quad/triangle subdivision, nonhomogeneous refinement equation and polynomial reproduction

the quad/triangular subdivision, whose control net and refined meshes consist of both quads and triangles, provides better visual quality of subdivision surfaces. while some theoretical results such as polynomial reproduction and smoothness analysis of quad/triangle schemes have been obtained in the literature, some issues such as the basis functions at quad/triangle vertices and design of interpolatory quad/triangle schemes need further study. in our study of quad/triangle schemes, we observe that a quad/triangle subdivision scheme can be derived from a nonhomogeneous refinement equation. hence, the basis functions at quad/triangle vertices are shifts of the refinable function associated with a nonhomogeneous refinement equation. in this paper a quad/triangle subdivision surface is expressed analytically as the linear combination of these basis functions and the polynomial reproduction of matrix-valued quad/triangle schemes is studied. the result on polynomial reproduction achieved here is critical for the smoothness analysis and construction of matrix-valued quad/triangle schemes. several new schemes are also constructed.