Honeycomb and -fold Hermite subdivision schemes

We construct Hermite subdivision schemes for hexagon tiling and quadrisection / 3 refinement, which have applications in free-form surface design. Such subdivision schemes operate in such a way that when a “jet’’ of Hermite data is attached to each of the vertices in a coarse hexagon tiling, the subdivision rule is capable of defining Hermite data attached to the vertices of successively refined hexagon tilings, in such a way that the data converges to a smooth limit function which has Hermite data consistent with that generated by the subdivision process. Such a “vertex-based scheme on hexagon tiling’’ can be thought of as a “face-based schemes on triangular tiling’’. This simple connection allows us to put the construction under the mathematical framework of subdivision operators and refinement equations. the way, we introduce a general concept called k-fold Hermite subdivision, and analyze its properties with the help of the strong convergence theory of refinement equation. The case of k = 2 , together with an appropriate symmetry condition, can be used to handle the construction of honeycomb Hermite subdivision schemes. In particular, our framework allows us to construct smoother versions of two interesting honeycomb subdivision schemes in the literature.