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.
Highlights
Subdivision is an algorithmic technique to generate smooth curves and surfaces as a sequence of successively refined control polygons
We present some preliminary identities which play an important role in the construction of explicit formula for the mask of odd-points n-ary interpolating schemes for any odd n 3
We offered an explicit general formula to generate the mask of odd-points nary interpolating symmetric schemes for any odd n 3
Summary
Subdivision is an algorithmic technique to generate smooth curves and surfaces as a sequence of successively refined control polygons. The resulting shape is fed back into the subdivision scheme again and again until we get a reasonable level of detail. Mustafa and Rehman 6 generalized and unified existing even-point n-ary subdivision schemes for any n 2. It is natural to look for a general formula which generalize and unify existing odd-point n-ary subdivision schemes and provide the mask of higher arity schemes in a simple and efficient way. We introduce an explicit formula which generalizes and unifies all existing odd-point n-ary interpolating subdivision schemes. In 10 , Mustafa and Hashmi presented an algorithm to calculate error bounds of n-ary subdivision schemes.
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