Representations of Subdivision Schemes

There are four ways in which we can look at the weighted mean coefficients which define how a subdivision scheme behaves stencils, the subdivision matrix, the mask, and the generating function. 

Stencils The first obvious representation is the set of weight combinations for each of the types of vertex in the new polygon. 

Each set of weights is called a stencil, and the set of stencils taken together is a complete description of a uniform, stationary scheme. (The arity is implicit in the number of stencils.) 

Because the values in a stencil are often rational with a relatively small denominator, it saves space when writing to put a common denominator outside the brackets which enclose the values. Because the values are components of a weighted mean, the sum of the values in a stencil must be unity, so the denominator can always be derived from the set of values themselves. 

The stencils of example scheme 1 are thus [l,6,l]/8 for v-vertices and [4,4]/8 for e-vertices. There are two different stencils because the arity is 2. 

---