The objective of this paper is to introduce a direct approach for generating local averaging rules for both the √3 and 1-to-4 vector subdivision schemes for computer-aided design of smooth surfaces. Our innovation is to directly construct refinable bivariate spline function vectors with minimum supports and highest approximation orders on the six-directional mesh, and to compute their refinement masks which give rise to the matrix-valued coefficient stencils for the surface subdivision schemes. Both the C1-quadratic and C2-cubic spaces are studied in some detail. In particular, we show that our C2-cubic refinement mask for the 1-to-4 subdivision can be slightly modified to yield an adaptive version of Loop's surface subdivision scheme.
Chui, C. K., & Jiang, Q. (2003). Surface subdivision schemes generated by refinable bivariate spline function vectors. Applied and Computational Harmonic Analysis. Academic Press Inc. https://doi.org/10.1016/S1063-5203(03)00062-9