# Shape Preserving Piecewise KNR Fractional Order BiquadraticC2 Spline

Authors: Syed Khawar Nadeem Kirmani; Muhammad Bilal Riaz; Fahd Jarad; Hayder Natiq Jasim; Aytekin Enver
Journal: Journal of Mathematics
 (2021)

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## Abstract

In a recent article, a piecewise cubic fractional spline function is developed which producesC1 continuity to given data points. In the present paper, an interpolant continuity class C2 is preserved which gives visually pleasing piecewise curves. The behavior of the resulting representations is analyzed intrinsically with respect to variation of the shape control parameters t and s. The data points are restricted to be strictly monotonic along real line.


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## Body

## 1. Introduction

Among the various methods in computer aided geometric designing, piecewise spline-based techniques are the conventional methods. In many applications, one inclines interpolate or approximate univariate data by spline functions possessing certain geometric properties or shapes such as monotonicity, convexity, or nonnegativity. Due to the verity of spline algorithm, designers do not find any strain to adopt these techniques. Ample work has been done in this regard and researchers are still working on varied techniques by refining them to make it more and more diverse. The aim of spline interpolation is to get an interpolation formula that is continuous and smooth in both within the intervals and at the interpolating points. In recent past, a hatful of work have been done in the field of piecewise polynomial spline curve [1–4], rational spline [5], trigonometric spline [6], exponential spline [7], and spline-based surfaces which are used to preserve the C2 continuity. This paper is a continuation of a previous paper [8] in which piecewise C1 continuity is preserved. The fractional biquadratic spline is represented in terms of first and second order derivative values at the knots and provides an alternative to the ordinary spline. This paper is an attempt to embrace a novel technique on piecewise biquadratic polynomial.Fractional calculus has been an Annex of ordinary calculus that encapsulated integrals and derivatives that are defined for arbitrary real orders. The journey of fractional calculus commenced in seventeenth century and underscored different derivatives [1] with significant pros and cons ranging from Riemann–Liouville, Hadamard, and Grünwald–Letnikov to Caputo, and so forth. Selecting apt fractional derivatives is pertinent to its considered systems; therefore, fractional operators were also a prevalent focus of various research works. Concurrently, studying generalized fractional operators is also indispensable in the field of computer graphics [9–11].Fractional order derivatives are rapid emerging concept in different fields of mathematics, physics, and engineering in recent years [12–15]. Due to application of new approach of fractional order derivative, the computational cost is reduced. In this paper, an efficient and intuitive technique which is able to produce piecewise smooth curves in each given subinterval, [xi,xi+1], i=0,1,2,3,…n,∀xi∈ℝ, is adopted by combining both concepts of spline and Caputo–Fabrizio fractional order derivatives. With biquadratic piecewise polynomial assistance, higher accuracy is ensured.The paper is organized in the following way. In Section 2, the formula using continuity condition is established. In Section 3, all the results are included, and in Section 4, discussion related to the novel technique is highlighted.


## 2. Preliminaries

There are heaps of definitions of fractional integral and derivatives; among them, few are Riemann–Liouville, Riesz, Caputo [8], Riesz–Caputo, Hadamard, Weyl, Grünwald–Letnikov, Chen, etc. Here, we are discussing Riemann–Liouville and Caputo. The proofs of results may be found in [16, 17].Letg:a,b⟶ℛ be a function, α a positive real number, n the integer satisfying n−1≤α<n, and Γ the Euler gamma function [11]. Then, the left and right Riemann–Liouville fractional integrals of order α are deﬁned,(1)Iayαgy=1Γα∫ayy−τα−1gτdτ,Iybαgy=1Γα∫ybτ−yα−1gτdτ,respectively.The left and right Riemann–Liouville fractional derivatives of orderα are defined by(2)Dayαgy=dmdymIaym−αgy=1Γm−αdmdym∫ayy−τm−α−1gτdτ,Dybαgy=dmdymIybm−αgy=−1mΓm−αdmdym∫ayy−τm−α−1gτdτ.Therefore, the right and left Caputo fractional derivatives of orderα are defined by(3)DaCyαgy=Iaym−αdmdymgy=1Γm−α∫ayy−τm−α−1gmτdτ,DyCbαgy=−1mIybm−αdmdymgy=1Γm−α∫yb−1mτ−ym−α−1gmτdτ.Intrinsically, there exists a relation between Caputo fractional and Riemann–Liouville derivatives, and as a consequence, we have the following relations:Ifga=g′a=…=gm−1a=0, then DaCyαgy=Iayαgy;Ifgb=g′b=…=gm−1b=0, then DyCbαgy=Dybαgy.Ifg∈Cma,b, then the right and left Caputo derivatives are continuous on a,b. There are some properties which are valid for integer integration and integer differentiation which are also reflected in fractional integration and differentiation [18].


## 3. Piecewise KNR Fractional Order BiquadraticC2 Spline

LetPix, i=1,2,3,…,n, be a piecewise polynomial in a subinterval [xi,xi+1] for x∈ [xi,xi+1]:(4)Pix=aix−xi4+bix−xi3+cix−xi2+dix−xi+ei,i=0,1,2,3,…,n,x∈xi,xi+1,where ai,bi,ci,di, and ei are unknown constants which need to be calculated by means of the given continuity and differentiability conditions:(5)Pixi+1=Pi+1xi+1,Pi′xi+1=Pi+1′xi+1,Pi″xi+1=Pi+1″xi+1,Piαxi+1=−Pi+1αxi+1,1<α<2.The parameterα that appears in the above conditions is known as fractional order derivative. It is quite evident from the given conditions that the resulting piecewise curves will be smooth in each segment and will possess C2 continuity. The fractional order derivative of a function fx∈ACna,b such that f is absolutely continuous of order α with n−1<α≤n, where n denotes the order of derivative, which is(6)DCaαfξ=1Γn−α∫axfnξx−ξα−n+1dξ,x>a,where(7)Γα≔∫0∞e−uuα−1du.LetPix and Pi+1x be two piecewise spline polynomials with common point at x=xi+1. The application of the above continuity and differentiability conditions will result in ten unknown constants which need to be evaluated for practical applications. Since the spline curve passes through the given data points, it will result in ei=yi and ei+1=yi+1. The remaining eight unknowns can be calculated by applying Caputo fractional and derivative conditions.(8)1Γ2−α∫txi+1pi″τxi+1−τα−1dτ=−1Γ2−α∫xi+1spi+1″ττ−xi+1α−1dτ,1<α≤2.The given system of linear equations is of the form(9)aiAαj+biBαj+ciCαj=−ai+1Eαj+bi+1Fαj+ci+1Gαj,Aαj=−12−t+xi+12−αjK+L−M+N−4+αj−3+αj−2+αj,where(10)K=2xi+12+2xixi+1−4+αj,L=xi2−4+αj−3+αj,M=2txi+1+xi−4+αj−2+αj,N=t2−3+αj−2+αj,Bαj=−6−t+xi+12−αj−xi+1−xi−3+αj+t−2+αj−3+αj−2+αj,Cαj=−2−t+xi+12−αj−2+αj,Eαj=12s−xi+14−αj4−αj,Fαj=6s−xi+13−αj3−αj,Gαj=2s−xi+12−αj2−αj,j=1,2,3,and4.We will have four linear equations.The other four linear equations can be derived from continuity and differentiability conditions as follows:(11)aihi4+bihi3+cihi2+dihi=yi+1−yi,ai+1hi+14+bi+1hi+13+ci+1hi+12+di+1hi+1=yi+2−yi+1,4aihi3+3bihi2+2cihi+di=di+1,12aihi2+6bihi+2ci=2ci+1,where hi=xi+1−xi and hi+1=xi+2−xi+1.The above system of linear equations will give rise to a unique solution of unknownsai,bi,ci,di,ai+1,bi+1,ci+1,anddi+1.As an example, for a given set of data points, we have a piecewise biquadratic fractional spline curve. In Figures1 and 2, we have two kinds of curves: one is concave while the other one is convex. The fractional order derivatives used in both curves are given by Table 1. These figures also indicate the potency of the technique at the bending points. We also have a liberty to control the bending due to the introduction of two parameters denoted by t and s.(12)t∈xi,xi+1,s∈xi+1,xi+2.Figure 1
Convex function.Figure 2
Concave function.Table 1
Order of fractional derivatives used for both curves.
α1α2α3α41.871.91.921.95They both will serve as shape control parameters. Different choices of these parameters will cause changes in the final shapes. The piecewise curve (Figure3) shows a C2 KNR biquadratic fractional spline curve, whereas Figure 4 indicates the exact location of the points and Figure 5 indicates the concentration of the points.Figure 3
C2 KNR biquadratic fractional spline curve.Figure 4
Location of the points.Figure 5
Concentration of the points.In this method, we have the liberty to modify the path of the curve. Figures6–9 are good examples of different values of shape parameters t and s. As these parameters move away from the connecting point xi+1, the curve starts to flatten at the point and will have effect on the final shape of the curve.Figure 6
An example of different values of shape parameters t and s.Figure 7
Impact of shape parameterst and as it moves away from connecting point.Figure 8
Impact of shape parameterst and as it moves away from connecting point.Figure 9
Impact of shape parameterst and as it moves away from connecting point.Figures10–12 indicate the evidence for the effectiveness of the novel technique. The data equally reflect back after application of the newly adopted technique. The straight lines can also be graphed accordingly. Constant function (in y-values) as shown in Figure 11 and monotone increasing data as shown in Figure 12 can also be preserved, which indicates the accuracy of the technique. In all these shapes, Table 1 is used. Effect on final shape can also be observed if the fractional order derivatives are changed.Figure 10
After application of the newly adopted technique.Figure 11
Constant functions are preserved.Figure 12
Monotone increasing data are preserved.


## 4. Comparison of KNR Biquadratic Fractional Spline with Ordinary Cubic Spline

Since ordinary cubic spline is a conventional tool for curve generation, the given comparison indicates that the newly adopted technique coincides with the ordinary one.For different choices of shape parameterst and s, Figures 13––15 show that the given piecewise curves can be manipulated by the choice of shape parameters. The slight adjustment of the shape parameters can give rise to different shapes. It also indicates that a small change can be made in final shape by altering these parameters.Figure 13
Piecewise curves can be manipulated by the choice of shape parameters 1.Figure 14
Piecewise curves can be manipulated by the choice of shape parameters 2.Figure 15
Piecewise curves can be manipulated by the choice of shape parameters 3.Geometrically, we havet∈xi,xi+1 and s∈xi+1,xi+2, which gives us better control on curve’s path. Different values of these parameters can change the whole geometry/pattern of the curves. Although the given fractional spline curve will pass through the given data points, but still we can have improved control on the curve.


## 5. Application of Fractional Spline ton Data Points

Letxi,yi, i=0,1,2,…,n, be a set of n data points. Using first three data points, we can find two patches of curves as defined in this paper above. Since all the unknown constants of these two patches are already known, they can be used to find three or more patches of the curves.By applying continuity and differentiability conditions, we have the following system of linear equations in three unknowns, namely,ai+1,bi+1, and ci+1.(13)ai+1hi+14+bi+1hi+13+ci+1hi+12=yi+2−yi+1−di+1hi+1,ai+1Eαj+bi+1Fαj+ci+1Gαj=−aiAαj+biBαj+ciCαj,j=1,2,where hi+1=xi+2−xi+1, di+1=4aihi3+3bihi2+2cihi−di, Aαj,Bαj, Cαj,Eαj,Fαj, and Gαj are already calculated in the previous section.The above system involves three linear equations for two values ofj. In each subsequent segment of curves, we will repeatedly solve the above system for n−1 segments of curve. Hence, the above system is true for i=1,2,…,n−1.In Figure16, curve segments in [x0,x1] and x1,x2 intervals can easily be calculated by the algorithm as defined prior, whereas the curve segment in interval [x2,x3], in which x2 is the connecting point, can be evaluated by the following way:(14)P2x2=P3x2,P2′x2=P3′x2,P2″x2=P3″x2,P3x3=y3,P2αx2=−P3αx2.Figure 16
Curve segments.Here, in polynomialP3x, we have five unknowns which can easily be calculated by the abovementioned conditions. Similarly, in Figure 17, one more curve segment is included by aforesaid way.Figure 17
Another curve segment is included.



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