In S E 2.CaturulClark variant of Doo-Ssbin subdivision m e ur 'lark variand of con 巨=+k1+，+h =*六n++++云王 =0.. 1) Fig 3) Non anihrmqutintkBspiatsbdioa ThetheshCamll-Cark prepod in ->1 subdivision. Section 3 describes the decomposition of the non￾uniform quadratic subdivision into one non-uniform linear sub￾division step and one averaging (dual) step. In Section 4, we propose the refinement rules for NURDSes, discuss the con￾tinuity of this scheme and give some examples. Finally, we conclude the paper with some suggestions for future work. 2. Catmull-Clark variant of Doo-Sabin subdivision This section briefly reviews Doo-Sabin subdivision surfaces, especially the Catmull-Clark variant of Doo-Sabin subdivision. 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 Figure 1: Doo-Sabin subdivision. The Doo-Sabin subdivision algorithm is a generalization of the subdivision scheme for uniform biquadratic B-splines to control meshes of arbitrary topology [1]. The initial control mesh may consist of faces and vertices with arbitrary valence. During each Doo-Sabin subdivision step (see Figs. 1 and 2), for each face with n vertices P0, . . . , Pn−1, the corresponding new vertices P0, . . . , Pn−1 are computed by Pi = Xn−1 j=0 wi, jPj , i = 0, . . . , n − 1. (1) Then a new face of type F is created by connecting P0, . . . , Pn−1 to replace the old one. For each edge, a new four-sided face of type E is formed by connecting the images of the new points that have been generated for the faces sharing this edge. For each vertex, a new face of type V is formed by connecting the new points that have been generated for the faces surrounding the vertex. The weights in Eq. (1) have two forms. One is suggested by Doo and Sabin in [1] as follows wi j =    n+5 4n , i = j 3+2 cos(2π(i−j)/n) 4n , i , j The other is the Catmull-Clark variant proposed in [2] wi j =    1 2 + 1 4n , |i − j| = 0 1 8 + 1 4n , |i − j| = 1 1 4n , |i − j| > 1 bc bc bc bc bc b b b b b Pi Pi−1 Pi+1 Pi Ei−1 Ei F rs rs rs rs rs rs Figure 2: One linear subdivision step followed by a dual step produces the Catmull-Clark variant of Doo-Sabin subdivision. The latter one has a more intuitive geometric interpretation in terms of repeated averaging [3, 4]. As illustrated in Fig. 2, for an n-sided face, one linear subdivision step inserts a new edge point Ei at the midpoint of each edge PiPi+1 and a new face point F at the centroid of each face; then it inserts edges by connecting the face centroid with each of the surrounding edge midpoints. Applying one dual step to the linearly refined mesh, one obtains the same control mesh by performing one step of the Catmull-Clark variant of Doo-Sabin subdivision on the ini￾tial mesh. The dual of a mesh is a new mesh whose vertices are the centroids of old faces and whose edges join centroids of faces that share a common edge. Since the linearly subdi￾vided mesh consists of only quadrilateral faces, the new control vertex Pi corresponding to Pi in an n-sided face is computed as Pi = 1 4 (Pi + Ei−1 + Ei + F) = ( 1 2 + 1 4n )Pi + ( 1 8 + 1 4n )(Pi+1 + Pi−1) + 1 4n X |i−j|>1 Pj . where Ei = (Pi + Pi+1)/2 is the edge point on PiPi+1, and F = Pn−1 j=0 Pj/n is the centroid of the n-sided face. For both Doo-Sabin subdivision schemes, the extraordinary points are at the ”centers” of n-sided faces with n , 4, and their limit positions are exactly at the centroids of the faces. Fig. 3 shows the effect of applying the Catmull-Clark variant of Doo-Sabin subdivision to a cube. Fig. 3(b) is the result of linear subdivision. Fig. 3(c) is the result of next applying dual averaging and corresponds to one round of subdivision applied to the initial cube. Fig. 3(d) is the limit mesh via applying one more dual step. 3. Non-uniform quadratic B-spline subdivision In this section, we show that non-uniform quadratic B-spline subdivision schemes can be also decomposed into one non￾uniform linear subdivision step and one averaging (dual) step. Non-uniform B-spline curves are specified in terms of a set of control points, a knot vector, and a degree. A knot interval is the difference between two adjacent knots in a knot vector, 2