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 C 1 -quadratic and C 2 -cubic spaces are studied in some detail. In particular, we show that our C 2 -cubic refinement mask for the 1-to-4 subdivision can be slightly modified to yield an adaptive version of Loop’s surface subdivision scheme.