Ternary Schemes and Higher Arities

Both of the above examples approximately double the number of vertices in the polygon with each step of refinement. They are binary schemes. It is also possible to have schemes in which the number of vertices trebles or quadruples or is multiplied by a still higher factor. As mentioned above, we call that factor the arity, so that binary schemes have an arity of 2, ternary of 3, quaternary of 4 etc. Some of the mathematics applies to all arities, and in such cases we will denote the arity by the letter a. 

One particular ternary scheme, called the ternary quadratic B-spline for 

Clearly the old vertices have labels which are multiples of 3. The new e-vertices get labels of integers not divisible by 3. Such a scheme is both primal and dual, because vertices map into vertices and also edges map into edges. We call it a both scheme. 

This particular scheme turns out to have exactly the same limit curve as the second example above. 

It is also possible to have more complicated schemes in which vertices map under the labelling into edges and edges into vertices, so the scheme is neither primal or dual. 

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