Abstract
We present a new variant of Lane-Riesenfeld algorithm for curves and surfaces both. Our refining operator is the modification of Chaikin/Doo-Sabin subdivision operator, while each smoothing operator is the weighted average of the four/sixteen adjacent points. Our refining operator depends on two parameters (shape and smoothing parameters). So we get new families of univariate and bivariate approximating subdivision schemes with two parameters. The bivariate schemes are the nontensor product schemes for quadrilateral meshes. Moreover, we also present analysis of our families of schemes. Furthermore, our schemes give cubic polynomial reproduction for a specific value of the shape parameter. The nonuniform setting of our univariate and bivariate schemes gives better performance than that of the uniform schemes.
Highlights
Introduction and Related WorkSubdivision is a process of generating curves/surfaces by iteratively refining a set of control points according to some specific refinement rules
Linear, and uniform subdivision schemes, these refinement rules are same at each refinement level
We have shown the refinement of only two Chaikin points which have been inserted around point fik−1
Summary
Subdivision is a process of generating curves/surfaces by iteratively refining a set of control points according to some specific refinement rules. They combined the symbols of 4-point interpolatory subdivision scheme and odd stencil of 4-point interpolatory subdivision scheme as refining and smoothing operators, respectively Their univariate family gives cubic polynomial reproduction but by increasing smoothing stages continuity of their family may or may not be increased. They combined the symbol and odd stencil’s symbol of Dubuc Deslauriers 6-point interpolatory scheme [14] as refining and smoothing operators, respectively Their univariate family gives quintic polynomial reproduction but level of continuity of their family does not increase in general by increasing the smoothing stages. Mustafa et al [13] proposed a family of univariate subdivision schemes by using the symbol of 4-point interpolatory subdivision scheme [15] as the refining operator and the symbol of even stencil of the 4-point approximating scheme [16] as the smoothing operator Their univariate family gives cubic polynomial reproduction but level of continuity of their family does not increase in general by increasing the smoothing stages.
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