Abstract
A subdivision scheme defines a smooth curve or surface as the limit of a sequence of successive refinements of given polygon or mesh. These schemes take polygons or meshes as inputs and produce smooth curves or surfaces as outputs. In this paper, a class of combine refinement schemes with two shape control parameters is presented. These even and odd rules of these schemes have complexity three and four respectively. The even rule is designed to modify the vertices of the given polygon, whereas the odd rule is designed to insert a new point between every edge of the given polygon. These schemes can produce high order of continuous shapes than existing combine binary and ternary family of schemes. It has been observed that the schemes have interpolating and approximating behaviors depending on the values of parameters. These schemes have an interproximate behavior in the case of non-uniform setting of the parameters. These schemes can be considered as the generalized version of some of the interpolating and B-spline schemes. The theoretical as well as the numerical and graphical analysis of the shapes produced by these schemes are also presented.
Highlights
The refinement schemes known as subdivision schemes are widely used in the design of curves and surfaces
If we have an initial sketch of any shape obtained by joining the 2D points fi0, i ∈ Z to refine the sketch, we suggest the following refinement scheme αfik−1 + (1 − β(1 − α)fik−1
Since the limit curves produced by the refinement schemes do not have closed form so the traditional methods fail to compute the points on the curve
Summary
The refinement schemes known as subdivision schemes are widely used in the design of curves and surfaces. A class of 4-point subdivision schemes with two parameters was presented in 2004 by [2]. Mustafa et al [12] introduced the families of interpolating schemes with parameters in 2014. Tan et al [14] presented a new 5-point binary approximating scheme with two parameters in 2017. In 2018, Asghar and Mustafa [15] presented a family of a-ary univariate subdivision schemes with single parameter. Another trend to introduce the combined schemes was evoked. Rehan and Siddiqi [17] presented a combined binary 6-point scheme with a parameter in 2015.
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