## Journal of Applied Mathematics

### 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 $n{\geqslant}3$, 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.

#### Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 205863, 20 pages.

Dates
First available in Project Euclid: 2 January 2013

https://projecteuclid.org/euclid.jam/1357153573

Digital Object Identifier
doi:10.1155/2012/205863

Mathematical Reviews number (MathSciNet)
MR2997246

Zentralblatt MATH identifier
1274.65355

#### Citation

Mustafa, Ghulam; Deng, Jiansong; Ashraf, Pakeeza; Abdul Rehman, Najma. The Mask of Odd Points $n$ -Ary Interpolating Subdivision Scheme. J. Appl. Math. 2012 (2012), Article ID 205863, 20 pages. doi:10.1155/2012/205863. https://projecteuclid.org/euclid.jam/1357153573