• International Journal of CAD/CAM
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• 제5권1호
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• pp.29-38
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• 2005

A non-uniform 3-point ternary interpolatory subdivision scheme with variable subdivision weights is introduced. Its support is computed. The $C^0$ and $C^1$ convergence analysis are presented. To elevate its controllability, a modified edition is proposed. For every initial control point on the initial control polygon a shape weight is introduced. These weights can be used to control the shape of the corresponding subdivision curve easily and purposefully. The role of the initial shape weight is analyzed theoretically. The application of the presented schemes in designing smooth interpolatory curves and surfaces is discussed. In contrast to most conventional interpolatory subdivision scheme, the presented subdivision schemes have better locality. They can be used to generate $C^0$ or $C^1$ interpolatory subdivision curves or surfaces and control their shapes wholly or locally.

• Journal of applied mathematics & informatics
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• 제18권1_2호
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• pp.553-562
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• 2005

It is well-known that the smoothness of 4-point interpolatory Deslauriers-Dubuc(DD) subdivision scheme is $C^{1}$. N. Dyn[3] proved that 4-point interpolatory subdivision scheme is $C^{1}$ by means of eigenanalysis. In this paper we take advantage of Laurent polynomial method to get the same result, and give new way of strict proof on Laurent polynomial method.

• 호남수학학술지
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• 제27권4호
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• pp.665-672
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• 2005

By its simplicity and efficiency, subdivision is a widely used technique in computer graphics, computer aided design and data compression. In this paper we prove a regularity theorem for ternary interpolatory subdivision scheme that can be applied to non-stationary subdivision. This theorem converts the convergence of the limit curve of a ternary interpolatory subdivision to the analysis of the rate of the contraction of differences of the polygons.

• 대한수학회논문집
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• 제31권2호
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• pp.395-414
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• 2016

In this work, we study on subdivision schemes reproducing polynomials and build a symmetric subdivision scheme reproducing polynomials of a certain predetermined degree, which is a slight variant of the family of Deslauries-Dubic interpolatory ones. Related to polynomial reproduction, a necessary and sufficient condition for a subdivision scheme to reproduce polynomials of degree L was recently established under the assumption of non-singularity of subdivision schemes. In case of stepwise polynomial reproduction, we give a characterization for a subdivision scheme to reproduce stepwise all polynomials of degree ${\leq}L$ without the assumption of non-singularity. This characterization shows that we can investigate the polynomial reproduction property only by checking the odd and even masks of the subdivision scheme. The minimal-support condition being relaxed, we present explicitly a general formula for the mask of (2n + 4)-point symmetric subdivision scheme with two parameters that reproduces all polynomials of degree ${\leq}2n+1$. The uniqueness of such a symmetric subdivision scheme is proved, provided the two parameters are given arbitrarily. By varying the values of the parameters, this scheme is shown to become various other well known subdivision schemes, ranging from interpolatory to approximating.

• East Asian mathematical journal
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• 제38권3호
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• pp.365-377
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• 2022

In this paper, we present a new class of interpolatory Hermite subdivision schemes of order 2 reproducing polynomials. Each member in this class, denoted by H_n for n ≥ 1, preserves polynomials of degree up to 4n + 1 admitting the approximation order of 4n + 2. Furthermore, it has free parameters which provide flexibility in designing curves/surfaces. H₁, the simplest and the most attractive scheme in this class, achieves C⁴ smoothness with the parameters in certain ranges, and its performance is demonstrated with numerical examples.