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We present a general formula to generate the family of odd-point ternary approximating subdivision schemes with a shape parameter for describing curves. The influence of parameter to the limit curves and the sufficient conditions of the continuities from C⁰ to C⁵ of 3- and 5-point schemes are discussed. Our family of 3-point and 5-point ternary schemes has higher order of derivative continuity than the family of 3-point and 5-point schemes presented by [Jian-ao Lian, On a-ary subdivision for curve design: II. 3-point and 5-point interpolatory schemes, Applications and Applied Mathematics: An International Journal, 3(2), 2008, 176-187]. Moreover, a 3-point ternary cubic B-spline is special case of our family of 3-point ternary scheme. The visual quality of schemes with examples is also demonstrated

A new 4-pointC3quaternary approximating subdivision scheme with one shape parameter is proposed and analyzed. Its smoothness and approximation order are higher but support is smaller in comparison with the existing binary and ternary 4-point subdivision schemes.

This study deals with the numerical solution of the non-linear differential equations (DEs) arising in the study of hydrodynamics and hydro-magnetic stability problems using a new cubic B-spline scheme (CBS). The main idea is that we have modified the boundary value problems (BVPs) to produce a new system of linear equations. The algorithm developed here is not only for the approximation solutions of the 10th order BVPs but also estimate from 1st derivative to 10th derivative of the exact solution as well. Some examples are illustrated to show the feasibility and competence of the proposed scheme.

In this paper, we analyze shape-preserving behavior of a relaxed four-point binary interpolating subdivision scheme. These shape-preserving properties include positivity-preserving, monotonicity-preserving and convexity-preserving. We establish the conditions on the initial control points that allow the generation of shape-preserving limit curves by the four-point scheme. Some numerical examples are given to illustrate the graphical representation of shape-preserving properties of the relaxed scheme.

In this paper, we propose a three point approximating subdivision scheme, with three shape parameters, that unifies three different existing three point approximating schemes. Some sufficient conditions for subdivision curve C 0 to C 3 continuity and convergence of the scheme for generating tensor product surfaces for certain ranges of parameters by using Laurent polynomial method are discussed. The systems of curve and surface design based on our scheme have been developed successfully in garment CAD especially for clothes modelling.

Subdivision schemes are extensively used in scientific and practical applications to produce continuous geometrical shapes in an iterative manner. We construct a numerical algorithm to estimate subdivision depth between the limit curves/surfaces and their control polygons after k-fold subdivisions. In this paper, the proposed numerical algorithm for subdivision depths of binary subdivision curves and surfaces are obtained after some modification of the results given by Mustafa et al in 2006. This algorithm is very useful for implementation of the parametrization.

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