Jet subdivision schemes on the k-regular complex

We introduce a new family of subdivision schemes called jet subdivision schemes. Jet subdivision schemes are a natural generalization of the commonly used subdivision schemes for free-form surface modeling. In an order r jet subdivision scheme, rth order Taylor expansions, or r-jets, of functions are the essential objects being generated in a coarse-to-fine fashion. Standard subdivision surface methods correspond to the case r=0. Just as the standard free-form subdivision surface schemes, jet subdivision schemes are based on combining (i) a symmetric subdivision scheme in the shift-invariant setting with (ii) an extraordinary vertex rule. We formulate the notions of stationarity, symmetry, and smoothness for jet subdivision schemes. We then extend some well-known results in the theory of subdivision surfaces to the setting of jet subdivision schemes.By incorporating high order data into the subdivision rules, jet subdivision schemes offer more degrees of freedom for the design of extraordinary vertex rules than do standard subdivision schemes. Using a simple 2-point stencil, we construct an order 1 jet subdivision scheme, which is interpolatory, C1 everywhere, and free from polar artifacts at extraordinary vertices of any valence. It is similar to the popular butterfly scheme, but unlike the butterfly scheme, it produces surfaces that can be explicitly parameterized by spline functions. In addition, our jet subdivision scheme allows for explicit control of gradient information on the resulting surface.We address the problem of constructing curvature continuous subdivision schemes at extraordinary vertices by giving an example of a 'flexible C2 scheme' for extraordinary vertex of valence k = 3. The stencils for this subdivision scheme coincide with those of the popular Loop scheme. We discuss the mathematical structure of this example of a C2 scheme.

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