Boupding the Distance betweena Loop Surface and Its Limit Mesb 1 where 5=s(（@ayl,0ssA-1,A2 34567890 m;9设名1招引8至 7 Conclusions and Future Work Bounding the Distance between a Loop Surface and Its Limit Mesh 13 where lj =  log 1 rλ(n)  rj (n)Cλ(n)M ǫ  , 0 ≤ j ≤ λ − 1, λ ≥ 1 . For regular Loop patches, k =  log4 ￾M 2ǫ
. Table 3 shows the comparison results of subdivision depths. The error toler￾ance ǫ is set to 0.0001, and the second order norm M is assumed to be 1. As can be seen from the table, the limit mesh approximation has a 20% improvement over the control mesh approximation in most of the cases. Table 3. Comparison of subdivision depths n 3 4 5 6 7 8 9 10 control mesh 8 10 14 7 19 21 22 23 limit mesh 7 9 12 6 16 17 18 18 7 Conclusions and Future Work In this paper we investigate the distance (error) between a Loop subdivision surface and its limit mesh. The maximal distance between a Loop patch and its limit triangle is bounded in terms of the second order norm of the initial control vertices and a constant that depends on the valence of the patch. An efficient subdivision depth estimation technique is also proposed. Test results show that a limit mesh approximates the limit surface better than the corresponding control mesh in general. The bounds achieved are still upper bounds, not necessarily strict. In future work we hope to derive an accurate error measure with a technique similar to Wu et al’s [4, 5]. Besides the parametric distance, the bounds of other distances, such as the Hausdorff distance, are yet to be investigated. Acknowledgments. We would like to thank the anonymous reviewers for their comments. This work was supported by the 973 Program of China (2004CB719403), NSF of China (60573151, 60473100), the 863 Program of China (2006AA01Z334, 2007AA01Z318), and China Postdoctoral Science Foundation (20060390359). References 1. Loop, C.T.: Smooth Subdivision Surfaces Based on Triangles. M.S. Thesis, De￾partment of Mathematics, University of Utah (1987) 2. Lanquetin, S., and Neveu, M.: A Priori Computation of the Number of Surface Sub￾division Levels. In: Proceedings of International Conference on Computer Graphics and Vision 2004 (Graphicon’2004), pp. 87–94 (2004)