Abstract
The (2s-1)-point non-stationary binary subdivision schemes (SSs) for curve design are introduced for any integer sgeq 2. The Lagrange polynomials are used to construct a new family of schemes that can reproduce polynomials of degree (2s-2). The usefulness of the schemes is illustrated in the examples. Moreover, the new schemes are the non-stationary counterparts of the stationary schemes of (Daniel and Shunmugaraj in 3rd International Conference on Geometric Modeling and Imaging, pp. 3–8, 2008; Hassan and Dodgson in Curve and Surface Fitting: Sant-Malo 2002, pp. 199–208, 2003; Hormann and Sabin in Comput. Aided Geom. Des. 25:41–52, 2008; Mustafa et al. in Lobachevskii J. Math. 30(2):138–145, 2009; Siddiqi and Ahmad in Appl. Math. Lett. 20:707–711, 2007; Siddiqi and Rehan in Appl. Math. Comput. 216:970–982, 2010; Siddiqi and Rehan in Eur. J. Sci. Res. 32(4):553–561, 2009). Furthermore, it is concluded that the basic shapes in terms of limiting curves produced by the proposed schemes with fewer initial control points have less tendency to depart from their tangent as well as their osculating plane compared to the limiting curves produced by existing non-stationary subdivision schemes.
Highlights
1 Introduction The importance of subdivision schemes (SSs) cannot be denied because they play a vital role in computer-aided geometric designing (CAGD), geometric modeling, computer graphics, medical image processing, scientific visualization, reverse engineering, robotics, etc
SSs can be distinguished in various types: they can range from uniform to non-uniform; from binary to an arbitrary arity; from interpolatory to approximating; from stationary to nonstationary
4 Results and comparisons Here we discuss visual quality of limit curves obtained by the proposed SSs
Summary
The importance of SSs cannot be denied because they play a vital role in computer-aided geometric designing (CAGD), geometric modeling, computer graphics, medical image processing, scientific visualization, reverse engineering, robotics, etc. SSs can be distinguished in various types: they can range from uniform to non-uniform; from binary to an arbitrary arity; from interpolatory to approximating; from stationary to nonstationary. It seems that stationary SSs have interesting features, but reconstruction of special types of limit curves of various shapes, including polynomial functions, conic sections such as circles, ellipses, and spiral curves, could not be accomplished without the non-stationary SSs. In literature, several articles have been published during the last couple of decades. In 2009, Daniel and Shunmugaraj [6] introduced a 6-point binary interpolating non-stationary SS that is C2 limit curve.
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