In this paper, an algorithm is defined to construct 3n-point quaternary approximating subdivision schemes which are useful to design different geometric objects in the field of geometric modeling.We are going to establish a family of approximating schemes because approximating scheme provide maximum smoothness as compare to the interpolating schemes. It is to be observed that the proposed schemes satisfying the basic sum rules with bell-shaped mask go up to the convergent subdivision schemes which preserve monotonicity. We analyze the shape-preserving properties such that convexity and concavity of proposed schemes. We also show that quaternary schemes associated to the certain refinable functions with dilation 4 have higher order shape preserving properties. We also calculated the polynomial reproduction of proposed quaternary approximating subdivision schemes. The proposed schemes have tension parameter, so by choosing different values of the tension parameter we can get different limit curves of initial control polygon. We show in the table form that the proposed schemes are better than the existing schemes by comparing them on the behalf of their support and continuity. The visual quality of proposed schemes is demonstrated by different snapshots.
This article deals with univariate binary approximating subdivision schemes and their generalization to non-tensor product bivariate subdivision schemes. The two algorithms are presented with one tension and two integer parameters which generate families of univariate and bivariate schemes. The tension parameter controls the shape of the limit curve and surface while integer parameters identify the members of the family. It is demonstrated that the proposed schemes preserve monotonicity of initial data. Moreover, continuity, polynomial reproduction and generation of the schemes are also discussed. Comparison with existing schemes is also given.
We present an efficient and simple algorithm to generate 4-pointn-ary interpolating schemes. Our algorithm is based on three simple steps: second divided differences, determination of position of vertices by using second divided differences, and computation of new vertices. It is observed that 4-pointn-ary interpolating schemes generated by completely different frameworks (i.e., Lagrange interpolant and wavelet theory) can also be generated by the proposed algorithm. Furthermore, we have discussed continuity, Hölder regularity, degree of polynomial generation, polynomial reproduction, and approximation order of the schemes.
A new class of shape preserving relaxed 5-point n-ary approximating subdivision schemes is presented. Further, the conditions on the initial data assuring monotonicity, convexity and concavity preservation of the limit functions are derived. Furthermore, some significant properties of ternary and quaternary subdivision schemes have been elaborated such as continuity, Hölder exponent, polynomial generation, polynomial reproduction, approximation order, and support of basic limit function. Moreover the visual performance of schemes has also been demonstrated through several examples.
We derive three-step algorithm based on divided difference to generate a class of 6-point n-ary interpolating sub-division schemes. In this technique second order divided differences have been calculated at specific position and used to insert new vertices. Interpolating sub-division schemes are more attractive than approximating schemes in computer aided geometric designs because of their interpolation property. Polynomial generation and polynomial reproduction are attractive properties of sub-division schemes.Shape preserving properties are also significant tool in sub-division schemes. Further, some significant properties of ternary and quaternary sub-division schemes have been elaborated such as continuity, degree of polynomial generation, polynomial reproduction and approximation order. Furthermore, shape preserving property that is monotonicity is also derived. Moreover, the visual performance of proposed schemes has also been demonstrated through several examples.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.