三进制Loop细分曲面的C~1连续性分析

本文利用计算机代数系统,通过对三进制Loop细分算法的细分矩阵和特征映射的构造与分析,证明Loop给出的掩模设计能够保证细分曲面在奇异点是C1连续的,还给出细分矩阵的次优势特征值的一个取值范围,在此范围内利用三进制Loop细分算法生成的细分曲面都是C1连续的.最后给出一种三进制Loop细分算法的新的边点掩模设计方法,在保证细分曲面是C1连续的前提下,比Loop给出的计算公式更简单,细分算法在奇异点附近收敛更快.

The analysis of C^1 regularity of ternary Loop subdivision surface

In this paper, by using a computer algebra system, we prove that ternary Loop subdivision scheme can guarantee C^1 regularity at the extraordinary point of the subdivision surface based on the construction and analysis of its subdivision matrix and the characteristic maps. Besides, an interval is given for the sub-dominant eigenvalue of the subdivision matrix, which can guarantee that ternary Loop rules always generate C^1 subdivision surface. Finally, we propose a new edge rule. This can guarantee the C^1 regularity of the subdivision surface. And also comparing with Loop rules, this scheme has simpler form and faster convergence speed around the extraordinary point.