝜃)}=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. The𝐶¹piecewise rational cubic function (2.1) preserves the convexity of convex data if in each subinterval𝐼_𝑖=[𝑥_𝑖,𝑥_𝑖+1],𝑖=0,1,2,…,𝑛, the free parameters satisfy the following sufficient conditions: 𝑤_𝑖>max0,𝑑^𝑖+1𝜈_𝑖𝑑_𝑖+1−Δ_𝑖,𝑑^𝑖+1𝜈_𝑖Δ_𝑖−𝑑_𝑖,2𝑢^𝑖𝜈_𝑖𝑑_𝑖+1−Δ_𝑖𝑑_𝑖+1𝜈_𝑖−Δ_𝑖𝑢_𝑖,2𝑢^𝑖𝜈_𝑖Δ_𝑖−𝑑_𝑖Δ_𝑖𝜈_𝑖−𝑑_𝑖𝑢_𝑖,𝑢^𝑖𝜈_𝑖𝑑_𝑖+1−𝑑_𝑖𝑢_𝑖Δ_𝑖𝑢_𝑖+𝜈_𝑖,,𝜈_𝑖>0.(4.1) The above constraints are rearranged as 𝑤_𝑖=𝑙_𝑖+max0,𝑑^𝑖+1𝜈_𝑖𝑑_𝑖+1−Δ_𝑖,𝑑^𝑖+1𝜈_𝑖Δ_𝑖−𝑑_𝑖,2𝑢^𝑖𝜈_𝑖𝑑_𝑖+1−Δ_𝑖𝑑_𝑖+1𝜈_𝑖−Δ_𝑖𝑢_𝑖,2𝑢^𝑖𝜈_𝑖Δ_𝑖−𝑑_𝑖Δ_𝑖𝜈_𝑖−𝑑_𝑖𝑢_𝑖,𝑢^𝑖𝜈_𝑖𝑑_𝑖+1−𝑑_𝑖𝑙_𝑖Δ_𝑖𝑢_𝑖+𝜈_𝑖,≥0,𝑢_𝑖,𝜈_𝑖>0.(4.2)

Proof. Let{(𝑥_𝑖,𝑓_𝑖),𝑖=0,1,2,…,𝑛} be the given set of convex data. For the strictly convex set of data, so Δ₁<Δ₂<Δ₃<⋯<Δ_𝑛−1.(4.3) In similar way for the concave set of data, we have Δ₁>Δ₂>Δ₃>⋯>Δ_𝑛−1.(4.4) Now, for a convex interpolation𝑆_𝑖(𝑥), necessary conditions on derivatives parameters𝑑_𝑖 should be in the form such that 𝑑₁<Δ₁<⋯<Δ_𝑖−1<𝑑_𝑖<Δ_𝑖<⋯<Δ_𝑛−1<𝑑_𝑛.(4.5) Similarly, for concave interpolation, 𝑑₁>Δ₁>⋯>Δ_𝑖−1>𝑑_𝑖>Δ_𝑖>⋯>Δ_𝑛−1>𝑑_𝑛.(4.6) The necessary conditions for the convexity of data are Δ_𝑖−𝑑_𝑖≥0,𝑑_𝑖+1−Δ_𝑖≥0.(4.7) Now a piecewise rational cubic interpolation𝑆_𝑖(𝑥) is convex if and only if𝑆^𝑖(2)(𝑥)≥0,∀𝑥∈[𝑥₁,𝑥_𝑛], for 𝑥∈[𝑥_𝑖,𝑥_𝑖+1] after some simplification it can be shown that; 𝑆^𝑖(2)(𝑥)=∑^8𝑘=1𝜃^𝑘−1(1−𝜃)^8−𝑘𝐶_𝑖𝑘ℎ_𝑖𝑞_𝑖(𝜃)³,(4.8) where 𝐶_𝑖1=2𝐶𝜈^2𝑖_𝑖3𝑤_𝑖𝑑_𝑖+1−Δ_𝑖+𝑑_𝑖𝑢_𝑖−𝑑_𝑖+1𝜈_𝑖,𝐶_𝑖2=4𝐶_𝑖1+6𝜈^2𝑖𝑑_𝑖+1𝜈_𝑖−Δ_𝑖𝑢_𝑖,𝐶=𝐶_𝑖2−𝐶_𝑖1+6𝜈_𝑖𝑤_𝑖𝑑_𝑖+1𝜈_𝑖−Δ_𝑖𝑢_𝑖−2𝑢_𝑖𝜈_𝑖𝑑_𝑖+1−Δ_𝑖_𝑖4,𝐶_𝑖5=𝐶_𝑖3+𝐶_𝑖1−𝐶_𝑖2+2𝑤_𝑖𝑤_𝑖Δ_𝑖𝑢_𝑖+𝜈_𝑖−𝑢_𝑖𝜈_𝑖𝑑_𝑖+1−𝑑_𝑖+14𝑢_𝑖𝜈_𝑖𝑑_𝑖+1𝜈_𝑖−𝑑_𝑖𝑢_𝑖,=𝐶_𝑖6+𝐶_𝑖8𝐶−_𝑖6𝐶_𝑖7+2𝑤_𝑖𝑤_𝑖Δ_𝑖𝑢_𝑖+𝜈_𝑖−𝑢_𝑖𝜈_𝑖𝑑_𝑖+1−𝑑_𝑖+14𝑢_𝑖𝜈_𝑖𝑑_𝑖+1𝜈_𝑖−𝑑_𝑖𝑢_𝑖,𝐶=𝐶_𝑖7−𝐶_𝑖8+6𝑢_𝑖𝑤_𝑖Δ_𝑖𝜈_𝑖−𝑑_𝑖𝑢_𝑖−2𝑢_𝑖𝜈_𝑖Δ_𝑖−𝑑_𝑖_𝑖7,=4𝐶_𝑖8+6𝑢^2𝑖Δ_𝑖𝜈_𝑖−𝑑_𝑖𝑢_𝑖,𝐶_𝑖8=2𝑢^2𝑖𝑤_𝑖Δ_𝑖−𝑑_𝑖+𝑑_𝑖𝑢_𝑖−𝑑_𝑖+1𝜈_𝑖.(4.9) All𝐶_𝑖𝑘’s are the expression involving the parameters𝑑^′_𝑖𝑠,Δ^𝑖𝑠,𝑢^𝑖𝑠,𝑣^𝑖𝑠,and𝑤^′_𝑖𝑠.
A 𝐶¹piecewise rational cubic interpolant (2.1) preserves the convexity of data if𝑆^𝑖(2)(𝑥)≥0.
𝑆^𝑖(2)(𝑥)>0 if both ∑^8𝑘=1𝜃^𝑘−1(1−𝜃)^8−𝑘𝐶_𝑖𝑘>0andℎ_𝑖(𝑞_𝑖(𝜃))³>0.
Since𝑢_𝑖,𝜈_𝑖,𝑤_𝑖 are positive free parameters, soℎ_𝑖(𝑞_𝑖(𝜃))³>0must be positive ⁸_𝑘=1𝜃^𝑘−1(1−𝜃)^8−𝑘𝐶_𝑖𝑘>0if𝐶_𝑖𝑘>0,𝑘=1,2,3,4,5,6,7,8.(4.10) Hence,𝐶_𝑖𝑘>0,𝑘=1,2,3,4,5,6,7,8if we have the following sufficient conditions on parameter𝑤_𝑖: 𝑤_𝑖>max0,𝑑^𝑖+1𝜈_𝑖𝑑_𝑖+1−Δ_𝑖,𝑑^𝑖+1𝜈_𝑖Δ_𝑖−𝑑_𝑖,2𝑢^𝑖𝜈_𝑖𝑑_𝑖+1−Δ_𝑖𝑑_𝑖+1𝜈_𝑖−Δ_𝑖𝑢_𝑖,2𝑢^𝑖𝜈_𝑖Δ_𝑖−𝑑_𝑖Δ_𝑖𝜈_𝑖−𝑑_𝑖𝑢_𝑖,𝑢^𝑖𝜈_𝑖𝑑_𝑖+1−𝑑_𝑖Δ_𝑖𝑢_𝑖+𝜈_𝑖.(4.11) The above constraints are rearranged as 𝑤_𝑖=𝑙_𝑖+max0,𝑑^𝑖+1𝜈_𝑖𝑑_𝑖+1−Δ_𝑖,𝑑^𝑖+1𝜈_𝑖Δ_𝑖−𝑑_𝑖,2𝑢^𝑖𝜈_𝑖𝑑_𝑖+1−Δ_𝑖𝑑_𝑖+1𝜈_𝑖−Δ_𝑖𝑢_𝑖,2𝑢^𝑖𝜈_𝑖Δ_𝑖−𝑑_𝑖Δ_𝑖𝜈_𝑖−𝑑_𝑖𝑢_𝑖,𝑢^𝑖𝜈_𝑖𝑑_𝑖+1−𝑑_𝑖Δ_𝑖𝑢_𝑖+𝜈_𝑖,𝑙^𝑖≥(40.1,2) whereΔ_𝑖=(𝑓_𝑖+1−𝑓_𝑖)/ℎ_𝑖.

5. Determination of Derivatives

Usually, the derivative values at the knots are not given. These values are derived either at the given data set{(𝑥_𝑖,𝑓_𝑖),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 𝑑_𝑖=⎧⎪⎨⎪⎩0ifΔ^𝑖−1ℎ=0orΔ_𝑖=0,_𝑖Δ_𝑖−1+ℎ_𝑖−1Δ_𝑖ℎ_𝑖+ℎ_𝑖−1otherwise,𝑖=2,3,…𝑛−1,(5.1) and the end conditions are given as 𝑑₁=⎧⎪⎨⎪⎩0ifΔ¹Δ₁=0orsgn𝑑₁≠sgnΔ₁,+Δ₁−Δ₂ℎ₁ℎ₁+ℎ₂𝑑otherwise,_𝑛=⎧⎪⎨⎪⎩0ifΔ^𝑛−1Δ_𝑛−1=0orsgn𝑑_𝑛≠sgnΔ_𝑛−1,+Δ_𝑛−1−Δ_𝑛−2ℎ_𝑛−1ℎ_𝑛−1+ℎ_𝑛−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 parameters𝑢_𝑖=0.02,𝜈_𝑖=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.

Table 1 

Convex data set.

Table 2 

Numerical results of Figure 2.

Figure 1 

Cubic Hermite spline scheme.

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 parameters𝑢_𝑖=0.02,𝜈_𝑖=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.

Table 3 

Convex data set [11].

Table 4 

Numerical results of Figure 4.

Figure 3 

Cubic hermite spline scheme.

Figure 4 

Convexity shape-preserving rational cubic Interpolation.

7. Conclusion

In this paper, we have constructed a𝐶¹piecewise 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.