• 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.

• 한국전산구조공학회:학술대회논문집
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• 한국전산구조공학회 2003년도 봄 학술발표회 논문집
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• pp.253-260
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• 2003

In computer-aided geometric modeling(CAGD), subdivision surfaces are frequently employed to construct free-form surfaces. In the present study, Loop scheme and Catmull-Clark scheme are applied to generate smooth surfaces. To be consistent with the limit points of target surface, the initial sampling points are properly rearranged. The pointwise errors of curvature and position in the sequence of subdivision process are evaluated in both Loop scheme & Catmull-Clark subdivision scheme. In partcural, a general subdivision method in order to generate considering extraordinary points are implemented free from surface with arbitrary sampling point information.

• 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.

• 한국CDE학회논문집
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• 제8권1호
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• pp.1-9
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• 2003

Recently, designing 3D objects using various modeling techniques become getting more important issues in related industrial fields. The subdivision scheme is a technique that generates a smooth sur-face through many times of refinement processes that split polygons of control mesh into several smaller polygons. In this paper, we propose a new subdivision algorithm that preserves sharp-edges of control mesh after several refinement processes in the Doo-Sabin subdivision scheme. Using the pro-posed algorithm, the Doo-Sabin subdivision scheme can be well applied to modeling 3D objects with sharp-edges.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제13권4호
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• pp.307-314
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• 2009

In this work, we introduce a new quatnary approximating subdivision scheme for curve and deal with its analysis (convergence and regularity) using Laurent polynomials method. We also discuss various properties, such as approximation order and support of basic limit function.

• 한국전산구조공학회:학술대회논문집
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• 한국전산구조공학회 2002년도 가을 학술발표회 논문집
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• pp.237-243
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• 2002

Recently, in computer-aided geometric modeling(CAGD), subdivision surfaces are frequently employed to construct free-form surface. Subdivision schemes have been very popular in computer graphics and animation community, but the community of CAGD adopts this tool only recently to handle surface geometry. In the present study, Loop scheme and Catmull-Clark scheme are applied to generate smooth surfaces. To be consistent with the limit points of target surface, the initial sampling points are properly rearranged. The pointwise curvature errors and coordinate value errors between the points in the sequence of subdivision process and the points on the target surface are evaluated In the numerical examples in both Loop scheme ＆ Catmull-Clark subdivision scheme.

• 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.

• International Journal of CAD/CAM
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• 제6권1호
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• pp.139-148
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• 2006

Catmull-Clark subdivision scheme provides a powerful method for building smooth and complex surfaces. But the number of faces in the uniformly refined meshes increases exponentially with respect to subdivision depth. Adaptive tessellation reduces the number of faces needed to yield a smooth approximation to the limit surface and, consequently, makes the rendering process more efficient. In this paper, we present a new adaptive tessellation method for general Catmull-Clark subdivision surfaces. Different from previous control mesh refinement based approaches, which generate approximate meshes that usually do not interpolate the limit surface, the new method is based on direct evaluation of the limit surface to generate an inscribed polyhedron of the limit surface. With explicit evaluation of general Catmull-Clark subdivision surfaces becoming available, the new adaptive tessellation method can precisely measure error for every point of the limit surface. Hence, it has complete control of the accuracy of the tessellation result. Cracks are avoided by using a recursive color marking process to ensure that adjacent patches or subpatches use the same limit surface points in the construction of the shared boundary. The new method performs limit surface evaluation only at points that are needed for the final rendering process. Therefore it is very fast and memory efficient. The new method is presented for the general Catmull-Clark subdivision scheme. But it can be used for any subdivision scheme that has an explicit evaluation method for its limit surface.

• Journal of applied mathematics & informatics
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• 제24권1_2호
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• pp.95-104
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• 2007

We study stationary subdivision schemes based on radial basis function interpolation. Each scheme has a tension parameter, say ${\lambda}$, which actually belongs to the radial basis function. In particular, adapted subdivision rules on bounded intervals are developed.

• 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.