Interpolation Scheme for Planar Cubic G 2 Spline Curves
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Interpolation Scheme for Planar Cubic G2 Spline Curves Marjeta Krajnc
Received: 3 February 2010 / Accepted: 12 October 2010 / Published online: 3 December 2010 © Springer Science+Business Media B.V. 2010
Abstract In this paper a method for interpolating planar data points by cubic G2 splines is presented. A spline is composed of polynomial segments that interpolate two data points, tangent directions and curvatures at these points. Necessary and sufficient, purely geometric conditions for the existence of such a polynomial interpolant are derived. The obtained results are extended to the case when the derivative directions and curvatures are not prescribed as data, but are obtained by some local approximation or implied by shape requirements. As a result, the G2 spline is constructed entirely locally. Keywords G2 spline · Polynomial curve · Geometric interpolation · Existence of solution · Shape · Algorithm Mathematics Subject Classification (2000) 65D05 · 65D07 · 65D17 1 Introduction One of the problems encountered often in CAGD applications is to find a smooth curve of a proper shape that interpolates given data points. It may be important that the interpolant depends on geometric quantities such as data points, tangent directions and curvatures only. In this case, the geometric interpolation schemes are considered as a proper tool to be used. The main property of such schemes is that the parameters at which the points are interpolated, the lengths of tangents, etc., are not prescribed in advance but are considered as unknowns. The additional freedom is used to interpolate more data which results in a higher approximation order and a more desirable shape obtained. However, geometric interpolation schemes include nonlinear problems. The questions of the existence of the solution, the approximation order and an efficient implementation may turn out to be a hard nut to crack. Most of the results are thus obtained by the asymptotic approach ([3, 7, 9, 10, 12–14, 22, 25–27], etc.). But, in practical applications the results M. Krajnc () FMF and IMFM, Jadranska 19, Ljubljana, Slovenia e-mail: [email protected]
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obtained by the asymptotic analysis are not adequate. For robust algorithms geometric conditions for the existence of the interpolant should be known in advance. Some of results of this nature can be found in [8, 15–21, 24], etc. In this paper the interpolation of planar data points by a cubic G2 spline curve is considered. On each polynomial segment two points, tangent directions and curvatures are interpolated. Directions and curvatures may not be explicitly prescribed as a data. If not, a way they can be chosen so that the spline exists is presented. For this reason we first revisit a well known interpolation problem that started the theory of geometric interpolation, the BHS scheme proposed by C. de Boor, K. Höllig, and M. Sabin in [3]. Here, the analysis is carried over based entirely on geometric considerations rather then by the original asymptotic approach. The existence ana
