The EH Interpolation Spline and Its Approximation

Home , Cubic Hermite spline

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 745765, 6 pages http://dx.doi.org/10.1155/2014/745765

Research Article The EH Interpolation Spline and Its Approximation

Jin Xie1 and Xiaoyan Liu2

1 Department of Mathematics and Physics, Hefei University, Hefei 230601, China 2 Department of Mathematics, University of La Verne, La Verne, CA 91750, USA

Correspondence should be addressed to Jin Xie; [email protected]

Received 28 February 2014; Accepted 15 June 2014; Published 30 June 2014

Academic Editor: Grzegorz Nowak

Copyright © 2014 J. Xie and X. Liu. 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.

A new interpolation spline with two parameters, called EH interpolation spline, is presented in this paper, which is the extension of the standard cubic Hermite interpolation spline, and inherits the same properties of the standard cubic Hermite interpolation spline. Given the fixed interpolation conditions, the shape of the proposed splines can be adjusted by changing the values ofthe parameters. Also, the introduced spline could approximate to the interpolated function better than the standard cubic Hermite interpolation spline and the quartic Hermite interpolation splines with single parameter by a new algorithm.

1. Introduction computation will increase dramatically if the length of the interpolating intervals decreases. On the other hand, the Spline interpolation is a useful and powerful tool for curves approximation accuracy of these new splines may not be and surfaces modeling. Standard cubic Hermite spline is better than the standard cubic Hermite spline. one of those interpolation functions. But, for the given In [7–17], many rational form interpolation splines with interpolation condition, the cubic Hermite interpolation multiple parameters were presented. For the given interpo- splineisfixed;thatistosay,theshapeoftheinterpolation lation data, the change of the parameters causes the change curve or surface is fixed for the given interpolation data [1– of the interpolation curve. Nevertheless, the computation of 6]. Since the interpolation function is unique for the given the splines with multiple parameters is very complicated. interpolation data, to modify the shape of the interpolation Several kinds of rational splines with a single parameter were curve to approximate the given curve seems to be impossible presented in the papers [18, 19], which is simple to compute, and it is contradictory to the uniqueness of the interpolation but its approximation accuracy is not good for the given function for the given interpolation data. For the given curves and surfaces. In general, polynomial-form splines interpolation condition, how to improve the approximation aresuitabletobeusedtodesignandcompute.In[20], a accuracy of the interpolation spline is an important problem polynomial-form spline, called quartic Hermite spline with in the computer aided geometric design. In recent years, single parameter, is presented as the extension of the standard many authors have presented some new method to modify Hermite spline. The quartic spline has a simple form, and its theshapeoftheinterpolationcurvetosatisfytheindustrial approximation rate to the given curves and surfaces is not product design with several kinds of new interpolation high. splines with parameters [7–20]. In this paper, a class of new quartic splines with two These new splines all have similar properties of the parameters is developed, which is the extension of the standard cubic Hermite spline. For example, for the given standard cubic Hermite interpolation spline and inherits the interpolation data, if interpolation interval approaches zero, samepropertiesofthestandardcubicHermiteinterpolation theoretically speaking, these splines can approximate the spline. For the given interpolation conditions, the shape of given curve and surface well. However, there exists a problem, theproposedsplinescanbeadjustedbychangingthevalues in the process of the actual calculation; the amount of of the parameters. Furthermore, the introduced splines could 2 Abstract and Applied Analysis approximate to the interpolated functions better than the 3.1. The Basis Functions of the EH Interpolation Spline standard cubic Hermite interpolation splines and the quartic 𝜆 ,𝜇 0≤𝑡≤1 Hermite interpolation splines with single parameter. Definition 1. For arbitrary real number 𝑖 𝑖 and , 𝜆 ,𝜇 The remainder of the paper is organized as follows. the following four functions with parameters 𝑖 𝑖, Section 2 introducesthestandardcubicHermitesplineand 2 3 4 some of its properties. A kind of interpolation spline with two 𝑒𝛼0 (𝑡) =1+(𝜆𝑖 −3)𝑡 +2(1−𝜆𝑖)𝑡 +𝜆𝑖𝑡 , parameters is presented in Section 3. Section 4 discusses the approximation of the introduced spline curve with numerical 2 3 4 𝑒𝛼1 (𝑡) =(3−𝜆𝑖)𝑡 +2(𝜆𝑖 −1)𝑡 −𝜆𝑖𝑡 , examples. In the end, a summary and conclusions are given. 2 3 4 𝑒𝛽0 (𝑡) =𝑡+(𝜇𝑖 −2)𝑡 +(1−2𝜇𝑖)𝑡 +𝜇𝑖𝑡 , 2. The Standard Cubic Hermite Spline 2 3 4 and Its Basis Functions 𝑒𝛽1 (𝑡) =−(𝜇𝑖 +1)𝑡 +(1+2𝜇𝑖)𝑡 −𝜇𝑖𝑡 , (4) Generally, for 𝑡∈[0,1],thefollowingfourbasisfunctions,

2 3 𝛼0 (𝑡) =1−3𝑡 +2𝑡 , are calledbasis functions of the EH interpolation spline, briefly EH bases. 2 3 𝛼1 (𝑡) =3𝑡 −2𝑡 , The EH bases are the extension of the standard cubic (1) Hermitebases.When𝜆𝑖 =𝜇𝑖 =0, the bases are the standard 𝛽 𝑡 =𝑡−2𝑡2 +𝑡3, 0 ( ) cubic Hermite bases. The bases have the similar properties of 2 3 thestandardcubicHermitebases. 𝛽1 (𝑡) =−𝑡+𝑡 , By computing directly, we have 𝑒𝛼0(0) = 𝑒𝛼1(1) = 󸀠 󸀠 󸀠 1, 𝑒𝛼0(1) = 𝑒𝛼1(0) = 0, 𝑒𝛼1(0) = 𝑒𝛼1(1) = 0, 𝑒𝛼0(0) = are called the standard cubic Hermite bases. 󸀠 𝑒𝛼0(1) = 0, 𝑒𝛽0(0) = 𝑒𝛽0(1) = 0, 𝑒𝛽1(0) = 𝑒𝛽1(1) = These bases satisfy 󸀠 󸀠 󸀠 󸀠 0, 𝑒𝛽0(0) = 𝑒𝛽1(1) = 1, 𝑒𝛽1(1) = 𝑒𝛽1(0) = 0,and𝑒𝛼0(𝑡) + 𝛼0 (0) =𝛼1 (1) =1, 𝛼0 (1) =𝛼1 (0) =0, 𝑒𝛼1(𝑡) = 1,0 𝑒𝛽 (𝑡) + 1𝑒𝛽 (1 − 𝑡) =. 0 󸀠 󸀠 󸀠 󸀠 𝜆 =𝜇 𝛼1 (0) =𝛼1 (1) =0, 𝛼0 (0) =𝛼0 (1) =0, When 𝑖 𝑖,theEHbases(4)arebasisfunctionswith single parameter in [20]. 𝛽 (0) =𝛽 (1) =0, 𝛽 (0) =𝛽 (1) =0, 0 0 1 1 (2) Figure 1 shows the four EH bases, where the solid lines are 󸀠 󸀠 󸀠 󸀠 the standard Hermite bases, the parameters 𝜆𝑖 =2,𝜇𝑖 =−2 𝛽0 (0) =𝛽1 (1) =1, 𝛽0 (1) =𝛽1 (0) =0, are for the dot-dash lines, and 𝜆𝑖 =−2,𝜇𝑖 =2are for dashed 𝛼0 (𝑡) +𝛼1 (𝑡) =1, 𝛽0 (𝑡) +𝛽1 (1−𝑡) =0. line. So, we may construct the Ferguson curve with two pa- For given knots 𝑎=𝑥0 <𝑥1 < ⋅⋅⋅ < 𝑥𝑛 =𝑏and data rameters based on the EH bases as follows: {(𝑥𝑖,𝑦𝑖,𝑑𝑖),𝑖 = 0,1,...,𝑛},where𝑦𝑖 and 𝑑𝑖 are the values of the function value and the first-order derivative value of the 󸀠 󸀠 EH𝑖 (𝑡) =𝑒𝛼0 (𝑡) 𝑝𝑖 +𝑒𝛼1 (𝑡) 𝑝𝑖+1 +𝑒𝛽0𝑝𝑖 +𝑒𝛽1 (𝑡) 𝑝𝑖+1, (5) function being interpolated, let ℎ𝑖 =𝑥𝑖+1 −𝑥𝑖,𝑡=(𝑥−𝑥𝑖)/ℎ𝑖 andthenthestandardcubicHermitesplineintheinterval [𝑥 ,𝑥 ] 󸀠 󸀠 𝑖 𝑖+1 can be defined as follows: where 𝑝𝑖,𝑝𝑖+1 and 𝑝𝑖 ,𝑝𝑖+1 are two interpolation points and their tangent vectors, respectively. For the given two interpolation points and tangent 𝐻 (𝑥) =𝛼 (𝑡) 𝑦 +𝛼 (𝑡) 𝑦 +𝛽 (𝑡) ℎ 𝑑 +𝛽 (𝑡) ℎ 𝑑 , 𝑖 0 𝑖 1 𝑖+1 0 𝑖 𝑖 1 𝑖 𝑖+1 vectors, with the different parameters 𝜆𝑖,𝜇𝑖,wemayobtain 𝑖=0,1,...,𝑛−1. different shape of the Ferguson curve with two parameters accordingly. (3) Figure 2 shows the Ferguson curves with different parameters, where the solid line is the standard Ferguson spline 󸀠 Obviously, we have 𝐻𝑖(𝑥𝑖)=𝑦𝑖,𝐻𝑖(𝑥𝑖+1)=𝑦𝑖+1,𝐻𝑖 (𝑥𝑖)= curve with 𝜆𝑖 =𝜇𝑖 =0, the parameters 𝜆𝑖 =2,𝜇𝑖 =−2 󸀠 𝜆 =−2,𝜇 =2 𝑑𝑖,𝐻𝑖 (𝑥𝑖+1)=𝑑𝑖+1. are for the dot-dash line, and 𝑖 𝑖 are for dashed 1 The standard cubic Hermite spline is C continuous. line. However, if interpolation data is given, the shape and approximation of the spline cannot be changed. 3.2. The EH Interpolation Spline {(𝑥 ,𝑦,𝑑),𝑖 = 0,1,...,𝑛} 3. The EH Interpolation Spline Definition 2. Given a data set 𝑖 𝑖 𝑖 , where 𝑦𝑖 and 𝑑𝑖 are the values of the function value and the In order to overcome the disadvantage of the standard cubic first-order derivative value of the function being interpolated Hermite spline, we extend its basis functions firstly. and 𝑎=𝑥0 <𝑥1 < ⋅⋅⋅ < 𝑥𝑛 =𝑏is the knot spacing, let Abstract and Applied Analysis 3

1.0 1.0 0.20 0.00

0.8 0.8 0.15 −0.05

0.6 0.6

0.10 −0.10

0.4 0.4

0.05 −0.15 0.2 0.2

0.0 0.0 0.00 −0.20 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

(a) 𝑒𝛼0(𝑡) (b) 𝑒𝛼1(𝑡) (c) 𝑒𝛽0(𝑡) (d) 𝑒𝛽1(𝑡)

Figure 1: The graph of the four EH bases.

2 0.5 𝐶 -continuous in the interval [𝑥0,𝑥𝑛]. In fact, denote Δ 𝑖 = (𝑦𝑖+1 −𝑦𝑖)/ℎ𝑖,andlet 0.4 p󳰀 p󳰀 󸀠󸀠 󸀠󸀠 0.3 i i+1 EH (𝑥𝑖+) = EH (𝑥𝑖−), 𝑖=1,2,...,𝑛−1; (8)

0.2 then the equations connecting the parameters 𝜆𝑖 and 𝜇𝑖,

0.1 ℎ𝑖 [Δ 𝑖−1 (𝜆𝑖−1 +3)−(𝑑𝑖 (2 − 𝜇𝑖−1)+𝑑𝑖−1 (𝜇𝑖−1 + 1))] p 0.0 pi i+1 0.0 0.2 0.4 0.6 0.8 1.0 =ℎ𝑖−1 [Δ 𝑖 (𝜆𝑖 −3)+(𝑑𝑖+1 (1 + 𝜇𝑖)+𝑑𝑖 (2 − 𝜇𝑖))] , Figure 2: The Ferguson curve with different parameters. 𝑖=1,2,...,𝑛−1, (9)

ℎ𝑖 =𝑥𝑖+1 −𝑥𝑖,𝑡 = (𝑥−𝑖 𝑥)/ℎ𝑖; then the EH interpolation may be obtained. If the successive parameters (𝜆𝑖−1,𝜇𝑖−1) spline in the interval [𝑥𝑖,𝑥𝑖+1] canbedefinedasfollows: and (𝜆𝑖,𝜇𝑖) satisfy (9)at𝑖 = 1,2,...,𝑛,thenEH −1 (𝑥) ∈ 2 𝐶 (𝑥0,𝑥𝑛). Furthermore, if the knots are equally spaced and EH (𝑥)|[𝑥 ,𝑥 ] 𝑖 𝑖+1 𝜆𝑖 =𝜇𝑖 =0,then(9) becomes the well-known tridiagonal system for a cubic spline =𝑦𝑖𝑒𝛼0 (𝑡) +𝑦𝑖+1𝑒𝛼1 (𝑡) +𝑑𝑖ℎ𝑖𝑒𝛽0 (𝑡) +𝑑𝑖+1ℎ𝑖𝑒𝛽1 (𝑡) ,

𝑖=0,1,...,𝑛−1, 𝑑𝑖−1 +4𝑑𝑖 +𝑑𝑖+1 =3(Δ 𝑖−1 +Δ𝑖) , 𝑖=1,2,...,𝑛−1. (10) (6) Hence, if given the parameter values 𝜆0,𝜇0 in the interval [𝑥0,𝑥1],by(9), we may obtain the 𝜆1 and 𝜇1 andsoon.Thus, where 𝑒𝛼0(𝑡), 1𝑒𝛼 (𝑡), 0𝑒𝛽 (𝑡), 1𝑒𝛽 (𝑡) are the EH bases. 2 we can construct a 𝐶 -continuous interpolation curve. Obviously, for the data set {(𝑥𝑖,𝑦𝑖,𝑑𝑖),𝑖 = 0,1,...,𝑛}, EH(𝑥) satisfies 4. The Approximation of the EH 󸀠 RH (𝑥𝑖)=𝑦𝑖, RH (𝑥𝑖)=𝑑𝑖, Interpolation Spline (7) 𝑖=0,1,...,𝑛. According to the interpolation reminder of cubic Hermite spline, when interpolation interval approaches zero, the cubic If 𝜆𝑖 =𝜇𝑖 =0,itisjustthestandardcubicHermite Hermitesplinecurvecanapproximatewelltothefunction spline. It is of interest that, for suitable selected parameters being interpolated. However, for the EH interpolation spline 𝜆𝑖,𝜇𝑖, the piecewise interpolation function EH(𝑥) can be we constructed, it can approximate well to the function being 4 Abstract and Applied Analysis

Table 1: The parameters 𝜆𝑖 and 𝜇𝑖 for EH interpolation spline and the max error.

𝑥𝑖 𝑦𝑖 𝑑𝑖 𝜆𝑖 𝜇𝑖 𝑅𝐻𝜀𝑖 𝐻𝜀𝑖 − − 0.0000 1.0000 1.0000 0.0421 0.0412 0.2569 × 10 4 0.9062 × 10 3 − − 0.5000 1.2071 −0.1107 0.0146 0.0129 0.2111 × 10 4 0.2569 × 10 3 − − 1.0000 1.0000 −0.5708 0.2451 0.2783 0.3769 × 10 4 0.2569 × 10 3 − − 1.5000 0.7955 −0.1451 0.0188 0.0192 0.2974 × 10 4 0.1069 × 10 2 − − 2.0000 1.0000 1.0000 0.0108 0.0108 0.1735 × 10 4 0.8647 × 10 3 2.5000 1.7929 2.1107

interpolated without interpolation interval approaching zero, and 𝜇𝑖 are given for every interval [𝑥𝑖,𝑥𝑖+1] in Table 1.The and it can approximate to the interpolated functions better errorcurvesoftheEH(𝑥) and 𝐻(𝑥) to 𝑦(𝑥) are shown in than the standard cubic Hermite interpolation spline. Figure 3. Firstly,we give the definition of the “good approximation.” By using the tensor product method, we can construct the Definition 3. Let 𝐻𝑖(𝑥) be the standard cubic Hermite EH interpolation spline surfaces, which has the similar EH spline, EH𝑖(𝑥) be the EH interpolation spline, and 𝑦(𝑥) interpolation spline curve. be the function being interpolated. Denoting EH𝜀𝑖 = Ω [𝑎, 𝑏] × [𝑐, 𝑑] max𝑥 <𝑥<𝑥 |EH𝑖(𝑥) − 𝑦(𝑥)|, 𝐻𝜀𝑖 = max𝑥 <𝑥<𝑥 |𝐻𝑖(𝑥) − Definition 5. Let : be the plane region 𝑖 𝑖+1 𝑖 𝑖+1 𝑓(𝑥, 𝑦) Ω 𝑦(𝑥)|,thenifEH𝜀𝑖 <𝐻𝜀𝑖,wecancallRH𝑖(𝑥) has “good and a bivariate function defined in the region and approximation” to the interpolated function𝑦(𝑥) better than let 𝑎=𝑥0 <𝑥1 < ⋅⋅⋅ < 𝑥𝑚 =𝑏and 𝑐=𝑦0 <𝑦1 <⋅⋅⋅<𝑦𝑛 = 𝑑 ℎ =𝑥 −𝑥 ,ℎ =𝑦 −𝑦 𝐻𝑖(𝑥). be the knot sequences. Denote 𝑖 𝑖+1 𝑖 𝑗 𝑗+1 𝑗, 𝑢=(𝑥−𝑥𝑖)/ℎ𝑖, V =(𝑦−𝑦𝑗)/ℎ𝑗; then the EH interpolation According to the Definition,ifEH 3 𝜀𝑖 <𝐻𝜀𝑖,wecanget spline surface on the region [𝑥𝑖,𝑥𝑖+1]×[𝑦𝑖,𝑦𝑖+1] can be defined the range of the parameters value, 𝜆𝑖 and 𝜇𝑖.Intherange as follows: of the parameters value, selecting the arbitrary values of the 󵄨[𝑦 ,𝑦 ] 𝜆 𝜇 (𝑥, 𝑦)󵄨 𝑖 𝑖+1 parameters 𝑖 and 𝑖,wehavea“goodapproximation”curve. EH 󵄨[𝑥𝑖,𝑥𝑖+1]

Example 4. Given the function 𝑦(𝑥) = 𝑥+ cos((𝜋/2)𝑥) and 𝑒𝛼0 (V) knots 𝑥𝑖 = (𝑖/2)(𝑖 = 0,1,...,5),namely,ℎ𝑖 = (𝑖/2) (𝑖 = 𝑒𝛼1 (V) =(𝑒𝛼0 (𝑢) ,𝑒𝛼1 (𝑢) ,𝑒𝛽0 (𝑢) ,𝑒𝛽1 (𝑢))𝑀( ), 0,...,4). According to the inequality EH𝜀𝑖 <𝐻𝜀𝑖,wemay 𝑒𝛽0 (V) get the range of the parameters 𝜆𝑖 and 𝜇𝑖.Forthefixed 𝑒𝛽1 (V) interpolation condition, the max error and the parameters 𝜆𝑖 (11) where

󸀠 󸀠 𝑓(𝑥 𝑖,𝑦𝑖)𝑓(𝑥𝑖,𝑦𝑖+1)ℎ𝑗𝑓V (𝑥𝑖,𝑦𝑖)ℎ𝑗𝑓V (𝑥𝑖,𝑦𝑖+1) 󸀠 󸀠 𝑓(𝑥 𝑖+1,𝑦𝑖)𝑓(𝑥𝑖+1,𝑦𝑖+1)ℎ𝑗𝑓V (𝑥𝑖+1,𝑦𝑖)ℎ𝑗𝑓V (𝑥𝑖+1,𝑦𝑖+1) 𝑀=( 󸀠 󸀠 󸀠󸀠 󸀠󸀠 ). (12) ℎ𝑖𝑓𝑢 (𝑥𝑖,𝑦𝑖)ℎ𝑖𝑓𝑢 (𝑥𝑖,𝑦𝑖+1)ℎ𝑖ℎ𝑗𝑓𝑢V (𝑥𝑖,𝑦𝑖)ℎ𝑖ℎ𝑗𝑓𝑢V (𝑥𝑖,𝑦𝑖+1) 󸀠 󸀠 󸀠󸀠 󸀠󸀠 ℎ𝑖𝑓𝑢 (𝑥𝑖+1,𝑦𝑖)ℎ𝑖𝑓𝑢 (𝑥𝑖+1,𝑦𝑖+1)ℎ𝑖ℎ𝑗𝑓𝑢V (𝑥𝑖+1,𝑦𝑖)ℎ𝑖ℎ𝑗𝑓𝑢V (𝑥𝑖+1,𝑦𝑖+1)

−3 Given the end-points, the first order partial derivative and the max error of the EH(𝑥, 𝑦) − 𝑓(𝑥, 𝑦) equals 0.5069 × 10 , −1 the second-order blending partial derivative of the function but the max error of the 𝐻(𝑥, 𝑦)−𝑓(𝑥, 𝑦) equals 0.1061 × 10 . interpolated, with proper parameters, the EH interpolation spline surfaces could approximate to the bivariate functions Figure 4 shows the error surface of the EH(𝑥, 𝑦)−𝑓(𝑥, 𝑦). being interpolated better than the standard cubic Hermite Figure 5 shows the error surface of the 𝐻(𝑥, 𝑦) − 𝑓(𝑥, 𝑦). spline surfaces. 5. Conclusion Example 6. Given the bivariate function being interpolated 𝑓(𝑥, 𝑦) = sin(𝜋/2)𝑥 cos(𝜋/2)𝑦,let𝑎=0<1<2=𝑏and 𝑐= This paper introduced a kind of EH interpolation spline, −1<0<1=𝑑be the knot sequences. Denote ℎ𝑖 =𝑥𝑖+1 −𝑥𝑖, which is the extension of the standard cubic Hermite inter- ℎ𝑗 =𝑦𝑗+1 −𝑦𝑗, 𝑢=(𝑥−𝑥𝑖)/ℎ𝑖,andV =(𝑦−𝑦𝑗)/ℎ𝑗.Byselecting polation spline. The shape of the proposed splines can be 𝜆0 =𝜆1 = 0.3208, 𝜇0 =𝜇1 = 0.6995,wecanworkoutthat adjusted by changing the values of the parameters for the Abstract and Applied Analysis 5

0.0010 Conflict of Interests H(x) − y(x) The authors declare that there is no conflict of interests 0.0005 EH(x) − y(x) regarding the publication of this paper.

0.0000 Acknowledgments The work was funded by the Natural Science Foundation −0.0005 of Anhui Province of China under Grant no. 1208085MA15, the Key Project Foundation of Scientific Research, Edu- cation Department of Anhui Province under Grant no. 0.0 0.5 1.0 1.5 2.0 2.5 KJ2014ZD30, and the Key Construction Disciplines Founda- tion of Hefei University under Grant no. 2014XK08. Figure 3: The error curves of the EH(𝑥) − 𝑦(𝑥) and 𝐻(𝑥) − 𝑦(𝑥). References

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