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SMOOTH CURVE FITTING WITH SHAPE PRESERVATION USING OSCULATORY QUADRATIC SPLINES

IP.com Disclosure Number: IPCOM000128147D
Original Publication Date: 1978-Dec-31
Included in the Prior Art Database: 2005-Sep-15
Document File: 4 page(s) / 20K

Publishing Venue

Software Patent Institute

Related People

L. E. Deimel: AUTHOR [+5]

Abstract

Algorithms are presented for calculating an osculatory quadratic spline f. The spline f preserves the monotonicity and convexity of the data if the first derivatives are consistent with the monotonicity and convexity of the data. The knots of.the spline are the data points and at most two additional knots between each adjacent pair of data points. The specification of the spline between two adjacent data points depends only on the points and the first derivatives at these points.

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

SMOOTH CURVE FITTING WITH SHAPE PRESERVATION USING OSCULATORY QUADRATIC SPLINES*

L. E. Deimel D. F. McAllister J. A. Roulier**

TR 78-03

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Abstract Algorithms are presented for calculating an osculatory quadratic spline f. The spline f preserves the monotonicity and convexity of the data if the first derivatives are consistent with the monotonicity and convexity of the data. The knots of.the spline are the data points and at most two additional knots between each adjacent pair of data points. The specification of the spline between two adjacent data points depends only on the points and the first derivatives at these points.

1. Introduction

There are several techniques for constructing smooth functions which interpolate data such as the method of ordinary polynomial interpolation [4). cubic spline interpolation [21, trigonometric inter-polation (4]. and convex spline interpolation [6.6.71. There are also methods which use osculating cubic poly-nomials where the derivatives at each point are esti-mated using quadratic interpolation or a method proposed by Akima [1) which uses thi slopes of the lines passing through adjacent points. In all of the above methods. except for the convex spline methods, there is no guarantee that the monotonicity of the data is preserved by the Interpolating function. The convex spline methods have the drawback that the data must first be divided into increasing, constant and decreasing segments and then further subdivided into convex and concave segments before the method can be applied. Although they do not require derivatives, they can be easily modified to use derivatives if they are available; however. the degree of the splines or the number of knots may be increased. In this paper we propose a method for computing an osculating quadratic spline which interpolates data and first derivatives and preserves the monotoni-city and convexity of the data if the first derivatives are consistent with the monotonicity and convexity. Moreover, the knots of these splines will be the data points and at most two additional knots between each adjacent pair of data points. The method is useful in cases where first derivatives are available or can be estimated. It has the advantage over convex spline methods in that it uses only "local" information to compute the quadratic spline between two adjacent data points, namely the two points and the first derivatives associated with those points. For this reason it Is easy to add and delete points and modify derivatives without having to recalculate all information required to specify the spline.

1 *Work by Dei.mel supported i,n part by Rome Air Development Center, Contract F30602-75-C-0118. Work by McAllister and Roulier supported in part by NSF Grant MCS76-0403.

2 **Department of Mathematics, N. C. State University

North Carolina State University Page 1 Dec 31, 1978

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SMOOTH CURVE FITTING WITH SHA...