Interpolation or approximation schemes that preserve the shape of the data are referred to as shapepreserving schemes. Read p 1 reproducing shape preserving quasiinterpolant using positive quadratic splines with local support, journal of computational and applied mathematics on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Shape preserving interpolation using 2 rational cubic spline. Usually, a shapepreserving scheme is based on a suitable space of functions like exponentials, rationals, variable degree polynomials, limits of subdivision schemes. 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. This book aims to develop algorithms of shape preserving spline approximation for curvessurfaces with automatic choice of the tension parameters. Journal of computational and applied mathematics 29. Approximation by shape preserving interpolation splines.
I have this data and i dont seem to find the appropiate function in r to do this. It is natural to apply this principle also if shape preserving is added as a constraint, although the construction process is then nonlinear. Shape preserving interpolation by quadratic splines sciencedirect. C2 cubic polynomial splines interpolating convex and monotone data while. Here, the construction of the quadratic x spline interpolation becomes straight forward. Journal of computational and applied mathematics 39. Shape preserving interpolation using rational cubic spline. Shape preserving interpolation by quadratic splines. In other words, it is impossible to interpolate convex data by a convex polynomial spline of bounded degree for general data and knots. In particular, they proposed a newtontype metho d and, based.
Algorithms for computing shape preserving spline interpolations to data. Ali abstract this study was concerned with shape preserving interpolation of 2d data. Shape preserving planar quadratic bezier interpolation spline. On shape preserving quadratic spline interpolation. Pdf an algorithm is presented for calculating an osculatory quadratic spline that preserves the. Refer to the scatteredinterpolant, griddata, and tpaps functions for more information about surface interpolation. The idea of a spline the general idea of a spline is this. An algorithm for computing a shapepreserving osculatory quadratic spline. Most of the shape preserving spline interpolation schemes including those based on classi cal quadratic spline either alter the derivative parameters or introduce additional nodes and construct polynomials. Shape preserving rational cubic spline fractal interpolation a. Schumaker 1983 and mcallister and roulier 1981 have proposed algorithms for shape preserving interpolation using quadratic splines.
Figure 4 is produced by the convexitypreserving rational cubic spline interpolation developed in section 4 with the values of free parameters 0. Refer to the spline function for more information about cubic spline interpolation. Ocr errors may be found in this reference list extracted from the full text article. In many situations the values of a function are known only at some fixed set of points and in addition something may be known about the shape of the function. Shapepreserving polynomial interpolation scheme request pdf. This chapter concentrates on two closely related interpolants. Shapepreserving interpolation of spatial data by pythagorean. Because the c 1 quadratic bezier interpolation spline can be determined only by the second control point of the first interpolation curve, we give the method for determining the second control point of the first interpolation curve by. The present approach finds the derivatives rni which satisfies the shape preserving criteria by selecting suitable values for oi, pi. Refer to the pchip function for more information about shape preserving interpolation, and for a comparison of the two methods. Is it there a way to adjust a quadratic spline instead of a cubic one to some data.
Shape preserving curves using quadratic trigonometric splines. The prototype problem consists in finding the interpolating or approximating function preserving some convex constraints such as monotonicity or convexity of given data. This should produce a much more satisfactory graph and the shape preserving spline should be even better. Some progress has been made in the last decade for shape preserving piecewise polynomial interpolants for data sets that are either monotone or convex. Linear interpolation the simplest form of interpolation is probably the straight line, connecting two points by a straight line. On the basis of this information one has then to form an approximation to the function. Algorithms are presented for computing a smooth piecewise polynomial interpolation which preserves the monotonicity andor convexity of the data.
Two parameters are constrained to preserve shape of the data whereas third parameter is left free for shape modification of the shape preserving curve. This research is a contribution towards achieving shape preserving curves and surfaces for positive data. This book aims to develop algorithms of shapepreserving spline approximation for curvessurfaces with automatic choice of the tension parameters. Farouki department of mechanical and aerospace engineering, university of california, davis ca 95616, usa. Local convexity shapepreserving data visualization by spline. In this note, we use a new approach to define the quadratic xsplines and then examine it for preserving shape when applied to strictly convex data. A univariate rational quadratic trigonometric interpolating spline. A newton method for shapepreserving spline interpolation. Anderson, for his advice and guidance during my years as a graduate student. Pdf an algorithm for computing a shapepreserving osculatory. In 6 mcallister and roulier, and in schumaker, have studied quadratic splines which preserve monotonicity and convexity. This study was concerned with shape preserving interpolation of 2d data. Here we show that, for a particular slope estimation technique, the two methods are identical, and that in this case the schumaker algorithm automatically.
We discuss two algorithms for the construction of the cubic spline interpolant under the constraint of positivity or monotonicity, and give a detailed convergence analysis. A univariate rational quadratic trigonometric interpolating. We have seen that it is suitable to consider some additional shape preserving conditions in order that the interpolation spline function preserves some shape. Following a precise definition of shapepreserving interpolating functions to data, we construct in a new and elementary manner such a cubic spline s2. A code for computing shape preserving bivariate spline interpolation, submitted to acm trans, on math. Shape preserving surfaces for the visualization of positive. A method for constructing shapereserving planar c 1 quadratic bezier interpolation spline by minimizing the stretch energy is proposed in this paper. Analysis of two algorithms for shapepreserving cubic spline.
Example 2 this example presents the curves drawn by. Such a scheme for quadratic spline interpolation was described by mcallister 12 and was further developed by schumaker 15 using piecewise quadratic polynomial which was very economical, but the method generally inserts an extra knot in each interval to interpolate. In this dissertation, we propose two new algorithms for comonotone in. Therefore, in this section, the shape preserving techniques are developed to interpolate shape preserving data. Sarfraz and hussain 17 used the rational cubic function with two shape parameters. An interpolating quadratic spline was constructed which preserves the shape of data. Shape preserving approximations by polynomials and splines. Pdf algorithms for computing shape preserving spline. Schumaker 1983 and mcallister and roulier 1981 have proposed algorithms for shapepreserving interpolation using quadratic splines. In this paper we will discuss a rational spline solution to the problem of shape preserving interpolation based on references 3, 4, 7 and 8.
We examine numerical and theoretical questions related to constrained interpolation and smoothing. Note that in this example, s must have an inflection point in the interval t1. Convexity preserving interpolation by splines is the topic of section 3. Shapepreserving interpolation of spatial data by pythagoreanhodograph quintic spline curves rida t. The rational spline is represented in terms of first derivative values at the. Shapepreserving rational interpolation for planar curves.
I want to use a shape preserving piecewise cubic interpolation on it similar to pchip in matlab. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. There is a unique straight line passing through these points. The rational quadratic spline function of sarfraz 11, which was used to achieve monotony preserving curves for monotonic data, has been extended to a rational bi quadratic spline function. In this note, we use a new approach to define the quadratic x splines and then examine it for preserving shape when applied to strictly convex data. In this paper we will discuss a rational spline solution to the problem of shape preserving. Acm has opted to expose the complete list rather than only correct and linked references. Shape preserving interpolation using quadratic xsplines. A new gc 1 trigonometric quadratic spline is developed.
The former requires the user to provide and perhaps to adjust estimates of the slope at the data points. Mar 01, 2007 comonotone shape preserving spline histopolation comonotone shape preserving spline histopolation fischer, malle. The developed piecewise trigonometric quadratic spline function involves three parameters in its construction. Comonotone shapepreserving spline histopolation, journal of. On shape preserving quadratic spline interpolation jstor. Numerical results of figure 4 are determined by developed convexity preserving rational cubic spline interpolation shown in. An algorithm for computing a shapepreserving osculatory. The second scheme preserved the shape of the data by the insertion of a new interpolation point. In this paper, we give a survey of some shape preserving approximation methods.
A onepass algorithm for shapepreserving quadratic spline. On shape preserving quadratic spline interpolation siam. Acknowledgements i am indebted to my major professor. The idea has been extended to shape preserving interpolation for positive data using the constructed rational cubic spline interpolation. The resulting curvessurfaces retain geometric properties of the initial data, such as positivity, monotonicity, convexity, linear and planar sections. Hermite and spline interpolation algorithms for planar. In 5 fritsch and carlson have studied cubic splines that preserve monotonicity. Automatic generation of shape preserving quadratic splines by. On shape preserving quadratic spline interpolation siam journal. This paper discusses the construction of new rational cubic spline interpolant with cubic numerator and quadratic denominator. P 1 reproducing shapepreserving quasiinterpolant using.