单变量均匀静态细分格式的连续性分析和构造 进 《中国科华技术大早数学系，家莹合厘23026 Continuity Analysis and Construction of Uniform Stationary Univariate Subdivision Schemes HANG Zhang-Jn aa ISSN 1000-9825, CODEN RUXUEW E-mail: jos@iscas.ac.cn Journal of Software, Vol.17, No.3, March 2006, pp.559−567 http://www.jos.org.cn DOI: 10.1360/jos170559 Tel/Fax: +86-10-62562563 © 2006 by Journal of Software. All rights reserved. 单变量均匀静态细分格式的连续性分析和构造∗ 黄章进+ (中国科学技术大学 数学系,安徽 合肥 230026) Continuity Analysis and Construction of Uniform Stationary Univariate Subdivision Schemes HUANG Zhang-Jin+ (Department of Mathematics, University of Science and Technology of China, Hefei 230026, China) + Corresponding author: Phn: +86-551-3601009, Fax: +86-551-3601005, E-mail: zjh@mail.ustc.edu.cn, http://math.ustc.edu.cn Huang ZJ. Continuity analysis and construction of uniform stationary univariate subdivision schemes. Journal of Software, 2006,17(3):559−567. http://www.jos.org.cn/1000-9825/17/559.htm Abstract: With the necessary and sufficient conditions for Ck -continuity of uniform stationary subdivision schemes, the range of free parameter in several classical interpolating curve schemes is presented. For the first time, this paper points out that the arity-2 interpolating 6-point scheme is C3 -continuous in certain range. A new C3 -continuous arity-3 interpolating 6-point scheme is also proposed. Key words: subdivision; interpolating scheme; continuity analysis 摘 要: 利用单变量均匀稳定细分格式 Ck 连续的充要条件,分析了已有的插值曲线格式各阶连续时参数的取 值范围.首次指出了六点二重插值格式可以达到 C3 连续,并构造了一种新的 C3 连续的六点三重插值细分格式. 关键词: 细分;插值格式;连续性分析 中图法分类号: TP391 文献标识码: A 由于具有算法简单、易于实现和高效性等优点,细分方法在计算机图形学和计算辅助几何设计中越来越受 到关注.最早的单变量细分格式要追溯到 Chaikin 于 1974 年提出的用于生成二次 B-样条的算法[1] .20 世纪 80 年代,Dubuc 以及 Dyn,Gregory 和 Levin 独立地提出了四点二重插值格式,并证明该格式是 C1 连续的[2,3]. Weissman 构造了一种六点二重插值格式,并给出了 C2 连续时参数的取值范围[3] .Deslauriers 和 Debuc 从多项式 插值出发,分析了 2N 点 b 重插值格式[4] . Kobbelt 引入了一种称为 3 格式的曲面细分方法[5] ,该方法在两步细分后得到三重格式.该类细分方法的 边界曲线是单变量三重格式生成的.受此启发,Hassan 和 Dodgson 等人提出了 C1 的三点三重插值格式[6] 和 C2 的四点三重插值格式[7] . ∗ Supported by the National Natural Science Foundation of China under Grant Nos.10201030, 60473132 (国家自然科学基金); the National Grand Fundamental Research 973 Program of China under Grant No.2004CB318000 (国家重点基础研究发展计划(973)); the National Science Fund for Distinguished Young Scholars of China under Grant No.60225002 (国家杰出青年科学基金); the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of the MOE (教育部高校优秀青年教师教学 科研奖励计划) Received 2005-04-13; Accepted 2005-08-25