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2003. A factored approach to subdivision surfaces. Submitted to Comput. Graph. Applicat. WARREN, J. AND WEIMER, H. 2001. Subdivision Methods for Geometric Design. Morgan Kaufmann. ZORIN, D. 2000. Smoothness of subdivision on irregular meshes. Construt. Approx. 16, 3, 359–397. ¨ ZORIN, D. AND SCHRODER , P. 2001. A unified framework for primal/dual quadrilateral subdivision schemes. Comput. Aided Geomet. Design 18, 5, 429–454. Received September 2003; revised May 2004; accepted August 2004 ACM Transactions on Graphics, Vol.

In this pass, the authors provided a different extension from Stam and Loop’s [2003] scheme to extraordinary vertices contained by both quads and triangles using a centroid averaging approach. For each vertex in the input mesh, averaging repositions that vertex to its final location after one round of subdivision. To reposition a vertex, averaging finds all polygons containing that vertex and computes the set of centroids shown in Figure 4 for those polygons. For quads, the centroid is simply 14 of each of its vertices, summed together.

2 Sufficient Conditions To analyze the smoothness of the subdivision scheme that we present, we use a sufficient test described by Levin and Levin [2003]. 1. Furthermore, the subdivision scheme along the edge must satisfy a joint spectral radius condition. To perform the joint spectral radius test, we require two subdivision matrices (A and B) that map an edge L on the boundary to two smaller edges (L1 and L2 ) after one round of subdivision. The matrices A and B should contain all of the vertices that influence the surface over the edges L1 and L2 .

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