oth slbdivision scheme,an einanalysis c achs to both NURBS 一之w ark surfa ed out tha tructured as follows T 2 Quadratic NURSSes Fig.1:Doo-Sabin subdivision. 2 Z. Huang et al. /Journal of Information & Computational Science 7: 1 (2010) 1–6 stationary smooth subdivision scheme, an eigenanalysis can be performed for each configuration of knot intervals to compute the limit point and corresponding limit normal vector [7]. However, explicit limit point and limit normal rules for quadratic NURSSes or NURCCs have not yet become available. M¨uller et al. recently presented two different approaches to generalize both bicubic NURBS and Catmull-Clark surfaces. ESubs (Extended Subdivision Surfaces) [8] offer limit point rules but are non-stationary. DINUS [9] is a stationary scheme and provides limit point as well as limit normal rules. Both schemes use the refinement rules for Catmull-Clark surfaces [2] near control points with valence unequal to four. Quadratic NURSSes [3] are the only subdivision surfaces that extend both non-uniform bi￾quadratic B-spline surfaces and Doo-Sabin surfaces [1]. In Doo-Sabin like schemes, the extraordi￾nary points are at the ”centers” of n-sided faces with n 6= 4. In [10], Qin et al. pointed out that quadratic NURSSes converge for n ≤ 12, but may diverge when n > 12. As a stationary scheme, a detailed eigenanalysis has been performed for quadratic NURSSes [11]. But no closed-form limit point or limit normal rules are known for this scheme up to now. This paper presents a modification of quadratic NURSSes called EDSes (Extended Doo-Sabin Surfaces): For four-sided faces, the quadratic NURSS refinement rules [3] are applied; for n-sided faces with n 6= 4, the Doo-Sabin refinement rules [1] are used. As a result, if the knot intervals are all positive, the EDS scheme always generate G1 continuous limit surfaces. And explicit limit point and limit normal rules are derived as well. The remainder of the paper is structured as follows. The next section briefly reviews quadratic NURSSes. Sect. 3 describes the refinement rules for EDSes, and derives limit point and limit normal rules. Several examples are given as well. Finally, we conclude the paper with some suggestions for future work. 2 Quadratic NURSSes bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc b b b b b b b b b b b b b b b b b b b b b b b b b V F F E Fig. 1: Doo-Sabin subdivision. Quadratic NURSSes are non-uniform Doo-Sabin surfaces that generalize non-uniform biquadratic B-spline surfaces to meshes of arbitrary topology [3]. Quadratic NURSSes have the same topo￾logical split rules as Doo-Sabin surfaces [1]: At each subdivision step, new vertices are generated