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 of the 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
Spline interpolation is a useful and powerful tool for curves and surfaces modeling. Standard cubic Hermite spline is one of those interpolation functions. But, for the given interpolation condition, the cubic Hermite interpolation spline is fixed; that is to say, the shape of the interpolation curve or surface is fixed for the given interpolation data [1–6]. Since the interpolation function is unique for the given interpolation data, to modify the shape of the interpolation curve to approximate the given curve seems to be impossible and it is contradictory to the uniqueness of the interpolation function for the given interpolation data. For the given interpolation condition, how to improve the approximation accuracy of the interpolation spline is an important problem in the computer aided geometric design. In recent years, many authors have presented some new method to modify the shape of the interpolation curve to satisfy the industrial product design with several kinds of new interpolation splines with parameters [7–20].
These new splines all have similar properties of the standard cubic Hermite spline. For example, for the given interpolation data, if interpolation interval approaches zero, theoretically speaking, these splines can approximate the given curve and surface well. However, there exists a problem, in the process of the actual calculation; the amount of computation will increase dramatically if the length of the interpolating intervals decreases. On the other hand, the approximation accuracy of these new splines may not be better than the standard cubic Hermite spline.
In [7–17], many rational form interpolation splines with multiple parameters were presented. For the given interpolation data, the change of the parameters causes the change of the interpolation curve. Nevertheless, the computation of the splines with multiple parameters is very complicated. Several kinds of rational splines with a single parameter were presented in the papers [18, 19], which is simple to compute, but its approximation accuracy is not good for the given curves and surfaces. In general, polynomial-form splines are suitable to be used to design and compute. In [20], a polynomial-form spline, called quartic Hermite spline with single parameter, is presented as the extension of the standard Hermite spline. The quartic spline has a simple form, and its approximation rate to the given curves and surfaces is not high.
In this paper, a class of new quartic splines with two parameters is developed, which is the extension of the standard cubic Hermite interpolation spline and inherits the same properties of the standard cubic Hermite interpolation spline. For the given interpolation conditions, the shape of the proposed splines can be adjusted by changing the values of the parameters. Furthermore, the introduced splines could approximate to the interpolated functions better than the standard cubic Hermite interpolation splines and the quartic Hermite interpolation splines with single parameter.
The remainder of the paper is organized as follows. Section 2 introduces the standard cubic Hermite spline and some of its properties. A kind of interpolation spline with two parameters is presented in Section 3. Section 4 discusses the approximation of the introduced spline curve with numerical examples. In the end, a summary and conclusions are given.
2. The Standard Cubic Hermite Spline and Its Basis Functions
Generally, for t∈[0,1], the following four basis functions,
(1)α0(t)=1-3t2+2t3,α1(t)=3t2-2t3,β0(t)=t-2t2+t3,β1(t)=-t2+t3,
are called the standard cubic Hermite bases.
These bases satisfy
(2)α0(0)=α1(1)=1,α0(1)=α1(0)=0,α1′(0)=α1′(1)=0,α0′(0)=α0′(1)=0,β0(0)=β0(1)=0,β1(0)=β1(1)=0,β0′(0)=β1′(1)=1,β0′(1)=β1′(0)=0,α0(t)+α1(t)=1,β0(t)+β1(1-t)=0.
For given knots a=x0<x1<⋯<xn=b and data {(xi,yi,di),i=0,1,…,n}, where yi anddi are the values of the function value and the first-order derivative value of the function being interpolated, let hi=xi+1-xi,t=(x-xi)/hi and then the standard cubic Hermite spline in the interval [xi,xi+1] can be defined as follows:
(3)Hi(x)=α0(t)yi+α1(t)yi+1+β0(t)hidi+β1(t)hidi+1,hhhhhhhhi=0,1,…,n-1.
Obviously, we have Hi(xi)=yi,Hi(xi+1)=yi+1,Hi′(xi)=di,Hi′(xi+1)=di+1.
The standard cubic Hermite spline is C1 continuous. However, if interpolation data is given, the shape and approximation of the spline cannot be changed.
3. The EH Interpolation Spline
In order to overcome the disadvantage of the standard cubic Hermite spline, we extend its basis functions firstly.
3.1. The Basis Functions of the EH Interpolation SplineDefinition 1.
For arbitrary real numberλi, μiand0≤t≤1, the following four functions with parameters λi, μi,
(4)eα0(t)=1+(λi-3)t2+2(1-λi)t3+λit4,eα1(t)=(3-λi)t2+2(λi-1)t3-λit4,eβ0(t)=t+(μi-2)t2+(1-2μi)t3+μit4,eβ1(t)=-(μi+1)t2+(1+2μi)t3-μit4,
are calledbasis functions of the EH interpolation spline, briefly EH bases.
The EH bases are the extension of the standard cubic Hermite bases. When λi=μi=0, the bases are the standard cubic Hermite bases. The bases have the similar properties of the standard cubic Hermite bases.
By computing directly, we have eα0(0)=eα1(1)=1,eα0(1)=eα1(0)=0, eα1′(0)=eα1′(1)=0,eα0′(0)=eα0′(1)=0, eβ0(0)=eβ0(1)=0,eβ1(0)=eβ1(1)=0,eβ0′(0)=eβ1′(1)=1,eβ1′(1)=eβ1′(0)=0, and eα0(t)+eα1(t)=1, eβ0(t)+eβ1(1-t)=0.
When λi=μi, the EH bases (4) are basis functions with single parameter in [20].
Figure 1 shows the four EH bases, where the solid lines are the standard Hermite bases, the parametersλi=2,μi=-2 are for the dot-dash lines, andλi=-2,μi=2 are for dashed line.
The graph of the four EH bases.
eα0(t)
eα1(t)
eβ0(t)
eβ1(t)
So, we may construct the Ferguson curve with two parameters based on the EH bases as follows:
(5)EHi(t)=eα0(t)pi+eα1(t)pi+1+eβ0pi′+eβ1(t)pi+1′,
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 vectors, with the different parameters λi, μi, we may obtain different shape of the Ferguson curve with two parameters accordingly.
Figure 2 shows the Ferguson curves with different parameters, where the solid line is the standard Ferguson spline curve with λi=μi=0, the parametersλi=2,μi=-2 are for the dot-dash line, andλi=-2,μi=2 are for dashed line.
The Ferguson curve with different parameters.
3.2. The EH Interpolation SplineDefinition 2.
Given a data set {(xi,yi,di),i=0,1,…,n}, where yi anddi are the values of the function value and the first-order derivative value of the function being interpolated and a=x0<x1<⋯<xn=b is the knot spacing, let hi=xi+1-xi, t=(x-xi)/hi; then the EH interpolation spline in the interval [xi,xi+1] can be defined as follows:(6)EH(x)|[xi,xi+1]=yieα0(t)+yi+1eα1(t)+dihieβ0(t)+di+1hieβ1(t),hhhhhhi=0,1,…,n-1,
where eα0(t),eα1(t),eβ0(t),eβ1(t) are the EH bases.
Obviously, for the data set {(xi,yi,di),i=0,1,…,n},EH(x) satisfies
(7)RH(xi)=yi,RH′(xi)=di,hhi=0,1,…,n.
If λi=μi=0, it is just the standard cubic Hermite spline. It is of interest that, for suitable selected parameters λi,μi, the piecewise interpolation function EH(x)can be C2-continuous in the interval [x0,xn]. In fact, denote Δi=(yi+1-yi)/hi, and let
(8)EH′′(xi+)=EH′′(xi-),i=1,2,…,n-1;
then the equations connecting the parameters λi and μi,
(9)hi[Δi-1(λi-1+3)-(di(2-μi-1)+di-1(μi-1+1))]=hi-1[Δi(λi-3)+(di+1(1+μi)+di(2-μi))],hhhhhi=1,2,…,n-1,
may be obtained. If the successive parameters (λi-1,μi-1) and (λi,μi) satisfy (9) at i=1,2,…,n-1, then EH(x)∈C2(x0,xn). Furthermore, if the knots are equally spaced and λi=μi=0, then (9) becomes the well-known tridiagonal system for a cubic spline
(10)di-1+4di+di+1=3(Δi-1+Δi),i=1,2,…,n-1.
Hence, if given the parameter values λ0,μ0 in the interval [x0,x1], by (9), we may obtain the λ1 and μ1 and so on. Thus, we can construct a C2-continuous interpolation curve.
4. The Approximation of the EH Interpolation Spline
According to the interpolation reminder of cubic Hermite spline, when interpolation interval approaches zero, the cubic Hermite spline curve can approximate well to the function being interpolated. However, for the EH interpolation spline we constructed, it can approximate well to the function being interpolated without interpolation interval approaching zero, and it can approximate to the interpolated functions better than the standard cubic Hermite interpolation spline.
Firstly, we give the definition of the “good approximation.”
Definition 3.
Let Hi(x) be the standard cubic Hermite spline,EHi(x) be the EH interpolation spline, and y(x) be the function being interpolated. Denoting EHεi=maxxi<x<xi+1|EHi(x)-y(x)|, Hεi=maxxi<x<xi+1|Hi(x)-y(x)|, then if EHεi<Hεi, we can call RHi(x)has “good approximation” to the interpolated functiony(x) better than Hi(x).
According to the Definition 3, if EHεi<Hεi, we can get the range of the parameters value, λi and μi. In the range of the parameters value, selecting the arbitrary values of the parameters λi and μi, we have a “good approximation” curve.
Example 4.
Given the function y(x)=x+cos((π/2)x) and knots xi=(i/2)(i=0,1,…,5), namely, hi=(i/2)(i=0,…,4). According to the inequality EHεi<Hεi, we may get the range of the parameters λi and μi. For the fixed interpolation condition, the max error and the parameters λi and μi are given for every interval [xi,xi+1] in Table 1. The error curves of the EH(x) and H(x) to y(x) are shown in Figure 3.
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
The error curves of the EH(x)-y(x) and H(x)-y(x).
By using the tensor product method, we can construct the EH interpolation spline surfaces, which has the similar EH interpolation spline curve.
Definition 5.
Let Ω: [a,b]×[c,d] be the plane region andf(x,y) a bivariate function defined in the region Ω and let a=x0<x1<⋯<xm=b and c=y0<y1<⋯<yn=d be the knot sequences. Denotehi=xi+1-xi,hj=yj+1-yj, u=(x-xi)/hi,v=(y-yj)/hj; then the EH interpolation spline surface on the region [xi,xi+1]×[yi,yi+1] can be defined as follows:(11)EH(x,y)|[xi,xi+1][yi,yi+1]=(eα0(u),eα1(u),eβ0(u),eβ1(u))M(eα0(v)eα1(v)eβ0(v)eβ1(v)),
where(12)M=(f(xi,yi)f(xi,yi+1)hjfv′(xi,yi)hjfv′(xi,yi+1)f(xi+1,yi)f(xi+1,yi+1)hjfv′(xi+1,yi)hjfv′(xi+1,yi+1)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)).
Given the end-points, the first order partial derivative and the second-order blending partial derivative of the function interpolated, with proper parameters, the EH interpolation spline surfaces could approximate to the bivariate functions being interpolated better than the standard cubic Hermite spline surfaces.
Example 6.
Given the bivariate function being interpolated f(x,y)=sin(π/2)xcos(π/2)y, leta=0<1<2=b and c=-1<0<1=d be the knot sequences. Denote hi=xi+1-xi, hj=yj+1-yj, u=(x-xi)/hi, and v=(y-yj)/hj. By selecting λ0=λ1=0.3208,μ0=μ1=0.6995, we can work out that the max error of theEH(x,y)-f(x,y) equals 0.5069 × 10−3, but the max error of the H(x,y)-f(x,y) equals 0.1061 × 10−1.
Figure 4 shows the error surface of the EH(x,y)-f(x,y). Figure 5 shows the error surface of theH(x,y)-f(x,y).
The error surface of the EH(x,y)-f(x,y).
The error surface of the H(x,y)-f(x,y).
5. Conclusion
This paper introduced a kind of EH interpolation spline, which is the extension of the standard cubic Hermite interpolation spline. The shape of the proposed splines can be adjusted by changing the values of the parameters for the fixed interpolation conditions. Also, the introduced spline could approximate to the interpolated function better than the standard cubic Hermite interpolation spline.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The work was funded by the Natural Science Foundation of Anhui Province of China under Grant no. 1208085MA15, the Key Project Foundation of Scientific Research, Education Department of Anhui Province under Grant no. KJ2014ZD30, and the Key Construction Disciplines Foundation of Hefei University under Grant no. 2014XK08.
ButtS.BrodlieK. W.Preserving positivity using piecewise cubic interpolation199317155642-s2.0-0027223776BrodlieK. W.ButtS.Preserving convexity using piecewise cubic interpolation1991151152310.1016/0097-8493(91)90026-E2-s2.0-0025842603DoughertyR. L.EdelmanA. S.HymanJ. M.Non-negativity, monotonicity, or convexity-preserving cubic and quintic Hermite interpolation19895218647149410.2307/2008477MR962209CarlsonR. E.FritschF. N.Monotone piecewise cubic interpolation19801722382462-s2.0-0022050882FritschF. N.ButlandJ.A method for constructing local monotone piecewise cubic interpolants19845230030410.1137/0905021SchumakerL. L.On shape preserving quadratic spline interpolation198320485486410.1137/0720057MR708462ZBLl0521.65009DuanQ.ZhangY.WangL.TwizellE. H.Region control and approximation of a weighted rational interpolating curves2006221415310.1002/cnm.797MR21943162-s2.0-29744434197DuanQ. I.DjidjeliK.PriceW. G.TwizellE. H.A rational cubic spline based on function values199822447948610.1016/S0097-8493(98)00046-62-s2.0-0032119650DuanQ.DjidjeliK.PriceW. G.TwizellE. H.The approximation properties of some rational cubic splines199972215516610.1080/00207169908804842MR1717924ZBLl0933.410072-s2.0-0038215491SarfrazM.Cubic spline curves with shape control199418570771310.1016/0097-8493(94)90165-12-s2.0-0028513034DuanQ.LiuA. K.ChengF. H.Constrained interpolation using rational cubic spline with linear denominators199961203215MR1670844HussainM. Z.SarfrazM.Positivity-preserving interpolation of positive data by rational cubics2008218244645810.1016/j.cam.2007.05.023MR24371182-s2.0-44649136115SarfrazM.HussainM. Z.HussainM.Shape-preserving curve interpolation2012891355310.1080/00207160.2011.627434MR28699192-s2.0-84857227072IbraheemF.HussainM.HussainM. Z.BhattiA. A.Positive data visualization using trigonometric function201220121924712010.1155/2012/2471202-s2.0-84871399184DuanQ.ZhangH.ZhangY.TwizellE. H.Error estimation of a kind of rational spline2007200111110.1016/j.cam.2005.12.007MR2276811ZBLl1106.410092-s2.0-33750944707TianM.GengH. L.Error analysis of a rational interpolation spline2011525–2812871294MR28359132-s2.0-80355137631BaoF.SunQ.PanJ.DuanQ.Point control of rational interpolating curves using parameters2010521-214315110.1016/j.mcm.2010.02.003MR2645926ZBLl1201.65021XieJ.TanJ. Q.LiS. F.Rational cubic Hermite interpolating spline and its approximation properties2010283385392MR2866459XieJ.Tan J. Q.LiS. F.A kind of rational cubic spline and its applications20102335847855LiuC.-Y.YangL.LiJ.-C.Quartic Hermite interpolating splines with parameters20123271868187010.3724/SP.J.1087.2012.01868