Interpolatory quad/triangle subdivision schemes for surface design

Recently the study and construction of quad/triangle subdivision schemes have attracted attention. The quad/triangle subdivision starts with a control net consisting of both quads and triangles and produces finer and finer meshes with quads and triangles. The use of the quad/triangle structure for surface design is motivated by the fact that in CAD modelling, the designers often want to model certain regions with quad meshes and others with triangle meshes to get better visual quality of subdivision surfaces. Though the smoothness analysis tool for regular quad/triangle vertices has been established and C1 and C2 quad/triangle schemes (for regular vertices) have been constructed, there is no interpolatory quad/triangle schemes available in the literature. The problem for this is probably that since the template sizes of the local averaging rules of interpolatory schemes for either quad subdivision or triangle subdivision are big, an interpolatory quad/triangle scheme will have large sizes of local averaging rule templates. In this paper we consider matrix-valued interpolatory quad/triangle schemes. In this paper, first we show that both scalar-valued and matrix-valued quad/triangle subdivision scheme can be derived from a nonhomogeneous refinement equation. This observation enables us to treat polynomial reproduction of scalar-valued and matrix-valued quad/triangle schemes in a uniform way. Then, with the result on the polynomial reproduction of matrix-valued quad/triangle schemes provided in our accompanying paper, we obtain in this paper a smoothness estimate for matrixvalued quad/triangle schemes, which extends the smoothness analysis of LevinLevin from the scalar-valued setting to the matrix-valued setting. Finally, with this smoothness estimate established in this paper, we construct C1 matrix-valued interpolatory quad/triangle scheme (for regular vertices) with the same sizes of local averaging rule templates as those of Stam-Loop’s quad/triangle scheme. We also obtain C2 matrix-valued interpolatory quad/triangle scheme (for regular vertices) with reasonable sizes of local averaging rule templates. ∗Research supported by UM Research Board 10/05 and UMSL Research Award 10/06.