The Mask of Odd Points n‐Ary Interpolating Subdivision Scheme

Abstract

We present an explicit formula for the mask of odd pointsn-ary, for any oddn⩾3, interpolating subdivision schemes. This formula provides the mask of lower and higher arity schemes. The 3-point and 5-pointa-ary schemes introduced by Lian, 2008, and (2m+1)-pointa-ary schemes introduced by, Lian, 2009, are special cases of our explicit formula. Moreover, other well-known existing odd pointn-ary schemes including the schemes introduced by Zheng et al., 2009, can easily be generated by our formula. In addition, error bounds between subdivision curves and control polygons of schemes are computed. It has been noticed that error bounds decrease when the complexity of the scheme decreases and vice versa. Also, as we increase arity of the schemes the error bounds decrease. Furthermore, we present brief comparison of total absolute curvature of subdivision schemes having different arity with different complexity. Convexity preservation property of scheme is also presented.

Generalized and Unified Families of Interpolating Subdivision Schemes

“…Now we discuss some important identities related to the Lagrange interpolant. We may refer to [1,16] for more detail about the proofs of these identities. For the given n, we define Lagrange fundamental polynomials of degree 2n -1 , corresponding to nodes by where j = -n, ■ ■ ■ ,( n -1),…”

“…Moreover, 3-point and 4-point ternary interpolating scheme of Hassan et al [5,6], Jian-ao Lian's 3-point, 5-point, 4-point, 6-point, (2m)point and (2m + 1)-point a-ary interpolating schemes [8][9][10], (2n -1)-point ternary interpolating schemes of Zheng et al [19], (2n -1)-point ternary interpolating scheme of Aslam et al [1] and odd points n-ary interpolating scheme of Mustafa et al [16] are also special cases of our family of scheme. We also have presented tensor prod uct version of the proposed generalized and unified family of interpolating schemes.…”

“…• By letting {p = 3, n = 2, Ul = v2, «i = + wi}, {p = 3, n = 3, wi = v2, vi = £ + wi}3 {p = 5, n = 2, ui = oi2 = and {p = 7, n = 2, V2 = <v3 = 3 §}, we get 3-point ternary, 5-point ternary, 3-point quinary and 3point septenary interpolating scheme of Mustafa et al [16] respectively. Similarly we can easily derive the mask of other odd-point n-ary interpolating schemes of [16].…”

“…Aslam et al [1] presented an explicit formula which unifies the mask of (2n -l)-point interpolating as well as approximating schemes. Mustafa et al [16] presented an explicit formula for the mask of odd-points n-ary (for any odd n > 3) interpolating subdivision schemes. This formula unifies the schemes of [9,10,19] and many other schemes.…”

A Family of 6-Point n-Ary Interpolating Subdivision Schemes

“…al. [8] gave a clear method to constract odd points n-ary, for every odd n 3 interpolation subdivision schemes. Both lower and higher arity schemes are generated by this formula.…”