3n-Point Quaternary Shape Preserving Subdivision Schemes

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.