Abstract
In this study, we are concerned with the interpolation problem that interpolates not only a given set of points but also derivatives at endpoints by using Bézier curves. The progressive iterative approximation (PIA) is proposed to interpolate the given points and derivatives. To speed up the convergence rate of PIA, we exploit the preconditioned PIA (PPIA), in which the diagonally compensated reduction is used to construct the preconditioner. The convergence of PIA and PPIA is also analyzed in this study. The proposed PPIA is applied to approximate higher order or/and rational Bézier curves. Numerical examples are given to illustrate the effectiveness of the proposed methods.
Highlights
We begin with the description of the data interpolation problem with constrains at endpoints
As stated in [11], the collocation matrix resulting from the Bernstein basis is usually ill-conditioned, and the convergence rate of progressive iterative approximation (PIA) is very slow. erefore, it is necessary to introduce a preconditioner to speed up the convergence rate of PIA
Polynomial approximation for higher order or rational Bezier curves has been a continuous hotspot in computeraided geometric design (CAGD)
Summary
We begin with the description of the data interpolation problem with constrains at endpoints. We need to control the smoothness of the composite curve when piecing curves together It would be great if the derivatives at the endpoints are known. A careful choice of end conditions is important because it determines the shape of the interpolating curve near the endpoints. We are especially interested in exploiting iterative methods for solving the interpolation curve (1) and do not consider the determination of constraints at the endpoints. We exploit the PIA format that interpolates a given set of points and derivatives at endpoints by using Bezier curves. Numerical examples are given to illustrate the effectiveness of our proposed methods in Section 5, and some conclusion remarks are given in the last section
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