This paper discusses deeply the extension of the quasiuniform bspline curves. Convexity preserving interpolation by splines is the topic of section 3. Chapter 5 spline approximation of functions and data uio. Of course, this text has no pretensions to be complete. This paper addresses new algorithms for constructing weighted cubic splines that are very effective in interpolation and approximation of sharply changing data. Shapepreserving rational bicubic spline for monotone. This is a complicated command that makes a sophisticated use of the underlying functions, so in these notes well sort out what is being done and in the process learn more about cubic splines and least squares. Then in section 4, a tool path quality evaluation tool set is presented, which use mathematical. Convexity preserving c2 rational quadratic trigonometric spline were presented in 4. Original article shape preserving rational cubic spline for positive and convex data malik zawwar hussain a, muhammad sarfraz b, tahira sumbal shaikh a a department of mathematics, university of the punjab, lahore, pakistan b department of information science, adailiya campus, kuwait university, kuwait received 3 may 2011. In section 3, the corresponding convergence analysis is discussed in detail.
In this paper, we give a survey of some shape preserving approximation methods. Third degree splines include two variants, namely, monotonisity preserving and positivity preserving splines. A function f interpolating at the set d is shape preserving or preserves the shape of d if it is piecewise monotone in the same way as d and if it is convex and concave in the same way as d. Furthermore decreases in the value of will pull the curves upward and vice versa. Pdf construction of a family of c1 convex integro cubic. This paper addresses new algorithms for constructing weighted cubic splines that are very effective in interpolation and approximation of sharply changing. These splines get shape preserviation at the cost of reducing smoothness till c1. A method for shape preserving approximation by circular splines based on. Pdf download curve and surface fitting with splines free. While many shape preserving interpolant methods with different approaches have been. The first scheme adopted the method of scaling weights and the second. Shape preserving interpolation using rational cubic spline. Convexity preserving interpolation by rational cubic spline. Discrete weighted cubic splines discrete weighted cubic splines kvasov, boris 20140430 00.
In mathematics, a spline is a special function defined piecewise by polynomials. Department of mathematical sciences norwegian university. Finally, we give some examples to illustrate the convex preserving properties of these splines. In 1 duan have presented the construction and shape preserving analysis of a new weighted rational cubic interpolation and its approximation. Box 12211, research triangle park, nc 277092211, usa received 1 march 2001.
Deepdyve is the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Researchers pay little attention to nonuniform bspline. A shape preserving approximation by weighted cubic splines. Shape preserving approximations by polynomials and splines. We show that shape preserving splines can avoid divergence problems while producing a smooth approximation to the value function. Journal of computational and applied mathematics, issn 03770427, vol. These methods are based on discrete weighted cubic splines.
Interpolation is a process for estimating values that lie between. For comparison with cubic l1 smoothing splines, cubic l2 smoothing splines with the integral in 3 discretized in the. The convexity preserving property and shape control analysis are shown. Meanwhile figure 2g shows the combination of figures 2a, 2b, and 2c, respectively clearly the curves approach to the straight line if, or. Methods of monotone and convex spline interpolation. Spline curves a spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces. Journal of computational and applied mathematics 102. Firstly, by introducing shape parameters in the basis function, the spline curves are. Shape controls are available to tighten the weighted. Approximation by shape preserving interpolation splines. Several numerical examples are given in section 4 to prove the worth of the new developed schemes. Below classical cubic spline is represented in a form, convenient to next transformations, and monotonisity or positivity preserving spline schemes are described as examples of tvd algorithms applications.
Weighted quasi interpolant spline approximation of 3d. Shape preserving interpolation by quadratic splines aatos lahtinen department of mathematics, university of helsinki, hallituskatu 15, sf00100 helsinki, finland received april 1988 revised 28 february 1989 abstract. Cubic bspline curves with shape parameter and their applications. This chapter concentrates on two closely related interpolants. The tension spline involves the use of hyperbolic functions and. Department of naval architecture and ocean engineering, research institute of marine systems engineering, seoul national university, seoul 151744, republic of korea. Positivity preserving using gc cubic ball interpolation. Theory and algorithms for shapepreserving bivariate cubic. We formulate the problem as a differential multipoint boundary value problem and consider its finitedifference approximation. A smooth curve interpolation scheme for positive, monotone, and convex data is developed. Lowess smoothing create a smooth surface through the data using locally weighted linear regression to smooth data. Pdf an algorithm for computing a shapepreserving osculatory.
The main difference is that to generate rational interpolating curves the first derivative parameter, is calculated by using arithmetic mean method amm. In this paper we consider the weighted cubic splines introduced in see also. Comonotone shapepreserving spline histopolation malle fischer, peeter oja. 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. Visualization of shaped data by cubic spline interpolation. Weighted quasi interpolant spline approximation of 3d point clouds via local re. Natural cubic splines arne morten kvarving department of mathematical sciences norwegian university of science and technology october 21 2008. One of them preserves the monotonicity of the data, while the other preserves the. Understand relationships between types of splines conversion express what happens when a spline curve is transformed by an affine transform rotation, translation, etc.
This shape control will be useful for shape preserving interpolation as well as local control of. Shapepreserving, multiscale interpolation by univariate curvaturebased cubicl1 splines in cartesian and polar coordinates john e. Weighted cubic and biharmonic splines springerlink. The coefficients of a cubic l1 smoothing spline are calculated by minimizing the weighted sum of the l1 norms ofsecond derivatives of the spline and the l1 norm of the residuals of the datafitting nist equations. For this example, the grid is a 51by61 uniform grid.
An interpolating quadratic spline was constructed which preserves the shape. This paper presents methods for shape preserving spline interpolation. The analysis results in two algorithms with automatic selection. In this paper we present a shape preserving method of interpolation for scattered data defined in the form of some constraints such as convexity, monotonicity and positivity. For such given data, the existence or nonexistence of such interpolating splines can. The analysis results in two algorithms with automatic selection of the shape control parameters. The construction of a leastsquares approximant usually requires that one have in hand a basis for the space from which the data are to be approximated.
As the example of the space of natural cubic splines illustrates, the explicit construction of a basis is not always straightforward. The present studies on the extension of bspline mainly focus on bezier methods and uniform bspline and are confined to the adjustment role of shape parameters to curves. Shapepreserving interpolation by parametrically defined. A shapepreserving approximation by weighted cubic splines. Shapepreserving rational interpolation for planar curves. This scheme uses rational cubic ball representation with four shape parameters in its description. Trigonometric splines serves as an alternative for polynomial splines for solving many problems of. Shape preserving leastsquares approximation by polynomial. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding runges phenomenon for higher degrees in the computer science subfields of computeraided design and computer graphics, the term. Kvasov, a shapepreserving approximation by weighted cubic splines.
This paper addresses new algorithms for constructing weighted cubic splines that are very effective in interpolation and approximation of. Piecewise rational cubic function to partially blended rational bicubic function coon patches with sixteen shape paramein each patchters. Monotone and convex interpolation by weighted cubic splines. We construct a family of monotone and convex c1 integro cubic. Leastsquares approximation by natural cubic splines matlab. Use csaps to obtain the new, smoothed data points and the smoothing parameters csaps determines for the fit. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding runges phenomenon for higher degrees. On shape preserving quadratic spline interpolation siam. Prefitting by shape preserving c1 b splines, reselect sample data points, fitting new sample data points by c2 continuous cubic b spline, and then blend these b splines by a new cubic b spline. We construct a family of monotone and convex c integro cubic splines under a strictly convex position of the dataset. The interpolation of discrete spatial dataa sequence of points and unit tangentsby g 1 pythagoreanhodograph ph quintic spline curves, under shape constraints, is addressed. Section 4 is concerned with fifth degree polynomial c 2 spline.
The authors derived the constraints the eight shape for parameters to preserve the shape of monotone data, while the remaining free parameters were left eight to user for the refinement of surfaces. A two parameter family of c1 rational cubic spline functions is presented for the graphical representation of shape preserving curve interpolation for shaped data. Weighted fifth degree polynomial spline science publishing. Shapepreserving interpolation of spatial data by pythagoreanhodograph quintic spline curves. Classical splines do not preserve shape well in this sense.
Shapepreserving, multiscale fitting of bivariate data by. C2 cubic splines play a very important role in practical methods of spline approximation. The weighted v spline is a c1 piecewise cubic polynomial interpolant that generalizes c2 cubic splines, weighted splines, and v splines. Fuhr and kallay 6 used a c1 monotone rational b spline of degree one to preserve the shape of monotone data. C1 convexitypreserving piecewise variable degree rational.
The cubic spline interpolant with so called natural end conditions solves an interesting. The problem of shape preservation has been discussed by a number of authors. Introduction shape preserving interpolation andor approximation methods appear of great interest when applied to graphical problems and to reconstruction of functions and curves according to the data. The aim of this section is to illustrate the shape features of weighted cubic interpolation splines with some popular examples. Dec 06, 2015 global fifth degree polynomial spline is developed. These splines may be considered as modifications of classical cubic spline and may be identical to this spline for good data.
They are a natural generalization of cubic splines, describing from a physical point of view, an inhomogeneous elastic beam supported at some points. Cartesiancoordinate cubicl1 splines preserve shape much better than analogous. Shape preserving interpolation by quadratic splines. Shape preserving rational cubic spline for positive and. Shape preserving data interpolation using rational cubic. To show the difference between rational cubic spline with three parameters and rational cubic spline of karim and kong 1719, we choose for both cases. Ideas applied in the field of high order weno weighted essentially non oscillating methods for numerical solving compressible flow equations are used to construct interpolation which has accuracy closed to accuracy of classical cubic spline for smooth interpolated functions, and which reduces undesirable oscillations often appearing in the. Request pdf a shapepreserving approximation by weighted cubic splines this paper addresses new algorithms for constructing weighted cubic splines that are very effective in interpolation and. Leastsquares approximation by natural cubic splines. Two algorithms for automatic selection of shape control.
The resulting curvessurfaces retain geometric properties of the initial data, such as positivity, monotonicity, convexity, linear and planar sections. Discrete weighted cubic b splines and control point approximation are also considered. Cubic l1 smoothing splines are compared with conventional cubic smoothingsplines based on the l2 and l2 norms. Cool simple example of nontrivial vector space important to understand for advanced methods such as finite elements. Next section is devoted to representation of weighted c 1 cubic spline. Theory and algorithms for shapepreserving bivariate cubic l1. Shape preserving interpolation using 2 rational cubic spline.
Oct 22, 20 algorithms for interpolating by weighted cubic splines are constructed with the aim of preserving the monotonicity and convexity of the original discrete data. Then, we find an optimal spline by considering its approximation properties. The analysis performed in this paper makes it possible to develop two algorithms with the automatic choice of the shape controlling parameters weights. Nonnegativity, monotonicity, or convexitypreserving cubic. Fit a smoothing spline to bivariate data generated by the peaks function with added uniform noise. In this paper we discuss the design of algorithms for interpolating discrete data by using weighted cubic and biharmonic splines in such a way that the monotonicity and convexity of the data are preserved. Discrete weighted cubic splines, numerical algorithms 10.
A shapepreserving approximation by weighted cubic splines core. This book aims to develop algorithms of shape preserving spline approximation for curvessurfaces with automatic choice of the tension parameters. Shape preserving, multiscale interpolation by univariate curvaturebased cubicl1 splines in cartesian and polar coordinates. Shape control 19, shape design 5 and shape preservation 1215 are important areas for the graphical presentation of data. The objective of this paper is to present two case studies of approximation of simulated urban structures by bivariate l1 smoothing splines. Figure 2f shows the combination of figures 2a, 2d, and 2e. The general approach is that the user enters a sequence of points, and a curve is constructed whose shape closely follows this sequence.
Brodlie and butt 2 preserved the shape of convex data by piecewise cubic. Simple approximation methods such as polynomial or spline interpolation may cause value function iteration to diverge. Pdf shape preserving interpolation by curves researchgate. Remarks and final conclusions advances in approximation. Shape preserving third and fifth degrees polynomial splines. This paper pertains to the area of shape preservation. In this paper we propose a method to construct shape preserving c cubic polynomial splines interpolating convex andor monotonie data. If the address matches an existing account you will receive an email with instructions to reset your password.
However, such splines do not retain the shape properties of the data, a. To achieve this, a local hermite scheme incorporating a tension parameter for each spline segment is employed, the imposed shape constraints being concerned with preservation of convexity. 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. These splines preserve shape for smooth data as well as for data. Such splines are c 1 piecewise cubic splines where weights are shape parameters. We want to notice that, before, to choose weight parameters we have used our algorithms described in section 4 and the formula 33 w i. Pdf an algorithm is presented for calculating an osculatory quadratic spline that preserves the monotonicity and convexity of the data when consistent. Keywords shapepreserving approximation integro spline. In this scheme, the spline is chosen so that its second derivative is zero at the end points of the interval a, 6. Convexity preserving c2 rational quadratic trigonometric.