Journal of Applied Mathematics
Volume 2012 (2012), Article ID 205863, 20 pages
Research Article

The Mask of Odd Points -Ary Interpolating Subdivision Scheme

1Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan
2Department of Mathematics, University of Science and Technology of China, Hefei 230026, China

Received 14 June 2012; Accepted 1 October 2012

Academic Editor: Debasish Roy

Copyright © 2012 Ghulam Mustafa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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