The Mask of Odd Points n -Ary Interpolating Subdivision Scheme

Abstract

We present an explicit formula for the mask of odd points $n$-ary, for any odd $n⩾3$, interpolating subdivision schemes. This formula provides the mask of lower and higher arity schemes. The 3-point and 5-point $a$-ary schemes introduced by Lian, 2008, and ($2m+1$)-point $a$-ary schemes introduced by, Lian, 2009, are special cases of our explicit formula. Moreover, other well-known existing odd point $n$-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.