In this work a new construction of a globally G1 smooth family of Bézier surfaces, defined by smoothing masks approximating the well-known Catmull-Clark (CC) subdivision surface, is presented. The resulting surface is a collection of Bézier (quad) patches, which are bicubic C2 around regular vertices and biquintic G1 around Extraordinary Vertices (EVs). Starting from the work of Loop and Shaefer, which provides a bicubic approximation of the CC limit surface with only C0 regularity around EVs, we improve this construction to reach a surface with global G1 smoothness; to impose G1 condition, we make use of quadratic gluing data functions around EVs which depends just on their valence. We present explicit formulas for G1 smoothing masks; moreover, these solutions possess degrees of freedom which can be fixed arbitrary. The entire system presents more than a way to be solved, and this yields a family of G1 solutions; between all the possible solving strategies, we identify four of them and we analyze each to determine the best solving strategies returning the smoothest surface. In order to assert the quality of the resulting surfaces and identify the ones that lead to the best result, both visually and numerically, we conduct curvature analysis on an extensive benchmark of meshes with different features. The resulting construction is described by explicit masks applied to the input control mesh, providing efficient computation and fast rendering of smooth piecewise polynomial surfaces of low degree and arbitrary topology.