Composite Primal/Dual Sqrt(3)-Subdivision Schemes

We present new families of primal and dual subdivision schemes for triangle meshes and √ 3-refinement. The proposed schemes use two simple local rules which cycle between primal and dual meshes a number of times. The resulting surfaces become very smooth at regular vertices if the number of cycles is≥ 2. The C-property is violated only at low-valence irregular vertices, and can be restored by slight modifications of the local rules used. As a generalization, we introduce a wide class of composite subdivision schemes suitable for arbitrary topologies and refinement rules. A composite scheme is defined by a simple upsampling from the coarse to a refined topology, embedded into a cascade of geometric averaging operators acting on coarse and/or refined topologies. We propose a small set of such averaging rules (and some of their parametric extensions) which allow for the switching between control nets associated with the same or different topologic elements (vertices, edges, faces), and show a number of examples, based on triangles, that the resulting class of composite subdivision schemes contains new and old, primal and dual schemes for √ 3-refinement as well as for quadrisection. As a common observation from the examples considered, we found that irregular vertex treatment is necessary only at vertices of low valence, and can easily be implemented by using generic modifications of some elementary averaging rules.