A Topological Lattice Refinement Descriptor for Subdivision Schemes

The topological step of a subdivision scheme can be described as a refinement on regular tiling lattices, or more generally as a local transformation descriptor. The former classifies all the regular lattice transformations which give rise to other regular lattices. These regular lattice descriptors are limited by the type of the control mesh faces, the subdivided mesh must be composed by faces of the same type. The latter category describes some local topological transformations as the insertion of vertices in each face, followed by the description of a connectivity; such a descriptor is called meta-scheme. But these meta-schemes cannot describe a large number of regular refinements. We propose a topological descriptor that generalizes meta-schemes. Our descriptor is locally defined by a triple of integers which describes the number of inserted vertices relatively to the components of each face: vertices, edges and the face center. It is combined with a flexible connectivity descriptor which enhances modeling capability. Our descriptor can build the schemes commonly used and it can describe a variety of others, including many regular rotative schemes. We ensure the conservation of the control mesh global topology. The subdivision operators described here can be concatenated, leading to more complex topological descriptions. In addition, we propose a general geometric smoothing step.