Face-based Hermite Subdivision Schemes

Interpolatory and non-interpolatory multivariate Hermite type subdivision schemes are introduced in [8, 7]. In their applications in free-form surfaces, symmetry properties play a fundamental role: one can essentially argue that a subdivision scheme without a symmetry property simply cannot be used for the purpose of modelling free-form surfaces. The symmetry properties defined in the article [8] are formulated based on an underlying conception that Hermite data produced by the subdivision process is attached exactly to the vertices of the subsequently refined tessellations of the Euclidean space. As such, certain interesting possibilities of symmetric Hermite subdivision schemes are disallowed under our vertex-based symmetry definition. In this article, we formulate new symmetry conditions based on the conception that Hermite data produced in the subdivision process is attached to the faces instead of vertices of the subsequently refined tessellations. New examples of symmetric faced-based schemes are then constructed. Similar to our earlier work in vertex-based interpolatory and non-interpolatory Hermite subdivision schemes, a key step in our analysis is that we make use of the strong convergence theory of refinement equation to convert a prescribed geometric condition on the subdivision scheme – namely, the subdivision scheme is of Hermite type – to an algebraic condition on the subdivision mask. Our quest for face-based schemes in this article leads also to a refined result in this direction. Mathematics Subject Classification. 41A05, 41A15, 41A63, 42C40, 65T60, 65F15