While shape preservingplanar interpolation can be traced to the introduction of the exponential tension spline in schweikert 1966, the development of shape preserving interpolation algorithms for spatial data is more recent see, for. The tension spline involves the use of hyperbolic functions and. The scheme has parameters that can be used either to preserve shape of. Goodman, ong and unsworth 8 presented two interpolating schemes to preserve the shape of data lying on one side of the straight line using a rational cubic. Shape preserving interpolation using rational cubic spline hindawi. Monotone and convex spline interpolation siam journal on. For such given data, the existence or nonexistence of such interpolating splines can. Journal of computational and applied mathematics 319, 277295. For example, the wind speed, solar energy, and rainfall received are always. Pdf shape preserving interpolation using c 2 rational cubic spline. On convexity preserving c1 hermite spline interpolation problem under consideration. The shape preserving interpolation problem can be very efficiently solved by using weighted splines. For this problem, we impose appropriate constraints on single shape parameter to conserve the shape. A globally c2 interpolatory cubic spline containing free parameters is derived and its properties established.
The c2 variable degree splines have been proven to be an efficient tool for solving the curve shape preserving interpolation problem in two and three dimensions. In this study, a new scheme for positivity preserving interpolation is proposed by using c 1 rational quartic spline of quarticquadratic with three parameters. Local convexitypreserving c2 rational cubic spline for. The sufficient condition for the positivity rational quartic interpolant is derived on one parameter meanwhile the other two are free parameters for shape modification. Conditions of two shape parameters are derived in such a way that they preserve the shape of the data, whereas the other two parameters remain free to enable the user to modify the shape of the curve. Data visualization using shape preserving c2 rational. We propose a general parametric local approach for functional c 2 hermite shape preserving interpolation. For full access to this pdf, sign in to an existing account, or. Visualization of positive and convex data by a rational cubic spline interpolation, information sciencesapplications. To restore c2smoothness fifth degree polynomial splines are considered, which are constructed as a sum of base cubic shape preserving splines and fifth degree terms, which are chosen to provide continuity of the spline second derivative.
A shape preserving cubic c 2 spline interpolation adapted from 16 has been then used to fit the obtained pdf curve. An algorithm for computing shape preserving cubic spline interpolation to data, calcolo 21, 295305. Furthermore, if g is a c2 cubic spline function satisfying the hypotheses of. Shape preserving interpolation using c 2 rational cubic spline article pdf available in journal of applied mathematics 20163. For example, in the case of the second derivative boundary. Finally, we give some examples to illustrate the convex preserving properties of these splines. Shape preserving c2 cubic spline interpolation oxford academic. Pdf construction of a family of c1 convex integro cubic. This work addresses the shape preserving interpolation problem for visualization of positive data.
The constructed interpolant is a parametric curve which interpolate values, first and second derivatives of a given function and reproduces the behavior of the data. This chapter concentrates on two closely related interpolants. C2 rational quartic interpolation spline with local shape. These l 1 splines preserve the shape of data even when the data have abrupt changes in. Cubic spline interpolation with c 2 continuity is not able to preserve the shape of the positive data. Shape preserving spline interpolation sciencedirect. A convexity preserving c2 parametric rational cubic interpolation. In this scheme, the spline is chosen so that its second derivative is zero at the end points of the interval a, 6.
Its application to shape preserving interpola tion is considered. On convexity preserving c1 hermite spline interpolation. In fact, the example data were generated from the polynomial x3. Shape preserving interpolation, rational cubic function, rational bi cubic function, monotone surface, monotone surface data, free parameters. Its application to shape preserving interpolation is considered. Shapepreserving, multiscale interpolation by bi and. Positivitypreserving c2 rational cubic spline interpolation. The method is detailed for parametric curves with piecewise cubic components. Simple data dependent conditions for a single shape parameter are derived to maintain. Suppose that f is piecewise c2 with increasing slopes, i. This work addresses the shape preserving interpolation. By introducing a new family of local shape parameters w i into the rational quartic interpolation spline given in, we give an explicit representation of a general c 2 rational quartic interpolation spline with three families of local shape parameters, which possesses local shape preserving property and includes several interpolation splines. Positivitypreserving c rational cubic spline interpolation.
Refer to the scatteredinterpolant, griddata, and tpaps functions for more information about surface interpolation. C2 cubic splines play a very important role in practical methods of spline approximation. Shape preserving third and fifth degrees polynomial. Local convexity preserving rational cubic spline interpolation. Siam journal on scientific and statistical computing 11. Analysis of two algorithms for shapepreserving cubic. Requirements for shape preserving interpolation by planar curves are discussed. Refer to the pchip function for more information about shape preserving interpolation, and for a comparison of the two methods. Rational spline interpolation preserving the shape of the monotonic data. The idea has been extended to shape preserving interpolation for positive data using the constructed rational cubic spline interpolation. The theory concerning natural cubic splines is elegant, but in practice they are less accurate than some other approaches. Hyman dedicated to professor eugene isaacson on the occasion of his 70th birthday abstract.
Shapepreserving, multiscale interpolation by univariate. Shape preserving rational cubic ball interpolation for positive data. In this section, we discuss the solution of convexity preserving problem by using c 2 rational cubic function with three shape parameters. In this paper we propose a method to construct shape preserving c cubic polynomial splines interpolating convex andor monotonie data. The shapepreserving interpolation problem consists of constructing a sufficiently smooth. Pdf shape preserving interpolation using c 2 rational. Interpolating c2 cubic spline, shape preserving, convex function, geometric mesh. We introduce a class of bi and multivariate cubic l 1 interpolating splines, the coefficients of which are calculated by minimizing the sum of the l 1 norms of second derivatives. Shape preserving piecewise cubic interpolation springerlink. Shape preserving c2 cubic spline interpolation researchgate.
Mathematical methods in computer aided geometric design ii, 343350. A curve interpolation scheme is developed using rational cubic ball basis functions, which involves three shape parameters. Shape preserving data interpolation using rational cubic. A smooth curve interpolation scheme for positive, monotone, and convex data is developed. Shape preserving c2 cubic spline interpolation ima.
For surveys on shape preserving interpolation the interested reader is referred, e. The constructed interpolant is a parametric cubic curve. Shape preserving approximations by polynomials and splines. The shape of the curve can be easily controlled via tension parameters which have an. Shape preserving positive c2 rational cubic curve interpolation in this section, we discuss the problem of getting a positivity preserving curve by using a c2 rational cubic spline 4. The next example figure 3 shows polynomial and cubic spline interpolation. Nonnegativity, monotonicity, or convexity preserving cubic and quintic hermite interpolation by randall l. This paper addresses new algorithms for constructing weighted cubic splines that are very effective in interpolation and approximation of sharply changing data. This paper presents ac 1 interpolation which preserves convexity to scattered convex data. Shape preserving interpolation using rational cubic spline. We construct a family of monotone and convex c integro cubic splines under a strictly convex position of the dataset. Fuhr and kallay 6 used a c1 monotone rational b spline of degree one to preserve the shape of monotone data. The paper proposes a method for the construction of a shape preserving c 2 function interpolating a given set of data. On shape preserving c 2 hermite interpolation springerlink.
Introduction we consider the interpolation of increasing and convex data with c2 cubic splines for an analysis of shape preserving c quadratic and cubic splines, see medinas thesis 4. Shapepreserving rational bicubic spline for monotone. The resulting curvessurfaces retain geometric properties of the initial data, such as positivity, monotonicity, convexity, linear and planar sections. Spline interpolation plays a significant role in computer graphics, computer aided geometric design. Box 12211, research triangle park, nc 277092211, usa received 1 march 2001. It is natural to apply this principle also if shape preserving is added as a constraint, although the construction process is then nonlinear. In this paper, a c2 piecewise rational cubic spline scheme with. How these may be satisfied is then illustrated by four different schemes. Due to the presence of the scaling factors in the rational cubic spline fractal interpolant, our approach generalizes the classical results on the shape preserving rational interpolation by.
Pdf algorithms for computing shape preserving spline. The focus is mainly on bivariate cubic l 1 splines for c 1 interpolation of data located at the nodes of a tensorproduct grid. In this paper, a c2 piecewise rational cubic spline. On shape preserving quadratic spline interpolation siam. Proceedings computer graphics international, 238244. It was shown in 16 that these splines are thirdorder accurate if f0 i.
Positivity preserving c2 rational cubic spline interpolation. Cubic spline interpolation with continuity is not able to preserve the shape of the. The c2cubic spline interpolant p, is displayed in figure 4. These splines get shape preserviation at the cost of reducing smoothness till c1.
This scheme uses rational cubic ball representation with four shape parameters in its description. This paper discusses the construction of new rational cubic spline interpolant with cubic numerator and quadratic denominator. Nearest neighbor, linear, cubic, shape preserving pchip for curves. I want to use a shapepreserving piecewise cubic interpolation on it similar to pchip in matlab. C2 cubic polynomial splines interpolating convex and monotone data while. Pdf shape preserving rational cubic ball interpolation. Approximation by shape preserving interpolation splines. Shape preserving interpolation by planar curves springerlink. Then, we find an optimal spline by considering its approximation properties.
Shape preserving c 2 cubic spline interpolation, ima journal of numerical analysis, volume, issue 4, october 1993. This book aims to develop algorithms of shapepreserving spline approximation for curvessurfaces with automatic choice of the tension parameters. Shape preserving interpolation with quartic c2 splines in one dimension the problem here of interest is to consider c2 splines s on a, which satisfy the interpolation condition. Shape preserving piecewise cubic interpolation semantic scholar. Shape preserving, multiscale interpolation by univariate curvaturebased cubicl1 splines in cartesian and polar coordinates john e. Shape preserving spline interpolation john a gregory a rational spline alternative to the spline undertension is discussed.
This paper discusses the construction of new c 2 rational cubic spline interpolant with cubic numerator and quadratic denominator. Shape preserving rational cubic spline fractal interpolation. Positive quartic, monotone quintic c2spline interpolation. Refer to the spline function for more information about cubic spline interpolation. Nonnegativity, monotonicity, or convexitypreserving.
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