In this work we present a parameter-dependent Refine-and-Smooth (RS) subdivision algorithm where the refine stage R consists in the application of a perturbation of Chaikin's/Doo–Sabin's vertex split, while each smoothing stage S performs averages of adjacent vertices like in the Lane–Riesenfeld algorithm (Lane and Riesenfeld, 1980). This constructive approach provides a unifying framework for univariate/bivariate primal and dual subdivision schemes with tension parameter and allows us to show that several existing subdivision algorithms, proposed in the literature via isolated constructions, can be obtained as specific instances of the proposed strategy. Moreover, this novel approach provides an intuitive theoretical tool for the derivation of new non-tensor product subdivision schemes that in the regular regions satisfy the property of reproducing bivariate cubic polynomials, thus resulting the natural extension of the univariate family presented in Hormann and Sabin (2008). 
A Chaikin-based variant of Lane–Riesenfeld algorithm and its non-tensor product extension